THE PROBLEMS OF THE INFINITE AND THE
CONTENUUM IN SOME MAJOR PHILOSOPHICAL
SYSTEMS or THE ENLIGHTENMENT

Thesis for ”To Dogma of DH. D.
MICHIGAN STATE UNIVERSITY

Rolf A. George
1961

{mass

This is to certify that the

thesis entitled

"The Problems of the Infinite and the
Continuum in Some Major Philosophical
Systems of the Enlightenment"

presented by

Rolf A. George

has been accepted towards fulfillment
of the requirements for

Ph.D. in Philosophy

degree

William J. Callaghan

 

 

Major professor

DueFebruary 22, 1961

 

0-169

 

 
     
    

LIBRARY

Michigan State
‘1 University I”-

 

 

w'wW—“l

ABSTRACT

THE PROBLEMS OF THE INFINITE AND THE CONTINUUM IN
SOME MAJOR PHILOSOPHICAL SYSTEMS OF THE ENLIGHTENMENT

by Rolf A. George

The philosophers discussed in this dissertation are
Leibniz, Berkeley, Bayle, Kant, and Bolzano. Its aim is to
show that certain difficulties connected with infinite and
continuous sets were recognized by these philosophers, and
that their systems were, at least in part,designed in such
a way that these difficulties did not arise in them. Notably
the so—called Paradox of Galileo played a major role in this
respect: Galileo had shown that a one-to-one correspondence
can be established between integers and their squares, and
Leibniz realized that it is a property of all infinite sets
that they have subsets the members of which can be brought
in such a biunivocal correspondence with the members of the
original set. Up to Bolzano, this was held to contradict the
"wholly reliable" Euclidean axiom that the whole is bigger
than its part, and was used to prove that infinite sets are
impossible. Berkeley, for one, was aware of precisely this
problem when he developed a metaphysic in which infinite sets

do not occur.

Rolf A. George

Leibniz' solution consisted in the following: He
assumed that the number of monads constituting or "giving
rise" to any given finite body is always larger than any
finite number, but that it is not permissible to speak of
these monads as forming a set.‘ When we nevertheless speak
of a given body as forming Qgg.thing, we are taking a liberty
that is excusable in everyday discourse. only‘ for reasons of
its pragmatic efficacy. Such statements cannot be tolerated
in a language that endeavors an ultimately reliable descrip-
tion of the universe. For, if a given body were an infinite
set of parts, then it would have contradictory properties, so
Leibniz believed.

Kant asserted that a phenomenal Object does not have
an infinite number of parts already in it. Its being extended
is the result of the form of outer sense. ‘Hence it cannot be
said that it has more parts than this sense distinguishes in
it. However, an operative decomposition of such an object
can be carried out, and a rule of reason guarantees that this
decomposition has no final Stage. But this is not the same
as to say that the object is a set of infinitely many members,
or a whole with infinitely many parts, and this distinction
supposedly forestalls the arising of paradox.“

Bolzano was the first to realize that the so-called

Rolf A. George

Paradox of Galileo is no paradox at all, but simply describes
a common property of all infinite sets.

As concerns the constitution of continua the problem
was that neither the assumption that a continuum ultimately
consists of unextended parts, nor that it consists of extended
parts seemed defensible. Against the former case it was
argued that unextended parts, no matter how many, cannot make
a finite extension, against the’latter that extended parts
are not ultimate, but are further diviSiblei ‘Bayle'held that
none of the logical alternatives are defensible, so that no
one need bother to change whatever opinion he happens to have
on the subject.

Berkeley argued that there is no extension in objects,
but only extension as perceived, and a phenomenal object
cannot be said to have parts smaller than a minimum sensibile.
Hence, any continuous shape will consist of a finite number
of smallest particles, and the difficulty disappears. BerkEIey_
held this to be one of the most important conséquences of the
"immaterial hypothesis". Kant's solution has already been
sketched in connection with the Paradox of Galileo.

Leibniz' position was that there is no aetualcontinuum,
but that any continuum is an "intellectUal cbnstrhct”.i It
must be analyzed as a concept, iue. into simpler concepts, not'

into smaller parts. A part of a given continuous entity is

Rolf A. George

considered to be a more complicated construct than the whole
of which it is a part, since it must be described by refer-
ence to that whole. Thus continua are said to be ”prior” to
their parts. Leibniz consequently held that the quest for
the ultimate spatial parts of continua is pointless.

Bolzano declared that in a continuum every point has
a neighbor within any distance, no matter how small. This
definition, although ultimately unsatisfactory, proved to
be of great help in discovering various important properties
of continuous sets.

The purpose of this dissertation is not to sketch
the evolution of thought on the subjects of infinite and
continuous sets, but to show how the problems connected with
them were no less important in the development of Enlightenment
philosophy than the epistemological predicaments customarily

discussed in histories of philosophy.

 

THE PROBLEMS OF THE INFINITE AND THE CONTINUUM IN SOME

MAJOR PHILOSOPHICAL SYSTEMS OF THE ENLIGHTENMENT

by

'0

\

{o
Rolf A5 George

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
' for the degree of
DOCTOR OF PHILOSOPHY
.Department of Philosophy

I961

I 7 ./_" /""

o... d .p‘

éfl/gd

To

Hellmut Keusen

 

ACKNOWLEDGEMENT

I should like to express my gratitude and
appreciation to Professors Rudner, Callaghan and Zerby
for all the help and encouragement they have extended to

me during the course of this investigation.

 

CONTENTS

Introduction.

Chapter I: Leibniz.
Chapter II: Berkeley.
Chapter III: Bayle.
Chapter IV: Kant.
Chapter V: Bolzano.

Conclusion.

.19

.63

111

133

192

211

 

 

INTRODUCTION

In 1662 Arnauld wrote the following about the pro-

blems of the infinite in his Port Royal Logic.l

The most compendious way to the full
extent of knowledge is not to toil ourselves
in the search of that which is above us, and
which we can never rationally expect to com-
prehend. Such are those questions that relate
to the Omnipotency of God, which it would be
ridiculous to confine within the narrow limits
of our Understandings; and generally, as to
whatever partakes of Infinity, and lies over-
whelmed under the multitude of thoughts, con-
tradicting one another.

Hence may be drawn the most convenient
and shortest solution of many questions, about
which there will be no end of disputing, so long
as Men are infected with the Itch of dispute,
in regard they can never be able to arrive at
any certain knowledge, whereby to assure
and fix the Understanding. Is it possible for
God to make a Body infinite in quantity, a
movement infinite in swiftness, a multitude
infinite in number? Is a number infinite even
or odd? Is one infinite more extensive than
another? He that should answer once for all,
I know nothing of it, may be said to have made
as fair a Progress in a moment, as he that had

_¥

l(Antoine Arnauld) Logic or the Art of Thinking,
Part 4, Ch. 1. (The quotation is taken from the 2nd
edition of the English translation, London 1693, p. 390 F.)

been beating his Brains twenty years, about

these Niceties. The only difference between

these Persons is, that he that drudges day and night

about these Questions, is in the greatest danger

of falling a degree lower than bare Ignorance;
which is, to believe he knows that which he knows
not at all.
Contrary to the spirit of this quotation. Arnauld then
proceeds to give various demonstrations for the infinite
divisibility of matter, which, following Descartes. he
considers to be continuous.

Before Arnauld. Descartes had similarly proclaimed
that there is not much point to an investigation of the
infinite division of matter, since it must be considered
to transcend our finite understanding,2 and much the same
reasons were advanced by Galileo when he cautioned "let
us remember that we are dealing with infinites and indi-
visibles, both of which transcend our finite understanding.
the former on account of their magnitude. the latter

. 3
because of their smallness."

These quotations express a sentiment apparently

quite widespread around the middle of the seventeenth

 

2Descartes. Les Principes de la Philosophie, Part
II. No. 35, Oeuvres. Ed.Victor Cousin, Paris 1824, vol. 3.
p. 150.

3Galileo Galilei, Dialogues Concerning Two New
Sciences, Ed. Henry Crew and Alfonso de Salvio, New York.
1914, p. 26.

 

century: a general resignation before the problems of
infinite sets and the logic of the continuum. But this
attitude changed very rapidly: a mere half century after

the Port Royal Logic, Collier remarks scornfully about

 

the passages in that work which pertain to our subject
that it is indeed a sign "that our understandings are very
weak and shallow, when such stuff as this shall not only
pass for common sense, but even look like argument."4 At
the beginning of the eighteenth century, the appeal to the
finitude of our understanding in discussing the apparently
incomprehensible properties of infinite sets was no longer
considered a philosophically tenable position. A vigorous
attack upon these problems had been initiated by Leibniz,
_ Bayle, Berkeley, Collier and others, and the belief that
the difficulties of the continuum and the infinite must be
capable of a rational solution, once gained, was never
relinquished until such a solution finally was accomplished
around the middle of the last century.

The present study is devoted to a discussion of

some attempts on the part of various philosophers to find

a solution to the problems of continuity and the infinite.

 

4Arthur Collier, Clavis Universalis, Ed. Ethel
Bowman, Chicago, 1909, p. 74.

4

In particular, Leibniz, Bayle, Berkeley, Collier, Kant and
Bolzano will be discussed. I wish to establish the thesis
that the philosophical systems developed by these philoso-
phers had as one of their primary objectives the resolution
of the indicated problems; and that this concern can,
accordingly, be said to be largely responsible for the
particular character which those philosophical systems took
on.

While we have many historical accounts of mathema-
ticians dealing with these and kindred problems, especially
problems in analysis, hardly any investigations have been
made into the role and importance of the problems of the
infinite and the continuum for the nature and structure of
metaphysical and epistemological systems in the seventeenth
and eighteenth century. This is all the more surprising as
the philosophical literature of that age contains so many
references to these difficulties, so many proposals for
their resolution that the sheer frequency of these remarks
would suffice to legitimate them as among the persistent
problems of philosophy in those centuries. A further fact
which should have drawn the attention of historians to
these philosophical labors is that the mathematicians of

the seventeenth and eighteenth century tended to regard

the problems of the continuum as "pre-mathematical": they
were to be resolved through metaphysical speculation, much
in the same way in which Plato wanted to establish the
truth of the axioms of geometry through dialectics. This

sentiment is described by Boyer in his Concepts of the

 

Calculus:

 

The attitude of most of the mathematicians

of the seventeenth century...was that of doubt.

They employed infinitesimals and the infinite

on the assumption that they existed. and treated

the continuous as though made up of indivisibles,

the results being justified pragmatically by

their consistency with Euclidean geometry. In

any case the attitude was not that of unpre-

judiced postulation and definition. followed

by logical deduction .5
One might add that the situation did not appreciably change
during the eighteenth century. and only in the nineteenth,
commencing with Bolzano. were the fundamental problems of
analysis. of the continuum and of infinity attacked in a
new spirit within the field of mathematics itself. Before
that time, problems in the foundation of mathematics were
relegated to the metaphysician. But it was not only the
case that the mathematicians were hesitant to attack the

foundation—problems of their science. and let their results

justify their assumptions; the philosophers of the age felt

 

5Carl B. Boyer. The Concepts of the Calculus,(New York),
1949. p. 305.

 

6
it incumbent upon themselves to supply what the mathemati-
cians avoided. Kant speaks of having to bring about a
reconciliation of geometry and metaphysics; Berkeley urges
the mathematicians to join metaphysics to their mathematics;
and Leibniz admonished the geometricians to leave well
alone the problems of the continuum:"...the geometer does
not need to encumber his mind with the famous puzzle of the
composition of the continuum..."6

In a sense. the philosophy of mathematics was not
only thought to have to describe mathematics, but also to
have to establish its first principles. Of course. the
notion that the latter should be the task of philosophy
could arise only if the former was not properly performed
In a way. all the philosophical investigations that were
undertaken in this direction stem from misconceptions con-

cerning the nature of mathematics. The nature of these

misconceptions. their genesis and consequences embodied in

 

6cf. George Berkeley. Of Infinites. The Works of
George Berkeley. ed. A. A. Luce and T. E. Jessup. London
1948 ff., Vol. 4. pp. 235-238. p. 238. In the sequel,
Berkeley will be quoted after this edition. which will be
referred to simply as The Works of George Berkeley. G. W.
Leibniz. Discourse on Metaphysics. section X. in Leibniz,
Selections ed Philip P. Wiener, New York 1951. (In the
sequel quoted as Wiener). p. 303.

 

 

these philosophical labors constitute the subject of the
presentidiscussion.

The methods of approaching the problems of the con—
tinuum were. for example. strongly influenced by the assump-
tion that geometry is the science whose task it is to
describe empirical space or phenomenal space. This had led
some of the philosophers under discussion to regard the
problem of the composition of the continuum as synonymous
with the problem of the composition of visually continuous
surfaces or lines. In others we find the problems of the
continuum discussed in connection with the compositiOn of
matter or of space or of time. Thus our problem was dis-
cussed under various guises and one must not expect to find
questions of mathematics divorced from considerations of
physics or from discussions concerning the structure of our
experience. After all. the enquiries concerning the nature
of the continuum are found embedded in larger treatises.
and it is characteristic of all these approaches that they
claim to make the puzzlements of the continuum disappear if
only the problem is seen in the right (empiricist or ideal-
ist or what not) context.

It is customary. in the introductory chapter of an

investigation such as the present one. to state precisely

the nature of the problem to be discussed in it. So far

I have offered not much more than vague generalities.
Evidently. in order to become more explicit I ought to
offer a definition of such key terms as "continuum" and
"infinite" if I wish to make the extent and concern of the
present dissertation perfectly clear. In trying to do so.
however. a perplexing difficulty presents itself: It has
often been claimed, especially by mathematicians, that the
first rigorous definition of "continuous" was given by
Dedekind. and that before this momentous event the word
"continuous" was associated with a more or less vague notion
of hang-togetherness.7 If this were so. then we would.
strictly speaking. have no assurances that the philosophers
we are about to investigate were in fact all dealing with
the same problem (as was. say. Dedekind) when they spoke of
the problem of the continuum. But the difficulty is not as

severe as it might seem. While no precise definition of

 

continuity was available. there was always the example of
the geometrical line. the points on which were always con—
sidered to form a continuous set. Thus the problem was not

simply to find a precise definition of a hitherto vague term,

 

‘7Cf. Boyer. op. cit, pp. 42. 291.

9
but to describe accurately the arrangement or order of the
points in a line in such a way that, for example, the axioms
of Euclid could be fulfilled. If it had been otherwise,
then there could be no reason why the definitions advanced
by Leibniz and later by Mach, namely that a line is con—
tinuous if between any two points on that line there lies
another point, should have to be rejected.8. We could
simply treat this definition as a hortatory one and
assume that it renders precise a concept that had so far
been vague. But the point is that linear point sets can be
produced which fulfill Leibniz' requirement but which do not
allow a development of geometry in Euclid‘s sense. For
example, conditions can be described in which "a straight

line meets two straight lines, so as to make the interior

 

8Leibniz defines "continuity" thusly: "There is con-

tinuOus extension whenever points are assumed to be so
situated that there are no two between which there is not

an intermediate point." (Bertrand Russell, A Critical
Exposition of the Philosophy of Leibniz, 2nd ed., London,
1937, p. 247). For Mach cf. Boyer, op. cit., p. 291: "The
scientist Ernst Mach likewise regarded this property of
denseness of an assemblage as constituting its continuity."
(Die Principien der Warmelehre, historisch-kritisch
entwickelt 2nd. ed. Leipzig, 1900, p. 71. Similar considera-
tions apply to Bolzano's definition of 'continuity'. Bolzano
claims that a point set is continuous if, and only if, the
points are situated in such a way that "every single one

of these points has at least one neighbor in the set within
any distance, no matter how small. (Bernard Bolzano, ggpgr
doxien des Unendlichen ed. Fritz Prihonsky, Hamburg 1955,
(lst. ed. Leipzig 1851), p. 73).

 

 

10
angles on the same side of it taken together less than two
right angles" where nevertheless the two straight lines do
not "at length meet" because one of them runs through a gap
in the other. i.e they do not have a point in common.

The defhfltions advanced by Leibniz. Bolzano and Mach
was at length discarded because they did not imply all the
properties associated with the preanalytic notion of con-
tinuity. But this only goes to show that this preanalytic
notion was a fairly precise one. What was lacking was a
definition of "continuous" in terms of the components of
the continuum, or. as the philosophers of the age were wont
to put it. the problem of the composition of the continuum
was unresolved, However. the mere fact that the term was
always applied to the same kinds of object assures us that
different philosophers. when speaking about the continuum,
were really dealing with the same problem. I have come to
this conclusion in spite of the fact that the descriptions
of the composition of the continuum vary from wildly meta—
phorical. as in the case of Galileo, to dryly sober in the

case of Bolzano. Galileo writes in the Dialogues Concerning

 

 

9Cf. Euclid's Elements. ed. Isaac Todhunter. London
1955. Axiom 12.

11

Two New Sciences.

 

Having broken up a solid into many parts.
having reduced it to the finest of powder and
having resolved it into its infinitely small ‘
indivisible atoms, why may we not say that this
solid has been reduced to a single continuum.
perhaps a fluid like water or mercury or even
a liquified metal?lO

It must be noted that the fact that Galileo was unable to
give an adequate description of the composition of the
continuum did not prevent him from employing the term
correctly, as his geometrical researches amply testify. It
is this similarity in use that assures us that Galileo
meant to deal with the same problems as later Bolzano and
Cantor.

The situation is slightly more difficult in the case
of the concept of infinity. Apparently the concept of
infinity has always been surrounded with certain romantic
notions. It is still customary to speak about somebody's
"infinite compassion." Infinity seems to have always been
considered by some people as an honorific attribute:

Descartes speaks of God's infinite and man's finite mind,

and similar examples could be adduced in great abundance.

 

loGalileo, o . cit., p. 39f.

12
Fortunately. the sense in which the term "infinity” is used
is usually made clear by the context. But let me give an
example of the confusion that this ambiguity of the term
frequently caused:

Collier. in his Clavis Unfive:salis attempts to show,

 

as Berkeley did before him, that there is no external
matter. After having done this to his satisfaction. he
tries to show that certain contradictions follow from the
contrary assumption, namely that there is external matter.
He writes: "External matter. as a creature. is evidently
finite. and yet as external is evidently infinite. in the

f
number of its parts. or divisibility of its substance.

"11
From this he concludes that a contradiction follows
from the assumption that there is external matter. namely,
such matter would have to be both finite and infinite ' But
this contradiction is only apparent. since the word 'finite'
is construed as the negation of an honorific attribute that
Tlas nothing to do with the mathematical notion of infinity.

Collier puts it thus:12

llArthur Collier, Clavis Universalis, ed. Ethel
Bowman. Chicago 1909. p. 69.

 

121bid., 02. cit., p. 68.

 

 

13
"Infinite is to be absolute. finite. to be not absolute."
Now since God is the only absolute being. infinity can
properly be attributed only to Him and not to anything that
depends for its being upon God‘s concurrence. In this sense
external matter is said to be finite because it is a crea-
ture. Now the term 'infinite' in the first quotation above
evidently denotes the number of members in a set and thus
can clearly not be construed as the contradictory of the
term "finite“ in the same quotation. as Collier would have
it. This example demonstrates amply the confusion caused

by the ambiguity of 'infinite'.13

 

13We have seen that it was frequently argued at the
time that man cannot understand the composition of the
continuum because his mind is finite. but that only God.
whose mind is infinite can have an adequate understanding
of it. Commentators have often argued that 'finite' and
'infinite' are here used in a non-quantitative way. They
speak of the "qualitative" infinite. Hegel even attaches a
Value—discrimination to this distinction when he condemns
‘the quantitative infinite as "das schlechte Unendliche".
iln.the text I have argued that the two conceptions should
the kept distinct, but it might be well to point out that
tlaere is a pretty obvious connection between these two con—
CHepts, especially among the Cartesians. I believe that
(tqualitative) infinity was thought to be that attribute of
tine understanding of God which allowed him to comprehend
.tJae nature of infinite sets. in particular the composition
.<>f the continuum. The postulation of this particular attri-'
ibute of God's understanding would preserve. after a fashion.
the rationality of the world. for if not even God could
reconcile the apparently contradictory properties of infin-
ite sets. then we must think of ourselves as somehow
surrounded by inconsistent facts. In addition, 'infinite'

 

14
For obvious reasons I have therefore made it a policy to
discuss only such occurrences of 'infinite' as seem to
refer to the number of elements in a set. For this reason
I have decided not to include a discussion of Spinoza.
Spinoza. as far as I can discern. defines "finite after its
kind" as a quantity that can always be increased. He says
"A thing is called finite after its kind. when it can be
limited by another thing of the same nature; for instance
a body is called finite because we always conceive another.
greater body.".14 A body is finite. then. if something can be
added to it. and it would seem that a body is infinite if
nothing can be added to it. Or. as Bolzano puts it.
Spinoza believed "only that to be infinite which is not
capable of further increase, or to which nothing can be

added."15

Bolzano points out that many other philosophers and

Vvas of course also used as an honorific term. However. the
Irelation between the ”qualitative" and the "quantitative"
linfinite in 17th century philosophy would bear some further
:anestigation-
l4 . . . . . . .

Benedict Spinoza. Ethics, Definition 21; in: Tp§_
Slgef Works of Benedict de Spinoza. ed. R. H. M. Elwes.
‘N. Y.. 1951.

15Bernard Bolzano, o . cit., § 12, No. 2.

 

 

15
mathematicians adhered to this definition of 'infinite'.
In the present paper I shall make no attempt to account for
any theories of this sort, i.e.. I shall iestrict myself to
only such accounts as do not regard an infinite number as
the greatest thinkable number. or give similar definitions.
As we shall presently see. writers who did not treat
infinity as an honorific attribute and who did not adhere
to a definition similar to that of Spinoza were able to
indicate. quite some time ago, several important properties
of infinite sets.

One difficulty seems to permeate all the discussions
of the continuum that we are about to study9and'that is the
sense of 'compose' in the question "of what is the con-
tinuum composed?" and we may just as well give it some
thought at this juncture. Apparently the continuum was
_thought to be "composed" of something in the sense in which
a brick wall is composed of bricks. The bricks are the
parts of which the brick wall is composed. but there are no
components that make up a continuum in this way. Certainly.
if we consider a continuous line. the points in this line
do not "compose" the line in this sense. We know that if
we have more bricks, then we can make a bigger wall, but a

larger number of points does not make a longer line than a

16
smaller number of points. Likewise. in a inck wall. a
brick has immediate neighbors (provided there is more than
one brick in the wall). This does not hold for the points
in a line. That the continuum would exhibit such features
was a fact not generally reckoned with. and the confusion
surrounding this concept can in a large part be explained
by assuming that what was sought after were parts of the
continuum that would compose it like bricks compose a wall.
The abandonment of this preconception was one of the most
important conditions. but evidently also one of the condi-
tions most difficult to attain before an adequate descrip-
tion of the continuum could be given.

At this place a few remarks are also in order con-
cerning the method of the subsequent enquiries. Our task
is the exposition. at least in part. of certain philosophi—
cal systems. Eor these expositions I have adopted the
following strategies: I attempt to identify the problems
which these systems Were designed to resolve, and try to
delineate the way in which this-resolution took place in
each case. At the danger of repeating myself needlessly. I
wish to emphasize again that the problems of the infinite
and the composition of the continuum hold a place of great

eminence among the problems to be reSolved in the systems

17
which will be discussed. This procedure of identifying the
initial problems of philosophical systems seems to have
certain advantages over other modes of exposition. It makes
clear. for one thing. that the philosophy of the enlighten—
ment must not be thought of as so much idle speculation: it
was to a great extent just as serious an endeavor to resolve
the problems indicated as were the later and more successful
investigations of mathematicians. Anothe. advantage of this
mode of exposition would seem to lie in the following: if it
can be shown that, for example. Leibniz adopted the divi-
sion between intelligible and phenomenal world with the
explicit purpose of resolving certain logical difficulties.
then the affirmation of this bifurcation among later ideal-
ists, who can think of no problem to be resolved thereby.
is just purposeless. ludicrous speculation. a senseless
repetition of a philosophical distinction the import of
which had been entirely lost.

It is a disgrace for the historians of philosophy
that the term 'idealist‘ should be applied indifferently to
both Leibniz and those who aped him. But a distinction
between the two can obviously only be found if one is aware

of the problem which precipitated‘the development of

‘ ‘1-

Leibniz' system, and the sad lack of such problems, for

18
example. with Fichte. The much touted "historical method"
would never be capable of finding this distinction, con-
centrating as it does on the lines of development of
philosophical ideas.

LetthEisuffice as an introduction. I shall take the
liberty to make further remarks concerning the mode of my

exposition in the body of this dissertation.

CHAPTER I

LEIBNIZ

In discussing the philosophy of Leibniz. formidable
difficulties present themselves. Leibniz did not leave a
definitive treatment of his philosophy. and his incidental
expositions are almost always adapted to the capacity of
his correspondent or his immediate audience. or are the
~outgrowth of polemics over certain restricted points. In
view of these facts it is necessary for the commentator to
adopt a strategy which will organize the material for him
and allow him to weave it into a coherent whole. In keep-
ing with a general practice, a historical approach to
Leibniz is most frequently chosen and the "development" of
his philosophy is discussed. ‘Various influences upon him
are cited and characteristic alterations of delivered
opinions are noted. Under this viewpoint. Leibniz appears
as a great synthesizer of previously held philosophical
opinions. Latta, for example,notes correctly that "the
philosophical work of Leibniz was an endeavor to reconcile
the notion of substance as continuous with the contrary

notion of substance as consisting of indivisible

20

elements".16

But while he recognizes the importance of the
subject to Leibniz. he thinks of him as primarily interes-
ted in affecting a synthesis of two positions that were in
fact held by his predecessors. namely Descartes on one
hand and the atomists on the other. A much more blunt
statement of the view that Leibniz was. in the main, a

synthesizer of previously held philosophical views is.

found in Maziarz' Philosophy of Mathematics. Maziarz

 

writes: "His (Leibniz') countless references to ancient,
medieval. and contemporary thinkers...reveal his basic
trend and tendency: to compromise and synthesize. to remove
individual differences. to harmonize speculative opposition

in his own center of perspective "17

The viewpoint here
characterized can easily be substantiated through quota-

tions from Leibniz himself. Thus in the Nouvaux Essais.

 

Leibniz writes: "This system appears to unite Plato and
Democritus, Aristotle and Descartes. the Scholastics with

the Moderns, theology and ethics with reason. It seems to

 

16Robert Latta, Introduction to: Leibniz. The
Monadology. ed. R. Latta. lst edition. Oxford 1898, p. 28.

17Edward A. Maziarz. The Philosophy of Mathematics,
New York, 1950, p 58. -

21
take the best from all sides..."18 It must be noted that
for a critical exposition of Leibniz it is really of
secondary importance to discover the historical sources of
various parts of Leibniz philosophy and to discuss the
psychological motivation for the development of his system.
It must be noted. and will be substantiated in the sequel.
that the views which Leibniz synthesized were each of them
held irrefutable by Leibniz without much regard for their
historical origin. Thus it is of primary importance to
undertake a reconstruction of his system. and it is only of
secondary philosophical interest to trace historical ori-
gins and psychological motivations. I believe that in
thus reconstructing Leibniz' system better justice is done
to the spirit of Enlightenment philosophy, which took its
metaphysical problems just as seriously as we take problems
of mathematics or methodology. and which was not given to
the transaction of speculations out of piety for received
opinions. Thus. in this chapter I do not wish to advance
a psychological or historical thesis concerning the genesis
of Leibniz' philosophy. ‘Rather. I wish to point out that

he was aware of certain problems connected with the con-

 

lBLeibniz. New Essays Concerning Human Understanding,
ed. Alfred Go Langley, La Salle, 1949, p. 66 (Book 1. Ch. 1).

22
tinuum. and that his philosophical system can be envisaged
as being designed. to a large extent, in order to resolve
these problems. Thus I shall not argue that the problems
of the continuum were the psychological starting point of
his speculations. On the other hand. the present chapter
can be taken as an attempt to show that there are no inner-
systematic reasons against this assumption. However. the
establishment of the psychological thesis itself would
require a good deal of additional detective work which I
do not consider to be the responsibility of the philosophi-
cal commentator.

A different strategy for the exposition of Leibniz'
system has been chosen by Bertrand Russell. Properly,
Russell does not pay much attention to the historical ori-
gin of Leibniz' views. but treats his philosophy, at least.
in its speculative parts, as a coherent system and claims
that he attempts to develop it from a small set of premises:
These premises are:

I Every proposition has a subject and a
predicate, II. A subject may have predicates

which are qualities existing at various times

(such a subject is called a substance). III.

True propositions not asserting existence at

particular times are necessary and analytic.

but such as assert existence at particular

times are contingent and synthetic. The latter
depend upon final causes. IV The ego is a

23

substance. V. Perception yields knowledge

of an external world, i e., of existents

other than myself and my states.19
One might expect that Russell would develop Leibniz' system
from these premises with the aid of the laws of logic. But
he does not give us the treat of demonstrating Leibniz'
philosophy as an axiomatic system, and one might excuse
his self-deception on this count by pointing out that his
book on Leibniz preceded his logical work. He claims, how-
ever. that "the first four of the above premises...1ead to
the whole or nearly the whole of the necessary propositions
of the system."20

I believe that a reconstruction of Leibniz' philoso—
phy as an axiomatic system would not present to us the
spectacle of a large number of theorems following from a
half dozen or so axioms. as one might expect from the
philosophical system of a man with Leibniz' logical acumen.
Notice that Russell merely claims to have identified the
“principal" premises of the system, although he fails to

make clear in what sense he takes certain premises to be

more important than others.

19Bertrand Russell. A Critical Exposition of the
Philosophyiof Leibniz. 2nd ed., London 1937, p. 4.

ZOIbid.

24

In keeping with the general procedures set forth in
the introduction I shall attempt to give an exposition of
parts of Leibniz' philosophy by a method different from the
above indicated two. I believe that Leibniz‘ major problem
lay in the fact that he adopted a number of propositions
each of which he had to consider as well established, but
which together seemed to form an inconsistent set. It
seems that through an identification of these propositions
a reasonable understanding of the problems of Leibniz‘ sys-
tem can be provided. I shall therefore proceed to point
out what these propostions were and we shall see that, in
order to harmonize them. a large part of Leibniz' system
had to be developed. In identifying the propositions in
question I shall occasionally refer to the history of their
discovery, but I want this taken as incidental information.
I do not wish to appear to make the same error that I have
criticized in Maziarz.

Leibniz knew that the members of an infinite
collection can be brought in a one to one correspondence
with the members of a proper subset of that collection.
This fact was already known to Galileo. who wrote in the

Dialogues Concerninngwo New Sciences:

25

Salv. Very well; and you also know that
just as the products are called squares. so
the factors are called sides or roots; while
on the other hand those numbers which do not
consist of two equal factors are not squares.
Therefore, if I assert that all numbers.
including both squares and nOn-squares. are
more than the squares alone, I shall speak the
truth, shall I not?

Simp. Most certainly-

Salv. If I should ask further how many
squares there are. one might reply truly that
there are as many as the corresponding number
of roots, since every square has its own root
and every root its own square, while no square
has more than one root and no root more than
one square.

Simp. Precisely so,

Salv. But if I inquire how many roots
there are, it cannot be denied that there are
as many as there are numbers, because every
number is the root of some square. This being
granted we must say that there are as many
squares as there are numbers, because they are
just as numerous as their roots. and all the
numbers are roots.

Galileo was quite puzzled with his discovery and he
assumes that the difficulty can in part be resolved by
assuming that "the attributes 'equal'. ‘greater'. and

'less', are not applicable to infinite, but only to finite.

21Galileo Galilei, Dialogues Concerning Two New

§£3§§g§§. Translated and ed. by Henry Crew and Alfonso

De Salvio. N. Y., 1914, p. 32.

26

quantities."22

Any difficulty that might remain must be
attributed to our finite understanding which cannot
entirely cope with the problems of the infinite.23

Leibniz was familiar with this result. He occasion-
ally claims to have discovered it himself. Thus he writes:
"Many years ago I proved that the number or sum of all,
numbers involves a contradiction (the whole would equal
the part)."24 In the Nouvaux Essais he writes: "But there
is no infinite number. neither line nor other infinite
quantity, if these are understood as veritable wholes, as
is easy to show."25 Here he obviously alludes to the same
result. It is noteworthy that Leibniz realized that the
paradox of Galileo not only applies to discrete sets such
as the set of natural numbers, for which alone Galileo pre-

sents a proof, but that the paradox is seen to apply also

to such point sets as lines. which are here considered as

 

22Ibid

23See Introduction. p. 2.
24Letter to Bernoulli, 1698, Wiener, o . cit.,

25Leibniz, Nouvaux Essais, p. 161, (Bk. II, Ch.
XVII);

27
just another kind of infinite quantity. The clearest state-
ment of the "Paradox of Galileo" is the following:

The number of all numbers implies a contra-
diction, which I show thus: To any number there

is a corresponding number equal to its double

Therefore the number of all numbers is not

greater than the number of even numbers 6i.e.

the whole is not greater than its part.

It is perhaps not unfair to Leibniz to state the
preposition in question as "If there is an infinite collec-
tion, then it has a proper subset which can be brought in
a one to one correspondence with it". This proposition is
clearly presupposed in the above presented arguments,
especially in the last one. But the arguments are of
modus tollens form: they deny the consequent, namely that
the part ever is as great as the whole. in order to deny
the antecedent, namely that there are infinite collections.

Thus we have discovered a second proposition which
Leibniz considered established beyond all doubt, namely
that the whole is greater than its part. This is the ninth

27

of Euclid's axioms. Leibniz frequently acknowledges his

unconditional acceptance of this proposition as for example

 

26Russell, 0 . cit., p. 244.

27Euclid, The Elements, ed. Issac Todhunter. London,
1955, p. 6.

 

28

in the Nouvaux Essais: "Euclid says that the whole is

 

greater than its part. a statement which is wholly trust—
worthy".28 On another occasion, Leibniz attempts to deduce
the principle of whole and part from a definition, He
writes: "If a part of a quantity is equal to the whole of
another quantity, then the first is called the greater,

the second the smaller. Whence the whole is greater than
the part."29 However. when Leibniz establishes that there
would be as many even numbers as integers. he does not make
use of the provisions of his definition, for, clearly, the
set of integers has a part which is equal in number to the
set of even numbers and that is the set of even numbers
itself. Hence by the above definition, the set of even
numbers must be called smaller than the set of integers.
Moreover, when we think of the set of integers as equal
with the set of even numbers, then the whole of which the
even numbers are a part must be greater than the set of
integers. But that whole is the set of integers itself.

-Hence, by the above definition, the set of integers is

greater than itself, which makes it impossible to consider

 

28
Leibniz, Nouvaux Essais, p. 471.

 

29 .
Metaphysical Foundations of Mathematics. Wiener,
op. cit., p. 205.

29
”greater than" as irreflexive, a very odd usage indeed
The upshot of this discussion is that Leibniz had very
pressing reasons to deny the existence of infinite sets.
If there are no infinite sets, then the above definition
becomes. of course. quite acceptable. since then the princi-
ple of whole and part would generally hold.

There are numerous other passages in which Leibniz
asserts the principle of whole and part. and occasionally
he calls it an axiom.3O The two propositions considered so
far. namely the paradox of Galileo and the principle of
whole and part could be reconciled with one another, but
will jointly lead to the consequence that there are no
infinite sets: The following, however, seems to assert
the existence of infinite sets and thus forced Leibniz to
all manner of speculation as to how it could be reconciled
with the first mentioned two propositions. It can be put
thus: There is external matter and it is infinitely divisi-
ble. It has infinitely many parts.

Obvious difficulties arise at once. Leibniz had

asserted that we cannot make any statements about infinite

 

30§pecimen Dynamicum. Wiener. op. cit.. p. 130.

30
sets considered as "veritable units" but that we can only
speak of infinite sets as what he calls distributive
wholes. Thus while we may form a sentence of the form
"Every even number has such and such a property", we are
not supposed to write "the class of even numbers is such
and such". This restriction becomes problematic once the
assumption is made that every piece of external matter
forms a whole of infinitely many parts. Then a sentence
like "this stone is of such and such a nature" violates
the rule that prohibits our speaking of infinite sets as
veritable wholes.

It seems then that the three propositions, namely
the paradox of Galileo, the principle of whole and part,
and the assertion of the actual infinite subdivision of
any given piece of matter form an inconsistent set. This
inconsistency is part of what Leibniz calls the "labyrinth
of the continuum". Actually, Leibniz does not consider
external matter to be continuous, but his denial of the
real continuity of matter is alreay part of the solutiOn of
the problem, as we shall see in the sequel. The reason why
the indicated problem must nevertheless be considered to

o .
Specimen Dynamicum, Wiener, op. cit., p. 130.

31
belong to the so-called labyrinth of the continuum is that
many of Leibniz predecessors subscribed to the notion that

matter is continuous and therefore infinitely divisible.

 

Leibniz agrees that matter is so divisible. but his views
on the continuity of matter are rather more complex and
shall be discussed later.

That Leibniz believed in the actual infinite is
shown in the following passage and numerous others.

I am so much for the actual infinite that

instead of admitting that nature abhors its, as

is commonly said, I hold that it affects nature

everywhere in order to indicate the perfections

of its author. So I believe that every part of

matter is, I do not say divisible, but actually

divided, and consequently the smallest particle

should be considered as a world full of an

infinity of creatures.31

That there should be external matter at all was
apparently not seriously questioned by Leibniz, at least
not in his later years. Russell remarks that only in his
earlier years did Leibniz regard the existence of matter
as a problem, but that later "he so far forgot his earlier

unresolved doubts that, when Berkeley's philosophy appeared,

‘Leibniz had no good word for it. 'The man in Irelandg Tue

 

3lspecimen calculi universalis. Wiener, op. cit.,
p. 99. ‘

32
writes,'who impugns the reality of bodies. seems neither to
give suitable reasons, nor to explain himself sufficiently.
I suspect him to be one of that class of men who wish to
be known by their paradoxesu32

The difficulty then is the following: there are
everywhere in nature infinite sets. Every piece of matter
can be thought of as such an infinite set. But this assump-
tion is not compatible with Leibniz' other results, namely
that there is a contradiction in the concept of an infinite
set. Since the problem here indicated is connected, his-
torically, with the assumption that matter is continuous,
let us call it the first problem of the continuum. Its
resolution, i.e. the reconciliation of the three proposi—
tions so far identified requires the generation of a large
part of Leibniz' system. and we can therefore think of
these propositions as "nuclear" propositions: the system
was generated for the most part to make their simultaneous
assertion tenable.

In saying that the paradox of Galileo, the principle
of whole and part, and the assumption of the actual infinv

ite were the propositions for the sake of which a large

 

32
Russell, op. cit., p. 72.

 

33
part of the system was developed, I do not wish to give the
impression that these propositions should be considered as
axioms. Rather. they have a relation to the fundamental
assumptions of the system similar to that of the better
known theorems of arithmetic to their axioms: they are
believed first and only subsequently axioms or premises are
sought from which they follow.

Before I try to show the ways in which Leibniz
attempts his resolution. let me point to a second problem
connected with the continuum. This second difficulty had
already engaged Descartes. It is the following: The'
ultimate parts of the continuum, or of matter when viewed
as continuous, can be neither points nor extended particles
(so it was argued). Not the former because extension
cannot be built up. it was believed, out of unextended
parts, not the latter. because extended parts or extended
atoms are not ultimate but can be further divided.33 Now
the number of parts, whatever their nature, must be infin-
ite since a finite number of divisions of a continuous
quantity will always result in extended parts which can be

further divided. Thus all that can be said is that any

 

33
Cf. Descartes. Principles. Part II. Principle 20.

34
continuous quantity consists of an infinite number of parts
of undetermined nature. This is where Descartes rests the
issue. He argues that we must not doubt this division.
although we cannot comprehend it.34 The reason why
Descartes did not want this division to be put in question
is. of course. that he wanted Euclidean geometry to be
applicable without restriction to extended objects. and
this geometry demands, among other things. that any line,
no matter how short. must be divisible. Let us call the
problem here indicated the second problem of the continuum.
I think that what Leibniz calls the "labyrinth of the
continuum" consists of the two problems here identified.

It will be noted that what I have called "the first pro-
blem of the continuum" attaches to all infinite sets. while
the second has to do with continuous sets only. We can
take Leibniz' word that his philosophy was to a large
extent designed to resolve both these problems. In the

beginning of the Theodicee he writes:

 

There are two famous labyrinths. in which
our reason often goes astray: the one relates
to the great question of liberty and necessity.
especially in regard of the production and
origin of evil; the other consists in the

 

 

34 ‘
Cf.. Ibid., Part II. Principle 35.

35

discussion of continuity and the indivisible
points which appear to be its elements. and
this question involves the consideration of
the infinite. The former of these perplexes
falmost all the human race, the latter claims
the attention of philosophers alone.35

 

 

 

Latta comments about this:

Leibniz makes the Theodicee an investigation

of the meaning of liberty and necessity, while
in others of his writings he offers a solution
of the problem which he describes as the special
perplexity of the philosophers.

 

Let us now proceed to sketch that part of Leibniz'
system which is required for the solution of the problems
of the continuum.

For reasons that need not concern us at this point,
Leibniz assumed that every well-formed proposition is con-
stituted of one subject and one predicate. Relations are
recognized as useful but merely "ideal things". In con-
sidering the relation "greater than" which is to hold
between M and L, he states that "is greater than L" can be
considered an accident of M, "is less than M" can be con-
sidered an accident of L. but he also realizes that there
is a third way of analyzing this proposition. namely by

treating the relation "as something abstracted from both".

 

35Latta, o . cit., p. 21.
36Ibid.

36

In this case
It cannot be said that both of them. L

and M together are the subject of such an

accident; for if so. we should have an acci-

dent in two subjects with one leg in one and

the other in the other; which is contrary to

the notion of accidents. Therefore we must

say that this relation, in this third way of

considering it, is indeed out of the subjects;

but being neither a substance, nor an accident,

it must be a mere ideal thing, the considera-

tion of which is nevertheless useful.37

Thus, while "is greater than M" refers to a real
accident, "is greater than" refers to an "ideal" thing.
It seems that the relations owe this shadow of an exis-
tence to the pragmatic justification that "their considera-
tion is useful", and indeed the knowledge of some of the
properties of relations is indispensable for certain calcu-
lations Nevertheless, the proper analysis of a statement
such as "M is greater than L" resolves it into one subject
and one predicate.

Thus the only sentences which are considered
strictly meaningful are ones that consist of one subject
and of an absolute predicate. Let us consider for the

moment only singular statements of this kind. Leibniz

held that such statements fall into two classes: those

 

37George Martin Duncan (ed.). The Philosophical
Works of Leibniz, New Haven 1890, pp. 266f. Cf. Russell,

0 cit., pp. 12f.

 

37
that are about what he calls true or substantial units,
and those whose subject refers to aggregates. Now the
statements about true units have a certain metaphysical
prerogative over the others: all others are meaningful if
and only if they can be replaced by statements about true
units. But what is a true or substantial unit? Leibniz
writes: "Substantial unity calls for a thoroughly indivisi-
ble being. naturally indestructible."38 Objects which are
not so indestructible are said to have no real existence.
"What is not really one being (pp etre) is not really a

. n 39 . .
being (un etre) . From true or substantial units
Leibniz distinguishes "beings by aggregation". As examples
he cites armies, flocks of sheep. piles of stone. More—
over, any extended physical object is considered such a
being by aggregation. He says that:
A block of marble is no more a thoroughly
single substance than would be the water in a
pond with all the fish included, even when all
the water and all the fish were frozen; or any.
more than a flock of sheep, even when all the
sheep were tied together so that they could only

walk in step and that one could not be touched
without producing a cry from all.40

 

38Letter to Arnauld, in: George R. Montgomery, (ed.)
Leibniz, Discourse on Metaphysics, Correspondence with
Arnauld. and Monadology, La Salle, 1902.

 

 

 

39Russell, op. cit., 242, Montgomery, op cit., 192.
40Letter to Arnauld, Montgomery, op. cit., p. 161.

38

It is the mind of the observer alone that gives
unity to such beings by aggregation. "This mass of sub-
stances", he says, does not form in truth one substance.
This is a result to which the soul by its perception and
. . . . . 41
its thought gives its last achievement of unity." Now
Leibniz holds that statements about aggregates can fre-
quently be replaced by statements about the components of
such aggregates. He points out:

It seems that what constitutes the essence

of a being by aggregation consists only on the

mode of the being of its component elements.

For example, what constitutes the essence of

an army? It is simply the mode of being of

the men who compose it.

Apparently, Leibniz was working on a method whereby
all statements about couples, trios, etc. can be translated
into statements about the members of such couples, trios,

43
etc. Now if such a translation can always be afforded,
then no harm can come from asserting something about an

aggregate, since such an assertion can always be replaced

by an equivalent set of statements about the members of the

 

41Nouvaux Essais, p. 235 (Bk II. Ch. XXIV).

 

42Letter to Arnauld, Montgomery, 0 . cit., p. 190.

43Ad specimen calculi universalis addenda, cf. R. M.
Yost, Leibniz and Philosophical Analysis, Berkeley 1954,
p. 11.

39
aggregate. The question whether or not the aggregate has
existence is then a purely metaphysical one, and in deny-
ing such existence, Leibniz shows himself as a nominalist.43
On the other hand, if there is a group of statements about
aggregates which cannot be translated. in principle, into
statements about true units, then the issue becomes some-
what more complicated. especially if any important state-
ments. such as the enunciations of physics or geometry are
included in this latter category. For all statements which
are in principle incapable of translation into statements
about true units must be considered meaningless in the
strictest sense. although practically speaking they may be
rather useful. For example, an assertion about a couple of
stones can perhaps be translated into statements about the
stones in that couple, but these latter statements are

themselves about aggregates, the stones being composed,

according to Leibniz, of an infinity of parts. Consequently

 

43In Leibniz, no clear distinction seems to be

drawn between the whole-part relation and the class mem-
bership relation. When I use the term 'aggregate' I wish
to use this term verbally so as to apply to wholes as well
as classes. Now Leibniz seems to have desired the exis-
tence of classes as well as that of wholes composed of
more than one metaphysically simple individual.

40
the statements about the stones are no more about true
units than were the statements about the couple. Now if
a set of statements about true units can be found equiva—
lent to the statements about the stones. then these latter
statements are shown to be meaningful in the ultimate meta-
physical sense. otherwise they are not. But for such a
translation it is not sufficient to develop a method where-
by statements about finite aggregates can be replaced by
statements about the members of such aggregates. rather. if
a precise and strictly meaningful language about physical
objects is to be developed. then a method of analysis must
be brought forth with which infinite aggregates can be
similarly handled. This is a consequence of the assumption
of the actual infinite. Does Leibniz hold that such an
analysis can, at least in principle, be carried out? I
believe that this question must be answered in the negative.
But in order to establish this result we will have to
examine Leibniz' views on matter.

Assuming that a statement about a stone is to be
analyzed, what can we say about the ultimate constituents
of that stone, and why should it be impossible to replace
the original statement by a series. (perhaps an infinite

series) about the ultimate parts of that stone? First of

41
all, what are the ultimate parts of which a physical object
is composed? These units cannot be particles of matter or
material atoms since "every part of matter is. I do not
say divisible. but actually divided. and consequently the
smallest particle should be considered as a world full of

an infinity of creatures."45

This amounts to saying that
while we can separate spatially the members of a class of
physical objects. we cannot carry out a spatial separation
down to the ultimate constituents. Thus while it is possi-
ble to divide. ad infinitum. any piece of matter. such a
division will not ultimately produce the units out of which
that piece of matter is constituted. Rather. the piece of
matter is given rise to by entities that do not partake of
spatial characteristics at all. Properly speaking, the

ultimate constituents of matter cannot even be said to have

location in space:

Space is the order of coexisting phenomena,
as time is the order of successive phenomena.
There is no nearness or distance, whether spatial
or absolute. among Monads, and to say that they
are collected together in one point or dispersed
throughout space, is to make use of certain
fictions of our mind. by which we try to repre-
sent to ourselves in imagination what cannot be.

 

45Specimen calculi universalis. Wiener. op. cit.,

p. 99.

42

imagined but only understood.46

In short, the true units or monads to which the
subjects of properly constructed sentences must refer, are
not bits of matter and are not amenable to any characteriza-
tion in terms of spatial properties. Rather. matter
"results from" and is constituted by. the ultimate indivi—
duals or monads. "Strictly speaking. matter is not com-
posed of constitutive unities, but results from them. for
matter or extended mass is nothing but a phenomenon founded
on things, like the rainbow or the parhelion, and all

reality belongs only to unities."47

Thus the analysis of
statements about physical objects runs into considerable
trouble since the monads are contained in a physical object
not in the same way in which an element of a set is "in"
that set. Thus it would seem that the rules developed for
the analysis of statements about couples. trios, etc. be-
come inapplicable not only because the sets here considered

are infinite. but because of the peculiar mode of contain-‘

ment of monads in their objects. This has as a consequence

 

46Letter to Des Bosses, Latta. op. cit.. p. 221.

47Russell, op. cit.. p. 243.

43
that the properties of phySical objects must. for the most
part. be considered emergent with respect to the properties
of the monads contained "in" that object, i.e. they are not
predictable from the properties of the contained monads.

This can be demonstrated in the following way. In
the Monadology Leibniz introduces a classification of
monads intosxnfls and other monads. the souls being dis-
tinguished by more distinct perceptions and by the posses-
sion of memory?8 Presumably. inanimate bodies. such as
stones. do not contain souls among their constituting
monads. Now in a letter to De Volder. Leibniz writes that
the essence of the pppl_is to represent bodies.

One gathers from the context that this is a dis-
tinguishing characteristic of souls. i.e. that other monads
do not represent bodies. On the other hand. monads are
defined as units of perception. Moreover. it is asserted
that all monads mirror the universe and therefore in parti—
cular also the body in which they are housed. One might
conclude that, if all the perceptions of any one monad in

any body were known. one would then be able to deduce the

properties which such a body would have when considered as

 

48Monadology, No. 19, Wiener, o . cit.. pp. 536f.

44

a physical or phenomenal object. But this does not have to
be the case. If a stone. for example. does not contain any
souls among its monads. then it does not contain any monad
which perceives the stone spatially in the way in which
souls perceive it. That is to say. since the modes of per-
ceiving the universe differ in the different classes of
monads. it may not be possible to deduce all the phenomenal
properties that the stone has in the experience of a soul
from the way in which it is perceived by the "bare monads"
in the stone.

Outside of perceptions each monad is said to have

materia prima. Materia prima is said to be that element

 

 

by virtue of which bodies resist penetration and locomotion.
As far as I can see. impenetrability and inertia are the
only properties that are found in monads as well as in the
bodies which arise from monads and hence are the only pro-
perties in bodies which are clearly not emergent with
respect to the monads out of which these bodies are con-
stituted.49

There is further evidence for the thesis just pro-

posed. In the Nouvaux Essais, Leibniz points out:

 

49Wiener, op. cit.. p. 161.

45

It is to be observed that matter, taken as
a complete being is nothing but a collection or
what results from it, and that every real collec—
tion presupposes simple substances or real unities,
and when we consider further what belongs to the
nature of these real unities, i.e. perception
and its consequences, we are transferred.:u3 to
speak, into another world, that is into the
intelligible world of substances, whereas beforpO
we were only among the phenomena of the senses.

The locution that we are "transferred ..into another we
world" suggests that Leibniz wished to distinguish as
sharply as possible between the phenomenal and the intelli-
gible world. But elsewhere he is even more explicit:

I believe the true criterion as regards
objects of sense is the connection of phenomena,
i.e. the connection of what happens in different
times and places. and the experience of different
men, who are themselves, in this respect, very
important phenomena to one another...But it must
be confessed that all this certainty is not of
the highest degree....For it is not impossible,
metaphysically speaking, that there should be a
dream connected and lasting as the life of a man.

 

51

Now if the characteristics of bodies were not emer-
gent with respect to the properties of the monads which
constitute the bodies,then the particular certainty which
Leibniz expresses with respect to his metaphysical scheme

would be transferred to statements about phenomena, the

 

50Nouvaux Essais, p. 428.

51Ibid., p. 422.

46
latter being deducible from the former. However. since
statements about the world of experience have only moral

certainty. the characteristics of matter as experienced

 

must be. for the most part. emergent relative to the pro-
perties of the monads in question. On the other hand.
wherever matter is experienced. and wherever such an ex-
perience occurs in an orderly connection with other experi-
ences. Leibniz assumes monads to be present "in" the matter
so experienced. This I take as the meaning of the dictum

that physical objects are phenomena bene fundata. Bene

 

fundata does not mean then that the properties of physical
objects are all inferable from the properties of their
constituting monads, rather. it is to say that physical
objects are not mere illusions, but that. wherever a physi-
cal object is present. there are monads.

It appears that in Leibniz‘ system statements about
physical objects were thought to have a function and
character akin to those attributed to value judgments by
the early positivists; the usefulness of value judgments
was never denied. but they were treated as some sort of
useful nonsense, irreducible to observation statements, i.e.
irreducible to the most fundamental kinds of assertion that

the positivists thought could be made.

47

Similarly with statements about physical objects in

Leibniz: we must make them in order to get around in the

world.

In point of fact. Leibniz is most emphatic in sta-

ting that the metaphysical analysis of matter contributes

nothing to the development of the science of physics and

is in no way required to put physics or common talk about

physical objects on a sure footing. Leibniz writes:

and

I grant that the consideration of these
forms (i.e. substantial forms or monads) is of
no service in the details of physics and ought
not to be employed in the explanation of
particular phenomena.

The physicist can explain his experiments,
now using simpler experiments already made, now
employing geometrical or mechanical demonstra-
tions without any need of the general considera-
tions which belong to another sphere, and if he
employs the cooperation of God, or perhaps of
some soul or animating force. or something else
of a similar nature, he goes out of his path
quite as much as that man, who, when facing an
important practical question, would wish to
enter into profound argumentations regarding
the nature of destiny and of our liberty.

This points again at the wide gap between the

phenomenal world of matter and physics and the intelligible

 

p.

52Discourse on Metaphysics, No. X, Wiener, o . cit.

53Ibid., p. 303.

48
world of substances and metaphysics. But why did Leibniz
introduce this bifurcation?

In Descartes we already find a similar distinction.
There. experience is divided into the veridical experience
of primary qualities and the illusionary experience of what
are called secondary qualities. Descartes undertook this
distinction partly in order to explain certain oddities of
direct experience: auditory experiences without external
stimuli, light sensations without external stimulation,
warm and cold sensations simultaneously from the same
external source etc. Leibniz, however, had different rea-
sons for distinguishing a phenomenal world from an intelli-
gible one. It will be noted that in Leibniz the bifurca-
tion is much more thorough, in that experiences of extension
and duration are also referred to the perceptor rather than
the perceived. While Descartes' distinction was to resolve
epistemological puzzles, in Leibniz the differentiation is
made in order to overcome logical difficulties. in partié
cular both the indicated problems of the continuum. How
is this result attained?

We have seen that statements about phenomenal
objects cannot be translated into statements about true

units or monads. Nevertheless, Leibniz assures us that it

49
is perfectly proper for the scientist to employ modes of
speech that are improper. metaphysically speaking. This
assurance, it must be noted again, is not based on the
assumption that such improper forms of speech can be
replaced by equivalent metaphysically correct expressions,
rather. it hinges on the fact that the phenomenal world
has a certain order in its own right, that the phenomena
occur in orderly sequences and can be described in a
practically satisfactory way without recourse to a descrip-
tion of the constituting monads. But this pragmatic justi-
fication does not guarantee that a description of the
phenomenal world will always be free of contradictions.
Thus, if bodies are phenomenally continuous, as Leibniz
admits, then they have infinitely many parts and thus have
proper parts with just as many parts as themselves. This
contradicts an accepted axiom. Leibniz argues that the
extended when conceived through itself alone. i.e. when
considered as a merely phenomenal object. contains a con-
tradiction. In order to prevent the arising of this con-
tradiction, all statements about physical (phenomenal)
objects are declared meaningless in the strictest meta-
physical sense. Hence strictly speaking. the contradiction

cannot be stated.

50

To summarize the results attained to this point:
Leibniz introduces a distinction of phenomenal and intelli-
gible world in order to remove a logical difficulty. For,
if any piece of material substance is considered as a
"veritable whole", then a contradiction arises since this
piece of matter is (at least phenomenally) continuous and
has therefore infinitely many parts. Consequently, under
this assumption it would have a proper part with just as
many parts as itself, which contradicts Euclid‘s ninth
axiom. Hence a piece of matter. or any extended substance
cannot. according to Leibniz, form a "veritable whole", its
wholeness is merely illusionary and depends on the parti-
cular mode of representation which characterizes souls.
Now any statement which presumes that a piece of matter
forms a whole, i.e. which makes an assertion about such a
piece of matter is. strictly speaking, nonsense, so that
the threatening contradiction cannot be stated in the pre—
cise language which requires that the subjects of sentences
must refer to veritable wholes.

While it cannot be said that Leibniz developed a
sort of type theory, the strategy is strikingly similar to
that employed by Russell. Russell, too, disallows certain.

statements which, at first sight seem quite inoccuous, for

51
example "the class of cats is not a cat", and his stipula-
tion to consider as nonsense utterly respectable sentences
of the natural language in order to keep contradictions
from arising seems already to have been employed by Leibniz.
But further parallels are hardly justifiable. For one
thing. Leibniz' theory contains a good deal of metaphysics.
His entities "of lowest type", the monads, are described
in some detail, while in Russell's theory it is not further
specified of what particular character the individuals on
the lowest level are; or even whether there is an absolutely
lowest type level. Furthermore. Leibniz' theory is dis-
tinguished from that of Russell by the fact that statements
which violate the prescriptions for the precision language
of monads nevertheless can be justified pragmatically,
which allows on the one hand free use of the language of
phenomena and on the other hand explains the ultimate
insufficiency and inconsistency of this language by its
essential imprecision.

It seems that what we have called the first problem
of the continuum is resolved in a very interesting way by
the provisions which have been discussed above. But I have
already pointed out that the so-called labyrinth of the

continuum actually consisted of two separate problems. The

52

second problem had to do with composition of the continuum
and its ultimate elements. Now in the intelligible world
of monads no continuum is to be found. Leibniz points out:
In actuals there is nothing but discrete

quantity, namely the multitude of monads or

simple substances, which is greater than any

number whatever in any aggregate whatever that

is sensible or corresponds to phenomena.54

Thus, continuity can be attributed only to phenomenal
objects, and investigations of these phenomenal objects
must be carried out in the imprecise language that is
inevitable with them. However, one might object that the
problem of continuity is really quite divorced from con-
siderations of what is real, and that it has only to do
with certain order types or with a discussion of the kind
of entity that fulfills the axioms of geometry. Under this
view the latter is also a purely abstract consideration
which is totally divorced from any question concerning the
nature of reality. I think that these objections are quite
well taken. Only, in the 17th and 18th century geometry
was not usually considered an axiomatic system in the

modern sense, but a description of space. For Leibniz this

meant that geometry had the task of describing the

 

54Russell, op. cit., pp. 245f.

53
phenomenal world in certain Of its aspects.

A word of caution must be uttered at this point.
"Phenomenal" for Leibniz did not mean the same as "per-
ceivable". When he defines in a passage already quoted in
the introduction a continuous line as one where there is a
point between any two points. he is speaking about the phenom-
enal world. But this does not mean that there is a visually
distinguishable point between any two other visually dis-
tinguishable points. Rather. this dictum describes a non-
perceivable feature of lines; in other words. not all charac-
teristics of phenomenal objects are perceptible. This assump—
tion. soon to be removed by Berkeley. seems to make it more
accurate to say that the phenomenal world is ordered accord-
ing to the principles of geometry. and that we can perceive
only the grosser features of this arrangement. This means
that the theorems of geometry. although descriptive of
phenomenal space, are not arrived at through the empirical
observation of the phenomenal world. Rather. it is asserted
apodicticafly that the phenomenal world conforms to the
theorems of geometry. even in those parts that are beyond
observation. Leibniz points out that "in order for there
to be any regularity and order in Nature. the physical must

be constantly in harmony with the geometrical."546

 

54aLetter to Varignon, Wiener, o . cit.. p. 185.

54
Thus the fundamental assumption that this universe
is well arranged, "the best of all possible", leads Leibniz
to conclude that geometry must apply without restriction to

the physical world, or. to be more specific, that Euclidean

 

geometry thus applies. Now the lines. surfaces and bodies
discussed in this geometry are said to have the property of
continuity: "Geometry is but the science of the continuous"55
Leibniz points out. Now since geometry and physics are in
harmony. physical bodies must also be considered continuous,
and this harmony. which is considered a priori certain. would
not obtain "if wherever geometry requires some continuation.
physics would allow a sudden interruption".56
It must constantly be borne in mind that geometry
does not describe the world as it Lg, but only as it appears.
or. as Leibniz puts it on another occasion: "continuous
quantity is something ideal. which belongs to possibles

and to actuals considered as possibles."57

It may
have seemed to Leibniz that the eviction of the problem

of the continuum from the realm of actuals robbed it of

 

551bid.

56Ibid.

57Russell. op. cit.. p. 246.

55
its metaphysical sting: there simply is no continuous enti-
ty and thus the question as to the composition of the con-
tinuum -- if there were one -- becomes rather "academic".
But this did not prevent Leibniz from addressing himself to
this problem: namely what sort of thing is it that is
demanded by Euclidean geometry or, to put it otherwise,
what is the composition of the phenomenal continuum of
physical bodies.

According to Leibniz, when a representation of a
continuous quantity takes place in a mind, there is no
corresponding continuous quantity "out there". This follows
from the distinction of the phenomenal from the intelligible.
Hence all that can be said when a mind has such a represen-
tation is that a "notion" is present in that mind which may
be occasioned through, or occur coincidentally with, the
presence of monads. Now one cannot speak of the parts of
a notion in the same sense in which one speaks of the parts
of an object in everyday discourse. A distinction must be
made between what may be called the analysantia of a nOtion
and the parts of an object. This Leibniz points out in
the following passage:

Several peOple who have philosophized, in

mathematics, about the point and unity, have
become confused, for want of distinguishing

56
between resolution into notions and division

into parts. Parts are not always simpler

than the whole,5§hough they are always less

than the whole.

It seems to be Leibniz‘ contention that any given
continuum, being nothing but a notion, has no parts which
are simpler than itself. This, it seems to me, amounts
to saying, that "continuum" cannot be defined through
terms denoting entities that are contained "in" continua.
This conviction, of course, is bound up with the notion
that in a definition the definiens must always consist of
concepts that are simpler in some sense than the concept
in the definiendum. Leibniz seems to have assumed that
parts of a continuum are never simpler notions than the
original continuum itself, supposedly because they have to
be described as l/2 or 1/4 of the original quantity, etc.
Thus the question as to what the phenomenally simple con-
stituents of the continuum are, is invalid if what is
asked for are entities simpler than the continuum itself;
there are no such things.

Nevertheless, divisions of a continuum can, of

course, be made. But Leibniz contends "there are no

 

58Russell, op. cit., p. 246.

57
divisions in it but such as are made by the mind, and the
part is posterior to the whole".59 This amounts to saying
that in a continuous quantity any assignable number of
divisions can be carried out, but the resultant parts will
never be the constituents of the continuous quantity, if
by "constituents" we mean parts that give rise to, or are
simpler than, the whole which they compose. This is the
strategem by which Leibniz wants to resolve the problem of
the composition of the continuum: the continuum is declared
not to have parts that can be said to compose it. Rather,
its parts are posterior to the whole. However. one might
justifiably ask what the difference is with respect to the
paradox of Galileo between infinite sets whose elements are
defined through the set and sets that are defined through
its elements. It seems that this distinction makes no
difference whatever. Thus. if a line has infinitely many
parts which are posterior to the whole, the paradox arises
just as much as it does if the parts are anterior to the
whole. It seems to me that it is not even necessary fully
to understand the anterior-posterior distinction here,

introduced by Leibniz in order to see that the stragegem

 

59
Russell, op. cit.. p. 245.

58
is of no avail. The paradoxof Galileo. successfully
defeated in the intelligible world rears its ugly head in
another world.

However, I believe that Leibniz can be defended. If
.the divisions of a continuum are, so to speak, man made,
then one cannot say that a continuum has any more parts
than are actually produced. and this number is, of course,
always finite. If this is what Leibniz had in mind, then
we may say that the problems of the continuum are, after a
fashion, removed. At any rate, the solutions which Leibniz
offers and which I have attempted to indicate on the pre-
ceding pages are more ingenious and comprehensive than any
subsequent ones down to the time of Bolzano.

One of the questions that remains to be resolved is
how Leibniz justified the fact that geometricians, he him—
self not excluded, availed themselves freely of the notions
that a line is infinitely subdivisible, that it contains
an infinite number of infinitesimals etc. The justifica-
tion for these procedures is entirely pragmatical, as I
have indicated above, and seemed to satisfy Leibniz fully

In the New Essays, Leibniz writes:

 

You (Philalethes -- Locke) deceive your-
self in wishing to imagine an absolute space,
which is an infinite whole composed of parts;

59

there is none such. it is a notion which implies

a contradiction. and these infinite wholes. and

there opposed infinitesimals, are used only in

the calculations of geometers. just like the
imaginery roots of algebra.

Of course, that these calculations are successful
is patent to every observer, and therefore Leibniz
seemed to believe the fictions contained in the calcula-
tions must be condoned. Some commentators seem to believe
that Leibniz, in claiming that there really are no
infinitesimals already had an inkling of the foundations

of the calculus that were to be laid much later with

Couchy and others. Boyer. in his Concepts of the Calculus

 

writes:

A rather inexact tradition would impute
to Leibniz a belief in actually infinitesimal
magnitudes. However Leibniz himself. in a
letter written about two months before his
death, said emphatically that he 'did not believe
at all that there are magnitudes truly infinite
or truly infinitesimal'. These conceptions he
regarded as 'fictions useful to abbreviate and
to speak universally'.61

The preceding investigation points at the sense in
which this remark is to be taken. Of course there are no

infinitesimals in reality. The intelligible. real world

 

6ONouvaux Essais. p. 163.

61Carl B. Boyer, o . cit.. p. 219.

60
contains no such things. But as far as the phenomenal
world is concerned they can be justified, so Leibniz seemed
to believe, just as much or just as little as other geo—
metrical entities, such as continuous lines. Thus the
repudiation of infinitesimals was not brought about
through predominantly mathematical considerations, but
followed from a certain metaphysical position, which in
turn, it must be pointed out again, was adopted in order
to overcome certain logico-mathematical difficulties.

Let me now briefly indicate the significance of
the foregoing investigation. I have attempted to show
that Leibniz' metaphysical system was adopted, to a large
extent, in order to overcome certain logical difficulties.
If I have succeeded in doing this, I have at the same time
shown that here is a system of metaphysics in which cer-
tain seeming logical paradoxes cannot arise. and I have
reason to suppose that other metaphysical systems of the
same and subsequent eras were developed with similar pur-
poses in mind.

Of course I have not attempted to give an exegesis
of Leibniz' entire system, but a great number of its
features have been discussed. and a great number of others

would seem to follow immediately from the points mentioned.

61

For example in the beginning of the Monadology Leibniz

 

writes:
There is also no way of explaining how

a monad can be altered or changed in its inner

being by any other creature. for nothing can

be transposed within it. nor can there be

conceived in it any internal movement which

can be excited. augmented, or diminished

within it, as can be done in composites.

where there is change among the parts. The

monads have no windows through which anything

can enter or depart.

Thus it can be seen that the assumption of actual
independence of the created monads from one another is an
outgrowth of the View that the monads have no parts. an
assumption which we saw above to be indispensible for the
resolution of the problems of the continuum. The lack of
windows in monads, in turn. requires the introduction of
the notion of the preestablished harmony in order to explain
the succession and presence of representations within the
individual monads. Thus other parts of Leibniz' system,
although not themselves required to resolve the problems
of the continuum, would seem to follow at least in Leibniz'

opinion, from the assertions which were made in order to

keep those problems from arising. This gives us all the

 

62Monadology. N). 7, Wiener, o . cit.. pp. 533f.

62
more reason for supposing that those parts of the system
which are connected with the problems of the continuum are
the ones which are closest to the problem for the sake of
which the whole scheme was developed.

The foregoing pages have shown Leibniz' philosophy
to be a very complicated and intricate system. In the
sequel. I shall examine solutions to the problems of the
continuum offered by Bayle and Berkeley which are much more

radical and also much simpler than Leibniz‘

CHAPTER II

GEORGE BERKELEY

I now wish to discuss the ways in which Berkeley
dealt with the problems of the composition of the continuum.
To this end it is necessary to remove certain prevalent
misconceptions concerning Berkeley's philosophy and its
development. For example, Berkeley's interest in mathe-
matics has sometimes been doubted despite the fact that a
very large proportion of his studies are concerned with
mathematical topics. A certain amount of mathematical
interest has sometimes been conceded to him, but his mathe-
matical ability has usually been considered extraordinarily
poor. Hand in hand with these attitudes has gone the
notion that Berkeley's motives for developing his imma-
terialist philosophy were largely of a moralist or theologi-
cal character. Occasionally, too, Berkeley is considered
to be one of the philosophers, in a long line of system
builders, who do not have much regard for extraphilosophical
problems or for the application of a philosophical scheme
to the world at large.

All of these interpretations I hold to be mistaken,

64
and I will try in the sequel to disprove them in detail.
In addition, I wish to show that the resolution of the
problem of the continuum. as far as we can now discern,
was one of the chief motivations for developint his
immaterialistic metaphysics. The consistent application
of his principles led him to develop a geometry that is
very odd. indeed, but the mere fact that the application
of his principles is consistent have led me to consider
his mathematical ability with greater respect than seems
customarily the case among his commentators.

Let us consider first the frequently held view that
Berkeley's immaterialism was inspired mainly by moral and
theologicaldesires. This notion is bluntly, if not very
aptly, expressed by Mr. Butler, who points out in his
Philosophy of Berkeley.

To Berkeley's mind it (i.e. the doctrine
of an external material world) is the main prop

for scepticism, atheism and materialism. As a

good Bishop he is therefore eager to establish
the View that only spirits and their ideas exist.

63
Iflr. Butler has here expressed in a singularly forthright

Inanner what seems to me a widely held opinion concerning

tflae origin of Berkeley's system. Ever since Kant (Critique

63
Benjamin Butler. PhilOSOphy of Berkeley, Boston,

l957,p.2.

66

themselves. since their doctrine has certain inconsistent
consequences which reduce the whole system to absurdity
and should leave the adherent of such a system suspended
in a state of doubt. i.e. should make him a skeptic. It
must be borne in mind that Belkeley's arguments contra
skepticism can be employed by any scientist who replaces
a given scientific generalization by another because the
former lead to inconsistencies or was inconsistent with
(zertain other statements believed to be true. But the fact
'that a scientist can use such an argument. and the fact
"that Berkeley did use it. explains in no way the form and
(zontent of the respective theories.

As to the second point, namely that Berkeley's
inmaterialism was advanced because the contrary doctrine
ssupports atheism, it must be agreed that Berkeley believed
1ihat a major result of his philosophy is the demonstration
<>f the untenability of atheism -- he gives a well-known
I>roof for the existence of God and in general expounds
tliis feature of his philosophy generously and repeatedly.
Eth one cannot sanely assume that this demonstration was
at) only, or even a major, motive for the development of
"tflle immaterial hypothesis." Moreover. many Christian

aPologists have made it their aim to disprove atheism

 

65

of Pure Reason B 71) mocking critics of Berkeley have felt

called upon to refer to him as "the good Berkeley" or "the

good Bishop Berkeley". It may not be out of place to point

at the exceptional inappropriateness of this faintly super-

cilious epithet where Berkeley's philosophical insight is

discussed. It must furthermore be pointed out that Berkeley

only became a bishop more than two decades after the

Principles were published. To be sure. at the time of the

publication of the Principles he was ordained. but the
ordination was a routine part of the academic career of
the fellows at Trinity.

Let us consider Butler's explanation of the origin
of Berkeley's immaterialism in some detail. First it is
said that the contrary doctrine, namely that there is an
external material world. furnishes support fOr skepticism.
One might be lead to think that skepticism was rampant at
lBerkeley's time, and that he set out to develop a theory

tfliat would not give rise to it. But such clearly was not

tflie case. The first dialogue between Hylas and Philonous
InaJces it amply clear that the man alleged to be a skeptic
VWBS jBerkeley himself and that, by way of a counter attack

he arttempts to establish that those adhering to the doc-

tririea of an outside material world ought to be skeptics

67

without thereby being led tova philosophy of immaterialism.
The third point made by Mr. Butler. namely that Berkeley
advocated his immaterialism because it would disprove
materialism is too trivial to need any comment. In summary,
the notion that Berkeley was posed against materialism be—
cause of certain religious and moral convictions concerning
its connection with skepticism and atheism gives no reason-
able explanation of the development of Berkeley's system.
More serious students of Berkeley have attempted
to "explain" the genesis of hissuetam by pointing to his
dependence upon Locke and Malebranche 64 These eminently
worthy pieces of historical scholarship suffer from the
one-sidedness of treating philosophy as something like an
intramural affair. a viewpoint that is perhaps appropriate
for a discussion of the German Idealists. but certainly out
(3f place where Enlightenment philosophers are under dis-
cuission. There can be no question that Berkeley was influ-
ernced by both Locke and Malebranche, but the mere tracing

of .lines of historical development will not answer the

Queustion why Berkeley differed so significantly from both

64Cf. R. I. Aaron, "Locke and Berkeley's Commonplace
300k", Mind, N.S. Vol. 40, 1931, pp. 439-459. A. A. Luce,
§2££§23gey and Malebranche, London, 1934.

68

these philosophers. Obviously, there were certain diffi-
culties which, to his mind, neither of these systems
resolved. A reconstruction of the context of the develop-
ment of Berkeley's philosophy can therefore not be complete
without an identification of these problems.

A third group of commentators who concerned them-
selves with the genesis of Berkeley's system is formed by
those philosophers who are reluctant to search at all for
a reason behind the adoption of the immaterial hypothesis,
who treat this adoption as some sort of spontaneous, uncon-
sidered act. This orientation leads to the assumption that
Berkeley's immaterialism must be considered as something
like an unmotivated fabrication whose author subsequently
tried, with considerable difficulty, to reconcile it with
111s experience and the body of knowledge that he had
:therited. Johnston expresses this viewpoint most clearly:

No harsh Socratic maieutic was needed to

bring it (the New Idea) to the birth; it came

to light easily and almost imperceptibly, and

as we scan the sentences in which Berkeley

indicated the process, it is easy to sympathize

with his joy and surprise as he gazes at the
child of his mind.6

65G. A. Johnston, The Development of Berkeley's
W, London, 1923, p. 20.

69

This naive view has at once the most adverse con-

sequences as regards the appreciation of Berkeley's system.

Immaterialism becomes an apergu, a mere playful invention,

which can hardly be justified in view of its results upon

geometry. Johnston writes:

His willingness to throw overboard the
solid achievements of the established geometry
simply because they did not accord with an

apergu of his own does not encourage to rate
his mathematical ability very highly. 6

I believe that Johnston's approach to Berkeley's

philosophy is misguided. His way of considering philoso-

phical systems would have been appropriate for a discussion

of some German philosophers of the early Nineteenth Century
whose systems can perhaps be described as free creations.

It is known that Schelling occasionally produced philosophi-

cal systems so rapidly that some of his products were

:superceded by others before they had a chance to appear

.in.print. The editor of one of the editions of Hegel's

Eflailosophy of Law attacks critics of Hegel who point out
trust Hegel's system does not agree with the facts by say-

ing; that "systems can only be refuted through other sys-

tenus "67 Implicit in this statement is the notiOn that

66Op. cit., p. 90.
67Hegel, Werke, 2nd. ed., Berlin 1840, Vol. 8, Ed.
Eduard Gans, p. xrv.

u,.

 

 

70
system-building philosophers are not concerned with problems
found in the sciences, in mathematics. or in everyday life.
and that therefore an appeal to facts cannot lead to a
refutation of their systems. For the philosophers of the
Enlightenment it can generally be said that their systems
had the purpose of resolving certain problems which are
encountered in everyday experience. in the sciences or in
mathematics. It is. therefore. of the turmost importance
that the commentator become aware of the extraphilosophical
problems which are the raison d'étre and mainspring of
these philosophical systems. A deficiency in this respect
is a very serious failure. Let me remark here parentheti-
cally that Cartesianism was thrown into its deepest crisis
‘Vhen it was discovered that a reasonable system of physics
rnust assume that a force is required to change the direction

(of a motion. The philosophers of that age took this extra-
pfliilosophical discovery seriouSAenough to abandon all attempts
of? rescuing Descartes‘ unsatisfactory attempts at explaining
the; interaction between mind and body.

Johnston's psychological thesis of spontaneous creation
has .its systematic counterpart in the notion that Berkeley's
fundennental assumptions have none or "few" arguments in its

favoir. This view is expressed by C. R. Morris, who writes:

71

"He (Berkeley) expounded his system boldly and shortly,
offering in the first place very few arguments in support
of it except that of its obviousness."68 Since Morris
never states what Berkeley does in the second place, and
since he never states the few arguments aside from obvious—
ness, we must assume that he was unable to find any. It
seems likely that Morris cannot find any such arguments
because he is, so to speak, looking in the wrong direction.
Let me insert at this point a general consideration
as to what kind of argument may be required in order to
make a philosophical system or thesis plausible, and where
one should look to find "arguments in favor" of a given
thesis. It seems that what Morris is looking for are ante-
<:edently established propositions of which the thesis or
asystem is a deductive consequence. But this procedure,
arlthough frequently employed by commentators. is not always
iri order. since it is not always possible to produce argu-

nmnits of this sort. But if no such premises are offered,

it cioes not. therefore. follow that the system in question

68C. R. Morris, Locke, Berkeley, Hume, London, 1931,
P. 625.

bus. huff-I - I. h... _.

 

72

has none or "few" arguments in its favor.69

I submit that Berkeley's system is to be envisaged
in much the same way as a high level scientific generaliza—
tion Its support is to be found in its deductive con-
sequences and their truth rather than in premises from
which it follows, There is much evidence in Berkeley's
writings to the effect that he wanted his philosophy to be
considered in this light. In the Philosophical Commentaries,
entry no. 207, he states:

My end is not to deliver Metaphysiques
altogether in a General Scholastique way, but

in some measure to accommodate them to the

Sciences and show how they may be useful in

Optiques, Geometry etc.

Generally, the statements in which Berkeley attempts
to justify the immaterial hypothesis always stress the fact
that the consequences of his thesis are more desirable,

explain more, or are less paradoxical, than the consequences

of the rival materialist thesis. For example, Berkeley's

 

69(Of course, I do not wish to claim that the pro—
cedure criticised apropos of Berkeley's system is always
out of order. If a philosopher claims that the evidence
for his system rests with a number of "self-evident” pro-
positions, then his claim will have to be examined. But
Berkeley never argues in this vein).

70A. A. Luce and T. E. Jessup, (eds.), The Works
of George Berkeley, London, 1948, Vol. I, p. 27.

 

73
claim that the problems of the continuum do not arise in his
system is well enough known. It must here be emphasized
that this alleged (ctmappearance of the problems of the
continuum is one of the results of the immaterial hypothesis
and is as such offered in support of the thesis.

The analogy between certain philosophical systems
and scientific generalizations will help us to clarify
another point. Scientific generalizations are generally
put forth in an attempt to remove certain problems, to ex-
plain certain lower level propositions. These problems, or
the assertion of such propositions, can be said to occasion
the higher level generalization. While the occasioning of
such problems is, perhaps. of small logical importance, its
recognition is indispensible for the reconstruction of the
context of discovery. For the philosophy of science such
reconstructions may be dispensible.‘ Not so in the history
of philosophy. Berkeley is a case in point. On a success-
ful reconstruction of the context of his discovery of the
immaterial hypothesis hinges the answer to such questions
as "was the immaterial hypothesis an ppegpp_that was only
subsequently and with difficulty reconciled with experience
and mathematics?“ or "was it a hypothesis specifically

introduced in order to resolve certain difficulties in our

74
account of experience or mathematics?" I am convinced that
the latter is the case. and that the problem of the con-
tinuum was the particular occasion for Berkeley's introduc-
tion of the immateralist hypothesis.

It is now incumbent upon me to demonstrate in what
way the problem of the continuum influenced the development
of Berkeley's metaphysics and to what solution the problem
was brought. Berkeley's solution is familiar in kind to
those of some modern analytic philosophers: he declares
the problem of the continuum to be a pseudo-problem. Con-
tinuous quantities had been described as being infinitely
divisible i.e. as having more than a finite number of parts.
Galileo had shown, and Berkeley knew, that infinite sets
have proper subsets with as many members as themselves.
That Berkeley was familiar with this result is attested

to in the following passage from the Philosophical

 

Commentaries. "The infinite divisibility of matter makes
the half have an equal number of equal parts with the

'71 , , . . . . .
whole.‘ This is at variance w1th Euclid's ninth aXiom.
Hence the difficulty. In Berkeley it is resolved, as we

shall see, through the assumption that there are no

 

71
Entry 322, The Works of George Berkeley, Vol. I,

p. 39.

75
entities which fit the description given above of continuous
quantities. Hence, if there are no entities of this sort.
then the question what it is that they are composed of is
invalid, is a pseudo—question.

Berkeley's solution may here be sketched in a few
words. The problem is construed as being concerned with
the composition of extended sensible objects. These can
be said to be continuous in appearance. "continuous" here
referring to a certain phenomenal characteristic. I
hesitate to hazard a definition of "continuous", but by
way of example it can be said that a figure is visually
continuous if it is uniformly colored. Similar criteria
inay be set up for phenomenal continuity in the tactile
field etc. and eventually one may arrive at a definition
c3f phenomenal continuity in general. However, since
Iierkeley nowhere attempts to produce such a definition, a

rweconstruction of his system runs into rather formidable
<djgfficulties on this point. We have to be satisfied with
a :rather vague notion of phenomenal continuity.

According to Berkeley, any phenomenally continuous

fisnire, line, etc.. is composed of a finite number of

SmaJJlest phenomenal units, called minima sensibilia.

 

flifléiflg sensibilia can be divided into minima visibilia,

. v. I c h f‘ I) 7"
n — w 5 fl» n.s

:5-

 

 

76

minima tangibilia etc, Since space and extension are con-

structions dependent only upon the senses of sight and
touch. the minima Visibilia and tapgibilia are the only

kinds of smallest unit of perception with which we need be

concerned at this point.

It is a consequence of the principle that esse est

Dercipi that entities smaller than minima sensibilia cannot

The said to exist. Now since the number of minima sensibilia

in a figure is finite. and since no figure can be said to

'have parts smaller than a pinimum sensibile it cannot be

asserted within the framework of Berkeley's system that

eany figure has more than a finite number of parts. Thus

tlie paradox of Galileo cannot arise. Moreover, in Berkeley's

s3zstem geometry is described as a science whose task it is

tc> describe the various relations obtaining between con-

tiJauous figures and lines. etc. Geometry is thereby made

ari emmflxical science. It is to discuss certain aSpects of

serrscuy'experience. It does not concern itself with

ixufiJiitely divisible lines and figures. but only with the
Phenomenal pseudocontinuum. It seems that Berkeley took
serixbusly the theory that geometry is concerned with des-
Cril>idig the space of our experience, and it is perfectly

JUStifiable to say that this space contains nowhere any

I1

. ‘
~\u

 

 

77

subspace of which it can be empirically ascertained that it

it has infinitely many parts.

A serious criticism of Berkeley's views on this sub-

ject can only proceed from the assumption that it is not

the aim of geometry to describe empirical space. The

construction of such a geometry, of course, is in part mm.

arbitrary and, in particular, it can be so constructed that

it speaks of spaces. surfaces and lines which have infin—

itely many parts. It speaks well for Berkeley's mathe-

matical ability that he was willing to countenance such a

geometry, but, since it would largely consist of asser-

tions not referring to matters of empirical fact, he places
aagainst it a moral injunction, to wit. that it is not worth-
vihile to investigate matters without practical value. If

ciifficulties and inconsistencies arise in such a geometry,

tliey can no longer be said to arise from an inability to

h‘o

arialyze matters of empirical fact. We could summarize

Berkeley's opinions on this subject by saying that it was
Of small concern to him that some mathematicians have con-

StIWJCted a system called geometry, the difficulties of

Whitih.they were unable to resolve. On the other hand, he

ClaiJns that:

Whatever is useful in geometry, and

78

promotes the benefit of human life, does still
remain firm and unshaken on our Principles;

that science considered as practical will rather
receive advantage than any prejudice from what
has been said...For the rest, though it should
follow that some of the more intricate and
subtle parts of Speculative Mathematics may

be paired off without any prejudice to truth,
yet I do not see what damage will be thence

derived to mankind.
The above gives in essence Berkeley's solution to

the problem of the continuum. In summary, what is con-

sidered is not the continuum as later defined e.g. by
Dedekind, but a phenomenal continuum not further defined.

.Any figure that is continuous in this sense is said to con-

sist of a finite number of minima sensibilia, smaller than

Vihich nothing can be. This result follows from the princié

gale that esse est percipi. This being the case there can

hug no actual infinite, and the paradox of Ce ileo cannot

.airise with respect to actuals. Thus geometry as the science

c>fF empirical space is vindicated. Against "speculative

geometry" Berkeley places a moral injunction. We thus find

ill lBerkeley the same attitude that we could observe in

LGiJJniz: the problems of the continuum were felt to be

72Principles of Human Knowledge, No. 131. Wherever
Ber1<erley has provided a numbering or the paragraphs, I
Shalml quote by paragraph number, without stating edition

or Page.

79

severe only as long as they arose in connection with a
description of the world. Not nearly as much significance
was attached to them as problems of pure mathematics.

After having thus outlined Berkeley's solution, I
will now endeavor to demonstrate that among those problems
which Berkeley initially attempted to solve with his rue
immaterial hypothesis the problems of the continuum and
of the infinite divisibility of matter have a prominent
place.

Let us. first of all. turn to a passage in the hr

Principles in which Berkeley sets forth some of the

 

"innumerable consequences. highly advantageous to true
lphilosophy as well as to religion" which flow from his
Iarinciples. He points out:

If by distinguishing the real existence

of unthinking things from their being perceived.

and allowing them a subsistence of their own,

out of the minds of spirits, no one thing is I
explained in nature, but on the contrary a ‘
great many inexplicable difficulties arise;

if the supposition of matter is barely pre-

carious, as not being grounded in so much as

a single reason; if its consequences cannot

endure the light of examination and free
‘enquiry, but screen themselves under the dark

and general pretence of infinites being incom-

prehensible; if withal the removal of this

matter be not attended with the least evil

consequence; if it be not even missed in the

world, but everything as well, nay much easier

conceived without it; if. lastly, both sceptics \\

 

80

and Atheists are forever silenced upon supposing
only spirits and ideas, and this scheme of things
is perfectly agreeable to both Reason and Religion:
methinks we may expect it should be firmly em-
braced, though it were proposed only as an hypo-
thesis, and the existence of matter had been
allowed possible: which yet I think we have
evidently demonstrated that it is not.

I think that we can expect this passage not only to
supply us with arguments for the acceptance of the imma-

terial hypothesis. but also with a reason why it was

adopted in the first place. Three such reasons are given

here. First, the contrary assumption. namely that there is

matter independent of mind. leads to difficulties with the

infinite and thereby to contradictions concerning the
(:omposition of the continuum. Secondly. the adoption of

tihe immaterial hypothesis has no evil consequences. And

-t}airdly. the adoption of this hypothesis will silence the

.sczeptics and atheists forever. Clearly. not much weight

Gian.be attached to the second of these reasons.

Fflaeer will concede that his system has evil consequences,

EUiC! one can hardly be expected to accept inoccuous nonsense

Imerwely because it is harmless. .The third of the stated

reasons has customarily been brought forth to explain

 

73Berkeley. Principles of Human Knowledge. No. 133.

No philoso—

81

Berkeley's motivation for developing his metaphysics. How-
ever. while it stands to reason that Berkeley would not
have accepted or developed a doctrine that runs counter to
the tenets of received religion. I believe that it suffi-
ciently established that the fortification of religious
arguments was a welcome consequence rather than the problem
originally to be resolved. There remains the third point
mentioned, and I believe that from a consideration of
Berkeley's early writings it can be shown that the immaterial
hypothesis was adopted primarily in order to cope with the
problem of the composition of the continuum.

Some questions of chronology have to be considered
at this juncture. Three of Berkeley's early writings have
to be discussed to settle the issue. namely his Arithmetica

and Miscellanea Mathematica.74 his small essay Of Infinites75

 

 

(and his Commonplace Book or as Luce has renamed it. ghiloso—

Eflgical Commentaries.76 The composition of all three of

74The Works of George Berkeley, Ed. A. A. Luce and
'P; E. Jessop, London 1948 ff., Vol. IV, pp. 159ff. (Where-
evwer Berkeley has not supplied paragraph numbers, I shall
(TUCJte according to the Luce - Jessop edition).

751bid.. Vol. IV, pp. 235—238.

761219.. Vol. I. pp. 1-139.

82

these probably falls into the years 1704—1708. The

Arithmetica and Miscellanea Mathematica were published in

 

1707. but supposedly composed three years earlier. 9;

Infinites is a small paper that was presumably read before

 

a philosophical discussion group. It is undated. The
Philosophical Commentaries are a diary in which notes are uw~
taken, obviously in preparation of the writing of the New

Theory of Vision and the Principles. There are no dates

 

in this diary either. The immaterial hypothesis is found
almost at the outset of it. and one can assume that it was
started with the idea of developing the consequences of
this thesis.

I think that it can be shown that the essay 9;

Infinites and the Arithmetica etc. preceded the diary and

 

‘the inception of the immaterial hypothesis. and that

IBerkeley's concern with the questions of geometry and the

p.10.

<:ontinuum were not resultant upon the development of his
tihesis but preceded it. Luce writes the following about
tine Arithmetica and Miscellanea Mathematica:

The work has the look of a fellowship thesis,
specially designed to remove the impression that
he disliked mathematics. We know from the
Philosophical Commentaries that Berkeley about
this time held very scornful views about some
mathematics and some mathematicians, and if he
did not keep those views to himself. College ~\

83
gossip would soon make him out ignorant of
mathematics or prejudiced against them, and
such gossip might weaken his chances of election.
We see that even Luce. otherwise a very competent

commentator, had considerable doubts whether Berkeley had
any genuine interest in Mathematics. In this case the
immaterial hypothesis is said to force him. for reasons
having to do with his career. to undertake a study of
mathematical subjects. But Luce's assumption clearly rests
on very shaky grounds. We have no reason to mistrust

Berkeley when he states in the introduction to the essay

that most of the parts of the work "triennium in scriniis

 

delituerint".78 Now the fellowship which Berkeley wished

 

to fill fell vacant in 1706 and was to be filled in 1707.
We are hardly justified in assuming that he wrote an essay
in 1704 in order to convince the dons in 1707 that he did
not dislike mathematics. It seems much more natural to
assume that the work was composed out of true interest for
mathematical subjects. The publication date may have been

decided upon for career reasons, but it is not likely that

 

77Editor's‘Introduction to the essay in The Works
9; George Berkeley, Vol. IV, pp. 159f.

 

78The Works ofTGeorge Berkeley. Vol. IV, p. 167.

 

 

84

the same holds for the content. The essay itself
I do not wish to discuss here. It is nowhere concerned
with the problems of the continuum. I merely mentioned it
to lend emphasis to the theory that Berkeley showed interest
in mathematics prior to the inception of the immaterial
hypothesis.

Let us now turn to the second of the above mentioned
writings. I think that a strong case can be made for the

assumption that Of Infinites also preceeded the immaterial

 

hypothesis. Luce again believes the contrary. Apparently.
to some commentators Berkeley is unthinkable without
immaterialism. Luce writes in his "Editors Introduction"
to the essay:
Here is no abstract mathematical problem

accidentally connected with Berkeley's studies.

It touches the heart of his philosophy. and is

vitally connected with the massive argument for

immaterialism.

and in his Life of George Berkeley he points out:

 

The mathematical doctrine of infinite
divisibility was commonly regarded as furnishing
evidence for the existence of matter; hence
Berkeley's special interest in infinitesimals.

79Ibid., p. 233.

80A. A. Luce, The Life of George Berkeley, London.

-l949, p. 36.

 

 

‘10‘V.4W0‘t .OMO‘H -. . % J..d.un~_.£.

 

85

It is clear from these two quotations that Luce
assumes the small essay to have been written as a result
of Berkeley's concern with immaterialism. However, there
is no indication whatever in the essay that Berkeley so
much as toyed with the notions that were to form the core
of his philosophical system. In fact. the theories expounded
in the essay are in part incompatible with his later argu—
ments. as I will now show.

Of Infinites shows Berkeley's disinclination to
accept the thesis that the continuum is infinitely divisible,
but beyond this it merely indicates in a vague way that the
problems of the continuum might be removed through a certain
terminological distinction. To begin with he quotes Locke
to the effect that a distinction must be made between the
idea of infinity of space and of space infinite. With
Locke, Berkeley holds that we are able to form the former.
but not the latter. He continues:

Now if what Mr. Locke says were, mutatis
mutandis applied to quantities infinitely small.

it would, I doubt not, deliver us from that

obscurity and confusion which perplexes other-

wise very great improvements of the Modern

AnalySis. For he that. with Mr. Locke, shall

duly weigh the distinction there is betwixt

infinity of space and space infinitely great

or small, and consider that we have an idea

of the former, but none at all of the latter.
will hardly go beyond his notions to talk of

86

parts infinitely small or partes infinitesimae
of finite quantities, and much less of infinitesi-
mae infinitesimarumy and so on.

Berkeley then points out that the "supposition of

quantities infinitely small is not essential to the great

improvements of the Modern Analysis"82 and concludes the

essay by saying:

Now I am of the opinion that all disputes
about infinites would cease and the consideration
of quantities infinitely small no longer perplex
Mathematicians, would they but join Metaphysics
to their Mathematics, and condescend to learn
from Mr. Locke what distinction there is betwixt
infinity and infinite.83

Locke's notion was that we can have no idea of an
infinitely large space or infinitely long time but that,
by contrast, we can have an idea of infinitey, which is

84 Similarly, and Berkeley

an "endless growing idea".
repeats here only another contention of Locke's. we can
liave no idea of an infinitely small extension but.only the

ixiea of a perpetually diminishing or dividing of finite

‘3}(tensions which can never come to an end since any result

81The Works of George Berkeley. Vol. IV, p. 235.

821bid., p. 237.

83Ibid.

84Locke, An Essay Concerning Human Understanding,
Bk. II. Ch. XVII, No. 7.

87

of such a division will still be a finite extension.85

Now the proof that Berkeley must have written this
essay before he conceived of the immaterial hypothesis is

simple enough. After all, the whole doctrine of the

minima sensibilia amounts to saying that the process of
division cannot be continued ad infinitum as Locke pro—

posed and Berkeley suggested in Of Infinites. The sugges-

tion that extension can be divided ad infinitum Was one

of the first doctrines to be rejected in the development
of the immaterial philosophy as recorded in the Philosophi-
cal Commentaries. These considerations show the little
essay quite at variance with Berkeley's later philosophy
and indicate clearly that the thesis must be rejected that
Berkeley concerned himself with the problem of infinite
divisibility in this essay because he was trying to work
out the consequences of the immaterial hypothesis. Luce
luimself states that the doctrine of infinite divisibility
Vvas supposed to furnish evidence for the existence of
nnatter. How can we assume that Berkeley the immaterialst,
<2cmld embrace such a doctrine as he obviously does in Q;

Ikafinites? On the other hand, Of Infinites shows Berkeley

 

85 ‘
Cf. Ibid., Ch. XVII, No. 12.

88
actively engaged in an attempt to resolve the problems of
the continuum. The above considerations should suffice
not only to establish the essay as written before the
Philosophical Commentaries were begun, but also to show
that a solution for the problems of the continuum was
actively sought by Berkeley before the inception of the
immaterial hypothesis.

I have stated above that in the Philosophical

Commentaries the immaterialist hypothesis is found almost

 

at the outset. It is of great importance to notice, how—
ever, that at the very beginning of this diary Berkeley
seems to assume that aside from phenomenal extension and
time there is also a real extension and time. Now
phenomenal extension quite clearly does not admit of
infinite subdivision. and neither does phenomenal time.
¢Dn the other hand, Berkeley seems to assume that real
(Extension and time permit such a treatment, as I shall
Eiubsequently document. He left this position shortly
Eifterwards in favor of his immaterialism. Now if we assume
tflnat real extension and time were rejected solely on the
Etrounds that their existence cannot be empirically demon—
Ertrated or perhaps for no reason at all, then we cannot

eEasily account for the particular form Berkeley's philosophy

 

 

89
took, since a rather simpler form of immaterialism suggests
itself.

It is generally recognized that upon rejection of
"real" extension and time Berkeley had to harmonize his
notion of phenomenal extension with geometry, since the
latter can now no longer be said to have "real" extension
as its subject matter. But unlike the Gospels. immaterial-
ism and geometry can be made to harmonize through the
following device: assume that any idea of visual perception
is produced by God with infinitely many parts of which our
visual acuity can distinguish only a finite number. We can
then pursue geometry as before, since we can always claim
that the teachings of Euclid apply to our ideas as pro-
duced by God, although not as seen through us.

I do not wish to claim that this is a very reasonable
philosophical stand to take, but I believe that a case can
be made for a brand of immaterialism that allows for this
sort of direct reconciliation with geometry. However, I
IDelieve that it would have been quite unacceptable for
13erkeley, since it inherits all the problems of geometry,
Lin.particular the difficulties in connection with the
ilifinite divisibility of the continuum. That is to say

tlaat if we assume that it is the removal of the latter

 

90
which is Berkeley's main concern, then we can understand
why Berkeley adopted his particular type of immaterialism:
immaterialism does not in and by itself prevent the
occurrence of the problems of the continuum, but immaterial-
ism together with the doctrine of minima sensibilia.
smaller than which nothing exists, avoids them.

That Berkeley‘s reasoning followed these lines can
be shown through a discussion of the beginning sections of

86
the Philosophical Commentaries.

 

I have already pointed out that at the beginning
of the 18th century the question_of the composition of the
continuum was rarely considered in abstracto. but was
usually couChed in terms of a problem concerning the nature
of time or physical space. At the outset of the Philosophi-
cal Commentaries Berkeley is concerned with precisely these
problems viz., “what is the nature of time?" and "what is
the nature of extension?". The fourth entry reads "time
train of ideas succeeding each other", and the third, which

861 shall quote the entries in the Philosophical

lgfiflmmentaries according to the numbers given them by A. A.
Ifllce in The Works of George Berkeley, Vol. I, pp. 1-139.

I Shall assume throughout that the order in which Luce
giVes the entries reflects correctly the temporal sequence
of their composition.

91

is obviously also about time. says "Whether succession of

ideas in ye divine intellect."87 Apparently, the flow of

'time is here considered to be formed by a series Of

succeeding ideas. This notion of time is familiar from

the Principles, where Berkeley says:

For my own part, whenever I attempt to
frame a simple idea of time, abstracted from
the succession of ideas in my mind, which
flows uniformly, and is participated by all
beings, I am lost and embrangled in extri-
cable difficulties. I have no notion of it
at all: only I hea~ others say it is infinitely
divisible, and speak of it in such a manner as
leads me to harbor odd thoughts of my existence:
since this doctrine lays one under the absolute
necessity of thinking, either that he passes
away innumerable ages without a thought. or else
that he is annihilated every moment of his life;
both which seem equally absurd.

This quotation from the Principles shows not only
that the characterization of time sketched in the Commen-
taries is retained in the Principles but also that the
(difficulty was with the notion of the infinite divisibility
.itself.. Berkeley recognizes that infinite divisibility
Irresupposes infinitely many parts, and these he does not

CKDncede for finite extensions or time spans. He holds

87The Works of George Berkeley, Vol. I, p. 9.

88Principles of Human Knowledge, No. 98.

92
that if there is any time-span between two thoughts, and
if this time span is infinitely divisible, then it must be
infinitely long. Thus the notion that I must pass innumer—
able ages without a thought. On the other hand, if for
the spirit esse cst percipere, then the assumption that a
spirit does not think at any one time leads to the conse-
quence that he is annihilated at that time. Hence Berkeley
assumes that time for a spirit is measured in terms of
succession of ideas in that spirit's mind. This makes time
private, and it cannot be assumed that all beings partici-
pate in the same time. In the beginning of the Commentaries
Berkeley does not seem to have been entirely satisfied with
this making private of time.‘
So aside from the concept of time in the above sense
he introduces the notion of duration. Entry No. 8 reads
"duration infinitely divisible, time not so."89 There is
no definition given of "duration", but it stands to reason
‘that by this term he meant the intersubjective Newtonian
‘tinmrin contradistinction to the phenomenal time discussed
above. A similar dualism seems to be intended with respect
t<> extension. Entry No. ll says "Extension not infinitely

89The Works of George Berkeley, Vol. I., p. 9.

93
divisible in one sense.90 I would presume that Berkeley
held at the time that there was another sense in which one
could speak of extenSion as infinitely divisible. This
means that time and extensionl are not infinitely divisible,
while duration and extensionz are. Time and extensionl
would be phenomenal time and space, duration and extensionz
would be the abstract time and space of Newtonian physics.
The puzzlements connected with the continuum can occur in
the latter, but they are exorcised from the former, simply
because neither phenomenal time nor phenomenal space form
a continuum. But the problem is not thereby resolved. It
is merely extradited from the phenomenal sphere, and
retains all its severity in the realm of absolute space
and time. It is not until these latter are denied any
existence that the problem of the continuum is made to
disappear. Entry No. 26 states “Infinite divisibility of
extension does suppose ye external existence of extension,
but the latter is false, ergo ye former also.\"91 This

terse modus tollens argument announces the solution that

Berkeley gave to the problem of the continuum: he denied

9°Ibid.

91
Ibid., p. 10.

94
that there is absolute space or duration, and assumed that
the phenomenal time and space are not infinitely divisible.

However, the doctrine here brought forth is not with-
out its difficulties, as Berkeley well realized. For one
thing, its repercussions on classical geometry become at
once apparent. Entry No. 29 says "Diagonal incommensurable
with ye side Quaere how this can be in my doctrine?"92

This entry bespeaks the fact that at this point
Berkeley still meant to bring about a reconciliation of
his doctrine with classical geometry. The entry in fact
asserts the incommensurability of the diagonal with the
side, but the inconsistency of this theorem with his doc-
trine is realized. How does he resolve the inconsistency?.
A large number of entries deal with the problem,93 but of
especial interest are entries 263 and 264. 263 states:
"Mem: To enquire most diligently Concerning the Incommen-
surability of Diagonal and side. Whether it does not go on
the supposition of unit being divisilbe ad infinitum, i.e.,

of the extended thing spoken of being divisible ad

 

92Ibid.

_ 93See Nos. 249, 250, 258, 263, 264, 276, 340, 457,
'469, 470, 481, 500, 510, 516.

95
infinitum..."94 and the next_entry states flatly: "The
Diagonal is commensurable with the Side."95

Here the decision is made. To achieve consistency
classical geometry is declared in error. Of course, this
not only holds for particular theorems of geometry but for
the whole mode of approach associated with Euclidian geo-
metry. ACCOlding to Berkeley geometry is to be transacted
as an empirical science. This becomes evident in entry 249:

Particular Circles may be squar'd, for

the circumference being given, a Diameter

may be found betwixt which and ye true there

is not any perceivable difference, therefore

there is no difference. Extension being a

perception and perception not perceived is a

contradiction, nonsense, nothing. In vain to
allege the difference may be seen by Magnifying

Glasses. For in that case there is ('tis true)

a difference perceived but not between the same

ideas but others much greater entirely different
therefrom.96

This passage is remarkable not only because of the
assertion concerning the ratio between the diameter and
circumference of'a given circle, but also because it
explains how the assumption of external matter could lead

94 '

The Works of Gegrge Berkeley, Vol. I, p. 33.
951pid.

96Ibid., p. 31.

96
to the thesis of infinite divisibility. For Berkeley,
the sensation that we have when we look at a circle through
a magnifying glass cannot be said to be "of the same object"
than when we look at the "same" circle with the naked eye.
Hence magnifying glasses will not allow us to discern parts
in a minimum visibile that we sensed in a previous sight
experience with the naked eye. If we assumed the contrary,
then we would have to agree that more powerful magnifying
instruments might produce still larger images of the fsame"
object, and that we could then discern parts that were not
seen before, etc. Unless an absolute limit of magnifica-
tion can be demonstrated, this would lead to precisely
the problems that the immaterial hypothesis was meant to
remove.

The above analysis has shown that Berkeley's solu-
tion to the problem of the continuum depended crucially
upon the notion of the minimum sensibile (visibile,
tangibile, etc.). Whenever a specific problem of a geo-
metrical nature arises, e.g., the squaring of a particular
circle or the ascertainment of the length of the diagonal
of a given square, then we ought to proceed, according to

Berkeley, by counting the minima sensibilia out of which

 

these figures are said to be composed. The minima

97

sensibilia, being the smallest perceptible units, have no

 

perceptible parts and therefore, according to Berkeley,

no parts at all. But since they are perceivables, they
have some magnitude so that a line or a shape is always
made up of a finite number of them. But a peculiar diffi-
culty arises at this juncture. Warnock puts it in the
following way:

He (Berkeley) says that there is an idea
nothing that is not actually discerned in it.
But certainly a line drawn on paper is not
gegg_as composed of a definite number of points;
it looks continuous. It is not only that one
has no idea how to make even a reasonable guess
at the number of points in a line; it does not
look as if it were made up of points at all.

And should not Berkeley have concluded that
it i§_not made up of points?

The difficulty here exposed cannot be resolved by
referring to Berkeley's theory of abstraction. It is well
enough known that he countances only one mode of "abstrac-
tion” and that it is the imagining of parts of complex
ideas of perception. He says:

I find indeed I have a faculty of imagining,
or representing to myself, the ideas of those
particular things I have perceived, and of
variously compounding and dividing them...To be

plain, I own myself able to abstract in one
sense, as when I consider some particular parts

97G. J. Warnock, Berkeley, London 1953, p. 219.

98
or qualities separated from others, with which,

though they are united in some object, yet it
is possible they may really exist without them.

98

As concerns the number of points in a line it is precisely
the question how many parts there are to the line that can
really exist (i.e., exist in perception) without the rest
being around, and before this is ascertained no subdivision
of the line into smallest parts can be carried out in
imagination. As far as I can see, Berkeley nowhere suggests
a solution to the difficulties here indicated. A solution

might be found if some kind of operation were indicated

through which the division of a line or surface into

 

minima visibilia or pangibilia might be affected. But even
then considerable oddities would remain.
Berkeley repeatedly uses "point" and "minimgm

visibile" in the same sense. For example in the New Theory

 

of Vision he says:

No exquisite formation of the eye, no
peculiar sharpness of sight, can make it (i.e.,
the minimum visibile) less in one creature
than another; for, it not being distinguishable
into parts, nor in anywise, consisting of them
it must necessarily be the same to all. For
suppose it otherwise, and that the minimum
yigibile of a mite, for instance, be less than
the minimum visibile of a man; the latter

8
Principles of Human Knowledge, Introduction, No.
LWO. lO.

99
therefore may, by detraction of some part, be
made equal to the former. It doth therefore
consist of parts, which is inconsistent with
the notion of a minimum visibile or point.
Assume that, as a matter of empirical fact the smallest
perceivable blotches of a certain color, say green, are

always oblong. Now if we were to say that green minima

visibilia are always oblong, we would be at variance with

 

Berkeley as quoted above, since it would be clearly false
that such an object does not in anywise consist of parts.
We can always find a statement that is true of one side
of such a shape but not of the other, and we thereby gain
a procedure of distinguishing parts in such a shape. If
this holds and yet there is no perceptible green speck
smaller than certain oblong shapes, we are forced to the

conclusion that there are no green mipima visibilia,

 

because the shapes described are no migime_since there is
a sense in which parts can be distinguished in them, and
they cannot be subtracted from since they would then no

longer be visibilia. Under the above assumptions a geo-
metry in Berkeley's sense of green objects would be quite

impossible, since no ascertainable relations of magnitude

99Essaytowards A New Theory of Vision, The Works
53f George Berkeley, Vol. I., pp. 159-239, No. 80., p. 204.

100
obtain between green objects, magnitudes being compared

by counting minima sensibilia. But this is clearly false.

 

We may, after all, measure the length of green lines as
well as the length of lines of any other color. Thus the
assumption that there are parts in the smallest visible
green specks must be false. In order to salvage the rest
of the system we shall have to say that green minima
visibilia are oblong. The consequences of this assumption,
however, are quite strange. For one thing, green geometry
would differ from the geometry of shapes of other colors.
It is conceivable, moreover, that a green square may be

produced the length of whose side is one minimum sensibile.

 

The difference between the length of the diagonal of this

square and a minimum visibile is smaller than one minimum

 

visibile, hence there is no difference. Another square
may be produced whose diagonal has the length of two
minima visibilia precisely. The difference between the
length of its sides and one minimum visible is smaller
than a minimum visibile, hence there is no difference. In
the first case 'a‘was equal to l, in the second case it
‘fias equal to 2. In other squares it will vary between
tihese two values. One can safely assume that this is

VVIeaking more havoc in geometry than Berkeley intended, but

101
if Geometry is to become an empirical science, it will have
to put up with all manner of empirical conditions. Surely,
since a set of statements that accurately describes the
world or part of it is always consistent, Berkeley's geo-
metry will be consistent. But one must hope that Berkeley's
method is not the only one of removing the apparent incon-
sistencies of the continuum: Were it not for the fact that
an inconsistent geometry always entails Berkeley's geometry,
one might even be tempted to put up with inconsistencies.

A consideration of the above sort has some merit in
that it throws doubt on the alleged generality of geometri-
cal proofs in Berkeley's system. A geometrical proof,
according to Berkeley, is conducted on some particular
figure, but it becomes general, i.e., holds for all
figures which have all those properties that are explicitly
mentioned in the given proof. In all other aspects they
may vary from the figure used in the demonstration.100
But it seems now that if Berkeley's program of turning geo-
metry into an empirical science is to be taken seriously,
certain empirical properties of the paradigm figures other.
than those customarily advanced in geometrical demonstra-

tions might have to be considered. It is a problem that

loogginciples of Human Knowledge, Iggroduction.

 

‘V"

 

102
must be empirically investigated whether or not these
empirical properties, e.g., color, do or do not make a
difference. Only within the limits indicated by such an
empirical investigation may we say that "any theorem may
become universal in its use."101

Let me include here a brief summary. I think that
I have shown the reasons for the adoption of the immateri-
alist hypothesis to lay with the fact that it avoids the
problems of the continuum. To paraphrase Berkeley: on one
hand, no demonstration can be given for the existence of
external matter, but on the other hand, its rejection

removed the difficulties of the infinite.102

But'Berkeley's
solution is obtained at the expense of having to put up
with a system of geometry of a rather strange nature. How-
ever, I believe that any attempt to construe geometry as an
empirical science in Berkeley's sense will run into

similar difficulties, a very good reason not to consider
geometry an empirical science.

With the proposal of immaterialism and the postula-

tion of a new basis for geometry Berkeley's task was not

 

Ibid , No. 128.

lpgg., No. 133.

103
completed. He had to concern himself with the obvious
fact that mathematical analysis, working with the assump—
tion of infinite divisibility of lines, etc., and with the
assumption of the existence of infinitesimals had been
overwhelmingly successful. According to Berkeley, analysis
as then conceived was premised upon erroneous assumptions.

This is at first dogmatically asserted in the Philosophical

 

Commentaries, the Principles and the Dialogues Between

 

 

103
Hylas and Philonous, and later on proved in the Analyst.

 

In the Analyst Berkeley presents two sets of arguments
against the Calculus. The first depends upon his own
theory of abstraction as put forth in the introduction to

the Principles. Query No. 4, at the end of the Analyst,

 

asks "whether men may properly be said to proceed in a
scientific method, without clearly conceiving the object

they are conversant about...?"104

and Query 9 reads
"Whether mathematicians do not engage themselves in disputes

and paradoxes concerning what they neither do nor can

 

103
The Analyst or A Discourse Addressed to an

Infidel Mathematician, first printed in 1734, The Works of
George Berkeley, Vol. IV, pp. 53-103.

104281., p. 96.

104

conceive?"105

According to Berkeley, what can be conceived are
ideas which must be considered in analogy to sense-data.
In this sense, of course, infinitesimals or velocities in
a point cannot be conceived. Therefore Berkeley's critical
attitude. I believe that these objections had a rather
salutory effect in that they made it quite clear that the
foundations of analysis did not lay, and should not be
sought, in the world of sense experience. But Berkeley's
error is quite apparent: whether or not the symbols used
in mathematical analysis in fact refer, or can be made to
refer, to sensory experience is of no concern to the
mathematician qua mathematician. It suffices that they
be well defined, except for the primitive terms, and that
all deductions proceed according to the laws of logic from
the accepted axioms. Thus these objections of Berkeley's,
while consistent with his philosophy and general approach
to mathematics were really beside the mark.

But there is a second kind of objection raised in

the Analyst which must be taken more seriously. They

consist in detailed studies and criticism of particular

 

105Ibid.

 

105
arguments put forth by Newton and Leibniz. Berkeley is
able to show that there is at least a good deal of im-
preciseness contained in the proofs which he considers.
Let me give one example:

Let the quantity x flow uniformly, and be
it proposed to find the fluxion of x“. In the
same time that x by flowing becomes x+o. the

power xn becomes (x+o)n, i.e., by the method
of infinite series

 

 

xn + noxn-l + nn - n ooxn_2 + etc.,
2
and the increments
o and noxn-l + nn-l ooxn"2 + etc.,
2
are to one another as
l to an-l 4 2? - n oxn"2 + etc.
2

Let now the increment vanish, and their last
proportion will be nxn-l. But it should seem
that this reasoning is not fair or conclusive.
For when it is said, let the increments vanish,
i e., let the increments be nothing, or let
there be no increments, the former supposition
that the increments were something, or that
.there were increments, is destroyed, and yet
a consequence of that supposition, i.e., an
expression got by virtue thereof is retained.106

Berkeley's reasoning is that the division undertaken
in order to obtain the third line of the above proof pre-

supposes that zero is not equal to zero, while later on,

 

106Ibid., No. 13, p. 71f.

106
when all but the first member of the infinite series are
cancelled, zero i§_assumed to be equal to zero. The point
I wish to make here is not that all computations of the
derivative of xn are open to Berkeley's objection, but that
in point of fact the status of the differential zero in
the calculation was not clear at the time, and that some
mathematicians went so far as to claim that for the purposes
of the differential calculus a division by zero is permissible.

Berkeley brings forth arguments similar to the one
above, against other methods of arriving at derivatives,
and his criticisms are generally sound. The Analyst has
been said to mark "a turning point in the history of mathe-
matical thought in Great Britain",107 and Boyer points out
"Berkeley's criticism of Newton's propositions was well
taken from a mathematical point of view, and his objection
to Newton's infinitesimal conceptions as self-contradictory

108

was quite pertinent." It must be noted that Berkeley's

 

l
07Florian Cajori, A History of the Conception of

Limits and Fluxions in Great Britain from Newton to Wood-
house, Chicago 1919, p. 89. cf. Also The World of Mathema-
pipe, ed. James R. Newman, N. Y. 1956, Vol. I, p. 286. For
the importance of the Analyst in the history of mathematics
cf. also Gerhard Stammler, "Berkeley's Philosophie der
Mathematik", Kentstudien, Erganzungsheft No. 55, Berlin,
1922. ,

108Carl B. Boyer, The Concepts of the Calculus,
p. 226.

107
criticism was not so much directed against the mathematician
as an "artisan" whose procedures are justified by their
results, but against the mathematician qua logician. He
says:

To prevent all possibilities of your mis-
taking me, I beg leave to repeat and insist, that
I consider the geometridal analyst as a logician,
i.e., so far forth as he reasons and argues; and
his mathematical conclusions, not in themselves,
but in their premises; not as true or false, use—
ful or insignificant, but as derived from such
principles, and by such inferences. And, of as
much as it may perhaps seem an unaccountable
paradox that mathematicians should deduce true
propositions from false principles, be right in
the conclusion and yet err in the premises; I
shall endeavor particularly to explain why this
may come to pass, and show how error may bring 109
forth truth, though it cannot bring forth science.

The doctrine here indicated is that the results of compu-
tations in the differential calculus gain their truth and
value from a cancellation of erros, a doctrine that we need
not discuss here.

That Berkeley was correct in his objections was
finally borne out when the foundations of analysis could
be laid entirely without the introduction of actual in-
finitesimals solely through a consideration of limits. It

must be pointed out, however, that the later theories

 

109The Analyst, No. 20, p. 76f.

108
concerning the foundations of analysis, although doing
justice to Berkeley's objections against Newton and Leibniz,
did not incorporate any of his positive suggestions: none
of them assumes that a line or surface ultimately consists
of certain finitely extended particles.

What were Berkeley's positive suggestions concerning
the foundations of analysis? In his earlier writings he
asserts that on the adoption of his principles none of the
advantages of the modern analysis will be lost and he points
out that mathematicians really deceive themselves when they
think that they actually consider infinitesimals:

Whatever mathematicians may think of Fluxions,

or the Differential Calculus, and the like, a

little reflexion will show them that, in working

by those methods, they do not conceive or imagine

lines or surfaces less than what are perceivable

to sense. They may indeed call those little and

almost insensible quantities Infinitesimals....

But at bottom this is all, they being in truth

finite; nor does the solution of problems require

the supposing of any other.
In connection with this passage, emphasis must be put on
the phrase that mathematicians "in working with those
methods“ do not conceive, etc. Berkeley seems to mean by

this that the mathematician, as soon as he employs the

methods of the calculus in order to produce a drawing or

lloghe Principles of Human KnowledgeL No. 132.

 

109
in order to solve a practical problem must finally speak
of, and consider only, finitely extended quantities.
Berkeley seems to suggest that the calculations which pre-
cede this final interpretation are actually dispensible,
but his works contain no positive assertion on this point.
In the absence of such assertions I surmise that Berkeley
would be inclined to square circles and draw tangents to
curves entirely by gross visual estimation, assuming, as
it seems, that the results would not visually differ from
carefully drawn tangents that are produced after preliminary
calculation in the differential calculus. I can find no
evidence or suggestion for any other procedure in Berkeley.
But my surmise seems to be well inkeeping with Berkeley's
fundamental assumption that geometry, and likewise analysis,
are empirical sciences that must ultimately subject them-
selves to the judgments and estimates of the senses.

In conclusion let me once again quote from the
Philosophical Commentaries. Toward the end of the first
part of that diary he writes: "The Mathematicians think
there are insensible lines, about these they harangue,
these cut in a point at all angles,.these are divisible

ad infinitum. We Irish men can conceive no such lines."lll

 

44 111No. 393, lee Works of George Berkeley, V01- I-,
p. .

110
And a little later "I Publish not this so much for any—
thing else as to know whether' other men have the same
112

Ideas as we Irishmen." As it turned out, some other

men had different ideas.

112m. 398., Ibiq.

CHAPTER III

PIERRE BAYLE

My concern with Bayle in this chapter will be twofold:
I wish to investigate Bayle's contribution to the philoso-

phical discussion of the problem of the continuum and of

infinite sets, and secondly I want to make some comments on

the matter of Bayle's influence upon Berkeley. I have vio-

lated the historical sequence by discussing Bayle after
Berkeley, since it seemed more convenient to discuss Bayle's

.influence after the material on Berkeley had already been

assembled.

Bayle‘s main work and the one for which he is best

knaown is the Dictionaire,113 and I shall confine my discussion

t<> views contained in it. The Dictionary does not set forth

“filat one could call a philosophical system. Rather, it
discusses philosophical topics only incidentally and in

CNDrrnection with the biographies of the philosophers contained

 

113Pierre Bayle, Dictionaire historique et critique,
RCDtrterdam 1697. This is a two volume edition. A greatly
amended second edition in four volumes was published in 1702.
1“ 'the present paper, I shall quote Bayle after the second
English edition of the DiCtionaigg, London 1734-38. The anno-

tEi"tions will be made by citing the name of the article in

gu‘fiisstion, and, if required, the letter and number of the note

1 .
r1 (guestion.

112

in that work. Nevertheless, penetrating philosophical
insight reveals itself in these occasional remarks, and a
definite philosophical persuasion is quite apparent in them.

The Dictionaire exercised considerable influence, and

 

occupied a position of much greater importance in intellec-
. . 114
tual history than is the usual share of reference works.

I think that it is safe to assume that the magnitude
of Bayle's influence was, in part, due to the fact that he
entered the then topical dispute about the delineation of
the boundaries of faith and reason with extraordinarily
persuasive arguments, arguments that soon earned him the

name of a sceptic. Bayle‘s philosophical persuasion with

respect to these matters is, in a sense, a continuation of

 

114That the Qictionaire was widely used is shown by
the fact that upon the investigation of 500 private libraries
of the 18th century in France the work was found in 288.
(See Selections from Bayle's Dictionary, ed. E. A. Beller
and M. duP. Lee, Jr., Princeton 1952, p. XX). Forty-some
years after his death, Berkeley's and his son's and grand-
son's library was sold. According to Popkin, (Richard A.
Popkin, "Berkeley and Pyrrhonism", The Review of Metaphysics,
Vol. V., 1951-52, pp. 223-246) a copy of Bayle's Dictionaire
was contained in that library. Popkin bases his assertion
on the authority of A. A. Luce, (cf. Luce's edition of
Berkeley's Philosophical Commentaries, London 1944, p. 388)
who in turn trusted his own reading of an article by Aaron,
Ming, N.S. XLI, p. 465 ff, which discusses the content of
the library auctioned off. Aaron mentions only Boyle's, not
Bayle's works as contained in Berkeley's library. That the
Encyclopedists paid great heed to Bayle's opinions is well
known, but it is obviously erroneous to assume that his ideas

were propagated exclusively through that circle.

 

 

 

 

113

‘ 1
that expressed in the Port Royal Logjc. In essence it

 

is the denial of the possibility that a consistent descrip-

tion of reality can ever be achieved. In the Port Royal_

 

Logic it had been asserted that "there are some things which
are incomprehensible in their manner, yet certain in their
existency, we cannot comprehend how they are, however it

is certain, they are."116

and Bayle notes with approval that
the Port Royalists had already pointed out that researches
concerning the nature of the infinite have their only use
in forcing the understanding "however unwilling, to own that
some things exist though it is not capable of comprehending
them... All the force of human understanding cannot compre-
hend the smallest atom of matter, and is obliged to own that
it clearly sees that such an atom is infinitely divisible,
117

without being able to see how that can be."

This insufficiency of reason over against reality was

supposed by Bayle as well as the Port Royal Logicians to

lead the thinker to an unquestioning acceptance of the

 

115See Introduction to this dissertation, p. 1f.

116
(Antoine Arnauld), Logic or the Art of Thinking,
p. 392.

117Bayle, pp. ci§., Article geno, End of remark G.

Bayle took the quotation from the Port Royal Logic, loc. cit.

 

114‘
doctrines of religion, or at least to wrest from him the
argument that these doctrines cannot be accepted because
they are contrary to reason, since not even the description
of physical reality can be carried out without finally lead-
ing to inconsistencies.

There is some doubt, however, whether or not Bayle
was ultimately sincere with respect to the persuasion that

I have delineated above: In the article Leucippus Bayle

 

makes the following statement:

Leucippus, Epicurus, and the other Atomists
might have guarded against several unanswerable
objections, if they had bethought themselves of
giving a soul to every atom... I know they .
could not have avoided all difficulties by
ascribing it to them: they might still have been
pressed with invincible objections. Yet there
had been some glory in parrying a thrust here

and there.118 '

This quotation is apt to substantiate to some extent
the claim that I made above, namely that Bayle did not be-
lieve that an ultimately consistent and trustworthy meta-
physic or science could be developed. When sufficiently

hard pressed, any position would lead to absurd consequences,

so Bayle contended. Now part of the task of the Dictionaire

 

clearly was to investigate various philosophical positions,

118
Ibid., Article_Leucippus, end of note E.

115
not in order to accept one or the other, but to show that
in the final analysis they are all untrustworthy. I do not
wish to decide here, and it is quite unnecessary for the
purpose at hand to do so whether this undertaking was
designed to show forth the limitations of reason and to
support the position of faith, or whether it was performed
for the glory that there is dealing many a thrust. Thus,
while the over-all aim might have been to display’ brilliance
in criticism or to demonstrate the weakness of reason for

the benefit of faith, the result is that the Dictionaire

 

hardly ever expounds or establishes a doctrine without
criticising or demolishing it in some other passage or per-
haps even some other volume of the work.

As a case in point let us consider some arguments
connected with naive realism and immaterialism as they

occur in the Dictionaire. We find the following situation.

 

In the article Anaxagoras, Bayle criticizes that philosopher

 

who seemed to have assumed that the ultimate components of
bodies are destructible. Bayle points out:

Compound Bodies alone are born and die and
pass through a thousand Vicissitudes of Genera-
tion and Corruption; but Principles retain their
Nature unchangeably under all the Forms which
‘are successively produced. Anaxagoras could not
say this of HIS Principles.

119Ibid., Article Anaxagoras, Remark C, No. I.

116

and
Wood, when destroyed by Fire, ceases not

to exist as Matter, or extended Substance.

Thus there is a great Defect in the System of

Anaxagoras.

A little later he criticizes Moreri for attributing
a certain doctrine to Anaxagoras by exclaiming:

Here is Earth, there Air and Water; here

a Meadow, and there a Wood. Anaxagoras would

have been more extravagant, than even the most

absurd Visionary that was ever put in a Mad-

House, had he entertained any Doubts about it.121

The point at issue was the opinion, which Moreri
had attributed to Anaxagoras, that the universe is homo-
geneous. So in the last quoted passage Bayle did not want
to defend the existence of external matter. but the hetero—
geneity of the universe. Nevertheless, the language is
not that of a phenomenalist or immaterialist, but rather
that of a realist, a position which was clearly expressed
in the quotations preceding the last one, and which clearly
underlies Bayle's criticism of Anaxagoras.

Now the doctrine that is here used, or at least pre-

supposed, does not tally in the least with the immaterialism

that is proposed in the articles Pyrrho and Zeno. as we

 

120113161.

121 .
Ibid., remark C, No. IX.

118

that bodies are coloured, and yet it is a mistake.

I ask, whether God deceives men with respect to

those colours? If he deceives them in that respect,

what hinders but he may deceive them with respect

to extension. This latter illusion will not be

less innocent, nor less consistent. than the for-

mer. with the most perfect being.121

And further on Bayle points out "God does not force
you to say. that it does ex‘st, but only to judge that you
. . . "122
feel It, and that It appears to you to eXist.

While Bayle here merely puts in doubt the existence
of external matter on the grounds that there is not a
sufficient reason for believing it, he produces very force-
ful reasons for denying its existence in the article Zeno.
Before proceeding to a discussion of these arguments, let
me again caution against the assumption that these dis-
quisitions are intended to establish the foundations for
an immaterialist or idealistic philosophy. They are only
part in the general plan of «demonstrating the unten-
ability of any system of metaphysics. Thus, while the
existence of external matter is put in doubt in Pyrrho, and

a demonstration against this existence is offered in Zeno,

external matter is nonchalantly assumed in Anaxagoras, as

 

 

121Ibid., Article Pvrrho, remark B.

122Ibid.

117
shall presently see. It is clear, then, that Bayle's
critical endeavours do not proceed from a well formulated
and systematically developed philosophical position, but
that they use whatever means are available, or can be pro-
vided ad hoc, in order to demonstrate the untenability of
any given philosophical system. It seems then that we
cannot attribute to Bayle a philosophical position or sys-
tem in the ordinary sense. but rather some sort of meta-
position, which maintained the futility of any philosophical
commitment. This explains. and perhaps was meant to excuse.
the eclectic and inconsistent method of criticism found in
the Dictionaire.

But let me now try to document the phenomenological
or immaterialist views that are set forth in the articles
Pyrrho and gang. In Pyrrho the existence of external matter
is put in doubt, and in ggng space and external matter are
denied outright, and forthful reasons are presented in
support of this denial.

In Pyrrho, Bayle writes:

I have...not one good proof for the existence
of bodies. The only good proof they can give me

for it, is, that God would deceive me, if he

imprinted in my soul the ideas I have of body,

if there were no bodies, but that proof is very

weak; it proves too much. Ever since the

beginning of the world all men, except, perhaps,
one in two hundred millions, do firmly believe

119
we have seen. The demonstrations in g322_are not expressions
of what Bayle affirms. but propositions which he contemplates.

A peculiar feature of relation between Bayle and
Berkeley now becomes apparent: if there was any influence
of Bayle upon Berkeley. we cannot say that Berkeley was
persuaded by Bayle's convictions, but rather, that Berkeley
asserted some of the propositions which Bayle merely dis-
cussed. Thus any discussion of the relation between these
two men will have to acknowledge that, while Berkeley pro-
bably accepted some of Bayle's demonstrations. he was con-
vinced, unlike Baylethat a consistent and rational account
of reality can be given. and that the tenets of religion.
are not well served through attempts at demonstrating the
fundamental irrationality of the world.

How could Berkeley have come to such a position if the
the_Dictionaire offered convincing arguments against every
philosophical position which is discussed there? The
answer is that the Dictionaire actually falls far short of
achieving this goal. On one hand, there are very good
arguments against the existence of external matter, but
the immaterialism that is thus established is nowhere con-

vincingly refuted. Rather. as in Anaxagoras. realism is

 

assumed, but not proved to be tenable.

120

It must be noted that the position delineated in
the articles Pyrrho and Zggg taken in itself does not con-
stitute a philosophy that could properly be called sceptical.
After all, the affirmation that there is no external matter
is an affirmation. I believe that Bayle wanted to point
out that there are very good arguments against naive
realism, but that an immaterialist philosophy is not a good
alternative. In fact. a decision between immaterialism and
realism would either run counter to the arguments in Eggg.
or contradict what everybody knows, namely that external
matter "obviously" exists. Thus, no matter how we proceed.
an inconsistency is immediately forthcoming.

It is to Berkeley's credit to have realized for the
first time that the one alternative. namely epistemological
realism, is not really supported by any arguments, a fact
that Bayle completely overlooked. This is one of the
reasons why any assertion that Bayle anticipated Berkeley
must be accepted with great reservations.

I realize that the interpretation which I have here
put on the passages in Bayle is not the customary one.
Traditionally, the fictional personage who presents the

above quoted passages in the article Pyrrholz3 is called

123See above, pp. 117f.

121

the Abbe Pyrrhonien, indicating what is here stated is

 

Pyrrhonist or sceptical position. Popkin asserts that the
argument expresses "a brilliant conception of the ultimate

in scepticism."124

However. the viewpoint expounded by
the Abbe Pyrrhonien is not a sceptical. but an immaterialist

or, if you will, phenomenological one. Notice again the

passage "God does not force you to say that it does exist,

 

but only to judge that you feel it, and that it appears to
you to exist."125 Again it is Berkeley's merit to have
emphasized time and again against a welter of misunderstand-
ing that the phenomenological vieWpoint must not be con-
founded with the sceptical one, that they have nothing in
common. It is, in a way, the most important point of his
system. The arguments in gy££h2_can be used for the support
of a sceptical philosophy only, if a naive realism is main-
tained at the same time, and if the resulting inconsistency
is turned into an argument for the whithholding of judgment.
It is now time to turn to an examination of the

arguments which Bayle brings forth in support of his

 

124Richard H. Popkin, "Pierre Bayle's Place in 17th
Century Scepticism" in Pierre Bayle, Le Philosophe de

Rotterdam, Paris 1959, pp. 1-19, p. 6.

1ZSSee above, p. 118.

122
"irrationalism". They are of two kinds. The arguments
offered in Pvrrho point, in a rather naive way, to the so-
called mysteries of religion, pointing out that the pro-
positions put forth there are some of them contrary to all
reason and should nevertheless be accepted. But in Pyrrho
it is also suggested that extension is a secondary quality.
If this assumption could be proved, i.e. if the external
extended matter could be shown to be illusionary, then a
contradiction with naive realism would arise. I think that
in Bayle's opinion this would make both naive realism and
phenomenalism doubtful, and would show that they are in no
better position than the revelations of religion as far as
plausibility is concerned. The argument for the non-exis-
tence of external matter is presented in 2232- It is, as

. we shall see, of the reductio ad absurdum type. Let me

 

remark, in passing, that the employment of this type of
argument is not in keeping with the general tenets of
Bayle's undertaking: if the inconsistency of an assumption
does not necessarily discredit it (see the assumptions of

revealed religion), then reductio ad absurdum is of little

 

value.
Let us have a look at the argument. Bayle argues in

support of Zeno's contention that there is no motion, and

123
he writes the following:

There is no extension, therefore there is
no motion. The consequence is good, for what has
no extension fills no space, and what fills no
space cannot possibly pass from one place to
another, and consequently move. This is incon-
testable; the difficulty is then to prove that
there is no extension. Zeno might have argued
thus: Extension cannot be composed either of
mathematical points, or of atoms, or of parts
divisible ad infinitum; therefore it's existence
is impossible. The consequence seems certain,
by reason it is impossible to conceive more than
these three modes of composition in extension;
wherefore the antecedent alone remains to be
proved. A few words shall suffice as to mathe-
matical points; for a man of meanest capacity
may apprehend with the utmost evidence, if he is
but a little attentive, that several nothingnesses
of extension joined together will never make an
extension...Wherefore...let us take it to be
impossible, or at least inconceivable. that
matter should be composed of them...

Nor is it less impossible or inconceivable
that it should be composed of Epicurean atoms,
that is, of extended and indivisible corpuscles;
for every extension, how small soever, hath a
right and a left side, an upper and lower side:
therefore it is a conjunction of distinct bodies;
and I may deny of the right side what I affirm
of the left, for these two sides are not in the
same place: a body cannot be in two places at
once and consequently every extension which fills
several parts of space contains several bodies...
whence it follows that if there be an extension,
its parts are divisible in infinitum. But on the
other side, if they cannot be divisible in infini-
tum, we ought to conclude the existence of exten-
sion impossible, or at least incomprehensible.

An infinite number of parts of extension, each
of which is extended, and distinct of all others,
as well with respect to its entity as to the space
which it fills, cannot be contained in a space
one hundred thousand million times less than a

 

 

124
hundred thousands of a barley corn....If there
be no body but what contains an infinity of
parts, it is evident that each particular part
of extension is separated from all others by
an infinity of parts, and that the immediate
contact of two parts is impossible, wherefore
the existence of extension necessarily requiring
the immediate contaCt of its parts, and that
immediate contact being impossible in an exten-
sion divisible in infinitum, it is evident that
the existence of such an extension is impossible.

126
Of the three arguments here presented, only the
case against Epicurean atoms holds water. They cannot be
the ultimate constituents of extended bodies since they
are themselves constituted of parts. But while Bayle
argues, in the case of these atoms, against their ultimacy,
in the other two cases it is the mode of composition which
provides the difficulties. Now none of these latter argu-
ments are cogent, at least not unless important qualifica-
tions are added. In the first argument it was asserted that
several points "joined together" will not make a finite
extension. The difficulty seems to lie in the phrase
"joined together". In the sequel we shall see that one of
the most important advances in the analysis of continuous
quantities was Bolzano's realization that an infinity of
points, when "joined together" in a certain way, can

126Bayle, op. ci§., Article geno, Remark G.

 

125
constitute a finite (continuous) extension. I take Bayle's
remark to mean that several points, no matter how many,
when joined together, no matter in what way, will not make
a finite extension. This statement is clearly false. Thus
the argument against points is not valid. I presume that
it is precisely the intuitive accompaniment of such a
phrase as "joined together", or the mental image that goes
with it, which misled Bayle as it did so many of his
contemporaries.

The argument against infinite divisibility is like-
wise invalid. First, it is not the case that an infinite
number of finitely extended parts of extension cannot be
contained in a finite extension. If a half inch is added
to an inch, and a quarter inch to the sum of the two etc.,
the total will not exceed two inches, even if infinitely
many of these finitely extended lengths are added in accord-
ance with the law of the series. To be sure, I cannot
actually produce all these different lengths first, and
then join them together; all I can do is to give the law
‘according to which they must be produced and joined, and I
believe that it was this restriction which lead Bayle to
believe that an infinity of finite extensions cannot be

contained in some other finite extension.

 

126

Finally, it has been recognized for quite some time,
that it does not make sense, given a point on a line, to
speak of the "next" or neighborning point, as it also does
not make sense, given a moment, to speak of the "next"
moment. If two points are distinct, then there must be
another point which lies between them, from which it follows
that there must be an infinity of points which lie between
them. I believe that this fact was first recognized and
emphasized by Boscovich127 and later on accepted by
Bolzano, as I shall show. Bayle apparently recognized this
feature of continuous sets. but it seemed so incredible to
him that he preferred to reject the notion that the con-
tinuum is infinitely divisible. However. it must be noted
that in the last argument given above the concept of exten—
sion is arbitrarily restricted by the assertion that it
requires the immediate contact of its parts. If, as seems
to be the case, by these parts are meant the ultimate parts
or points, then this requirement would in fact lead to the

elimination of continua, or extension. since two points

127Ruggiero Guiseppe Boscovich. Theoria Philosophiae
Naturalis, Vienna 1758, 8 30-33.' See Ernst Cassirer, Das
EEkenntnisproblem, IBVOls. 3rd. ed., Berlin 1922, Vol. 2,
P- SlOf. '

 

127
that are in "immediate contact" are identical. It is there-
fore this arbitrary restriction which must be rejected, and
_may be rejected, since it can be shown that immediate con-
tact of points is not necessary. or even possible. in the
type of serial order which we call a continuum.

We see then that Bayle's entire argument derives its
superficial plausibility merely from the difficulties
connected with continuous sets. and. generally. from the
counterintuitive features of infinite sets. However, as may
be expected. arguments of this sort were the rule rather
than the exception at Bayle's time, and they exerted a tre-
mendous influence upon philosophical speculation.

It has been suggested that these discussions of
Bayle's exercised considerable influence upon the develop-
ment of Berkeley's philosophy, and it is this claim which
I shall now examine.128

Popkin tries to rake together all the evidence that
would support the supposition at issue. He points out that

Berkeley's repeated and vigorous insistence that his

128Both Popkin and Luce have asserted that Bayle
strongly influenced Berkeley. Cf} Richard H. Popkin,
"Berkeley and Pyrrhonism", The Review of Metaphysics, Vol.
V-, 1951-52, PP. 223-246 and A. A. Luce's note on p. 388
Of his editio diplomatica of Berkeley's Philosophical
Commentaries, London 1944.

 

~128
philosophy is not a sceptical one can be interpreted as an
attempt to distance himself from Bayle and the reputation
that the latter had acquinal. He points out that Berkeley
had obviously acquainted himself with some of the arguments

in Zeno. since entry no. 358 of the Commentaries reads

 

"Malebranche's and Bayle's arguments do not seem to prove
against Space. but only Bodies."129 Considering the wide

distribution and the popularity of the Dictionaire, and the

 

material accumulated by Popkin, there seems little doubt
that some influence took place, especially as regards the
treatment of primary qualities. However. in their zeal
for connecting Berkeley with Bayle. both Popkin and Luce
have overlooked that there is a rather considerable dis-
agreement between the two philosophers, especially as con-
. . . . 130
cerns the topic of this dissertation.
After having presented his case in the form quoted
above, Bayle summarizes his results in the following fashion:
All those who argue an extension are
determined in their choice of an hypothesis no

otherwise than by the following principle: I;
there are but three ways of explaining_a subject,

 

129The Works Cf George Berkeley, V01- I. P- 43-

1301 have already pointed at another divergence

concerning scepticism, above, pp. ll9f.

129

the truth of the third necessarily follows from
the falsity of the other two. A zenonist might
tell those who choose one of these three hypo—
theses: you do not argue right, you make use
.of a disjunctive syllogism...The fault of your
argumentation lies not in the form, but in the
matter: you ought to lay aside your disjunctive
syllogism, and make use of this hypothetical
one: If extension existed, it would be composed
either of Mathematical points, or of Physical
points, or of parts divisible in infinitum.

But it is not composed either of Mathematical
points, nor of Physical points. nor of parts
divisible in infinitum. Therefore it doth not
exist.

 

 

 

 

Compare this with entry no. 26 of the Philosophical

 

Commentaries, which says: "Infinite divisibility of exten-

 

sion does suppose ye external existence of extension but
the latter is false. ergo ye former also."132 Now Luce
asserts that "entry no. 26...relates the infinite divisi-
bility to external extension exactly as the Zeno article
does;"133 and Popkin fully agrees with this result.134
However, a moments consideration will show that such is not

at all the case. For one thing, the consequent in Bayle's

hypothetical syllogism is an alternation of three members.

 

l3lBayle, op. cit.. Article Zeno, remark G.

q
l‘2The Works of George Berkeley. Vol. I.. p. 10.

133A. A. Luce. loc. cit.

134
Richard H. Popkin, op cit., p. 243.

130
One could,however assume that Bayle listed all three possi-
bilities only in order to serve the tenets of his encyclope—
dic enterprise, and that the case against atoms was already
so well established that Berkeley deemed their discussion
superfluous. Furthermore. Berkeley might have realized that
the infinite divisibility and constitution of points.amounts
to the same thing: But even if one were to agree to all
this. a fundamental difference would still remain. If we
consider only the case of infinite divisibility. Bayle's
argument would read: 'if extension exists, then it mustlxg
infinitely divisible: But it is not infinitely divisible.
Therefore it does not exist.‘ On the othe: hand. Berkeley's
argument could be phrased thus: 'If something is infinitely
divisible, then it exists externally. But nothing exists
externally. therefore, nothing is infinitely divisible.‘
Thus, to claim that Berkeley relates infinite divisibility
to external extension exactly as the Zeno article does is
patently false. Rather. in Bayle the denial of external
extension is the final result of the argument, in Berkeley
it is a premise. Actually, Berkeley's argument is similar
to the supporting proofs of Bayle's in which he attempts to
demonstrate the impossibility of infinite division, but

while Bayle‘s arguments suffer from logical inadequacies,

131
Berkeley puts forth a metaphysical assumption which in fact.
if true. would remove the difficulties concerning the compo-
sition of bodies, as I have shown above.
The differences between the two arguments can be

summed up as follows: the texminus ad guem of Bayle's demon-

 

stration is the proposition that extension does not exist.

In order to show this, a (fallacious) proof is offered to

the effect that extension is not infinitely divisible. In
Berkeley's argument. the proposition to be proved is that
infinite division is impossible. To show this. it is assumed

as a hypothesis (not proved) that there is no external

 

extension.

Thus, here again we must meet the claim that Berkeley
was strongly influenced by Bayle with great reservations.
Their arguments differ essentially as their aims were different.
Berkeley was primarily concerned with the problems of the
infinite, while Bayle's aim. at least in gppp_was to dis—
credit naive epistemological realism.

Nevertheless. it is a likely supposition that some
influence took place. Let me conclude this chapter with
another quotation from Bayle which tends to substantiate
this latter assumption.

Since the same bodies are sweet to some men

132

and bitter to others, it may reasonably be inferred
that they are neither sweet nor bitter in their
own nature, and absolutely speaking. The modern
Philosophers, though they are no Sceptics, have

so well apprehended the foundations of the epoch
with relation to sounds, odours, heat, and cold,
hardness and softness, ponderosity, and lightness,
savours and colours etc. that they teach all these
qualities are perceptions of our mind. and do not
exist in the objects of our senses. Why should we
not say the same thing of extension? If a being,
void of colour, yet appears to us under a colour
determined as to its species. figure and situation.
why cannot a being, without any extension, be
visible to us, under an appearance of determinate

extension, shaped and situate in a certain manner?135

 

35
Bayle, op. cit., Article Zeno, Remark H.

CHAPTER IV

IMMANUEL KANT

So far we have discussed three approaches to the
problem of the composition of the continuu. and Kant was
apparently familiar with all three of them. In the preface

to the first edition of the Critique of Pure Reason he

 

asserts that he has "found a way of guarding against errors
which have hitherto set reason, in its non‘empirical employ—
ment. at variance with itself"; and he continues: "I have not

evaded its questions by pleading the insufficiency of human

reason."136
The difficulties which set reason against itself are.
of course. those which are recorded in the "antinomies". of

which the second is of especial importance for our present

investigation.137 It seems to me that the above quotation

 

136I shall quote from Norman Kemp Smith, A Translation

 

of Kant's Critique of Pure Reason, London 1929, but, as cus-
tomary. I shall give the pagination of the first (A) and
second (B) editions of 1781 and 1787. All other works will
be quoted after the Akademieausgabe, Berlin 1902 ff, but I
shall consult and indicate already existing translations.
The present quotation is from A 12.

137 , . . .

The Second Antinomy (Critique of Pure Reason,

A 434, B 462) is stated thusly: "Thesis: Every composite

134
indicates Kant's familiaritvaith earlier approaches such as
Bayle's which blaime the arising of contradictions in specula-
tion upon the insufficiency of human reason.

According to a recent commentary. the problem of the
constitution of matter and Bayle's account of the issue were
frequently discussed in the Wolfian school. so that there is
sufficient reason to suppose that Kant did not merely mean to
attack a hypothetical position in the above quoted passage.138

But not only was Kant familiar with the school of
thought that blamed the insufficiency of human reason for
the arising of antinomies. he also seems to have known Berkeley
quite well. However, it is doubtful that he envisaged
Berkeley's philosophy as an attempt at solving or removing
the problem of the constitution of (continuous) matter. He
says of Berkeley:

He (Berkeley) maintains that space. with all

the things of which it is the inseparable condi-

tion, is something which is in itself impossible;

and he therefore regards the things in space as

merely imaginary entities (Einbildungen) Dogmatic
idealism is unavoidable. if space is interpreted

 

 

substance in the world is made up of simple parts, and
nothing anywhere exists save the simple or what is composed
of the simple. Antithesis: No composite thing in the world
is made up of simple parts. and there nowhere exists in the
world anything simple."

138Cf. Gottfried Martin, Kant's MetaPhYSiCS and Theory

of Science. Manchester 1955, p. 47.

135

as a propert that must belong to things in
themselves." 39

If this passage is interpreted to say that Berkeley developed
his philosophy because he recognized contradictions in the
assumption of continuous bodies which fill space, then Kant
recognized precisely the starting point of Berkeley's
philosophy. I have some doubts about this. Nevertheless,
Kant here, in a more or less vague fashion. seems to link
Berkeley's problem with that of the second antinomy.

That Kant was familiar with Leibniz is commonplace
and does not need much further substantiation. Kant was an
avid student of Leibniz and began his philosophical career
entirely within the tradition of the Leibniz-Wolff school.
The problem of the composition of the continuum was much
discussed in these circles and several of Kant's first
philosophical essays concern themselves directly or indir-
ectly with it.

Thus it is clear that Kant, in addressing himself
to the problems of the second antinomy. consciously became

part of a long line of attempts at a solution of this

 

l39Critique of Pure Reason. B 274.

 

136

problem.140

What distinguishes Kant from some of his predecessors.
notably Bayle and the Port Royalists. and from many of his
successors, is a wholesome fear of contiadictions. Time and
again he declares that one of the most important features of
his Critical Philosophy is the avoidance of antinomies that
have hitherto bedevilled speculation. Thus in the preface
to the first edition of the gritique of Pure Reason he
bemoans the fact that through the uncritical acceptance of
"principles which overstep all possible empirical employment...
human reason precipitates itself into darkness and contradic-

"141

tions. The Critique, of course, is to be the cure for

this malady. In the preface to the second edition he makes

142

a similar point, and the body of the work contains many

related remarks. Finally. an explicit statement of the

 

140According to Erich Adickes, Kant als Naturforscher.

2 Vols., Berlin 1924/25, Vol. I, p. 172, Kant in the Monadolo-
gia Physica, Akademieausgabe. Vol. I. pp. 473-487. asks him-
self the question, "what is it that makes it happen that
matter, stuff, occupies a space and cannot be removed from
that space?" Adickes claims that it was Kant's "immeasurable
merit" to have seen "the necessity for that question." On

the evidence of our previously collected material it is
evident that to claim originality for Kant on that score is
patently absurd.

 

141Critique of Pure Reason, A VIII.

142Ibid., B xx.

137

difficulties and contradictions which beset reason in its

143
"uncritical" use is found in the Antinomies.

 

In the second half of the last century the importance
of the Antinomies for the development of Kant's philosophy
came to be realized. Alois Riehl. for one. pointed out
that the essential features of Kant's philosophy sprang
from the desire to avoid the occuiience of the antinomies.
and Erdmann. Vaihinger, Adickes and many others accepted his
opinion. so that now it is the subject of scarcely any'
further dispute.144

Thus I cannot claim any originality in elaborating
this point again. Nevertheless, several reasons prompted
me to include a consideration of Kant in the piesent disserta-
tion. One is that previous commentators have ielied largely
on testimony contained in letters and posthumously found
notes in order to establish the importance of the problem of
the antinomies for Kant's philosophy, i.e. they have chosen

a historical approach. I shall confine myself to a discussion

of his published work in order to establish the same point.

 

4
1 3Ibid., A 407-567. B 435-595.

144Cf. Klaus Reich in the introduction of his edition
of Immanuel Kant, De mundi sensibilis atgue intelligibilis
forma et principiis. Hamburg (Meinei) 1958, pp. VIII ff

 

138

This procedure will have the advantage of making clear the
inner-systematic importance of the problem of the antinomies.
in particular the second antinomy, and will thus differ
markedly from a mere ascertainment of historical sequence

A second reason why I thought it necessary to include
a discussion of Kant lies in the fact that many commentators.
particularly Riehl. do not seem to have been aware at all
of the implications oftha paoblem at hand. Thus Riehl
feels justified in claiming that "opposite assertions, based
on entirely different presuppositions. do not contradict one

anothei".145

and that

the proofs for and against the infinite divisi-
bility of matter are not conducted from the same
or similar standpoints. The reasons offered

on either side are not homogenious, so that there
can be no real contradiction between them. The
thesis is proved ongologically from the conception
of a composite reality. while the antithesis is
proved for perception from the idea of space 146

Now clearly, for the occurrence of a contradiction it
can be of no importance whateverlxwveach of the contradictory
statements has first been established. Nor did Kant doubt

for a moment that he had contradictions on his hands, merely

 

145Alois Riehl, Introduction to the Theory of Science
and Metaphysics, London 1894, p. 270.

146 _
Ibid., p. 271.

139
because thesis and antithesis had been established in
different ways. But Riehl committed another major blunder.
With reference to the third and fourth antinomy he points
out:

In order to understand the proof of the
theses of these dynamical antinomies, it is
necessary first to forget the doctrines es-
tablished by Kant in the Transcendental Analytic.
This evident contradiction in Kant's system
(gig) can only be explained by assuming that
the antinomies are the oldest part of the
Critique, or rather that they preceded it.147

 

There could be no clearer testimony that Riehl‘s
interest and ability were merely historical. not systematic.
The passage shows a misunderstanding of the position of the
antinomies in Kant's system while, at the same time, it
establishes a historical thesis that is probably correct.

It was Kant's precise objective first to establish in each
antinomy both the thesis and the antithesis from a pre-
critical viewpoint. i.e. forgetting the doctrines established
in the Transcendental Analytic. and then to discuss and
remove the ensuing difficulties by bringing to bear on them

the findings of the preceding parts of Critique. i.e. the

 

Transcendental Esthetic and Analytic. On the other hand,

there is a good deal of historical testimony to vindicate

 

147Ibid.

140
Riehl‘s thesis that the antinomies (possibly even in their
final form) preceded much of the rest of the Critique.148
I find it generally the case that commentators either
did not pay much attention to the systematic significance
of the antinomies for the rest of the critical enterprise,149
or else were determined largely by historical considerations
and preoccupied with the establishment of such philological

theses as the so—called patchwork theory. according to which

the CJitique of Pure Reason was pieced together out of a

 

large number of preexisting writings.150

It is for all these reasons that I propose to investi—
gate again the importance of the antinomies, in particular

the second one. for the development and character of Kant‘s

 

148A letter from Kant to Garve has been found dated
the let of September, 1798, in which Kant says, "It was not
the investigation as to the existence of God, but the anti-
nomies of pure reason which first wakened me from a dogmatic
slumber, and impelled me to a critique of reason itself."
Cf. also the grolegomena, beginning of § 50. Akademieausgabe.
Vol. IV, p. 338.

 

149E.g. Norman Kem Smith, A Commentary on Kant's
Critique of Pure Reason, : New York 1950, and T. W. Weldon,
Kant's Critique of Pure Reason, é Oxford 1958.

150Cf. Alois Riehl, Der philosophische Kritizismus
und seine Bedeutung_fur diGngSithe Wissenschaft. 3 Vols.,
Leipzig 1876-87, The above quoted work, Introduction to the
Theory of Science and Metaphysiq§_(above p. 115) is a trans-
lation of the third volume of yer philosophische Kritizismus.
-fihe charge of preoccupation with historical and philological

141
. 151
philosophy.
Before I proceed to analyze Kant's theories in detail.
let me try to indicate briefly the epistemological status
that Kant ascribed to his proposed solutions. It will be

remembered that Berkeley described the main facet of his

philosophy as the immaterial hypéthesis. In keeping with

 

this characterization. supporting arguments were sought, as
we have seen, not in propositions of which the immaterial
hypothesis is a consequence, but in propositions which it
entails. In opposition to this procedure. Leibniz delivers

his system as a priori certain, and does not ordinarily

 

attribute hypothetical character to it.

How did Kant regard his own doctrines? In the preface
to the second edition of the Critique of Pure Reason, when
describing in a preliminary way his "Copernican revolution"

with the aid of which certain contradictions can be avoided,

 

questions can be levelled even against Vaihinger, whose
commentary is generally held to be above reproach. Hans
Vaihinger, Kommentar zu Kant, 2 vols. End. ed. Stuttgart

1922. It must be said, in vindication of Vaihinger that

his commentary does not proceed beyond the Transcendental
Esthetic, and that he discusses our present problem in an
avowedly historically oriented Excurgz "The Historical Origin
of the Kantian Doctrine of Space and Time", Vol. II, p. 422ff
which follows the systematic exposition of the Transcendental
Esthetics.

1511 have found the best discussion of the importance

of the antinomies in Kant in Gottfried Martin, Kant's Meta—
physics and Theory of Science, Manchester 1955.

142

he points out,

The (Newtonian attraction) would have remained
forever undiscovered if Copernicus had not dared,
in a manner repugnant to common sense, (wider-
sinnisch) but yet true, to seek the observed move-
ments not in the heavenly bodies, but in the
spectator. The change in point of view, analogous
to this hypothesis, which is expounded in the
Critique, I put forward in this preface as an
hypothesis only, in order to draw attention to
the character of these first attempts at such a
change, which are always hypothetical. But in
the Critique it will be proved apodeictically
not hypothetically, from the nature of our repre-
sentations of space and time and from the ele-
mentary concepts of the understanding.152

 

It seems that Kant considered the introduction of a
proposition as a hypothesis merely as a preliminary matter.
and that any hypothesis, in order to be permanently acceptable
must be "proved". In this respect his outlook did not
change from his early rationalist days to the time of the

Critiques. In one of his earliest writings, the Monadologia

 

Physica, he criticizes those approaches to natural philosophy
which do not accept any information but What is immediatelyi
evident through experiment. He says: “Ex hac sane via leqes
naturae exponere profecto possumus, legum originem et causas

non possumus."153 Thus. through observation and experiments

 

152Critique of Pure Reason, B XXIII.

153Meta2h¥§jg§ cum geometriae iunctae usus in philoso-

phia naturali, cuius specimen I continet monadologiam physicam,
Konigsberg 1756, Akademieausgabe Vol. I, pp. 473-487, p. 475.

143
laws can indeed be made evident, but their origin and causes
cannot be found in this way. This seems to be an invitation
to speculation, to the formation of hypotheses which would
explain the laws found through experimentation - a perfectly
legitimate enterprise. However, the body of the Monadologia
supposedly proves every hypothesis which is introduced there.
Needless to say, many of these proofs are invalid. but the
attempt is there. just as in the Critique of Pure Reason

The difference between Kant and Berkeley could not be
more striking. I am quite convinced that Berkeley did not
think his hypothesis demonstrable in the same sense in which
Kant sought to prove his. The immaterial hypothesis is said
to be as good as its explanatory merit. Kant on the other
hand. always thought of his doctrines as requiring some sort
of deductive proof, or. to put it more strongly. he attempted
to. and presumably believed that he did, prove all_proposi-
tions which he offered. Interestingly enough, it is those
parts of the Critique of Pure Reason which offer these proofs,
namely the Esthetic and Analytic, which must be considered
dated’although even they command interest at every turn.

In offering a reconstruction of parts of Kant's
philosophy I shall do some violence to Kant's own intentions

by neglecting. for the most part, the alleged proofs that he

144
offers in support of his positions - otherwise I would have

to discuss the entire Transcendental Esthetic and Analytic,

 

 

a task much too formidable for the small scope of this
paper. Thus I shall discuss certain of his propositions as
though they had been intended merely to function as hypotheses,
The above considerations point to a remarkable and
perhaps fundamental distinction between the British empiri-
cist school and the continental rationalism and idealism.
Concerning the epistemological status of his doctrines, Kant
did not differ at all from his rationalist predecessors.
The earliest work in which Kant attempted to offer
a reasonable solution to the problem of the constitution of

matter was his Monadologia Physica. To understand fully the

 

problem as it was envisaged in that little essay, let us

first turn to a much later work, namely the grolegomena

 

to Any Future Metaphysics. Here Kant writes:

 

It will always remain a noteworthy phenomenon
in the history of philosophy that there was a time
when even mathematicians, who were at the same
time philosophers, began to doubt. though not the
correctness of geometrical theorems as far as they
concern mere space, but at least the objective
validity (Gflltiqkeit) and application of this
concept to nature, since they feared that a line
in nature might consist of physical points, hence
the true space in the object consist of simple
parts, although the space which the geometrician
has in mind cannot possible consist of them.154

 

 

154Akademieausgabe, Vol. IV, pp. 253-383.§13, Note I.
p. 287f. '

145
It is quite clear that the object of this criticism
is Christian Wolff, who was a mathematician and at the same
time a philosopher. ’Wolff had assumed that the physical
world consists of bodies which are extended, and which are

made up of simple elements called 'atomi naturae‘ and some-

 

times also 'monads'. Wolff seems to have subscribed to
Leibniz' views that whatever is extended in space, and hence
composite, must consist of simple, indivisible elements. But
while Leibniz categorically denies the spatiality of these
simple elements, Wolff was not quite certain what character
he should ascribe to them. He believed that his atoms could
combine to produce larger extended bodies, which he believed
to be continuous, but since he could not explain the nature
of this composition, he claimed that human notions of the
composition of matter must remain unclear. However, in one
way or another, a finite number of atoms was believed to
produce a body of perceptible size, and hence it was concluded
that the atoms must themselves have a finite extension. Hence
any piece of matter could be divided down to the atoms, but
not further.

Geometrical space, on the other hand, was said to be
infinitely divisible, hence, so it was argued, can have no

simple parts. These considerations led Wolff to distinguish

146
between physical or natural Space (i.e. space occupied by
a bodyL and geometrical or pure space.155

It is this distinction between two kinds of space
that Kant has in mind when he claims that geometry and
metaphysics are at variance, or at least were before his
critical system. Once one realizes this distinction, and
sees that geometry and metaphysics were thought to make con-
flicting assertions about space, one understands the urgency
of the Kantian question 'How is it that geometry can apply to
experience?‘ which, of course, means the same as ‘How is it
that geometry can be used in physical space when supposedly
it describes accurately only its own special kind of space?’
For Kant, the puzzlement consisted in the fact that the
application of geometry was successful despite the conflict
between the metaphysical and geometrical description of
(occupied) space — for, that there was but one space, and
that Wolffs distinction was untenable, Kant assumed from the
beginning. 1

This is a preliminary discussion of the problems that

Kant attempted to solve in the Monadologia Physica. The

 

155Cf° Ueberweg, Grundriss der Geschichte der

Bhilosophie, 3 V018,, Berlin 1905-1909, VOl. III, p. 228.

 

147
title of the little work already suggests its aim: £2327
physicae cum geometriae iunctae usus in philosophia naturali
specimen. Both geometry and metaphysics are to be used jointly
in natural philosophy. This means that on one hand the
'metaphysical' demonstration of the existence of simple bodies
is accepted, while on the other hand the geometrical demon-
stration of the infinite divisibility of space is also
granted, even if that space is occupied by a body.

First the term "monad" is introduced by definition:
"Substantia simplex, NOD18 dicta, est guae non constat
156

pluhalitateypartium, guirum ura absque aliis existere pptest."

Then a theorem is offexed, namely "Corpora constant

 

monadibus".157 This is to be proved by the statement that

if all composition were suspended, then the remaining parts
obviously do not have composition, and hence are simple.

This alleged proof is similar to one which was time and again
offered by Leibniz. Its difficulty lies in the notion of
'suspending composition‘. I'am not at all clear what could be
meant by this, and I cannot see that this "proof" demonstrates

in any way the existence of ultimate parts.

 

156Akademieausgabe, Vol. I, p. 477.

157Ibid.

148
After thus having satisfied himself that there are
ultimate parts to bodies - and Kant means finitely extended
ultimate parts, as we shall see - he gives a demonstration
that the space which a body occupies is infinitely divisible,

and hence does not consist of simple parts: "Spatium, guod

 

coxpora implent, est in infinitum divisibile, neque igitur

 

l
constat partibusgpgimitivis atqpe simplicibus." He offers

 

the following pioof: Conside. a line ef159

of "indefinite"
length, i.e. one that can be extended at will, and let it

be filled with monads. (linea partibus materiae primitivis

 

conflata) Call such lines "physical lines". On §£_draw at

 

right angles another physical line ggJ and likewise at right
angles to §£_another physical line ax; of the same length as
9g and different from 9g, Now mark points on 3f, call them
g, h, i, k, etc. and connect these points through physical

lines with c.

 

1581bid., p. 478.

159Notice that 'e' and 'f' do not denote points, but

this is of no consequence for the proof. See diagram.

149

The point in which gg_inte1sects §b_is called 0. Now any
line which is drawn from a point to the right of g to c will
intersect gb_above<>, and as we choose points farther and
farther to the right on ef, the intersections approach a more
and more closely. In fact, Kant concluded, since we can ex-
tend gf_at will, we can make as many divisions in §9_as we
wish. According to Kant, this proves that §g_and hence space
is infinitely divisible, and hence cannot consist of simple
parts.

As far as the mathematics of the situation is concerned,
'me proof did not offer any novelty, and Kant admits as much.
Why then did he include it? The only innovation that it
incorporates is the stipulation that the lines be "physical"
lines, 1 e. filled with monads. Now it must be understood
that the number of monads in any physical body is always
considered finite, hence that a monad occupies a finite space
(We shall see later on what it means for a body to be "filled"
with monads). Cleaaly, this innovation makes the proof
invalid, since the lines drawn between c and the points on
§f_have finite width, so that only a certain number of them
can be accomodated in the proximity of c. Kant apparently

did not realize this.

It is clear that the proof was to demonstrate something

150
over and above the similar geometrical proof, and while in
fact it proves nothing, we can nevertheless inquire what it
was supposed to prove. This is actually a quite simple
matter. If a certain finite number of monads were located
on abJ then for any monad m there is always a point n on g:
such that gg_cuts through m. Or, to put it geometrically,
for any finite part il_of §b_there is a point n on e; such
that the intersection of gn_and ab_lies between i and j.
This is supposed to show that the space which a monad occupies
is always divisible, although the monad itself is simple.

Or to put it otherwise: simple, finitely extended monads

are located in a space which is infinitely divisible, although
the monads themselves are not.

Clearly, the demonstration is directed against Wolff,
Who claimed that the space occupied by an indivisible body
is itself indivisible, and who was therefore forced to intro-
duce a distinction between geometrical and physical space.

This situation is completely misunderstood by Adickes,
who writes:

Only the space which they (the monads) occupy

is divided to infinity. Of the monads, on the other
hand, only a finite numbe is present. However, if
this is the case, then there is no reason why the

monads had to be dragged into the proofs of the
infinite divisiblity of space.

159Adickes, op. cit., Vol. I, p. 135.

 

151
That there was a yeason for considering the monads
in this proof is indicated by the passage cited from the

Prolegomena, in which Kant attacks the view that "the true

 

space of the object" consists of simple parts. The demon-
stration therefore must have been intended not only to show
that space is infinitely divisible, but that it is so divi-
sible no matter whether it is occupied by a monad or noto
Thus, the distinction between physical and geometrical space
is abandoned.

But this distinction had served a purpose. If the
space which a physical object occupied could not be divided
beyond the smallest parts of this object, then, it was
assumed, it did not make any sense to speak of matter as
being divisible beyond its atoms. Infinite divisibility
was restricted to geometrical space, which was considered a
merely ideal construct. designed to satisfy the postulates of
geometry In disregarding this attempt at solving the problem
of the constitution of matter, Kant at once faced again the
old problems that had been encountered, for example by
Cordemoy. Let us see how these problems are dealt with.

Consider theorem IV of the Monadologia Physica. It

 

states "Compositum in infinitlm divisibile non constat

 

Bartibus p.1mitivis s. simplicilus_"160 This, we know now,

160Akademieausgabe, Vol. I, p. 479.

152
needs the qualification that an infinitely divisible compound

does not consist of extended primitive parts. But theorem

 

IV gives rise to the corollary that if something does consist
of simple parts. then it is not infinitely divisible; accord-

ing to Proposition II, this holds of all bodies: "Corollarium:

 

Corpus igitur quodlibet definito constat elementorum

simplicium numero,"161 from which Kant concludes in Proposi-

tion V that the monad not only is in_space. but occupies

 

space. i.e. is extended. in spite of its postulated sim-
plicity.162

Here then arises the old predicament of atomism. If
the monad, or the atom. as case may be, is extended, how can
it be called simple. Or, in other w0rds. how can a body be
extended, but nevertheless indivisible?

Adickes has leveled the charge against Kant that he
confused physical divisibility with spatial extension. This
charge must be investigated; it will serve to bring the

difficulties of the atomist position into clearer focus. From

these considerations it will be seen first. that the problem

 

is

153

not nearly as simple as Adickes seems to have supposed.

and second that Kant is not guilty of any confusion in this

matter. Indeed. it will be shown that the second part of

the Monadologia Physica was. in fact. designed to cope with

 

the difficulties that arise from a distinction between exten-

sion and "physical divisibility".

We can best reach clarity about this matter by con-

sidering the various meanings of the term ‘divisible'. In

$0

to

It

in

to

doing I shall not investigate the difficulties that attach
this term insofar as it is a "dispositional predicate".
would be presumptuous to attack such a difficult problem
this context. Also. the term will be understood as having

do with division into spatial parts. Three meanings of

'divisible' can then be distinguished; namely. 1. x is

divisiblel= Df. a law of the form 'if such and such is done

to

x, then x will fall into parts' is known. 2. x is

divisiblezz Df. a law of the form ‘if such and such is done

to

x. then x will fall into parts' is not self-contradictory..

x is divisib1e3= Df. x occupies a finite space. This

third kind of divisibility should actually be attributed only

to

geometrical objects. Here it is no longer a question of

"taking something apart". One speaks of lines as being

bisected i.e. divided by other lines. surfaces by lines etc.

We may say that one line is divided by another line if the

154
two have a point in common. but both also have at least one
point which they do not share. Thus. in this third sense
of'divisiblefi,ve may say that a line or a surface or body
is divisible3 if it consists of more than one point, i.e. if
it is extended. Now since we are talking about division in
space, I should think that if a body is divisiblel, then it
is divisiblez, and if it is divisiblez. then it is divisible3.
However. both Descartes and Leibniz, as well as many others
believed that not only does divisibilityz imply divisibility3.
but that the converse also holds. An assertion frequently
found in these authors is one to the effect that an object
which is extended in space is also "divisible. at least in
thought". that is to say. if x is extended in space. than a
law of the form "if such and such is-done to x. then x will
fall into parts" is not self-contradictory. This then is
the perennial pJedicament of the classical atomists: they must
show that divisibility3 does not imply divisibilityz. In
fact, to be an atomist in the classical sense is to defend
precisely this point. But atomists have not generally
rested with the assertion that there are finitely extended
particles of matter which are nevertheless so indivisible.

They usually took it upon themselves to describe these parti-

cles in such a way that their view would become intuitively

155
plausible, a task from which they might better have refrained.
Thus they attached pseudo—phenomenal properties to their
atoms: Cordemoy called his "infinitely hard". Lambert called
his "solid", and Gassendi topped Lambert by ascribing
"absolute solidity" to his atoms. Now none of these attri-
butions can be said to add anything to the respective
author's central thesis that there are extended, but indivi-
sible2 particles of matter. On the contrary. these embellish-
ments were apt to cloud the issue. Leibniz for one did not
think that the properties of infinite hardness or absolute
solidity made it impossible to divide an atom "in thought",
and he seems to have realized that the ascription of these
properties did not add any force to, or make more plausible,
the proofs which were offered for the existence of atoms in
the first place.

Kant comes up against similar difficulties. He had
inferred. as we saw. that space is infinitely divisible,and
therefore does not consist of simple parts. Matter, it is
assumed, gges consist of simple parts, and therefore is not
infinitely divisible. Hence the physical monads are said to
,"occupy" finite spaces: Kant speaks of the "space of the
monad's presence". But if a monad occupies a finite space,

why is it not divisiblez? Kant makes a rather clever sugges-

156
tion by which the monad "occupies" a finite space in a certain
sense. without thereby becoming divisiblez. He contends that
the monad proper is indeed unextended. but that it is
enveloped in a field of forces which do not allow other monads

to approach beyond a certain limit. "Monas spatiolum

 

presentiae suae definit non plu alitate partium suarum
substantialumL‘sed sphera activitatis, qua externas utrinque
sibi praesentes arcet ab ulteriori ad se invicem appropinqua-

tione."163

That is to say the monad does not "define"
(circumscribe) the space of its presence through a plurality
of its substantial parts. but through a sphere of "activity"
which keeps the other monads from closer approach.
Proposition VII cautions that the radius of this
sphere of activity must not be confounded with the radius of

164

the monad propero The monad proper is altogether unex-

tended. The force which Kant introduced. and which keeps
monads from approaching one another beyond a certain limit
is said to be the same force that others have called impene-

trability.165

 

 

163gprg , p. 480.

1641b1g,, p. 481.

165Cf. prop. VIII. ibid., p. 482.

157
Thus the notion of impenetrability or repelling force be-
comes crucial. This force guarantees that any given piece
of matter is constituted of a finite set of monads, while
allowing the monads themselves to be unextended. Thus, thew
monad "occupies" a space not by being literally present in
it, but by exercising a force in it.

The repelling force which belongs to each monad is
not considered constant within a certain sphere but is
thought to decrease inversely proportional to the cube of
the distance from the monad proper. In close proximity to
the monad itself it is said to become infinite.

Much later, in the Metaphysische Anfangsgrfihde der

Naturwissenschaft, Kant makes it clear166 -- and in this respect

 

his views did not change -— that he thought of the impenetra-
bility of matter as what he calls "relative" impenetrability.
That is to say matter, and thus also the monads are capable
of being compressed to some extent. But since the repelling
force increases toward infinity in close proximity of the
monad proper, a total compression is assumed to be impossible.

In the Anfangsgrgnde he contrasts "relative" impenetrability

 

with "absolute" impenetrability, i.e. with the rigid

 

166Akademieausgabe, Vol. IV, pp. 465—565, p. 502.

 

158
occupation of space. A theory under which a space is filled
with absolutely impenetrable matter he calls a theory of
mathematical occupation of space, his own he calls a dynami-

cal theory.

The scheme presented in the Monadologia seems to work

 

out quite well. -The monad proper is not extended and there-
fore not divisiblez. But it nevertheless occupies space.
Hence the desired result is achieved.

But let us return again for a moment to the Anfangs-

grflhde. While Kant's views had undergone considerable change

 

between the Monadologia and this later work -- the Critique
of Pure Reason falls between the two -- there is much pro-
cedural similarity between the two. A passage from the
Anfangsgrghde is helpful in explaining how Kant thought this
kind of speculation to be justified. It seems that the
desired results are achieved ultimately by way of definition.
Thus, the system is developed in such a way that any critic

. »
eventually runs afoul of a definition. In the Anfangsgrunde

 

Kant makes the following point:

Lambert and others called that property by
virtue of which it (matter) occupies space 'soliditY'
(quite an ambiguous expression) and wanted it
assumed for everything that exists (substance), at
least as far as the world of outer awareness is
concerned. According to their notions, the presence
of something real in space should carry this
resistance with it by virtue of its concept, hence

159

by virtue of the law of contradiction, and should
thus bring it about that nothing can be co-present
with such a thing in the same space. However, the
law of contradiction does not repel a piece of
matter which approaches to occupy a certain space
in which another is found. Only when I attribute
to that which occupies space a force of repelling
any mobile matter which approaches can I understand
how there is a contradiction in (assuming) that a
thing penetrates into a space occupied by another
thing, 67 ‘

It is patent that the attribution of force is no more
successful in making impenetrability plausible, than is the
attribution of solidity. In the final analysis, impenetra-

bility is postulated in both cases, or, to put it otherwise:

 

the atom, matter, or monad, as case may be are defined in
such a way that the assumption of something else that
occupies the same space with that atom, that monad or that
piece of matter is contradictory, and it makes no difference
whether the notion of force or that of solidity is called

in to make the case plausible.

Essentially, the situation is no different in the
Monadologia. Here, too, it is the postulated properties of
physical monads that would make it contradictory to assume
that a monad is divisiblez, and it would seem that Kant has

overcome a major difficulty of atomism by postulating such

 

167
. Ibid., p. 497.

160
properties for his monads as would make them "occupy" -- in
a rather Pickwickian sense, to be sure -- a finite space,
while they are nevertheless indivisiblez.

Aside from impenetrability or repelling force, Kant
later on postulates a further property of monads: another
force is introduced, namelygravitation, which is said to
pull the monads together.168

As with impenetrability, Kant also assigns a definite
intensity to gravitation: it is said to decrease inversely
proportional to the square of the distance. I cannot under-
stand the reasons that led Kant to settle on the cube of the
distance for repulsion, and the square in this case, but the

following results from them:169

If the repulsion has a force
of, say, 1000 units of one kind or another at distance 1

from the monad proper, and attraction the force of 100, then
at distance 2, repulsion drops to 125, and attraction to 25.
At distance 10, the respectiVe forces balance one another,
and this distance then describes the limit of the"sphere

of activity” of the monad.

Adickes points out that Kant's spheres of activity

 

168Proposition X, Akademieausgabe, Vol. I, p. 483.

169
Cf. Adickes, o . cit., p. 175.

161
are literally spheres, in the sense that the monads cannot
come any closer to one another than is permitted by the cir-
cumference of these spherical entities, but that they also
cannot stay any further apart.170 He finds some difficulty
in the fact that Kant did not want to tolerate "empty spaces"

171 On this Adickes comments: "difficulties

within a body.
arise from the fact that his monads all have the same
spherical volume. If they touch one another with this, small
empty spaces must form in which the forces of repulsion do
not have the preponderance."172 However, I fail to see why
these small spaces deserve to be called empty. They can be
no more nor less empty than the spaces included ig_the ideal
spheres. Kant nowhere suggests that for a monad to occupy
space its repelling forces must outweigh the attractive ones,
Rather, he has developed a theory according to which matter
consists indeed of monads which are loosely scattered in
space, which are themselves unextended and hence indivisible,

but which have extended areas of activity and thereby "fill“

space. To say that monads fill a given space here merely

 

1
70 id., p. 161.

171
Cf. proposition VIII, ibid., p. 482.

172
Ibid.

162
means that the forces of attraction and repulsion of the
monads have values higher than some given value for any point
in that space. The value can be set in such a way that the
"pockets" between the spheres can.be considered filled.

This then is Kant's first attempt at a solution of
the problem of the composition of matter. The position he
takes here is frequently called one of dynamism as opposed
to atomism, but this distinction appears to me to be rather
artificial. I prefer to think of Kant's essay as an attempt
to describe atoms as both extended and indivisiblez. This

way of looking at Kant's Monadologia Physica shows very

 

clearly that he was not guilty of confounding spatial exten-
sion and what Adickes call "physical divisibility". The

entire problem of the Monadologia arises from this distinc-

 

tion, as I hope to have shown.
Kant's attempt is highly noteworthy, and finds its.
place in this dissertation as an attempt at reviving atomism.

But what is the significance of the Monadologia physica

 

in relation to the problem of the composition of the con-

tinuum itself? It seems that the little work constitutes

neither an advance, nor a regression in this respect. The
problem is not even considered. I believe that Kant was

completely unaware of, or did not recognize, the difficulties

163
concerning the continuum of abstract space itself; and this
in spite of his familiarity with Leibniz. He does not at
this point betray any knowledge of the paradox of Galileo,
nor of the fact that between any two points in a continuum
there is another point. both of which were known and pro-
vided considerable conceptual difficulties for some thinkers
of that age. Kant nowhere shows any apprehension when he
asserts that any finite space is infinitely divisible, nor
does he seem to realize the implications of that stand.
Rather. here as later on, he adheres to a definition of
continuity that is clearly inadequate. but which in Kant
never becomes the subject of any enquiry. This definition
is "A quantum is continuous which is not composed of simples."173
Many of the views espoused in the monadologia physica Kant ’
never altered in his later years. despite the profundity

of the change of his views from his precritical to his

critical period. However, this does not mean that the stand

 

l731h this form the definition is found in Kant's
inaugural dissertation De mundi sensibilis atque intelligi—
bilis forma etgprincipiis, Akademieausgabe. Vol. II, pp.
385-419. § 14.4, p. 399. The translation is taken from John
Handyside: Kant's Inaugural Dissertation and Early Writings
on Space, Chicago 1929, p. 540 I shall use Handyside's
translation throughout the discussion of Kant's dissertation.
Actually, the above definition does not occur in the
Monadologia Physica. Instead a Weaker statement is introduced:
Compositum in infinitum divisibile non constat partibus
primitivis s. simplicius. (Akademieausgabe, Vol. I, p. 479.

 

164

taken in the Monadologia remained unproblematic to him.

 

This early piece poses a number of problems which urgently
demand resolution. The most outstanding of these is that

matter as described in the Monadologia is not the same as it

 

appears to the senses - it is a congeries of monads and their
forces. while it appears, so Kant would hold. as continuous.
I do not think that Kant was very much disconcerted by this

fact when he wrote the Monadologia. The custom to regard

 

all sensory experience as confused was too well entrenched
among rationalists. I believe that it was Kant's encounter
with Hume that forced upon him the recognition that the
testimony of the senses cannot profittbly be discounted in
all cases when epistemological or metaphysical problems
arise. Thus Hume's influence made him recognize the pro-
blematic aspects of his earlier views.

The matter of Hume's influence upon Kant has often
been discussed. and Kant frequently acknowledges his indebted—
ness to Hume, especially in the introduction to the Prolego-
mena, This seems to be at variance with the views that I
have stated above. namely that the cosmological antinomies
were the most important factor in the development of Kant's

, 174
later philosophy.

 

174The problem here is not Hume's influence upon Kant

165

Unfortunately. the historical evidence on this point
is not sufficient to decide the case. Kant sometimes
attributes the awakening from his "dogmatic slumber" to his
encounter with Hume. and at other times to the antinomies.
What I wish to establish now is that the cosmological anti-
omies would not have arisen. had it not been for Kant's
acquaintance with Hume. The point is that the antinomies do
not seem to have been recognized befoze. As long as experi-
ence as a source of evidence is eschewed, the antithesis to
both cosmological antinomies cannot be established. In the
Monadologia we find a geometry which demands only a continuous
space, and a metaphysics which demands finitely extended
ultimate constituents of matter, but no evidence is recognized
which would demand that the same matter for which metaphysics
entails a finite number of constituents. should also have an
infinite number of ultimate parts. Now if it was Hume who
taught Kant the principles of empiricism, then it was also
Hume who made Kant recognize the problem of the antinomies.

Thus the dispute between Paulsen and Riehl toward the end of

 

as far as the latter's analysis of causality is concerned.
Influence in this respect is well nigh undeniable. Rather,
the point at issue is whether or not Kant's first writing
of his critical period. the Dissertation, which is not
concerned with an analysis of causality, is also partially
indebted to Hume.

166
the last century, whether it was Hume or the antinomies which
prompted the development of the critical vieWpoint is rather
pointless.175

Let us consider now how Kant's encounter with empiri-
cism could have led him to recognize certain problematic
aspects of his earlier philosophy. After the encounter with
Hume's philosophy, Kant no longer discounts sensory experience
lightly, but on the other hand, Hume's influence did not

reach deep enough to make him discard the rationalist scheme

advanced in the Monadologia. If such a half-way position

 

is taken, an account must be made of the discrepancy between
the rationalist account of matter on one hand. and the
sensory experience of matter on the other._ Here lies the
first problem. A second one becomes apparent when we con-

sider that in the Monadologia geomet y was accepted as an

 

ultimately trustworthy description of space. Now Kant must
have realized that geometry in its practical application
has to do with the world of sensory experience. But if
sensory experience is very likely to be misleading, then
geometry - whose certainty was unquestioned - could not

derive its validity from such experience. But if sensory

 

For this dispute cf. Klaus Reich, Introduction to
Kant, De mundi etc., Hamburg 1958, pp. VII ff.

167
experience is very likely to be misleading, then geometry -
whose certainty was unquestioned 4 could not derive its validity
from such experience. On the other hand. geometry could also
not be validated through a metaphysics whose principles run
counter to the doctrines of geometry. In the Monadologia,
metaphysics and geometry were considered. as it were. as two
independent and equally certain branches of speculation whose
reconciliation is attempted. But as Kant's attention was
drawn to the importance of sensory experience. the problem
of the application of geometry to sensory experience and its
validation -- not as a piece of speculation but as an applied
science -- becomes of pivotal importance.

That geometry could in fact be applied to empirical
objects was, of course, commonplace; but if geometry applied
universally and unrestrictedly to experienced objects, and
if these objects were thought to be continuous, then they
could not have simple extended parts, at least not as
experienced objects. But bodies as such did have Simple
parts. as had supposedly been shown in the monadolggia physica.

In summary, the difficulties that Kant realized after
his encounter with Hume stemmed largely from the fact that
sensory experience. relegated to insignificance in the early

writings. rose in status. and that geometry was recognized

168
not to apply to things as metaphysics showed them to be,
but to the world of sensory experience. We may say that it

is to Hume's credit that Kant recognized. and attempted to

 

resolve, the unfinished business of the monadologia physica.
This task was undertaken in the inaugural dissertation
of 1770, De mundi sensibilis acque intelligibilis forma et

principiis.176 Thus the Dissertation. as later the Trans—

 

 

cendental Esthetics. was meant to .esolve two problems at

the same time. namely the divergence between the metaphysical
and empirical accounts of matter. and the validation of geo-
metry. Moreoever, the suggestions made to this end are very
similar to those put forth in the Transcendental Esthetics.
In both accounts it is clearly stated that a distinction

must be made between things as they appear and things as

they really arel77 and. in addition, the Dissertation endea-

 

vors for the first time to explain why things appear as they
do. As far as the perception of objects in space is concerned,
it is asserted that:

The concept of space is a pure intuition...

It is not put together from sensations, but is
the fundamental form of all outer sensation. This

 

176Akademieausgabe. Vol. II, pp. 385-419. See also

note above.

Ibid., p. 392. Handyside. o . cit.. p. 44.

169

pure intuition can be readily observed in the
axioms of geometry. and in every mental construce
tion of postulates or of problems.178

and

Space is not something objective and real,
neither substance, nor accident. nor relation,
but subjective and ideal; and, as it were. a
schema, issuing by a constant law from the nature
of the mind, for the co-ordinating of all outer
sensa whatsoever.179

This theory does not only offer a ground for the
validation of geometry, but it also explains why geometry '
is applicable to the world of experience:

Thus, as regards all properties of space which
are demonstrated from a hypothesis not invented,
but intuitively given as being a subjective condi—
tion of all phenomena, nature is meticulously
conformed to the rules of geometry, and only in
accordance with them can nature be revealed to
the senses.

Here as later, Kant explicitly rejects the notion that
geometry is derived from experience, and thus places himself
in direct opposition to Hume. He points out:
Unless the concept of space had been given
originally through the nature of the mind, the use
of geometry in natural philosophy would be very

unsafe; for it would be poSsible to doubt whether
the notion of space obtained from experience will

 

179

 

Ibid., p. 403. Handyside op. cit.. p. 61.

190
Ibid»; p. 404,-Handyside, o . cit., p. 63..

 

170

sufficiently harmonize with nature, the determina-
tions from which it has been abstracted being per-
chance denied.181

In fact, the theory which would derive geometry from

experience is the only rival account of the validation of

geometry discussed in the Dissertation. It would indeed

 

explain why geometry applies to experience: because it is
abstracted from it. But it lacks the desired universality

and certainty which geometry de facto possesses, since any

 

generalization derived from experience has only a more or less
high level of confirmation, and can be invalidated “if the
determinations from which it has been abstracted...(are) per—
chance denied."

The theory of geometry so far expounded is well known,
being identical in substance with that put forth in the
Transcendental Esthetics. It differs widely from the con-
siderations offered in the monadologia. It must be noted
that in this new account sensuous knowledge is no longer
regarded as necessarily confused.

It is wrong to regard the sensitive as that

which is more confusedly known, and the intellectual
as that of which our knowledge is distinct...for

1811bid., p. 404f., Handyside, 09. cit.. p. 63. Cf.
David Hume, A Treatise of Human Nature. ed. Selby — Bigge,
Oxford 1955, p. 71.

171
that matter, the sensitive may be very distinct,
and the intellectual extremely confused. This
is shown, on the one hand by geometry, the pro-
,totype of all sensitive knowledge, and on the
other hand by metaphygécs, the instrument of all
things intellectual.
In the monadologia.the admission of the distinctness
of sensitive knowledge would have caused considerable diffi-
culties. While the monadologia assumes that differences
exist between the appearance and the real character of things,
the difference was put to the confusion of sensory experience.
Now, the difference is still recognized. but it is said to
be due to the fact that all sensitive knowledge is subject
to certain clearly recognizable forms. However, if objects
appear different from what they really are. how can the
recognition of their phenomenal character be called "know-
ledge"? Since things are different from their appearance,
does not the old charge that sensitive knowledge is confused
still hold? Would one not have to admit that beliefs formed
about phenomena are deceiving, since the things in themselves
do not agree with these beliefs? Kant answers this objection
in an exemplary fashion by observing that
to take judgments about what is known by sense,

the truth of the judgment consists in the agreement
of its predicate with the given subject. But the

 

182 ..
Ibid., p. 394f.. Handyside op. cit.. p. 47f.

172
concept of the subject, so far as it is a phenomen-
on, can be given only by its relation to the sensi-
tive faculty of knowledge; and it is by the same
faculty that sensitively observable predicates are
also given. Hence it is clear that the representa-
tions of subject and predicate arise according to
common laws, and so allow a perfectly true knowledge.
Thus the senses are capable of delivering perfectly

precise knowledge about phenomena. The passage does not
need any further explanation but its decidedly modern flavor
deserves to be noted.

However, now the difficulties come to a head. If
sensitive knowledge can be precise, and intellectual know-
ledge can also be precise, then the accounts of the composi-
tion of matter derived from both these sources must either
be in agreement, or. if they are not, an explanation must be
forthcoming why they differ. Unfortunately, the accounts
are not in agreement. Matter as perceived must obey the
laws of geometry, which asserts continuity of its bodies,
and to be continuous, according to Kant, is to have no ulti—

. . 184
mate parts, although the continuous 18 a compound. On
the other hand, matter when considered as a compound and
apart from the conditions of its sensory perception

183Ibid., p. 397, Handyside op. cit., p. 51.

184Ibid., p. 399, Handvside op. cit., p. 54.

183

173
must contain simple parts: “...the intellect proves that
given a complex of substances, there are given also elements
of composition, i.e. simples."185

It must be noted that in these two conflicting accounts
we have a statement of the second antimony. In order to see
how a resolution of this antinomy is attempted in the Disser—
tation, we must turn to the opening passages of that work.

Kant here makes a distinction between the operative and
the conceptual formation of the concept of a whole. Since he
does not distinguish between classes and wholes, he claims
that the composition in the latter case is achieved through
the class concept, but in the case of the operative formation
(synthesis) of the concept of a whole, the process depends
on the conditions of time. since it requires the successive
addition of part to part.

Thus, to venture an example, the conceptual formation
of a certain whole (class) e.g the class of brown horses and
blown cows, is achieved through the class concept, i.e. "pro-
duct of the class of brown things with the sum of the class

of horses and the class of cows." The operative formation

of the concept of that whole proceeds by "taking account" in

 

185
Ibid., p. 415. Handyside. op. cit., p. 79.

174
one form or another. of each member of that class, where the
whole finally emerges as "the class of all those things of
which I have here taken account (whnzh I have added together)".

For our purposes more impOrtant than the case of compo-
sition is the similar situation that obtains with respect to
decomposition. Here again Kant contrasts a conceptual
decomposition with an operative decomposition.

Given a complex of substances, we easily

reach the idea of simples by completely removing

the intellectual notion of composition; for it is
simples that remain when all conjunction is abolished.
But according to the laws of intuitive knowledge,

the case is different; composition is not removed
completely save by a regress from the given whole

to all possible parts whatsoever, in other words,

by an analysis which again rests on conditions of
time.186

Such a decomposition can be carried out fully only if
the process can be terminated within a finite time. But the
continuum is of such a character that its decomposition cannot

thus be brought to an end.

But since in the continuous quantum the
regress from the whole to its possible parts...
finds no end...the analysis...will be impossible
of completion. The whole cannot. in conformity
with the laws of intuition, be apprehended by an
exhaustive division of parts.

 

186Ibid., p_ 387, HandYSide, on. cit.. p. 36.

187 '
Ibid., p. 388. Handyside, o . cit.. p. 36.

175

In order to understand the force of this argument, it
must be recalled that both space and time are given as pure
intuitions and that, moreover, they are the forms under which
phenomena appear. This means that the conceptual decomposi-
tion of space or time, and of objects given in space and
time cannot be achieved; or, to put it otherwise, it is not
permissible to argue that an object given in intuition, i.e.
under the form of space. consists of simple parts, if it is
complex. This argument holds only for complexes given, not
th;ough intuition, but through the intellect. Thus, as far
as objects in space are concerned, one cannot "reason out"
what their ultimate parts are, nor that they have ultimate
parts at all. and the reason for this is that the primary
properties of space and time are, as it were. non-discursive.
Kantpoints out: “All primary properties of these concepts
(i.e. space and time) are beyond the juiisdiction of reason,
and so cannot in any way be intellectually explained."

Hence, if a division of space, or of an object in
space is to be undertaken, or if it is to be made out of what
space or objects in space are composed, one must avail oneself

of that method which is appropriate to objects given in

 

1881bid., p. 405, Handyside, o . cit., p. 64.

 

 

 

 

 

 

176
intuition. i.e. operative decomposition. Thus, according to
Kant. operative decomposition is the only way in which any—
thing can be made out about the parts of a continuum. But
such a decomposition can never come to an end in such a
case. and this is precisely what characterizes continua. The
reason why this decomposition cannot come to an end lies in
the following: Every division of a body or spatial region
:reguires a finite time. Hence, if the dividing game has been
<going on only for a finite time the parts of the body or
:spatial region which result upon a series of divisions will
ailways be finitely extended. Therefore, these parts are
:Eurther divisible. Kant presumably reasoned that if the
liltimate parts of the continuum were unextended, then they
CHDuld be reached only through a process of division extending
(Iver an infinite length of time. But the experience of an
infinite time, i.e. a length of time consisting of infinitely
Inany finite time-spans, is impossible. for. if I start at any
given moment, there is never a moment at which I can say
that now I have experienced such a time span. Hence to
Speculate what the ultimate constituents of the continuum
are is unwarranted, since the speculation transcends all
possible verification. «

In the Dissertation Kant merely asserts that it is

 

177
false that the continuum has ultimate parts The above
considerations, however, have shown that under his theory
he should have asserted that it is meaningless to say that
the continuum has ultimate parts. This position is expressed

in the Cjitigue of Pure Reason, in the proof of the antithe-

 

sis of the second antimony. There Kant says:

The second proposition of the antithesis that
nowhere in the world does there exist anything
simple, is intended to mean only this, that the
existence of the absolutely simple cannot be
established by any experience or perception, either
outer or inner, and that the absolutely simple is
therefore a mere idea, the objective reality of
which can never be shown in any possible experience,

and which, as being without an object. has no
application in the explanation of the appearances.

189
We may then say that because of the above described
limitations no operation can produce an intuition of ultimate
parts of the continuum. Thus, since ultimate parts of the
continuum cannot be known, a continuous body or surface or
line cannot be apprehenJed through the apprehension of its
ultimate parts.
It appears that although intuition provides us with
the notion of space as continuous, as well as with the know-

ledge of objects in space, it is not therefore capable of

delivering also all the implications of the concept of

 

189Critique of Pure Reason, A 436f, B 464f.

 

178
continuity which it has brought about One such implication
is that "in a continuous quantum the regress from the whole
to its possible parts...finds no end," which means that the

continuum cannot be gepiesented in intuition as the sum of

 

its parts. This leads to a rather thorny problem_ We have
seen that the discussion of the problems of the continuum

is outside of the domain of reason, and now it is asserted
that intuition, too, is incapable of asce taining much about
the continuous As a matter of fact, it is asserted that

some of the properties of the continuum, e.g. its divisibility
ad infinitum is contrary to all intuitive expectations, i.e.
it is counter-intuitive.

If this is the case, then continuous quantities ought
to be accepted as intuitively given, and one should forego
any analysis of that which is so given. But in point of
fact, an analysis of sorts had already been undertaken
when it was pointed out that a division of a continuous
quantity cannot come to an end.

It seems now that there are no grounds on which this
belief can be held, since both intuition and reason show
themselves incapable of substantiating such an assertion.
Here is the passage in which it is asserted that the infinite

division of the continuum cannot be apprehended in intuition,

179
but in which it is also asserted that this fact does not
marshal against the belief in this infinite division:

Since the unrepresentable and the impossible
are commonly regarded as meaning the same thing,
and since representation of the concepts of con-
tinuity and infinitude in accordance with the laws
of intuitive knowledge is clearly impossible, we see
how it comes about that these concepts are usually
rejected...It is, however, of the greatest importance
that those who follow this highly perverse line of
argument should be warned that in so doing they
fall into a most serious error. Whatever is
opposed to the laws of understanding and reason
is wholly impossible; but that which (as being an
object of pure reason) is at variance only with
the laws of intuitive apprehension is not necessarily
impossible. For this disagreement between sensi-
bility and intellect...shows no more than this, that
the mind is frequently unable to follow out in the
concrete, and translate into intuitions, abstract
ideas which it has received from the intellect.190

The passage is remarkable for its defense of counter-
intuitivity, but on the face of it it seems to raise more
issues than it settles. For one thing, it appears that all
of a sudden the continuous is characterized as arising from
the intellect, moreover it is indicated that this is the
reason why we have difficulties in intuiting it.

If this is what is meant, then the above passage
clearly introduces an inconsistency into the system of the

Dissertation. But I think that an interpretation is possible

 

190
pp. 37f.

fifademieausgebe Vol. I, pp. 153571:~ Handyside, o . cit.

180
which avoids this drastic course. To begin with, let me
reiterate that according to Kant, continua can be given only
in intuition. It is the attempt to figscribe_these continua
which soon introduces counter-intuitive aspects, i.e., it
is somewhere in this description that "abstract ideas...(are)

..received from the intellect" which the intuition cannot
follow out in the concrete. The only feature in the
description of continua which cannot be intuited is the fact
that they are capable of being divided indefinitely. Thus,
while reason cannot tell us what the ultimate parts of the
continuum are, or whether there are any such, it proves that
upon any operative division of any given continuum finitely
extended, divisible parts remain. That is to say, reason
does not give us the representation of continua, it is not
even responsible for the arising of the concept of the con—
tinuum in general, but it does elucidate certain aspects of
this concept which are incapable of being intuited. While
this view of the matter is not very clearly expressed in the

Dissertation, there can be no doubt that it is adopted in

 

the Critique of Pure Reason. Section eight of the Antinomy
of Pure Reason introduces what is called the Regulative
Principle of Pure Reason in its Application to the Cosmologi—

cal Ideas. There it is pointed out:

181

The principle of reason is thus only a rule,
prescribing a regress in the series of the condi-
tions of given appearances, and forbidding it to
bring the regress to a close by treating anything
at which it may arrive as absolutely unconditioned...
Thus it is a prin:iple of reason serving as a
rule, postulating what we ought to do in the regress,
but not_antigipating what is present in the object
as it is_in_itself,_prior to all regress. Accordipgly,

 

n—m—d

I entitle it a regulative principle of reason...

 

The regress that is generated through a series of
divisions of a given continuum falls under this rule of
reason: "If we divide a whole which is given in intuition, we

proceed from something conditioned to the conditions of its

"192

possibility. That this rule of reason prescribes a

proceSs which cannot, in its entirety, be represented in
intuition becomes clear from the following passage:

When...we have in mind the transcendental
division of an appearance in general, the question
how far it may extend does not await an answer
from experience; it is decided by a principle of
reason which prescribes that, in the decomposition

"of the extended, the empirical regress, in con-
formity with the nature of this appearance, be
never regarded as absolutely completed. 3

Thus, reason tells us that the operative decomposition
of continuous entities will never come to an end, and this

19lCritigue of Pure Reason A 508£,'B 536f.

lgzIbid., A 523, B 551.

l93Ibid., A 527, B 555.

182
is the counter-intuitive aspect which reason introduces in
the analysis of the concept of the continuum.

However, while reason provides us with this rule, it
cannot make out anything else about continua. To summarize,
since the concept of continuity has its origin not in the
intellect, but in intuition, reason cannot ascertain the
composition of the continuum. But reason dictates that any
division of the continuum will result in finitely extended
parts, no matter how many divisions are successively under-
taken. On the other hand, we cannot intuit such an unending
sequence of divisions, since they would demand an infinite
time, of which an intuition is impossible.

This is the point to which, according to Kant, the
analysis of the continuum can proceed, and any further
pronouncement about it, as I have tried to show is held to
be meaningless.

It is clear from the above that a continuous quantity
is not to be thought of as a complex of substances (for
as such it would alwyas have to composed of simples) but
only as a something which is given in intuition. Thus the
intellect, which demands that i:§_objects consist of simples
cannot pass judgment on the composition of such continuous

quantities. On the other hand, things in themselves,

183
according to Kant, must fulfill the requirement that, if
they are complex, they must consist of a number of simples,
either finite or infinite. This position is retained in

the gritique of Pure Reason, although neither there not in

 

Dissertation a proof is offered, unless one were to consider

 

the following passage a proof:

The assertion that if all compositeness of
matter be thought away nothing at all will remain,
does not appear to be compatible with the concept
of a substance which is meant to be the subject
of all compositeness, and which must persist in the
elements of the composite, even although the connec—
tion in space, whereby they constitute a body, be
removed. But while this is true of a thing in
itself, as thought through a pure concept of the
understanding, it does not hold of that which we
entitle substance in the field of appearance. For
this latter is not an absolute subject, but only
an abiding image of sensibility; it is nothing at
all save as an intuition, in which unconditionedness
is never to be met with.194

I interpret this passage in the following way: by
substance is here meant something which is absolutely simple.
Any complex of substances depends for its existence on the
existence of the (simple) substances of which it is composed.
frhus it is analytic that a complex of substances consists
c>f, and depends for its existence on, Simples. On the other

Tiand, not all complexes are complexes of substances, and hence

1
94Critique of Pure Reason, A 525, B 554.

 

184
the conclusion that all complexes consist of simples is
fallacious The latter holds in particular for continua.

Here then lies Kant's solution to the problem of the
continuum: As in Leibniz, an intelligible world of noumena
is distinguished from a phenomenal world. Continua occur
only in the latter, but since they have their origin in
intuition, not much can be made out about them discursively.
But intuition, being subject to the limitations of time is
also incapable of breaking down a continuum to its ultimate
parts, so that it becomes meaningless to speak of the
ultimate parts of the continuum. Noumena, or things in
themselves cannot be said to be continuous. Of them it can
therefore confidently be asserted that, if they are composite,
they have Simple parts.

A word of caution is here in place. When it is
‘asserted that a series of divisions of a given continuum
can, in principle, be infinitely extended, i.e. that the
operation of division cannot have a final member, then this
is not to mean that there is an infinity of parts already
given in that continuum. Any intuition is always given as
a whole, in one act of intuition. In the successive steps
of its operative decomposition more and more of its parts

are realized. But Since a given continuum is nothing but an

185
intuition, and nothing but ong_intuition, we cannot say that
these parts were already there before the operative decompo-
sition began. Thus it can be said that a continuum can be
divided ad infinitum, but not that it "has" infinitely many

parts.195

 

This point is made so often, from the Dissertation to

 

. O . . .
the MetaphySische Anfangsgruhde, and Kant is so inSistent

 

 

upon it that it is quite obvious that he attached a good deal
of importance to it. And justifiably so, for he regarded the
assumption of an actual infinite in a finite object as incon-

. .0
Sistent. In the Anfangsggunde he flatly asserts that "there

 

is no real infinite number (of parts) in the object (which

‘

would be an explicit contradiction)."196

In the Critique 0‘ Pure Reason he tries to give a ground

 

for this assertion by saying:

In the case of an organic body conceived as
organized in infinitum the whole is represented
as already divided into parts, and as yielding to
us, prior to all regress, a determinate and yet
infinite number of parts. This, however, is self-
contradictory. This infinite involution (Entwicklung)
is regarded as an infinite (that is never to be
completed) series, and yet at the same time as
completed in a complex (Zusammennehmung).

 

195
Cf. Critique of Pure Reason, A 513f, B 541f; and

A 524E, B 552f.
196

 

Akademieausgabe, Vol. IV, p. 507.

 

197Critique of Pure Reason, A 527, B 555.

186

That this argument is fallacious can easily be shown.
In order to avoid problems that arise from higher orders
of infinity let us consider the set of rational numbers
larger than or equal to l and smaller than or equal to 2.
This set has 1 and 2 as its smallest and largest member
respectively, and it contains infinitely many members. A
one to one correspondence between it and the set of positive
integers can be demonstrated. But to argue that there cannot
be infinitely many rational numbers between 1 and 2 because
the set of integers has no last member is clearly a non

sequitur. But this is what Kant's alleged proof amounts to.

 

One could perhaps surmise that Kant's denial of the
simultaneous infinite was brought about through Leibniz'
contention that there is no infinite number, which we have
discussed above. In that case we would find in Kant again
some ihfluence of Galileo's paradox. This surmise receives
some lilelyhood from the fact that Leibniz frequently
asserts that an infinite number of parts in an object is
impossible, while only occasionally stating his reasons for
this assertion. Be this as it may, Kant's denial of the

}-
actual, simultaneous infinite can be said to have had pro-

found influence upon the structure of his system, whatever

his reasons for this denial might have been.

187

The theory so far discussed is that represented int

the Dissertation. In my exposition I have made use of

 

material from the Critique of Pure Reason only for elucidatory

 

purposes, and I feel justified in having done so since I cannot
perceive any fundamental difference between the theories of

the Dissertation on one hand, and the Transcendental Esthetics

 

and the discussion of the first two antinomies on the other.

The first Critique goes beyond the dissertation in that it

 

gives a clearer statement of the antinomies to be resolved,
but the resolution, at least of the first two antinomies, is
undertaken entirely within the framework already provided by

the Dissertation. iat is left for us to do, then, is to

 

consider the Second Antinomy, and to see how, precisely, its
resolution is attempted. The Second Antinomy is stated thusly:

Thesis: Every composite substance in the
world is made up of simple parts, and nothing any-
where exists save the simple or what is composed
of the simple.

Antithesis: No composite thing in the world
is made up of simple parts, and there nowhere
exists in the world anYthing simple.198

By the term 'world' is here meant 'the sum Of all

199

appearancesg. which, according to Kant, makes the_thesis

 

19RCritique of Pure Reason, A 434, B 462.

199CE. Critique of Pure Reason, A 419, B 447.

188
false. The thesis would hold only of things in themselves,
but these do not belong to the world defined as the sum of
appearances.

On the other hand, the antithesis is intended to come
out true, with the understanding that 'world'. again, means
'the world of appearances.‘ Thus the conflict is removed
Since the thesis is true only for things in themselves,
the antithesis only for appearances.

y.

A difficulty arises from the fact that Kant seems to

 

claim, on at least one occasion, that both thesis and anti-

thesis are false. He points out that "the suit in which

reason is implicated...has been dismissed as resting, on
200

both Sides, on false presuppositions."
Now we have seen that the thesis is clearly meant to

be false, owing to the restriction of 'world' to appearances.

The antithesis, however, is meant to be true, if the term

'world' is used in the same restricted sense; it amounts to

the claim that among appearances simples can nowhere be found.
I believe that what Kant assails as resting on false

presuppositions is not the position stated in the antithesis,

\

nor, for that matter, that stated in the thesis, but doctrines

 

200Critique of Pure Reason, A 530, B 558.

189
that would arise, if by 'world' were meant, not 'the sum of
all appearances, but 'the sum of all there is'. In this
case, the thesis would be false because it is not considered
to hold of appearances, and the antithesis would be false,
since it supposedly does not hold of things in themselves.
Thus, a conflict arises only if the claims, either for thesis
or antithesis, are overextended, as would be the case if the
teim 'world' were meant to refer to all there is.

We have seen above what Kant's reasons were for accept-
ing the doctrine of the thesis for things in themselves, and
that of the antithesis for appearances, but a short summary
is here in place.

In the early Kant the problem of the composition of
matter was construed as a problem about the intelligible
world. Kant assumed then that any piece of matter, considered
in itself, is made up of a finite number of ultimate parts,
while the space in which such matter is located is infinitely
divisible. Since appearances do not agree with the meta—
physical description of matter, it must be assumed that
sensory experience is confused. This doctrine was put in
jeopardy by the realization that geometry applies unre-
strictedly to objects as they appear, and since geometry

demanded the infinite divisibility of its lines and bodies,

 

190
the same would have to hold of any and all objects in space.
The unrestricted application of geometry to spatial objects
was considered justified through the assumption that space
is the form of all outer experien e, and that the axioms
of geometry are an accurate description of the pure intui-
tion of space. As long as the doctrine of infinite divisi-
bility applied only to purely geometrical entities it was
perhaps not as pressing to the metaphysician as now, when
this same doctrine was to hold of any objects in space.

The solution was made more difficult through the assumption
that it is contradictory to assume an infinity of parts in

a finite object. The conflict was finally resolved through
the introduction of the notion of operative decomposition:
bodies in appearance are originally always given as gngJ and
they have only as many actual parts as are in fact realized
through successive steps of operative decomposition. That
there is no lest utep in this decomposition is guaranteed

by a so-called rule of reason. Thus bodies are said to

~ A

be divisible ed Sgginigufl without therefore having an infinity
of actual_parts. fi;s whole scheme required that bodies
are not something in themselves, because only if they are

not things in themselves can it be assumed, so Kant thought,

that the parts do not precede the actual division. If they

191

were things in themselves an infinite division in the object
would have to be assumed, which was held contradictory. Kant's
theory was not only meant to resolve the problems of the
continuum as far as the constitution of (continuous) bodies is
concerned, but it was intended also to settle the issue for
continua as they occur in pure geometry. For these the same
considerations would hold, since they, too, are objects that
occur in intuition. albeit in pure intuition.

Thus we can see how the doctrine of the Transcendental
Esthetics is a proposal which, in Kant's opinion, would
satisfactorily resolve the problems of the composition of the

continuum.

CHAPTER V
BERNARD BOLZANO

In the preceding chapters I have attempted to show
that some well-known philosophical systems of the Seventeenth
and Eighteenth century were constructed in such a way that
in them certain problems connected with the continuum and
with infinite sets did not arise. In all those systems such
problems were of central importance, as I hope to have shown,
but they were by no means the only problems to be discussed: .m»
they were always transacted in a larger context. We saw
that they arose in connection with the problem of the con-
stitution of matter, of space or of time, and that their
solution was attempted through metaphysical or epistemological
considerations.

In Bolzano this is, for the first time, no longer the
case. Bolzano addresses himself explicitly and directly to
the problem of infinite sets in general, and of continuous
sets in particular. Even through a casual perusal of Bolzano's
work one becomes impressed with the forthrightness and com-
parative precision with which he states his problems and

attacks them. It seems that this very fact earned him the

contempt of many of his contemporaries. I believe that this ‘\

193
is largely due to the influence of Kant's philosophy. Bolzano
actually addresses himself to several of the problems with
which Kant was originally concerned, and for which Kant
supplied his own particular solutions. But it was Kant's
solutions, rather than his problems, that directed the
philosophical enquiries of his recognized successors. A

passage from a review 0: Bolzano's Wissenschaftslehre bears

 

testimony to this:
The standpoint of the author is throughout
the old one, which is called strictly objective,
also dogmatic, in contradistinction to the con-
temporary one which is based on the psychological
self-consciousness of the thinking mind.
In the face of much derogatory twaddle, Bolzano
continued to deliver his philosophy from the outmoded objec-

tive standpoint, and one of the fruits of his labor is his

epochmaking, though by no means definitive Paradoxes of the

202

 

Infinite.

 

201?. Menelaos (probabl; a .om de plum), Review of
Bolzano's Wissenschaftslehre, Zeitschrift r3: Philosophie
und Katholische Theologie, Heft 25. Quoted from Dr. Bolzano
und seine Gegner, Sulzbach (Seidel) 1839, pp. 157E.

202

Bernard Bolzano, Paradoxien des Unendlichen, (ed.
Fritz Prihonsky), Hamburg (Felix Meiner) 1955, lst. ed.
Leipzig 1851. Translated as Paradoxes of the Infinite
(translated and ed. by Donald A. Steele, S.J.L London
(Routledge and Kegan Paul) 1950. This work was published
posthumously (Bolzano died in 1848), and there is a likely-
hood that Prihonsky did some editing which, at least on

 

 

194
This work is known chiefly for the fact that in it Bolzano
states again several examples of the Paradox of Galileo, and
generally embraces the position that all infinite sets have
subsets to which they stand in biunivocal correspondence.
Bolzano thought that he had, for the first time, discovered
this property:

We now pass on to consider a very remarkable
peculiarity which can occur in the relation between
two sets when both are infinite. Properly speaking
it always occurs, but to the disadvantage of our
insight into many a truth of metaphysics as well
as physics and mathematics, it has hitherto been
overlooked. Even now, when I come to state it, it
will sound so paradoxical that we shall do well to
spend some time over its investigation. I assert
the following: When two sets are both infinite, then
they can stand in such a relation to one another that:

 

 

one occasion, led to a palpable falsification. Bolzano him-
self had prepared for publication a manuscript on the theory
of functions which was published in 1930 (Functionenlehre,
Prague 1930), in which he gives an explicit example of a
continuous, non-differentiable function, while the paradoxes
declare (Footnote § 37) that all continuous functions are
differentiable, except for isolated points. The discrepancy
was first discovered by Jasek. Cf. Introduction to Paradoxes

 

 

of the Infinite, p. 54). It was generally held at the time
that all continuous functions are differentiable, so that it
can be unde stood how Prihonsky, a man of limited mathe-
matical background could have added an "explanatory" footnote
to that effect. On the other hand, in the present paper we
are concerned with doctrines, the ingenuity and novelty of
which point clearly to Bolzano as their author. Moreover,
there are a large number of points of agreement between the
Paradoxien and the Wissenschaftslehre (Bernard Bolzano,
Wissenschaftslehre, 4 vols., Sulzbach (Seidel) 1837). In
particular, the definition of "continuum" is identical in
both accounts.

 

 

 

 

195

(i) it is possible to couple each member of

the first set with some member of the second in

such a way that, on the one hand, no member of

either set fails to occur in one of the couples;

and on the other hand, not one of them occurs

in two or more of the couples; while at the same

time

(ii) one of the two sets can comprise the

other as a mere part of itself.203

Now we have already seen that Bolzano was mistaken
in assuming that he was the first to notice this relation,
since both Leibniz and Berkeley were aware of it in its
generality, while Galileo knew at least an example.204/ What
is new is that Bolzano did not think that this property
makes the existence of infinite sets impossible; in other
words, he no longer regards the sweeping acceptance of
Euclid's part-whole axiom as justified.

This attitude of Bolzano's, of course, rests on his
recognition that the assertion of the biunivocal corres-
pondence of an infinite set with one of its proper subsets
does not introduce any obvious contradiction into mathematics,
provided that Euclid's axiom is properly understood. Bolzano's

\

merit lies, then, not in his having discovered this feature

 

203Bolzano, Paradoxien, § 20, p. 27 f.

204Jourdain is consequently mistaken when he writes
that "this curious property of infinite aggregates was first
noted by Bernard Bolzano" (Philip E. B. Jourdain (ed.) 9927
tributions_tp.the,Eounding of the Thegry of Transfinite
Numbers, by Georg Cantor, New York (Dover) 1915, editor's
introduction, p. 41).

 

196:

tn

0 infinite sets, but in that he, for the first time,
placed enough trust in the procedures of mathematical
enquiry simply to accept it as a fact, rather than to con-
front it with bewilderment as his philosophical precursors

,-

had done. Thus we must criticize Fraenkcl ior assuming

that Bolzano's Paradoxien were to be "a catalogue of, as it

 

were lamentable, paradoxes which condemmsitself to fruit-
1essness."205

Actually, the Paradoxiea are a discussion of alleged

 

paradoxes, of which the Paradox of Galileo is one. But

 

their whole point is that here are only seeming paradoxes

rt

which a precise account of infinite se s will dispel.

Lamentable is at most the fact that other philosophers
found paradoxes were there are none.

Bolzano makes it quite clear where the oddity of the
Paradox of Galileo stems from:

As I am far from denying, an air of paradox
clings to these assertions; but its sole origin
is to be sought in the circumstance that the
mutual relation which we find between two sets
when we can pair off their parts (members) with
the previously mentioned result suff ces in every
case where these sets are finite to “tablish their
perfect equinumerosity of members.

F1.

0
O" (D

 

2
05Abraham A. Fraenkel, Meugenlehre und Logik,

Erfahrung und Denken, Vol. II, Berlin (Dunker und Humbiot),
1959, p. 10.

 

 

206Bolzano, Paradoxien, § 22, p. 31.

 

197

Let us consider, for the moment, only part of the
import of this statement. What is asserted is that it sounds
paradoxical to claim that infinite sets are in biunivocal
correspondence to proper substes of themselves, but only
because such a relation cannot obtain with finite sets.

The latter fact is seen as an obstacle to the acceptance
of the former, probably because we do not have any direct
acquaintance of an intuitive or sensory sort with infinite
sets.

A more important feature of the above passage is that
it adds to the assertion that all infinite sets stand in
this relation to some of their proper subsets the converse
of this assertion, namely that only infinite sets have this
‘property. However, this characteristic of infinite sets,
namely that all and_only infinite sets can be brought in
a one-to-one correspondence with proper subsets of them-
selves, is not taken to be the definitive characteristic
of infinite sets: Bolzano chose a different definition:

"I shall call an infinite multitude one that is larger than
any finite magnitude, i.e. one of which any finite set

represents only a part."207

L
fifi

207Bolzano, Paradoxien, § 9, p. 6.

198
I can see no objection against this procedure, which first

defines 'finite', and then employs the term 'finite' in

(A.

”he definition of 'infinite', provided that a satisfactory
definition of 'finite' is given. Balzano gives this defini—
tion in the following passage:

Let us consider a series whzse first term is
an individual of the species A, and whose every
subsequent term is derived from its predecessor by
joining a fresh individual with the equal of that
predecessor so as to form a sum. Then clearly
all terms which occur in this sequence, with excep-
tion of the first, which is a mere individual of
the species A, will be multitudes of the species
A. Such multitudes I call finite.208

 

 

w.

This definition poses more problems than it solves. Setting
aside minor inadequacies of expression, what is intended
is obviously the following: Consider a set A of objects.
Choose any member of A, for example k, and form the unit
set {kt of k. Then either take k, and form a set consisting
of k and some other element of A, say l. 0: else pick a
set of members of A which has as many elements asl kl , i
for example lm‘ , and form a set out of its members together
with precisely one other member of A. In general, if N is
a given set in the series, the next set is fcrmed either by
taking the elements of N together with precisely one other
element of A, or else by taking a set N' which is "equal"

' \

2081bid.. § 8, pp. 5 ff.

199
to N and which is also a subset of A, and adding to the
elements of N' another element of A. Clearly, the pro—
cedure depends upon a definition of 'equal'. which is nowiere
given, a fact that leads to other serious shortcomings, as
we shall presently see.

All sets in the series so generated are finite in the
general, preanalytic acceptation of that term, but we do
not therefore get a precise definition of 'finite', since
obviously there are sets which neither occur in the sequence,
nor fulfill the criteria for infinite sets. To remedy this
situation, we would have to consider not just one set (species)
A, but all setsthere are. and for each of these, except for
the unit sets, we would have to form more than one series,
since under the above described procedure for example.{l; ,
which is ordinarily considered a finite set, would nowhere
occur in the series we were describing.

‘but even if all these amendations were made. we would
still require a definition of equality, a proof that the
choices here described can always be made and assurances
that the consideration of all sets does not, in this case,
lead to inconsistencies. I cannot here undertake a detailed

discussiOn of these matters, but the above considerations

show that Bolzanofs definition of 'finite set' is not tenable

in the form in which it is given.

To return to the di3cussion of infinite sets: We saw
that Bolzano had recognized that all and only infinite sets
stand in biunivocal correspondence to some of their proper
subsets. It will be recalled that Cantor ascribed equal
"power" (Machtigkeit) to sets that stand in such a relation
to one another. Nowadays we generally call such sets equi—
valent, Presumably, Cantor chose the word 'pcwer' in order
to avoid such expressions as 'equal', or 'as large as', or
‘having as many members as', which might merely have involved
him in fruitless quibbles. Cantor, however. nowhere asserts
that of two sets which have equal power, one can be larger
than the other, no matter how "obvious" this might seem.
Bolzano had not reached this stage. He points out that if
we pair off the members of two finite sets, and none remains
in either set, then none of the two sets is larger than the
other. Not so with infinite sets. Here, he claims, two
sets can be in one-to-one correspondence, even though one
is larger than the other.209

The mere fact...that two sets A and B are

so related that every member a of A corresponds by
some rule to some member b of'B in such wise that

209Ibid., 5 22, p. 31.

201

the set of couples.(a~b) contains every member r*f

A or B once and only once, never justifies us...

to infer the equality of the two sets with respect

to the multiplicity of their members if these sets

are infinite.

The "obvious“ examples are the rational numbers
between 0 and 5 and O and 12 respectively, and the points
on a bounded line and on a proper part of that line.
Although in both cases biunivocal correspondences can be
established, Bolzano holds that there are "more" rational
numbers between 0 and 12 than there are between 0 and 5.
and a similar case is made for the two lines.

One might think that Bolzano perhaps differed only
in terminolognyrom the now generally accepted position.
But in order to establish this conclusively, one would have
to find out what, precisely, Bolzano meant by ‘equal in
number', 'larger' etc. To judge from his examples, what
he had in mind is the following: If A is a subset of B,
then A is smaller (has fewer members) than B. If A and B
have identical members. then they are equal. If this is
all that can be supplied in definition of 'equal' and
'smaller', then we are forced to the assumption that certain

sets cannot be compared with respect to the number of their

Inembers: and Bolzano in fact asserts "whether there are

 

210

Ibid. 8 21, p. 30.

/

202
more triangles or more syllogisms is indeed a question to
which no answer can be given other than that one does not
know how to compare these two infinite sets."211

The undesirability of Bolzano's approach becomes
clear when we ask whether there are "more" (in Bolzano's
sense) negative integers smaller than 0 than there are
positive integers larger than 1. While he would assert
that there are "more“ positive integers larger than 0 than
there are larger than 1. he cannot meaningfully answer the
first question at all. for if he were to pair off all
negative integers smaller than -l with their positive counter-
parts. in order to have -l left over. he would have estab-
lished equinumerosity through-one-to-one correspondence. a
procedure that he denies himself explicitely.212
Bolzano himself did not always heed his own injunction
against assuming equinumerosity for biunivocally correspond-

ing infinite sets. On one occasion he flatly asserts that

there are as many square of integers as there are integers,213

 

211Bernard Bolzano,'fiissenschaftslehre. Vol. 1, p. 4385.

 

212Cf. Bolzano, Paradoxien, § 22, p. Bl.

 

213
Ibid.. § 33. p. 54.

203
and on another he claims that there are as many circular
surfaces as there are circumferences.214

Let me now proceed to a discussion of Bolzano's views
on continuity. He gives the same definition of 'continuous'
‘in both the Wissenschaftslehre and the Paradoxien. He states
that a continuum is present when, and only when, there is
"a set (Inbegriff) of simple objects (of points in time or
space, or even of substances) which are so situated that
every single one of them has at least one neighbor for every

215
distance, however small."

P

In spite of its shortcomings, this definition enabled
Bolzano to recognize certain important properties<ofthe continuum,
and to dispel mistaken notions about its composition. He
points out that it had been known all along that the existence
of extended entities cannot be explained, without circularity,
by claiming that they are composed of parts that are them-
selves already extended. Nevertheless, he says, a contra-
diction was thought to be hidden in the assumption that
extended entities could be composed of parts that are

simple (points in time or space). He continues that the

 

214Bolzano,.Wissenschaftslehre, Vol. I, p. 439.

. 2lsBolzano, Paradoxien § 38, p. 73. Cf. also
Wissenschaftglehgg, Vol. III p. 252.

204

reason why this last assumption was considered contradictory
was sometimes given in the claim that a property which is
lacking in the parts cannot be found in the whole. This
Bolzano rejects on the grounds that every whole mu§t_have
several properties which are not found in any of its parts.
According to Bolzano, a second objection was seen in the
fact that any two points in time or space must always have a
distance between them, and thus cannot form a continuum.

This Bolzano takes to be a non-seguitur. He agrees that

 

between any two points in time or space there must always
lie another point, in fact infinitely many other points, but
he denies that there is a contradiction in this assumption.
All it proves is that a finite set of points can never form
a (continuous) extension. Even an infinite set of points
does not always form a continuum; it does not if the points
do not occur in the proper order (Anordnung).216
Against the Charge that all this cannot be grasped,
he replies that "indeed, it cannot be grasped with the fingers,

and not be obserVed'with the eyes, but it is recognized

through the undefStanding, recognized as something that

 

216 .
Cf. Bolzano, ParadOXien, § 38, p. 72 f.

 

' ‘1

 

 

205
necessarily must be so and cannot be otherwise."217
Bolzano's description of continua, despite the in-

218 constitutes considerable

sufficiency of his definition,
progress in more than one respect. To begin with, it is
finally realized that continua can be described as point
sets, and this realization is occasioned by a reliance upon
the trustworthiness of purely logical analysis that had
hitherto not been equalled.

In the sequel, Bolzano describes several properties
of continuous sets, but they are generally ones which
continuous sets share with merely dense sets. Of particular
merit is the distinction that he draws for the first time
between open, half open, and closed intervals.219

Before I proceed to a discussion of the cosmology

which Bolzano bases on his theory of infinite sets, let me

p

 

217Ibid., p. 74.

218Cantor already pointed out that according to
Bolzano's definition of 'continuity', two entirely separate
bodies, e.g. two spheres, could form gg§_continuum. Cf.
George Cantor, in Mathematische Annalen, Vol. XXI, p. 576,
also Hans Hahn's note to § 38 of the Paradoxien, ibid.,
p. 148.

21
9Bolzano, Paradoxien, § 41, p. 82.

 

 

206
briefly recapitulate the main points that have so far been
made in this chapter.

We saw that Bolzano realized that biunivocal corres-
pondence of infinite sets with their proper subsets does not
demonstrate that infinite sets are impossible, but rather
must be accepted as-a general characteristic of all and only
such sets. On the other hand, Bolzano does not consider the
existence of such a correSpondence between two sets as a
proof for their "numerical equality". However, 'numerical
equality' is not defined, so that one cannot make out how such
equality is to be ascertained. A further shortcoming of
Bolzano's exposition was found in the fact that no acceptable
definition of 'finite set' is provided, the definition given
by Bolzano being totally unsatisfactory. We may say that
Bolzano's account despite its general precision suffers from
his lassitude in definition at crucial points. On one
occasion he writes‘that "if m is an infinite set, then

(I?

—j

00 03 . . . "220 . .
2 jf-: —_. are also infinite sets, Without haVing
8
given any indication what the expressions 3L etc. are to
2 c

mean .

We then considered his definition of 'continuity'

2
201bid, § 38, p. 74.

207
which was found to be too wide. Notwithstanding this fact,
Bolzano's analysis of the continuum must be considered a
major advance in the field. This holds especially for his
description of continua as points sets.

Bolzano fully realized that a description of the
properties of infinite sets does not in itself furnish a
proof for the existence of such sets, and he consequently
gave a number of examples of such sets. In doing so he
applied the curious ontological bifurcation that he had

already developed in the Wissenschaftslehre, i.e. he first

 

shows that there are infinite sets of objects "that do not
have reality”, and then he attempts to prove that there also
are infinite sets of "things that do have reality". In the
former class he mentions the set of all propositions (Satze
an sich), the set of integers, the set of rational numbers,
but also time and space, which he considers infinite point-
sets that are not actual, although they are determinations
'of the actual.221 It is interesting to notice that Bolzano
realized that in accordance with his theory of time as a
continuous set of moments it is false to say that there is a

' 222
moment which is the next to any given moment.

 

221Ibid., §§ 13-17, pp. 13-24.

222Cf. Bolzano, Wissenschaftslehre, Vol. I, p. 405.

 

 

208

Among those things that do have reality, Bolzano also
finds infinite sets, e.g. the set of all determinations
(Eggtimmgpgen) of a given object in any finite time. This
set is indeed infinite as can easily be shown since the
set of all properties of any given object is infinite at
any given moment.

A much more interesting example is given with physi-
cal bodies, which are said to be continuous and to consist
of an infinity of unextended atoms. Among these, some are
said to be the dominant atoms, others subordinate. The
former determine the characteristics of any given body, and
are finite and loosely scattered within that body, while the
latter form a sort of "ether". Where no dominant atoms are
present, we have mere ether, while in the bodies it acts as
a sort of "fill".

Ifind remarkable not so much this theory itself
(although it was revolutionary enough to embrace the notion
of the actual infinite in such a forthright fashion) but the
relentlessness with which intuitively startling consequences
are drawn from these asSumptions. One is that:

the same set of sugstances which at this

moment fills this cubic foot can at another time

be distributed in a million times larger space

and at still another time be compressed to one

thousandth of its Original volume without is being
the case that.during the expansion any point in

209

the larger space was empty, or during the

'compression any point had to accommodate two

or more atoms;223

Another remarkable insight concerns the limits of
bodies. By the limit of a body Bolzano means the set of
its outermost atoms. From the above considerations it follows
that if two bodies are immediate neighbors, then, if one _1
of them has a limit in the just defined sense, the other
cannot have such a limit where the two bodies border to one
another, which, I believe, is the same as to say that two
disjoint closedxintervals cannot be neighbors since there .c_
must always be some points between them.224

Let this suffice as a brief sketch of Bolzano's theory
of infinite sets andfithe continuum. It can be seen that
Bolzano's work constitutes an enormous step forward to a
final solution of the problems here discussed, although his
own exposition is by no means flawless in every detail.
The points of difference between Bolzano and his philosophi-
cal precursors as concerns the approach to these difficulties

are obvious: Bolzano relied wholly upon logical analysis,

developing his modest cosmological scheme subsequent to,

 

223Boizano, Paradoxien, B 59, p. 117.

224Ibid., § 66, pp. 125 ff.

210
and dependent upon, a solution to the problems of the con-

tinuum and of_infinite sets”that satisfied him. He realized

for the first time that'thedassumption of infinite sets does

‘I

.

not lead to inconsistencies, and, in particular, that the
so-called Paradox of Galileo does not constitute such an
inconsistency. He saw, furthermore, that continua can be
described as point-sets, one of whose properties it is that
between any two of their points there lies another point.

His predecessors, who did not have these insights, felt
dbliged to circumvent the seemingly paradoxical properties

of infinite and continuous sets by all manner of epistemologi-

cal and metaphysical assumptions, a procedure from which

Bolzano't logical acumen saved him.

 

211

CONCLUDING REMARKS

With the advent of modern analytic philosophy, much
aspersion was cast upon all but a few products of classical
philosophy. The positivist meaning criterion, for example,
even in its later and much weakened forms would, in fact,
reduce most older philosophems to nonsense. One of the
consequences of this increased rigor was that the word
"metaphysical" together with the systems that it describes
fell into general disrepute.

When I became acquainted with, and found myself
accepting, many of the procedures of this analytic philoso-
phy, I became concerned whether classical philosophy ought
to hold more interest for the modern student than a museum-
piece and, if so, how a well reasoned and honest system of
"metaphysics" might be distinguished from an irrelevant and
corrupt one. The second problem can probably be met through
careful interpretation, or, if you will, reconstruction of
the systems in question: but my concern was not only whether
the reconstruction of some philosophy would result in a sys—
tem of statements which are meaningful under some of the

present stringent meaning criteria, but also whether there

is a communality of purpose to be found between contemporary

 

 

 

 

 

~212
investigations and those of our philosophical precursors:
there had to be some assurance that, after the reconstruc-
tion had sifted out the pseudo-problems, more than a collec-
‘tion of trivia would remain. -

The foregoing dissertation was addressed to the second
of these concerns. I have not undertaken to provide a recon-
struction of the systems which I have considered, since each
such reconstruction would have to take into account a
philosophical system in its entirety. I felt justified in
taking this course since the importance and scope of the
problems which I found treated in these systems can be
recognized in them even though various of their aspects were
not clarified. I found that many facets of the philosophies
which I have considered had to do, in some way, with the dis-
quieting effect that the recognition of some of the properties
of infinite and continuous sets tend to have upon those who
concern themselves with such matters. In pursuing my task,

I found myself discussing more historical detail than had
initially been my intention, and also omitting certain topics
related to the general purpose of this paper, which would be
very worthwhile subjects for further inquiries. Of these,
the following seem to me the most interesting.

I have repeatedly stated that oftentimes the justifi-

 

 

 

213

cation of a philosophical system is to be sought in the
propositions which it entails, rather than in some allegedly
self-evident assertions which it contains or seems to pre-
suppose. I have claimed that inasmuch as this viewPoint is
adopted, a philosophical system is to be evaluated in much
the same way as a scientific theory or hypothesis. But
there is an important difference which I did not discuss:
the systems which I have considered have none of them
observable consequences, and it remains to be investigated
what kind of consequence it is that supports them. At this
point it seems to me that each bf them should be envisaged
as some kind of metatheory meant to provide certain rules
which a scientific system or theory can violate only at the
penalty of being inconsistent. I am far from clear on this
point. However, I should be satisfied if the merit is seen
that lies in judging a philosophical system by its con-
sequences, but also if the problems are recognized that arise
in connection with such an approach.

A second point that would warrant some further atten-
tion is of a more speculative nature: we have seen that
Bolzano was in the possession of two insights which his
predecessors had been denied, namely he realized that the

Paradox of Galileo is not really a paradox, and that the

214

sum of a finite or infinite number of extensionless parts
does not have to be extensionless. The question arises
what form the philosophical systems of his predecessors
would have taken, had they been familiar with these results.
In accordance with the results of this dissertation, I believe
that the differences would have been profound and extensive
to the point that any speculation about this issue is con-
demned to fruitlessness, irrespective of whether we take our
hypothetical assumptioninaipsychological or systematic sense.

A third, and more important, unfinished issue is the
following: We have seen, for example, that both Kant and
Berkeley disclaimed the existence of actual infinite sets,
at least for the world of appearances. It remains to be
investigated whether they are not otherwise .committed, with-
out intending it, to the assumption of such sets. In general,
since the aim of this paper was not a reconstruction of the
systems under discussion, the question never arose whether
these systems are consistent. I hope that any findings con-
tradicting my results can be put to such undiscovered incon-
sistencies.

The point that I hope to have made in the foregoing
dissertation -- and this may seem trivial to some -- is

simply this: that the philosophers whom I have investigated

215

were as far removed from purposeless speculation as from
mysticism, that they were attentive in the presence of
logic andfact, and that the modern analytic philosopher
is far fromhaving outgrown the problems with which they
concerned themselves.

It is ironical that those who proclaim themselves
the successors of these great men have often failed even
to graSp their problems, while many analytic philosophers,
unaware of the problems involved, have denounced a tradition

of which they are today's representatives.

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V

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' )

 

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