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A CRITICISM OF MEI'I-lODOWY IN
ALFRED NORTH WHITEHEAD'S AN ENQJIRY CONCERNING
THE PRINCIPLES OF NATURE: KNOWLEIIBE

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Rebecca A. Gardner

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A CRITICISM OF METHODOLOGY IN

ALFRED NORTH WHITEHEAD'S AN ENQUIRY CONCERNING
THE PRINCIPLES OF NATURAL KNOWLEDGE

By

Rebecca A. Gardner

A THESIS

Submitted to

Hichi an State University
in partial ful illment of the requirements
for the degree of

MASTER OF ARTS

Department of Philosophy

1992

7&3— WW

ABSTRACT
A CRITICISM OF METHODOLOGY IN

ALFRED NORTH VHITEHEAD’S AN ENQUIRY CONCERNING
PRINCIPLES OFgfléTURAL KNOWLEDGE

BY

Rebecca A. Gardner

This study investigated the natural philosophy of

Alfred North Whitehead in An Enguiry Concerning the

Principles of Natural Knowledge. The method of extensive
abstraction was the central tool employed by Whitehead to

reconstruct the physical and temporal theories of the
natural sciences. It is proposed that Vhitehead's use of
this method and his presupposition of Euclidean parallelism
distort the results of his work and defeat the purpose of
his enterprise. The entities derived from the method of
extensive abstraction do not possess the logical properties
necessary for their application to the mathematics of the
physical sciences. Vhitehead's entire project rests on the
applicability of these abstract entities and without their

use Vhitehead's results are called into question.

 

 

ACKNOWLEDGMENTS

I would like to thank the members of my committee, I
for their graciousness and support. for being very giving of
their time, and most of all, for helping make this project
an enjoyable endeavor. I appreciate their individual
contributions:

Dr. Peter Asquith, whose detailed comments and
questions on an earlier draft made me rethink and clarify
conceptual issues.

Dr. Mark Risjord, whose suggestions and comments on
the foundations of mathematics helped make the issues come
alive.

Dr. Winston Wilkinson, for his patience in reading
revisions of revisions. His thoroughness has improved each
one. I thank him also for his generosity with his time and
resources. I thank him especially for the freedom to make
this project my own.

I would like to thank a friend, Francesca Galarraga,

for sharing her time, and especially her computer skills.

 

TABLE OF CONTENTS

 

 

Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . vii i
PART I Ii
INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 '
Historical Overview 1
Aim of Paper . 2
Format 3
1. THE NATURAL VORLD . 4
The Whole of Nature . . . . . . 4
Experience and Process . 6
Relation of Extension 8
2. DEVELOPMENT OF TENPORALITY . . . . . . . . . . . 11
Durations . . . . . . . . . . . . . . . . . 11
Parallelism . . . . . . . . . . . . . . . . 13
Time-systems . . . . . . . . . . . . . . . . 15
3. METHOD OF EXTENSIVE ABSTRACTION . . . . . . . . . 19
Abstractive Sets . . . . . . . . . . . . . . 19
Primes and Anti-primes . . . . . . . . . . . 20
Abstractive Elements . . . . . . . . . . . . 22
Relations . . . . . . . . . . . . . . . . . 25
4. DEVELOPMENT OF SPATIALITY . . . . . . . . . . . . 29
Instantaneous Space . . . . . . . . . . . . 29
Event-particles . . . . . . . . . . . . . . 32
Solids and Volumes . . . . . . . . . . . . . 34
Components of the Three Types of Space . . . 36
Four-dimensional Space-time . . . . . . . . 39

iv

 

Routes and Stations . . .
Point- tracks and Null- tracks .
Matrices . . . . .
Three- -dimensional Time-less space

5. CONSTANTS OF EXTERNALITY
6. THEORY OF PERCEPTION

Immediate Data of Perception .

Principle of Relativity
Concept of Nature

Principles of Natural Knowledge
Re- examination of a Problem

PART II
POSSIBLE PROBLEMS WITH WHITEHEAD'S PHILOSOPHY

INTRODUCTION
1. PROBLEMS WITH THE THEORY OF PERCEPTION

Fundamental Thesis of Perception

Method of Extensive Abstraction and
Mathematical Limdting . . .
Review of Method of Extensive Abstraction

2. CRITICISNS OF METHOD OF EXTENSIVE ABSTRACTION .

Comments by C. D. Broad . . .
Re-definition: Irrationals and Integers
Re-definition: Points .

Criticism: Adolf Grunbaum

Denumerable and Non-denumerable

Infinite Sets . . . . . . .
Differentiation Between Points . . . . . .
Continuity of Inclusion . . .
Assumptions: Grunbaum . . .

Comments: V. Mays .
Comments: Arguments Against Method of.

Extensive Abstraction . . . . .

3. PROBLEM OF THE EXTENSIVE CONTINUUM . . . . . . .

Extensive Continuum of Events . .

General Properties of a Continuum
Properties of the Extensive Continuum
Continuity of a Finite Line Segment
Intuitionist Notion of Continuum . . .
Basic Properties: Intuitionist Mathematics .
Properties of Infinitely Proceeding Sequence

V

Page

41
44
45
49

51
55

55
56
59

60
62

64
64

66
66

68
74

97
98
99
101
107
109

109

 

5?

 

Page

Similarities of Properties . . . 110
Differences Between Philosophical Systems . 112
Conclusions: Extensive Continuum . . . . 115
Conclusions: Method of Extensive Abstraction 116
4. PROBLEMS WITH RELATION OF PARALLELISM . . . . . . 119
Parallelism and Temporality . . . . . . . . 120
Parallelism and Spatiality . . . . . . 121
Assumption of Euclidean Parallelism . . . . 122
PART III
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 123
LITERATURE CITED . . . . . . . . . . . . . . . . . . 124

vi

LIST OF TABLES

TABLE Page

1 Types of Space . . . . . . . . . . . . . . . 37

LIST OF FIGURES

FIGURE ' Page
1 Durations and Time-systems . . . . . . . . . 18
2 Properties of Covering and K-equality . . . . 23

3 Properties: Junction, Adjunction
and Injunction . . . . . . . . 27
4 Construction of a Matrix . . . . . . . . . . 48

vii

INTRODUCTION

Historical Overview

The time in which we live our lives and the space which
comprises our world weave together to form the basic fabric
of our existence. Throughout history mankind has sought to
understand these aspects of life. The passage of time and
the existence of space were once attributed to deities or
mythological creatures. Today the study of time and space
falls in great part to the physical sciences. From the
mathematics and philosophy of ancient Greece to our own
time, numerous theories have attempted to explain the nature
of the world and have been superseded by still other
theories. The seventeenth century saw tremendous growth in
the physical sciences and mathematics with the advent of
Newton's Laws of Motion and Gravitation and its accompanying
calculus. The next two centuries brought further
developments in classical mechanics and mathematics. With
the dawn of the twentieth century came the general
acceptance of relativity theory and the introduction of
quantum mechanics. However, the exact nature of time and

space remain an enigma.

Aim of Papgr
The philosophical career of Alfred North Whitehead

consisted in part in an effort to extract an understanding
of the world in which we live. Whitehead's notions of time
and space (indeed, his very conception of reality) differ
greatly from the science of his age. It is the purpose of
this paper to examine the natural philosophy of Whitehead as
it is presented in An Enguiry Concerping tpe Pzinciplgs of
Natural Knowledge.‘ In particular I will be concerned with
Whitehead's use of the method of extensive abstraction and
his theory of parallelism and the consequences of their
application upon his philosophy of nature.2

The method of extensive abstraction is of particular
interest because of its pervasive use by Whitehead as a tool
for abstracting and illuminating the basic relations which
occur among events in nature. Whitehead's use of the notion
of parallelism, especially in his early works, will be shown
to have unduly influenced the outcome of his natural

philosophy in a direction contrary to current opinions of

 

‘Alfred North Whitehead. An n u C ncern n he

Principles of Napural Knowledge. The University Press:
Cambridge, 1955. In the future references to this text will

be inserted by page number in parantheses in the main body
of this paper.

2The content of this paper will be strictly lindted in its
scope. I will deal with those aspects of events which are
subject to the relation of extension and will not cover
certain other aspects of Whitehead’s philosophy, such as his
Theory of Objects, which are of great importance but are not
directly affected by the relation of extension. The
technical terms referred to in this note will be explained
in detail as this paper progresses.

3

the physical sciences. It is noted that Principles 9f

Natppal Kpowlgdge is among the first in a series of

Whitehead's works in the philosophy of science and that his
notions have matured over time. It is, however, a
worthwhile enterprise to examine Whitehead's ideas at the
point of their conception. This prospective adds insight to
the development of later works as well as being of interest

in its own right.

Forppt of Pppgr
I will begin this paper with a detailed exposition of

Whitehead's natural philosophy as it appears in Ap_£ngpir1
Co 0 r in e rin i 1 of Na r w e. Once this
exposition has been completed I will embark upon a
discussion of the influence of the method of extensive
abstraction and the theory of parallelism on the work as a
whole and on the theories of time and space in particular.
It is my contention that the use of these theories result in
a natural philosophy which is not consistent with current
scientific_ beliefs and practices. I will present several
critical arguments concerning these aspects of Whitehead's
work and then add my own critical comments on these

arguments.

PART I

EXPOSITION: AN ENQUIRY CONCERNING
THE PRINCIPLES OF NATURAL KNOWLE E

 

 

CHAPTER 1
THE NATURAL WORLD
The Whole of Nature

The natural world, according to Whitehead, is not to be
considered in terms of either absolute time or absolute
space. Nor is it to be considered as being composed of
matter in some continual process of shifting or
rearrangement. Neither is it to be considered in an
idealist fashion, as something which exists only as a form
of thought or ideas. Instead of thinking of nature in any
of these ways, Whitehead asks us to drop the materialistic
point of view and adopt the perspective that what we
experience are events. He considers this view to be both

epistemologically and ontologically fundamental.=

 

“By indicating that the experiencing of events is both
epistemologically and ontologically fundamental I mean that
from an epistemological standpoint Whitehead justifies his
view that the fundamental structure of nature is expressed
by events and relations among events because such events and
relations are shown to be observable in any experience of
events. From an ontological standpoint the experiencing of
events is said to be fundamental because it is through such
an experience that the passage of nature becomes visible.
Through abstraction from this passage of nature Whitehead
shows the existence of those basic relations found among
events. The meanings of the technical terms used in this
brief explanation will be made clear in the development of
this paper.

5

The notions of events. experience. and process are
interrelated and their application in Whitehead's philosophy
can be somewhat confusing. Events are the basic entities
which make up nature. Experience is an essential aspect of
events, in the sense that events have the ability to
experience other events (through their relations). We (as a
certain type of event) gain our knowledge of nature through
our experiencing of other events. Process is the
fundamental activity in which events are involved. This
activity of events, their coming into being and passing out
of being, is the progressive advance of nature. Each of
these notions, in its own way, is a fundamental part of
Whitehead's philosophy of nature. They can be viewed as the
expression of nature from three different perspectives.
Because an understanding of these notions is fundamental to
my project, I will concentrate my attention on Part III of
An Epguiry Copcezning the Pripciples of Natural Knowlpgge,
which offers an explanation of the experiencing of nature.‘

Whitehead begins the development of his philosophical
system with an explanation of nature which differs
considerably from the way in which it is commonly conceived.
Because experience is considered to be one .of the
fundamental elements of nature, questions arise as to what

is meant by the term "experience" and how the more

 

‘I will, however, refer to comments and definitions offered
by Whitehead in other parts of the work.

6
complicated elements of nature are established from the
notion of experience. Whitehead wants to distinguish his
theory of nature from the classical Newtonian system of his
time and so he chooses a vocabulary which is decidedly
different from that generally employed in the description of
nature. We must take note, then, of his definitions of
terms as they are employed in the development of his natural

theory.

Egpgrience ang Process

If we were to consider the experiencing of nature in
terms of material things we would lose the Whiteheadian
view. When we drop the traditional point of view and try to
visualize nature as Whitehead does, we find that our very
notion of nature must change. Ordinarily when we speak of
experience we mean our experience of certain things:
objects, physical properties, and even emotions.
Whitehead's use of the term experience is something entirely
different when considered in its fundamental form. He
speaks of experiencing the progressive advance of nature
(13, also Mays‘ 38).

Consider experience as the first derivative of

process.“ "Process" can be defined as that act of becoming

 

‘W. Mays. Ihg Philppophy of Whitepeag. Macmillan: New York,
1959.

‘By derivative I mean a subsequent level of complication in
a series extending from the most simple to the most complex
(4) O

7
which is inherent in the creative advance of nature (14).
Events come into existence and are then replaced by other
events. Process is this basic activity. The experiencing
of the process is not a necessary part of the creative
advance of nature. In other words, the actuality of nature
is not dependent upon its having been experienced.

We are now in a position to consider two equally
correct perspectives. Both of these involve a discussion of
process as the creative advance of nature. First, we could
consider process to be an endless stream of events, i.e.,
the coming into existence and the passing out of existence
of an infinite number of events.7 The totality of these
events compose the creative advance of nature. This first
view of the term "process" involves events.

The second view exploits the idea of relations, where
the notion of an event is expressed in terms of the
interaction of relations (relationships) among one another.
Ordinarily we think of relations as occurring between
things, but Whitehead proposes the possibility that
relations can interact among themselves and not, at a basic
level, among material objects. This implies that events
themselves can be considered complexes of relations

interacting with other such complexes.

 

’Alfred North Whitehead. Copcept of Npturg. University
Press: Cambridge, 1964. p. 14. Concept of Ngturg is a
companion text to Pripciplgg of Natural Knowledgg and will
be used as such in this paper. Future reference to this
text will be designated by CN.

8

When we considered process in the first sense, we
arrived at a notion of events which is fundamental in the
development of nature. prior even to any experiencing of
nature and, so, prior to any experiencing of material
entities. By adding to this view the second notion (that
events can be conceived as a complex of interacting
relations), we have a view of the creative advance of nature
described in terms of events of a complex of interacting

relations.

Relation of Egtgpeion
It is in the Whiteheadian style to seek that which is

most fundamental and to build more complex systems from
these fundamental notions. If we accept this method, our
next step is to ask which relation is primary, in the sense
of being an aspect of all events. For Whitehead, this is
the relation of extension (61).

Extension among events can be described as consisting of
four possible cases. Given events A and B, the relation of
extension can produce the following states: either (i) A and
B are identical; (ii) A contains B either in part or
totally: (iii) B contains A either in part or totally; or
(iv) A and B are entirely separate. It is obvious from this
description that the relation of extension involves the
concept of containing, and the relation of part to whole.

Whitehead seems to use these notions in the ordinary sense.

9

The relation of extension. as it applies to a and b. is
represented by aKb (to be read "a extends over b"), and aKb
is to be interpreted as "b is a proper part of a". When we
speak of the relation of extension with respect to events we
say that the relation aKb means "event b is a part of event
a”. We find a certain similarity between this description
of the relation of extension and the description given by
classical set theory of a proper subset of a class, which is
represented in set notation as: b C a a b¢a.

By applying the method of extensive abstraction. to
events (utilizing the relation of extension) Whitehead
defines a space-time framework which will provide a physical
description of nature. The domain to which the method of
extensive abstraction can be applied is the domain of
events. By adding the relation of cogredience’ to that of
extension, it becomes possible for Whitehead to develop the
notion of an extensive continuum.‘° It 13 from the events
of the extensive continuum that the concepts of time and

space are abstracted or derived. To consider time apart

from space or space apart from time is itself an abstraction

 

'The method of extensive abstraction will be discussed fully
below.
’This concept will be explained in more detail later in this
per. Let it suffice for now to say that cogredience
indicates the here-nowbpresent of all events which are
simultaneous to a given event and, so, indicates the
position of these events in a space-time continuum (70).
‘°The extensive continuum refers to a continuum of events
(the totality of which gives nature as a whole) which is
formed through the relation of extension.

10
from the concept of the whole of nature. It is important to
remember, however, that although time and space can be
considered in isolation from one another, in nature they are

never separated and are dependent upon one another.

CHAPTER 2
THE DEVELOPMENT OF TEMPORALITY
Durations

At this point it becomes necessary to introduce some
technical terminology. Let us begin with durations. A
duration is an event and can be considered a piece of the
temporal advance of nature. When we consider a duration we
are speaking of nature as a whole, i.e., we perceive an
entire complex of events which occurs simultaneous to our
act of perceiving. This is all of nature at the time of our
perception. This complex of events forms the background for
our current perceptions and is called a "duration" (68).
Although it would seem obvious that all events constituting
a duration cannot be directly perceived by a percipient
event, the relation of extension, inherent in all events,
enables even the furthermost events in a duration to be
indirectly perceived. Since extension is the basic relation
involved in a duration, further explanation of the concept
of a duration will be helpful in understanding the notion of

extension.

A duration has a finite temporal dimension and also a
physical dimension, which represents a spatial part of the
extensive continuum that exists simultaneously to an event
within the given duration. These two dimensions of a
duration present the whole of nature taken from a particular

perspective (that of some percipient event). A duration is

11

12

infinitely divisible (each division creating another
duration) and the totality of these is nature from this

particular perspective. In the Whiteheadian sense, a
duration is not an abstract stretch of time but is a
temporal slab or section of nature.

Durations can vary in their degree of extension. If
the notion of duration is to be of use in a temporal theory
applicable to the natural sciences, we must be able to
abstract11 from our experience of durations a single moment
of time. This is accomplished by considering sets of
progressively diminishing durations until we reach an
approximation of a time-less instant. Whitehead refers to
this process as the method of extensive abstraction (104).
Because durations form a continuum of temporal extension, it
does not matter where we begin the application of the method
of extensive abstraction to a series of durations.

A duration extends over (overlaps) other durations
which are part of it, and, in turn. these durations extend
over still other durations which are parts of them. It is

in this sense that the series of durations mentioned above

 

“Whitehead's use of the term "abstraction" refers to a
process, involving the relations of events, where he attends
to certain relations while ignoring others. This results in
a method of simplification, where the relations attended to
illuminate an aspect of the entity being discussed. In this
particular case, temporal extension is being abstracted from
a duration, i.e., this temporal relation is being ignored.
When the relation of temporal extension is left out of the
consideration of a duration, we are left with an
extensionless moment of time.

13

is established. By continuing the process of diminishment
of a series of durations, we will eventually approach the
ideal of an extensionless moment of time. It is ideal
because an extensionless moment of time does not actually
exist in nature. This ideal abstraction is quite valuable
when we wish to examine spatial features of nature in the
absence of temporal change. Whitehead refers to a moment as
"a route of approximation to all of nature which has lost
its (essential) temporal extension; thus it is nature under
the aspect of a three-dimensional instantaneous space"(112).

A moment of time is established by the use of the
method of extensive abstraction applied to some duration
experienced from a particular standpoint. It is entirely
within the realm of reason that from a particular standpoint
(that of a some percipient event) indefinitely many
different divisions of durations can be made and so we can
abstract indefinitely many moments of time. The entire set
of moments established from a particular standpoint form
what Whitehead refers to as a "family” of moments (112-113).
Analogously we may speak of families of durations from which

these families of moments are derived.

Parallelism
Whitehead further explains the characteristics of these

families of moments/durations by asserting that all

14

moments/durations of one family are parallel to one another
(113-114). This is to say that a family of parallel
durations consists of all the durations parallel to a given
duration, including that duration itself. Because any two
members of a family of moments are parallel to one another,
'no duration or moment outside of a given family of
moments/durations can be considered parallel to any member
of the given family; a duration must either intersect the
given family or be parallel to it. If a duration is
parallel to the given family of durations/moments, it will
be included in that family (113).

The concept of parallelism12 is essential to
Whitehead’s notion of the abstraction of time and space from
events. It seems, in fact, that without the assumption of
Euclidean parallelism in his generation of the geometry of

the physical universe Whitehead does not feel he can succeed

 

'thitehead's use of the notion of parallelism is somewhat
ambigious. At times he makes particular reference to the
Euclidean geometry constructed by the use of entities he has
derived through the use of the method of extensive
abstraction. These entities require the property of
parallelism as a basic part of their construction. It would
be assumed in these cases that Whitehead is asserting

Euclidean parallelism. Palter, in Epitehead'g Philosophy of
Science, states that ”the particular concept of time which

[Whitehead] formulates is called ‘parabolic' ...because it
is assumed to lead ...to a ‘parabolic' (or Euclidean)
concept of space. Whitehead suggests the possible

usefulness to natural science of alternative types of time
and space,...both hyperbolic and elliptic space and time”
(Palter 60-61). This reference adds further evidence to the
idea that the type of geometry Whitehead is reconstructing
is Euclidean and so involves the notion of Euclidean
parallelism.

15
at his task of explaining the nature of our observations.
This statement will be considered in more detail later in
this paper.

Whitehead‘s system of time and space is dependent on
the assumption of parallelism. It is employed in the
description of durations when Whitehead states that "a pair
of durations both of which are parts of the same duration
are called "parallel”: and also a pair of moments such that
there are durations in which both inhere are called
"parallel"” (113). The importance of the concept of non-
intersection is evident in his statement that "two durations
which do not intersect are parallel: and parallel moments
which are not identical never intersect" (113). Throughout
his discussion of durations, moments, and time-systems
Whitehead continually refers to this type of parallelism.
We will return again to a discussion of the concept of

parallelism at the end of this paper.

Time-systgms
A time-system contains all durations which are parallel

one to another and thus all parallel moments associated with
these durations (114). It is possible for there to exist
more than one time-system because of the possibility that
there can exist sets of durations which are not parallel to
one another (see figure 1, p. 18). Indeed, this fact is
vital to the Whiteheadian theory of relative motion and his

explanation of physical change (176). A time-system is

16

considered to be a family of parallel moments which are
arranged in serial order. Whitehead establishes the

following criteria for serial order:

(1) A duration belonging to a time-system is
‘bounded' by a moment of the same time-system when
each duration in which that moment inheres
intersects the given duration and also intersects
events separate from the given duration:

(ii) Every duration has two such bounding
moments, and every pair of parallel moments bound
one duration of that time-system:

(iii) A moment B of a time-system ‘lies
between' two moments A and C of the same time-
system when B inheres in the duration which A and
C bound:

(iv) This relation of ‘lying between' has the
following properties which generate continuous
serial order on each time-system, namely,

(a) Of any three moments of the same time-
system, one of them lies between the other two;

(b) If the moment B lies between the
moments A and C, and the moment C lies between the
moments B and D, then B lies between A and D:

(c) There are not four moments in the same
time-system such that one of them lies between
each pair of the remaining three;

(d) The serial order among moments of the
same time-system has the Cantor-Dedekind type of
continuity'° (114-115).

 

1“'The Cantor-Dedekind continuum involves the key notions of
”an infinite set and a nondenumerable set. Dedekind defined
a set as infinite if. and only if, a one-one correspondence
can be established between its members and the members of a
proper subset of it. An infinite set is denumerable if, and
only if, it can be mapped in a one-one manner on the set of
natural numbers: otherwise it is indenumerable. A continuum
is a nondenumerably infinite set, say K, the elements of
which constitute a series that, apart from its serial order,
also conforms to the following postulates, as given by E. V.
Huntington in his Thg Cpppipppp (1917): (1) If K. and K, are
any two nonempty parts of K, such that every element of K
belongs either to K. or K; and every element of K. precedes
every element of K2. then there is at least one element K in
K such that any element that precedes X belongs to K., and
every element that follows X belongs to K2. (2)If a and b
are elements of the class K and a precedes b, then there

17

Because the members of any given family of moments are
all parallel to one another and because parallelism implies
the properties of transitivity, symmetry, and
reflexiveness,‘4 these properties also apply to all members
of a family of moments. In this way we are ,able to
establish serial order within a time-system.

A moment has been defined as an approximation of an
extensionless instant of time. It also has spatial

implications. An ideal moment of time is considered to be

 

exists at least one element x in K such that a precedes x
and x precedes b. The continuum is linear if in addition
(3) the class K contains a denumerable subclass R in such a
way that between any two elements of the class K there is an
element of R. For example, the class of real numbers
between 0 and 2 is a continuum. It is, moreover, a linear
continuum with R the class of rational numbers and J2 an

element of the continuum but not of R.” S. Korner.
"Continuity” in The Encyclopedia of Zhilgsgphy. Vol. 2. ed.
Paul Edwards. Kacmillan: New York, 1967. p. 206. This is a
long and rather technical explanation. However, it will

prove helpful when we consider certain objections leveled
against Vhitehead's notion of an extensive continuum.

“The properites of transitiveness, symmetry, and
reflexiveness can be defined as follows:

1) Transitive Property:

R is said to be transitive on A if and only if whenever a
first element [of A] is related to a second and the second
is related to a third, then it must follow that the first
and third are related - if this holds for all elements of A,
then R has the transitive property.

2) Symmetric Property:

R is said to be symmetric on A if and only if a pair of
elements in A is related via R, then it must follow that the
reverse-order pair is also related via R in order that R
have the symmetric property.

3) Reflexive Property:

R is said to be reflexive on A if and only if every
element in the set must be related to itself via the
relation R in order that R have the reflexive property.

These definitions were taken from John F Lucas.
Introductio to str ct Kat e tics. wadsworth Publishing
Co.: Belmont, 1986. p. 157.

18

the whole of physical nature“ taken from some particular

perspective. Here once again we find the inseparability of
space and time. Whitehead derives his theory of space from
his theory of time, that is, from durations, moments and

time-systems.

Figure 1"

Durations and Time-Systems

 

° The slab of nature forming a duration is limited in its temporal
dimension and unlimited in its spatial dimensions. Thus it repre-
sents a finite time and infinite space. For example let the horizontal

line represent the time; and assume nature to be spatially one-di- 6’ 5
measionsl, so that an unlimited vertical line in the diagram represents
space at an instant. Then the area between the unlimited parallel
lines AB and HG represents a duration. Also the area between CD

 

and EF represents another duration which is extended over by the
duration bounded by AB and HG. But in fig. 7 we have assumed
only one time-system, which is the Newtonian hypothesis. Suppose
there are many time-systems and consider two such systems a and 3.
These are represented by two lines inclined to each other. A dura-
tion of time-system a is represented by the area between A8 and Fig. ..
CD. and a duration of time-system fl is represented by the area
between EF and HK. Two such durations necessarily intersect
and also can neither completely extend over the other.

These diagrams are crude illustrations of some properties of

durations and are manv respects misleading as the sequel will
show. u\

 

 

 

 

(‘0‘
Timeeyltelle

4/ \
IL

“The physical nature of space is meant to be that of
instantaneous space. This is the space of all of nature
taken at an instant of time. It is an ideal condition and
does not occur in nature itself.

"Figure 1 is taken from Principles of Natural Knowledge,
pp. 111-112.

 

 

 

CHAPTER 3
THE METHOD OF EXTENSIVE ABSTRACTION

Abstractive Sets

Let us now introduce the notions of ”formative

condition," ”abstractive sets," "primes,” “anti-primes," and
the relations of "inhering," "adjoining," "covering," and
"intersection.” This discussion will help us to connect

Whitehead's technical definitions with the explanation of
events and durations given above and also demonstrate their
relationship to the method of extensive abstraction.

In Principles of Natural Knowledge an explicit
definition of "formative condition" is not given. However,
the notion of formative condition plays an important role in
the development of abstractive sets. A formative condition
provides a starting point for the identification of
particular groups of abstractive sets. Abstractive sets can
be seen as a series of nested durations which tend to
converge to some ideal approximation. Let us take, as an
example, a moment.17 There are a number of different

abstractive sets which can converge toward the same moment

and all such abstractive sets have some common feature which

is called their formative condition. We can, then, say that

the set of all abstractive sets having the same formative

condition will converge toward a single unique moment. When

 

‘7Here the events which comprise the abstractive set are
durations.

19

20
Whitehead speaks of some "formative condition a" in respect
to abstractive sets, o-primes, v-anti-primes, etc., the term
"a" refers to that common feature mentioned above.

An abstractive set is defined as a set of events for
which the following conditions hold: "(1) of any two members
one extends over the other and (ii) there is no event which
is extended over by every event in the set" (104). Because
the relation of extension is inherent in events, an
abstractive set forms a series of events whose converging
end “diminishes without linut" (104). This is to say that
the series of events which form the abstractive set has no

smallest member.

Ppipgs and Anti-primes

An abstractive set is called "prime" whenever all the
members of the abstractive set retain the formative
condition c and the abstractive set "is covered by every
other abstractive...[setl satisfying the same condition v”
(106). The reference to ”covered" means that every event in
one set is extended over" at least partially by some other
event in the other set (see figure 2, p. 23)(104).

When an abstractive set is referred to as being ”anti-
prime" it fulfills the following two conditions: "(1) it
satisfies the condition c and (ii) it covers every other

abstractive...[set] satisfying the same condition e" (106).

 

"To say event A extends over event B is to say A includes,
at least in part, B.

21

The reference to "covering” means that given two abstractive
sets every member of one set extends over some member of the
other set (104)-‘9 An abstractive set is considered an
"absolute anti-prime" when each set with the formative
condition w covers every abstractive set which covers it.
It is assumed that in each case the formative condition w is
represented in every event in the abstractive sets.

The reference to "primes" and ”anti-primes" is another
way of expressing the notion that, in the case of an c-prime
abstractive set, the convergence which occurs is the
sharpest with respect to the condition c. In the case of an
o-anti-prime abstractive set, the convergence which occurs
with respect to the condition r is the broadest that can
occur. This means that the neighborhoodzo established
around the abstractive element produced by an ”c-prime"
abstractive set is very narrow, while the corresponding
neighborhood around the abstractive element produced by an
”v-anti-prime" abstractive set is quite broad in comparison

(CN 88).

 

1fi'One distinctive difference between prime and anti-prime
abstractive sets is that a prime abstractive set is covered

by every other abstractive set satisfying some particular
formative condition. An anti-prime abstractive set covers
every other abstractive set satisfying some particular
formative condition. The properties of being covered by and
covering are being emphasized here.

2°leighborhood is being used here as a topological term
referring to the comparative area surrounding an abstractive
element located in the extensive continuum.

22

Abstractive Elements

Now that we have defined abstractive sets, primes, and
anti-primes we are in a position to discuss "abstractive
elements". An abstractive element is an example of the
ideal limit mentioned above in our discussion of durations
and moments. Abstractive elements can be identified "with
the whole set of abstractive sets which are K-equalz‘ (see
figure 2, p. 23) to any one of themselves"22 (Palter 53).
They are defined in terms of "primeness" and ”anti-
primeness” and are divided into two types, finite and
infinite.

"A ‘finite abstractive element deduced from the
formative condition 0' is the set of events of o-primes
where a is a formative condition regular for primes” (108).
A prime event (which possesses a particular formative
condition c and is then termed an w-prime) is said to be
regular "when (i) there are e-primes and (ii) the set of
abstractive...[setsl K-equal to any one assigned c-prime is
identical with the complete set of r-primes" (107). Regular

anti-primes are defined in an analogous manner.

 

Z‘For abstractive sets or classes K-equal means that each
infinite abstractive set or class in a set of such classes
covers and is covered by every other abstractive class in
that set (105).

22Robert K. Palter. Whitehead's Philosophy of Science.
University of Chicago Press: Chicago, 1960.

23

Figure 233 \

Properties of Covering and K-equality

 

[Nola Abstractive classes and the relation of ‘covering’ can
be illustrated by spatial diagrams, with the same caution as to
their possibly misleading character.

Consider a series of squares, concentric
and similarly situated. Let the lengths of
the sides of the successive squares, stated

in order of diminishing size, be . D

 

 

I... h... . .5... . ..

Then each square extends over all the
subsequent squares of the set. Also let

 

 

 

 

 

 

L h. - 0; Fig. 5.

I *fi
namely, let It. tend to zero as 2: increases indefinitely. Then
the set forms an abstractive class.
Again, consider a series of rectangles, concentric and similarly
situated. Let the lengths of the sides of the successive rectangles,
stated in order of diminishing size, be (4.11,), (a, h.),...(a, 11.)...”

Thus one pair of opposite sides is of the same length through-
out thc whole series. T hen each rectangle extends over all the

subsequent rectangles. Let It. tend to zero as 1: increases in-
definitely. Then the set forms an abstractive class.

Evidently the set of squares converges to a point, and the
set of rectangles to a straight line. Similarly, using three dimen-
sions and volumes, we can thus diagrammatically find abstractive
classes which converge to areas. If we suppose the centre of
the set of squares to be the same as that of the set of rectangles,
and place the squares so that their sides are parallel to the sides
of the rectangles, then the set of rectangles covers the set of
squares, but the set of squares does not cover the set of rect—
angles.

Again, consider a set of concentric circles with their common
centre at the centre of the squares. and let each circle be in-
scribed in one of the squares, and let each square have one of
the circles inscribed in it. Then the circles form an abstractive
class converging to their common centre. The set of squares
covers the set of circles and the set of circlr: covers the set of
squares. Accordingly the two sets are K-equal.]

 

2°Pigure 2 is taken from.

p.

105-106.

1

54*:
J

I.

 

 

 

DO.--COO-0------O----------dj

 

 

 

 

Fig. 6.

 

 

 

‘1:
L

Principles of Natural Knowleggg,

24

An ”infinite abstractive element” is defined in the
same manner as a "finite abstractive element” except c-anti-
primes are substituted for c-primes. The generic term,
"abstractive element,” refers to that set of elements which
contain all finite and infinite abstractive elements. This
combined set ”represents a set of equivalent routes of
approximation guided by the condition that each route is to
satisfy the condition v” (109).

Whitehead also uses abstractive elements to define
moments and durations. Those abstractive "elements of the
greatest complexity...which can cover [they are anti-primes]
elements of all types” will be moments (109). This
designates a particular type of infinite set of abstractive
elements (109). A moment includes all those abstractive
elements in a duration which are formed in reference to some
formative condition a. A duration is any member of an
absolute anti-prime, i.e., any member of an abstractive set
which covers every abstractive set which covers it (111).
In other words, these abstractive sets are K-equal.

A problem arises with this formulation of moments in

terms of anti-primes because durations are not the only

events which can satisfy the conditions for an anti-prime

(Palter 56). Whitehead himself notes this problem in Note

IV to the second edition of Principles of Ratupal gpowleggp:
"The attempt...to define a duration merely by means of its

25

unlimitednessfi4 is a failure. In a note to the QQEEEEE_—Qi

Nature, I point out that there is an analogous unlimitedness
through time, corresponding to the spatial unlimitedness of
a duration" (204).

Because a duration is not the only event which can be
considered an anti-prime, the application of the method of
extensive abstraction to an anti-prime would not necessarily
result in approximate convergence to a moment. This point
proves to be a problem for Whitehead. Later, in Concept of
EEEEEE. he considers duration to be a primitive concept (CN
59). In this way he tries to avoid the problem of the

derivation of moments from anti-prime abstractive sets.

Relations
When we consider relations such as "inhering,"
"adjoining," "Junction,” and "intersection," we are

referring to different ways in which events or abstractive
elements interact with one another (see figure 3, p. 27).
An abstractive element is said to inhere in an event when it

is extended over by some part of that event (109). An

abstractive element is called a member of an event when it
is a part of the set of abstractive elements which are

particular to that event (109). Abstractive elements

 

24The term ”unlimitedness" here refers to the property of an
anti-prime abstractive set to cover all other abstractive
sets which satisfy the same formative condition. This is
another reference to an anti-prime establishing the broadest
neighborhood around an abstractive element.

26

intersect if they are members of abstractive sets which
cover each other (109). The relation adjoining is defined
by Whitehead in the following manner: ”Two events which are
Joined have that relation to each other necessary for the
existence of one event which extends over them and over no
extraneous events. Two events which are both separate and
have a Junction are said to be ‘adJoined'" (103).

Whitehead depicts the continuity of nature by
considering the interaction of a variety of relations.
These relations are expressed in technical terms which, for
our purposes, need not be defined. It should be understood,
however, that these relations are, among other things, used
to explain the temporal and spatial boundaries of events and
so are important to Whitehead's philosophy of nature. They
are used in the development of the four-dimensional geometry
of events and analogous relations are employed to develop
three-dimensional geometry (103).

Let us take up anew the connection between abstractive
elements, anti-primes and the temporal system abstracted
from the events of the extensive continuum. As stated
earlier Whitehead attempts to define moments in terms of
abstractive elements and absolute anti-primes (110). But
not every event is an absolute anti-prime (111). Those

which are can be termed durations and only durations can

27

Figure 328

Properties: Junction, Adjunction, and Injunction

 

(i) Junction of intersecting events

(ii) Junction of separated events (or sdjunction)

(iii) Injunction of events 3 and y (s is separated from x and adjoined to y).

Fro. same. and bisection 0! events

 

2"Figure 3 is taken from Palter. Whitepeag‘s Ppilosophy pf
Science, p. 48.

28
extend over other durations. When we consider two time-
systems, each consisting of parallel durations (and all the
moments associated with a particular duration), it is
necessarily true that the durations of these two time-
systems will intersect. At this point we are able to
introduce the spatial system of Whitehead's three-

dimensional geometry.

CHAPTER 4
THE DEVELOPMENT OF SPATIALITY

Instantaneous Space
Whitehead begins the development of his philosophy of

nature with the examination of durations and their
corresponding time-systems. With a multiplicity of time-
systems at his disposal, he is now able to establish a
geometry of three-dimensional space as part of the four-
dimensional world.26 He does so by considering the various
intersections of moments defined by series of durations.
When two nonparallel moments intersect, a level is created
(116). In the Whiteheadian system a level is analogous to
the plane in Euclidean geometry. This new terminology is
necessary because of Whitehead's need to contrast his
philosophical system with that of the old classical system,
where time and space were considered to be separate
entities.

The result of the intersection of three nonparallel
moments is termed a "rect" (116). It is considered to be
analogous to the Euclidean straight line created from the
intersection of two planes. The intersection of a rect and
a level results in a ”punct" (117), which is analogous to

the Euclidean point. A punct can also be created by the

 

Q‘The four-dimensional world refers to the conjoining of
three-dimensional space with the dimension of temporality.

29

30

intersection of four distinct moments where "there is a
common intersection which is neither a rect nor a level"
(117). ”A rect either intersects a moment in one punct, or
is parallel to it, or contained in it" (118).

The characteristic of parallelism is essential in
Whitehead's derivation of space, as it was in his derivation
of time. We are forewarned that the characteristic of
parallelism in the derivation of space is not to be
separated from those properties to which it is applied (the
properties of levels, rects, and puncts) (119). Rects,
levels, and puncts are not natural entities but are logical
notions and as such cannot constitute routes of
approximation posited in sense-awareness” (Kneebone 349).
They are, however, indirectly related to events (through
their relationship with moments) and so provide tools for
use in scientific study.20

Because a rect intersects a moment in only one punct,
when we consider a whole time-system, it is clear that a
rect will intersect each moment in that time-system only
once per moment. The same can be said of levels. Since a
level intersects a moment in only one rect, when we consider

a whole time-system, a particular level will intersect each

 

3’0. T. Kneebone. Kathgmatical Logic and the Fopndapions of
Mathematicg. D. Van Nostrand Co. Ltd.: London, 1963.

"Levels, rects, and puncts are similar to the ideal
Euclidean point, straight line, and plane, in that they do
not actually exist in nature. Nevertheless, they all remain
valuable tools when applied to the task of understanding the
physical aspects of nature.

31
moment of that time-system, creating a rect, once only per
moment of the time—system. The puncts lying on a rect are
arranged in a particular order and that order is determined
by the order in which the rect intersects the moments of a
time-system of which it is a part (119).

Levels, rects and puncts are used to construct the
instantaneous space associated with moments. Instantaneous
space, like the moment itself, represents an ideal to which
we can only realize a close approximation. When we speak of
the domain of instantaneous space with its associated
levels, rects and puncts, we are also speaking of the space
to which the classical mechanical laws of Newton apply.
Whitehead insists that his system in no way violates these
Newtonian laws but goes beyond them to show a connection of
time and space which the Newtonian system cannot.

Through the use of levels, rects, and puncts Whitehead
determines a complete three-dimensional geometry of
instantaneous space (120). In a consideration of the
various theories of relativity Whitehead considers the
electromagnetic theory to be the more general (115).=’ This

electromagnetic theory of relativity provides instantaneous

 

z’Whitehead's use of the term ”electromagnetic theory of
relativity" is intended to imply opposition to the pre-
Einsteinian conceptions of relativity. Whitehead asserts
that the electromagnetic theory of relativity is that which
exhibits the physical electromagnetic field which is the
topic of physican science. He further asserts that it
establishes that some pairs of durations are extended over
by other durations but that some pairs are not (Palter 55).

32
space with "definitions of flatness, of straightness, of
punctual position,...[of time order and spatial order], of
perpendicularity, and of congruence, as well as of
parallelism" (115). These abstract concepts all arise from
"the nature of things as expressed by the fundamental
relations of extension and cogredience" (115).

Although events themselves do not change (61), it is
obvious that we observe changes in the physical state of
nature. Such changes include change in position, in
velocity, in motion, in mass, in density, in shape, in
direction and/or rotation, etc. Whitehead needs to explain
these changes in terms of events and relations among events.
We must refer to four-dimensional space-time for a
discussion of motion and change in velocity. Whitehead
begins this process with the introduction of abstractive

elements and event-particles.°°

Evgnt-pprticles

The intersection of time-systems introduces relative

motion into Whitehead's system. Thus, an event-particle

which is perceived as stationary in one time-system is
perceived by some event in another time-system.to be moving
in a uniform motion, following some definite route. The

converse of this statement is also true for Whitehead.

 

='°Event---particles are simple events illuminated by the use
of the method of extensive abstraction.

33

If we are to consider motion along a route of
approximation within the extensive continuum, it is
necessary to introduce the notion of event-particles. An
”event-particle" is defined by Whitehead as ”the route of
approximation to an atomic event,3‘ which is an ideal
satisfied by no actual event” (121). Thus an event-particle
is an event in its simplest form.

When Whitehead speaks of an event-particle following a
route of approximation, he does not mean that the particle
actually moves, intersecting durations, and following a
given path of motion. Instead he remdnds us that in the
intersection of time-systems every moment intersects each
duration of another time-system only once. By means of this
intersection puncts are established. An apparent route is
actually a series of puncts in sequential order, each punct
associated with a particular moment, along some possible
path. An event-particle is not a material-like entity
traveling from punct to punct. Rather, there exists an

event-particle which covers a punct at each moment along

some route of approximation.

 

"An atomic event is an event in its simplest form. It is
no longer divisible into other events. At this stage in
Whitehead's development of his philosophy of nature he
thought that all events were divisible. In this respect, an
atomic event could only be an idealization not found in
nature. In his later works Whitehead reformulated this
notion of event and took the position that all events are
atomic (non-divisible). The reference in this quote to an
event-particle as an approximation of an atomic event refers
to Whitehead's notion of event-particles as the simplest of
all actpal events involved in spatial representations.

34

§olid§ and Volumes

The move from event-particles to solids with density
and volume is made by the employment of a particular type of
”kinematic" route. Whitehead defines this route in the
following manner:

A ”kinematic route” is a route (i) whose end-
points are sequent and (ii) such that each moment,
which in any time-system lies between the two
moments covering the end-points, covers one and
only one event-particle on the route, and (iii)
all the event-particles of the route are so
covered (125).

Event-particles which lie on a kinematic route
represent a possible path for what Whitehead terms a
"material particle".°= He defines the term "solid" with a
reference to ”solid primes" (126). A solid prime is a prime
whose formative condition is being a simple abstractive
set°° which "covers all event-particles shared in common by

both boundaries of adjoining34 events” (126). A solid,

then, is the set of event—particles common to the boundaries

 

’2”Although Whitehead is dealing with event-particles,
he...stresses that each event-particle is related to other
event-particles in the four-dimensional (physical) continuum
by means of...elements of impetus or ‘atomic physical
fields'. They are the set of physical qualities referring
to the gravitational and electro-magnetic fields in the
neighborhood of E [where E refers to some definite event-
particle in the four-dimensional manifold]....0n his view,
then, material particles are pervaded tracks of event-
particles in the space-time continuum" (Hays 242). Bold type
is added by this writer.

°°A simple abstractive set is one in which there is no one
event-particle at which all members of the converging end of
the set have contact (123).

"For the definition of the term adjoining, refer to page 26
of this paper.

35
of two adjoining events. Such a solid is not equivalent to
a physical entity. It is a locus of event-particles, i.e.
it illustrates some quality of position in the space-time
manifold. A solid which is co-momentala‘ is termed a
”volume” (126). A volume, then, is "conceived as a locus of
event-particles" (127).

Whitehead indicates that a "concrete event...is defined
by...the event-particles inhering in it, and such a set of
event-particles defines only one event" (127). Such an
event can be said to be "uniquely defined by the set of
event-particles which form its boundary" (127). By a
"concrete event,” Whitehead means the locus of event-
particles which defines that event we perceive as an
ordinary physical entity, e.g., a chair.

To say that this physical entity is defined by the
event-particles inhering in it is to say that the event
which constitutes such a given set of event-particles is
unique and that it represents what is commonly thought of as
the volume of the entity (as a locus of event-particles); it
is a cross section of that part of nature36 which represents
the locus of event-particles that designates the physical
entity (the chair). To say that the physical entity, e.g.,

the chair, is an event ”uniquely defined by the set of

 

"Co-momental in this definition means that the locus of
event-particles defining the solid lies in a single moment.
"The moment which is associated with instantaneous space
provides position for the locus of event-particles.

36

event-particles which form its boundary” is to recognize the
particularity of that physical entity. That particular
locus of event-particles inheres in the event which is that
particular chair at that particular time.

Whitehead tells us that "the instantaneous volumes in
instantaneous space which are the ideals of our sense-
perception are volumes as abstractive elements" (CN 102).
This is not intended to provide an explanation of how we
perceive material objects but, rather, to explain how
material objects can be seen as having designated position
in the extensive continuum using the technical terms:

abstractive sets, event-particles, primes, anti-primes, etc.

The Co onents of the Three T s of ace

In Principlgs of Natural Knowledge Whitehead discusses
three different types of space. Only one of these three
terms refers to that space which we perceive, i.e.,
instantaneous space. The other two types of space are
required for the physical sciences, both the traditional
Newtonian science and the newer view of relativity. It is,
then, necessary for Whitehead to account for the types of
space required in these sciences in his theory of
spatiality. The three types of space discussed by Whitehead
are 1)instantaneous space, 2)four-dimensional space-time,
and 3)three-dimensional time-less space. The following

chart (Palter 55) is helpful in distinguishing the elements

37
in the geometries of these three types of space. Under. the
name of each type of space are listed those basic elements
from which the geometries of these spaces can be

constructed.

 

Tablg 1
Types of Space

 

 

Instantaneous space Space-time Time-less space
Punct Event-particle Point
Rect Point-track, Straight line
Null-track,

Set of co-rect
event-particles

Level Katrix, Plane

Set of co-level
event-particles

 

Whitehead is interested in space abstracted from those

events associated with our ordinary perceptions of the
natural world. Through the use of the method of extensive
abstraction he develops the notions of puncts, rects, and
levels needed to express the geometry of three-dimensional
space. Although these notions are ideal constructs, they
are developed from. durations/moments and so they retain,

through this indirect relationship, the essential relations

38

found in nature, i.e., those relations of perceivable
events.°7 Because of his desire to show that the basic
relations found in nature are preserved in these abstract
entities, Whitehead describes events in terms of abstractive
sets, abstractive elements, anti-primes, and primes and then
defines puncts, rects, and levels using these technical
terms.

Whitehead maintains that all three types of space
should be connected in some manner to the perceivable events
of the natural world. He feels this can be accomplished by
defining the basic geometrical elements of four-dimensional
space-time and three-dimensional time-less space in the same
manner he defined the elements of instantaneous space. He
employs the method of extensive abstraction and the
relations of cogredience and parallelism to define and
derive the basic elements of these other two types of space.

In the chart given above, the three groups of basic
elements correspond with one, two, and three dimensional
phenomena. Reading horizontally across the chart puncts,
event-partiCles, and points are considered to be one
dimensional. Rects, point-tracks, null-tracks, sets of co-

rect event-particles and straight lines are two dimensional,

 

“’The use of the method of extensive abstraction applied to
abstractive sets of events represents a basic tool in
Whitehead's natural philosophy. It is essential to
Whitehead because he believes that the end products of this
method retain just those relations which are found
throughout the entire series of events in the abstractive
sets.

39

and levels, matrices, sets of co-level event-particles, and

planes are three dimensional. In basic Euclidean geometry,
points are used to define straight lines and straight lines
are used to define planes. A similar move, proceeding in
the opposite direction, is used to define the components of

the three types of space (for example, in three-dimensional
instantaneous space, first levels are derived, then rects

from levels, and finally puncts from both rects and levels).

Four-dimensional Space—time

As stated above, Whitehead needs to express the
development of the elements of the three types of space in
terms of abstractive sets, abstractive elements, and primes
and anti-primes, in conjunction with various relations. By
following this procedure he maintains that each of these
types of space is related in some manner to the perceivable
natural world and, therefore, each type of space retains the
basic relations found in nature.

Event-particles have been defined in relation to
particular types of abstractive sets (via routes). Point-
tracks and null-tracks (necessary for the construction of
four-dimensional space-time) have yet to be introduced.
Point-tracks are defined by Whitehead as "the complete locus
of event-particles thus defined by the indefinite
prolongation of a station throughout its associated time-
system..." (130). The definition is of little value without

an understanding of the term "station”. Any detailed

40

description of the terms involved in the definition must
show their connection to abstractive sets, abstractive
elements, primes, and anti-primes. The derivation of these
terms is somewhat complicated but it is necessary to
consider them if we are to understand how the method of
extensive abstraction relates to the development of the
different types of space being discussed.

Let us briefly review some definitions already
established. Event-particles have been defined in terms of
absolute anti-primes which, in turn, are deduced from
abstractive sets. Abstractive sets and absolute-anti-primes
have also been defined in the preceding sections.39 We have
referred to puncts as the intersection of four nonparallel
moments. Event-particles which cover puncts are said to
have been given absolute position in the instantaneous space
of the moment in which they lie (121). Here the term
"absolute” is not used to indicate a position in absolute
space but rather means "absolute position" relative only to
that time-system in which the event-particle and the punct
lie.

Event-particles are considered dense within any
particular moment of a time-system; there is no punct in an
instantaneous moment which is not covered by an event-

particle and every position possible within a moment is

 

°°For these definitions see, respectively, pages 20 and 21
of this paper.

41

designated by some punct. A rect, then, since it can be
generated by the intersection of two levels in different
time-systems, can be said to cover a series of puncts or
event-particles. A rect can be considered a series of
event-particles, that is, a rect can be seen as being
composed of that series of event-particles which it

intersects.

Routes and §tations
The importance of event-particles lies in the fact that

they are able to connect one instantaneous space with
another and so they allow the identification of one
particular "point” at different times (Kneebone 350). This
identification is achieved by means of ”stations” which
correspond to various event-particles. A station is itself
a type of event-particle. As an intermediate step to an
understanding of stations, we must again consider the
meaning of the term "route”.

If an event-particle is considered as an "abstractive

element of atomic simplicity” (123), a route can be

considered as the next advance "towards increasing
complexity" (123). A route is defined by Whitehead as "the
abstractive element deduced from a linear prime” (124). It
is a class of events (a simple abstractive class) which
consists of some possible finite set of event-particles,
including end points (i.e., it covers two event-particles,

P1 and p2, which are the extreme event-particles in the

42

set), and "is such that no selection of the event-particles
which it covers can be completely covered by another simple
abstractive class, provided that the selection comprise all
the event-particles covered by [the simple abstractive
class)...” (124). The first part of the statement above
indicates the ”end-points" of a route are included as part
of the route and the second part of the statement indicates
that the abstractive class has "linear type [of]
continuity”3’ (124).

A "linear prime" can now be defined as "an abstractive
set which is prime in respect to the formative condition of
(i) being covered by an assigned linear abstractive set
which covers the two assigned end-points and (ii) being
itself a linear abstractive set covering the same assigned
end-points” (124). It must also be considered regular for
primes, i.e., K-equal to itself.

The complex definitions given above indicate that when
any two event-particles are designated as end-points (p, and
p2) of some series of event-particles, there can be found an
indefinite number of linear segments connecting these end

points. These linear segments constitute a route. A route

 

°’Vhitehead refers to linear type continuity as that type of
continuity expressed by a route (as a linear segment). He
also indicates that the continuity of events is derived from
the continuity of routes, i.e., linear type continuity
(124). The fundamental relation involved with this type of
continuity is that of "lying-between" (holding for triads of
points on the linear segment). Also associated with this
relation is the property of continuous serial order for the
particles on any route (125).

43

contains, i.e., covers, its own end-points and infinitely
many other event-particles. This is due to the "linear type
continuity" displayed by routes.

In order to connect the notion of routes to that of
stations, another ingredient is needed. This ingredient is
the relation of cogredience. Earlier in this paper I stated
that cogredience indicates an event's being "here-present".
This is to say that cogredience indicates position in the
extensive continuum and so is vital to the derivation of
space (since space is necessarily concerned with the
property of position). The term ”cogredience" is now
defined as being "here throughout a duration" or ”there
throughout a duration" (128).

A stationary prime has been defined by those conditions
required by the definition of all primes4° and it is also
cogredient to some particular duration. Therefore, "a
station within a duration is the absolute abstractive
element deduced from a stationary prime" (128). It is an
"absolute abstractive element" because, with the inclusion
of the relation of cogredience, the station is always
"here", this route in a duration.

Whitehead tells us that ”a station is a route [a linear
segment including its own end-points]; and also every

station in a duration intersects every moment of that

 

‘°The definition of a prime is given on page 20 of this
paper.

44
duration...in one and only one event-particle and intersects
no other moments of that time-system” (128). Any particular
station can be indefinitely prolonged within the duration in

which it resides.

Point-tracks and Null-tracks

A point-track, then, is defined by Whitehead as the
complete locus of event-particles found along a station in
some given time-system (129-180). Thus, a point-track can
be said to follow an indefinitely prolonged station
throughout the duration in which the station resides. The
theory of parallelism can be observed in the notion of
point-tracks. All point-tracks which are points in the
space of some time-system are said to be parallel to one
another“ (131). Since point-tracks involve the relation of
cogredience,42 they can be said to be there-now in
association with particular points of space within a time-
system. Therefore, Whitehead concludes that "a complete
family of parallel point-tracks is merely a complete family

of points in the space of some time-system” (131).

 

“A "point-track has a unique association with the time-

system in which the routes lying on it are stations. A
point-track is called a "point" in the ”space of its
associated time-system.” This space of a time-system is

called ”time-less" because its points have no special
relation to any one moment of its associated time-system”
(130).

‘zPoint-tracks establish their connection with cogredience
through the association of a point-track to some station.

45

The point-track in four—dimensional space-time is
analogous to the rect in instantaneous space and to the
straight line in three-dimensional time-less space. To
complete the set of straight lines required by the geometry
of four-dimensional space-time, we must include a set of
event-particles which occupy what is termed “the set of
null-tracks." The order of these null-tracks is derived

from routes (131). Although Whitehead only briefly

discusses null-tracks in Principles of Natural Knowledge,
Robert Palter. in Whiteheag's Philosophy of Science,

discusses them in greater detail.
He states:

...a matrix [to be discussed below] is
divided, relative to an event-particle in it, into
four mutually exclusive regions...which may
equally well be called rects or point-tracks -
these loci (and all others parallel to either one
of them) Whitehead terms ”null-tracks”. In any
matrix there are two families of parallel null-
tracks, exactly one member of each family passing
through each event-particle in the matrix. Thus,
through any given event-particle there is an
infinity of null-tracks which form two three-
dimensional ”conical” surfaces whose mutual vertex
is the given event-particle. The order of event-
particles on a null-track is derived from its
intersection with systems of non-co-momental
parallel rects or of parallel point-tracks, or
from the order of the event-particles on routes
lying on the null-track (Palter 74-75).

latzices

At this point we are in a position to introduce the
notion of "matrix” in four-dimensional space-time. A matrix
can be viewed as analogous to a level in three-dimensional

instantaneous space and a plane in three-dimensional time-

46

less space. The process of the generation of a matrix is
similar to that of the generation of a level. In order to
better understand the generation of a matrix, let us begin
with a discussion about the generation of levels.

A level can be generated in several ways: 1) by
considering some rect, r, and some event-particle, P, both
of which lie in the same moment and by constructing a locus
of event-particles lying on rects, where all the rects
intersect r and pass through P. Included in this locus are
the event-particles which lie on r and P itself. The
resulting locus of event-particles represents a level (133).
2) Given the same rect, r, and event-particle, P, the same
level can be generated by forming a locus of event-particles
which lie on rects intersecting r, with the qualification
that the rects intersecting r are parallel to some one rect
which passes through P and intersects r (133).

A matrix can be generated in four-dimensional space-
time in a similar manner. Matrices are defined in terms of
rects, event-particles, and point-tracks. It should be
noted that these terms have themselves been defined using
the notions of abstractive sets, abstractive elements,
primes, and anti-primes. Through this chain of development,
levels and matrices are indirectly established by the method
of extensive abstraction.

In the description of the generation of a level we

employed rects and an event-particle which were co-momental,

47
i.e., they lie in the same moment. Let us now consider a
rect r and an event-particle P which are non-coemnmental.
We can construct a matrix by taking the locus of event-
particles lying on rects which intersect rect, r, and pass
through event-particle P. These include the event-particles
lying on the rect parallel to r and passing through P.

A second method of construction of a matrix is
analogous to the second method given above for the
generation of a level. Here the matrix is obtained by "an
event-particle P and a point-track p not passing through P
(e.g., the event-particle p2' and the point-track p1...)"
(Palter 73). See the diagram below (Palter 72).

The connection between levels, matrices, and the
straight lines of time-less space is made by stating that
every matrix contains sets of parallel point-tracks and any
one of these sets can be considered "a locus of points" in
the space“ of some time-system. Such a locus of points is
called a ‘straight line' in the space of the time-system”
(136). Matrices can be associated with many different time-
systems but there is only one straight line in each matrix

associated with a particular time-system (136-137).

 

‘°See footnote 41, page 44 of this text, for a brief
discussion of the derivation of points in time—less space.
“This is still four-dimensional space-time.

48

Figure 4 ,

Construction of a matrix

 

 

 

W),

 

Flo. IJ.—Poinl-tm:h and points; mania: and straight lines. Each point-tuck p. define-
epointh time-lea spaced 8.; thefldpmllelpoint-mch his the matrix sdefinee

weightlincindme-Isupeudk.

 

49

The principle of parallelism can and is applied to
matrices. There exist families of parallel matrices and in
any one time-system the straight lines associated with a
family of parallel matrices are parallel to one another.
Whitehead asserts that levels and matrices combine to make
up what is equivalent to the complete set of planes in
Euclidean geometry, and that they retain all the properties
of this set (133). He does not, however, go into detail as

to how one might prove this statement.

Three-dimensional Time-less §pgce
When we consider the third type of space discussed by

Whitehead in Principles of Natural Knowledge, i.e., three-

dimensional time-less space, we find a direct correlation
between it and the instantaneous space previously discussed:

There is an exact correlation between the
time-less space of a time-system and any momentary
space“ of the same time-system. For any point of
the momentary space is an event-particle which
occupies one and only one point in time-less
space; and any straight line of the momentary
space is a rect which lies in one associated
matrix including one straight line of the time-

less space or (in other words) each straight line
of the momentary space occupies a straight line of
the time-less space (138).

Whitehead has attempted to reconstruct the properties
of space and time by purely logical means, employing only

these primitive notions provided by the immediate data of

 

“lomentary space is a reference to instantaneous space.

50

perception.4‘ Time and space, thus constructed, contain no
metrical properties. They only possess the relations of
order and extension. The correlation between the different
types of space which Whitehead discusses and his attempt to
deduce these by the use of the method of extensive
abstraction through the (not always direct) employment of
definitions in terms of abstractive sets, abstractive
elements, primes, and anti-primes emphasizes one of the main
aims of this paper, i.e., to give a careful examination of
the method of extensive abstraction and the implications of

its use which arise in Whitehead's philosophy.

 

“These primitive notions include events and the relation of
extension along with the property of cogredience.

CHAPTER 5
THE CONSTANTS 0F EXTERNALITY

In Principles ofggatural Knowledge Whitehead lists what

he terms "the constants of externality." He uses these to
define events in terms of their inherent properties and to
indicate the relationship of these properties to the passage
of nature (71). Externality and extension are seen as
essential properties of events. It is an event's
relationship to the other events which allows us to
experience what we commonly refer to as nature. Through an
examination of the properties illustrated in the constants
of externality Whitehead hopes to highlight those basic
concepts of nature which are presupposed in the physical
sciences. He considers only those relations which are
fundamental to the formulation of the concepts of time,
space, and physical material.

Whitehead discusses six constants of externality. I
will consider only the first three in this paper. The first
three constants of externality have to do with
specifications of qualities and discriminations among
various entities. The first constant of externality states
that the natural continuum apprehended through perception is
really a "potentially definite complex of entities for
knowledge" (74). By this Whitehead means that we are able

to perceive and understand the passage of nature because

51

52
that which we can know about nature is that which we can
directly perceive. There are, however, other aspects of
nature (beyond our direct apprehension) which can be
indirectly known through an awareness of various relations
existing between events.

The second constant of externality states that the
relation of extension holds among events. This relation
must obtain if events are to be considered as belonging to
the order of nature.47 The third constant of externality
states that ”an event as apprehended is related to a
complete whole of nature which extends over it and is the
duration associated with the percipient event of that
duration" (77). This refers to our common perception of
nature as having extension and that what we perceive at any
given time is that which exists within the duration we (as a
percipient event) presently occupy. Our perception
represents that standpoint from which this particular
duration is determined. It is here where we find the
concept of a duration as a cross-section of nature.

Whitehead attempts to conduct his study based on
observable nature. This aim would seem to give rise to a

number of problems. An example of such problems is seen if

 

"It might be noted that there could be no knowledge of
nature without an ordering of the events apprehended
(temporal ordering as well as spatial ordering is being
implied here). Such ordering is required for inferential
Judgments and without it we could not make the necessary
connections between similar events required for the process
of recognition.

53

we take another look at the description of events considered
from the perspective of perception.

If the concept of a duration of an event relies upon
the perception of that duration by some other event (the
percipient event), then such a perception and the
accompanying abstraction of time and space would seem to
imply the necessity of some type of consciousness.
Consciousness in a percipient event, it could be argued,
would also imply that the temporal and spatial structures of
nature have a subjective origin.4'

I believe Whitehead would respond to this argument with
an empathic denial that this would be the case. He would
maintain that every event is capable of ”feeling" all other
events in its presence." An event ”feels" those relations
which hold between itself and other events and also the
relations which hold among other events not directly
connected to itself. Such feeling is possible in even the
least complex of events, the event-particle.

However, he would remind us that the relations which
occur among events do not result from, a process of

abstraction but are a part of the structure of nature.

 

"They would be considered subjective in reference to the
consciousness inherent in the percipient event.

"This view is expressed by Whitehead in those works
beginning with his Process and Reality. Corrected Edition.
ed. David Ray Griffin and Donald W. Sherburne. The Free
Press: New York, 1978. Because of this, these views of the
"feelings” of events is not applicable in this paper,
although the foundations of these views are seen in

Epinciples of Hatpral Epowlpdgg and other early works.

54

Events are actual entities and they exist in, and only in,
relation to one another. Temporal or spatial abstractions
are established because they are inherent in this structure
of nature. The recognition of these abstractions,
apprehended by a consciousness, in no way influences the
natural structure of events. This gives a brief explanation
of what Whitehead means when he says that nature is closed
to mind, i.e., the actual existence of nature does not
depend on perception by some "knowing" subject. For this
reason, it is not a problem that perception, implying the
presence of consciousness, would interject a subjective
element into the structure of nature. Perception is
accomplished through those same relations which are inherent

in all events and so in nature itself.

CHAPTER 6
THE THEORY OF PERCEPTION

Immediate Data of Perception
Whitehead requires that his project start with the

immediate data of perception (5). In this statement
”immediate data" refers to our ordinary perception of the
natural world, e.g., our perception of physical objects, our
sense of spatial distance and temporal extension through our
perception of the passage of time. Since he considers the
perception of this immediate data as a starting point for
his investigation, Whitehead also needs to explain the
general characteristics of perception in terms of events and
relations of events. Only through an understanding the
characteristics of perception can we claim any knowledge of
the natural world.

From these immediately perceived data Whitehead
develops by the method of extensive abstraction those basic
notions we have been discussing, i.e., abstractive sets,
abstractive elements, event-particles, levels, rects,
puncts, etc. It should be noted that Whitehead does not
claim that entities established by means of the method of
extensive abstraction are actually perceivable. Whitehead
requires only that we begin our study with perceivable data.
However, these entities do retain the same basic relations

as the data from which they were abstracted and as such,

55

56

Whitehead claims that they are indirectly available for
perception. This point will be explained more fully below.
Having introduced these points, let us begin to examine
Whitehead's theory of perception. An understanding of what
Whitehead means by the term perception is not obvious. It
has been mentioned that Whitehead has the habit of taking
common terms and giving them uncommon lmeanings. This is
true with the term "perception." Its meaning has changed
throughout the development of his early works.‘° Paul F.
Schmidt, in Perception and Cosmology in Whitehead's
Philosophy,“ gives an exposition of this development. The
following account consists basically of Schmidt's comments
on the development of the meanings assigned to such terms as
"perception,” "awareness," ”cognition," "consciousness,”

etc.

The Principle of Relativity
Beginning with the more complete version of Whitehead's

theory of perception, I will move backwards through

Whitehead's works until I reach its sketchy formulation in

Princi les of Natural Knowled e. In The Principle of

E

l“Whitehead first begins his explication of the concept of

perception in The Organization of Thought and it is
developed systematically through Epinciplgs of Natpgal
We. Concept of Nature, and is given a more complete
formulation in The Principle of Relativity.

"‘Paul F. Schmidt. Perception ang Cosmology ip Whitepeag's

ihiloso h . Rutgers University Press: New Brunswick, 1967.

.,.' . Q ’1' Ir.. ‘

57
Relativity82 we find that Whitehead refers to perception
with respect to nature, i.e., in terms of fact and factors.
He tells us that "‘perception' will be the name given to the
consciousness of a factor when to full awareness cognitionsa
of it as an entity is also superadded” (PREL 19). To
understand this definition of perception, it is necessary to
determine the meanings of the terms involved. Fact is
defined as the totality of all that is (PREL 59). It is
seen as the whole of infinite and inexhaustible nature.
Factors are those elements (parts) of fact involved in the
relationships occurring within fact.‘4 Although factors are
always found within fact, there is no set of factors which
can completely exhaust fact. Awareness is then defined as a
factor in relation to other factors. This entails that
everything has awareness of everything else. It might be
argued that this is an absurd statement but this is exactly
what Whitehead means to imply. At the level of basic
awareness each event (factor) is either directly or

indirectly influenced by the relations of all other events.

 

“’Alfred North Whitehead. Th Princi le of elativit .
University Press: Cambridge, 1922. Here after in this paper
this work will be referred to as PREL. Schmidt's discussion
of Whitehead's theory of perception as found in 2p;

Principle of Relativity is found on pages 39-41 in
Pgrception and Cospplogy in Whiteheag's Philosophy.

I"'At this point in the development of Whitehead's theory of
perception, cognition (thought) is considered to be a part
of perception.

“The terminology is somewhat confusing unless it is
remembered that fact is considered to be the whole of nature
and factors represent some part or section of this whole.

 

 

 

I:

58
In this sense they have awareness. Consciousness is a
prerequisite for awareness of factors (events). For
Whitehead "consciousness" is considered a primary or
undefined term.

In the case of a percipient event much more than
awareness is needed to say that the percipient event
perceives. "Cognition" or "thought" is required, along with
awareness, to focus upon a factor in such a way that it is
seen as separate from other factors. It is the isolation of
a particular factor (event) from all other factors by means
of the awareness of this particular factor's particular
relations. In this respect, cognition adds a limiting
quality to awareness; it allows for the apprehension of a
factor, either by disregarding its relations to other
factors or by the apprehending of it in abstraction from its
own relations. When cognition is added to awareness,
perception is said to occur. "‘Full awareness’ occurs when
factors are clearly apprehended and their mutual relations
are jointly apparent" (Schmidt 40, PREL 84). "Perception"
is defined as the combination of basic awareness and the
recognition of a factor as some particular entity by means

of cognition.

59

Concept of Nature

In Concept of Nature,55 Whitehead replaces "perception"
with "sense-perception," and ”awareness" with "sense-
awareness”. Sense-awareness is considered to be
awareness,as from which the consciousness of logical,
aesthetic, and moral awareness of factors within fact has
been eliminated. Sense-perception is sense-awareness to
which cognition has been added. For this reason, sense-
awareness is considered more basic than sense-perception,
i.e., the former is necessary for the occurrence of the
latter.

Sense-perception is the apprehension of all events
which make up nature in respect to some duration. In an
initial statement concerning the purpose of his work,
Whitehead states that he is concerned with the objects of
perceptual knowledge and not with the knower of this
knowledge. He limits his study to this aspect of nature
(vii). Nature, it has been stated, is "closed to mdnd,”'7
i.e., it does not require perception for its existence.
Whitehead asserts that the natural sciences are interested
in that which is apprehended by sense-awareness and not with

cognition or sense-awareness itself. This exclusion of

 

“Schmidt's discussion of Whitehead's theory of perception
as it appears in Concppt of Nature is found on page 41 of
Schmidt's work.

“Awareness here is defined as consciousness of some factor
in contrast to its relationship to other factors in fact.
‘7For a discussion of the meaning implied by ”closed to
mdnd,“ see page 54 of this paper.

60
thought from nature does not mean the the two are separable.
Rather, ”[ilt means that in sense-perception nature is
disclosed as a complex of entities whose mutual relations
are expressible in thought without reference to mind, that
is, without reference either to sense-awareness or thought”
(CN 4-5). Nature, in this sense, is seen as one part of a

larger system of complex relations.

Principles of Natural Knowledge
In Principles of Natural Knowledge" Whitehead’s theory

of perception is less refined than in the former works,
although the same basic elements are evident. "Perception”
is defined in terms of the diversification of nature into
natural entities." The presence of thought in perception
is implied although not explicitly expressed. That
Whitehead does at this time recognize thought as a part of
perception is seen in his assertion that the vividness of
perception depends on stimulation from both awareness and
thought (”[Perceptionl is essentially apprehension of the
becomingness of nature....But equally perception requires
recognition" (98)). Those entities of nature which are made
known through perception are "events,” "percipient objects,"

"sense-objects," "perceptual objects,” and ”scientific

 

‘°Schmidt’s discussion of Whitehead’s theory of perception
in Principles pf Natpral gngwlpdgg is found on pages 41-42
in Schmidt's work.
'9Whitehead discusses perception in terms of those entities
involved in the process of perception rather than discussing
the process itself.

61
objects" (60). A discussion of the individual properties of
these entities is not needed to fulfill the purpose of this
paper, although the nature of events and percipient objects
are discussed in some detail in this paper.

In summery it is noted that a basic understanding of
perception is required in any study of Whitehead’s natural
philosophy because perception (at least in the form of basic
awareness of relations between events) is necessary for any
explanation of events. Events can be defined in terms of
their particular relations. In this sense, any interaction
in nature depends upon an awareness of the relations of
events by events. Without such awareness, Whitehead's
notion of events would be reduced to something resembling
the monads of Leibniz6° (Leclerc 120-121),

This is a mere sketch of Whitehead's theory of
perception. I have discussed how the concepts of awareness,
cognition, perception, etc., developed in Whitehead's early
works. It should be remembered that I began with
Whitehead's more complete formulation of these concepts and
then preceded backwards in the process of development,

ending with the more rudimentary explanations presented in

Principles of Natural Knowledge,

 

‘°Ivor Leclerc. ”Whitehead and the Problem of Extension."

in Alfred North Whitehead: Essays on His Philosophy. ed.
George L. Kline. Englewood Cliffs: Prentice- Hall, Inc.,

1963.

1...

62

Re-examination of a Problem

We can now attempt to relate Whitehead's theory of
perception as presented in Principles of Natural Knowledge
to the problem raised earlier in this paper. This problem
indicated that since perception seemed to require some type
of consciousness, the necessity of such consciousness could
interject a subjective element into Whitehead’s
interpretation of the basic relations of nature.“ We can
now conclude that not only is consciousness a necessary part
of perception but so are awareness and cognition, although
presence of the latter is only implied in Principles of
Natural Knowledgp. Whitehead believes he can sidestep the
problem of introducing subjectivity into the interpretation
of natural relations in the following manner.

Since nature is events, we are not dealing with two
different kinds of ”stuff" when we consider an entity in
nature (a concrete event) and the observer of that entity.
The percipient event (the observer) is itself a concrete
event. The same relations are possible among all events.
No problem arises in considering the perception of nature on
one hand and nature itself on the other, because these are
not different kinds of things (one subjective and the other

objective). This view is seen as consistent with the

 

“The answer to this problem discussed earlier was
formulated in terms of Whitehead's later works, in

particular, Ppocgss and Reality.

63

totality of Whitehead's natural philosophy, but it has been
an occasion of contention among Whitehead's critics.

The relationship between Whitehead's theory of
perception and the method of extensive abstraction becomes
evident when we consider that what is perceived (and, also,
the perceiver) are events, including those objects“2
inherent in events, and the various relations which occur
between events.‘° All three of these aspects of events are
brought into focus through the application of the method of
extensive abstraction.

I have not given a complete exposition of Whitehead's
work but I have indicated several areas where problems mdght
ultimately arise. Because of the complexity and scope of
Whitehead's writings in natural philosophy, it has been
necessary to limit the scope of my discussion and omdt
topics where points of contention might arise which could
assist my thesis. Such areas include the problems Whitehead
faces in explaining the measurement of space and time, and

those problems arising from his notions of cogredience and

significance.

 

‘2An explanation of Whitehead's theory of objects is not
necessary for the purpose of this paper. Suffice it to say
that objects make possible the permanence for the
recognition of some ”awareness" vital to cognition.

‘sFor a statement of Whitehead's fundamental thesis
concerning the importance of his theory of perception to his
entire project and his insistence that a philosophy of
nature be erected on a structure ultimately based on data
immediately perceived, see p. 66 below (Principlps of

Napural Knowledge, pp. 12-13.)

T*‘

PART II

POSSIBLE PROBLEMS WITH
WHITEHEAD'S PHILOSOPHY

INTRODUCTION

I now end my exposition of Principles of Natural
KBQELQQEQ. and turn my attention to several problems related
to the method of extensive abstraction and the relation of
parallelism. It will be necessary during the course of this
discussion to include some further explanation of certain
notions in Whitehead's works. However, these will be
related to the particular problems under consideration.

I intend to discuss three interrelated reasons why one
might choose to reject Whitehead's natural philosophy.
These reasons are related to specific topics in Whitehead's
early works. These topics include: 1) Whitehead's
fundamental thesis that natural philosophy should be
concerned with the perceivable phenomena of the natural
world, 2) the nature of Whitehead's extensive continuum of
events, and 3) Whitehead's application of the relation of
parallelism to his natural philosophy.

The particular problems I raise in respect to these
topics are: 1) Whitehead appears unable to maintain the
connection between the basic events derived by the method of
extensive abstraction and the entities of the perceivable

world. 2) The relations expressed in Whitehead's extensive

64

65

continuum of events do not meet the requirements of his

natural philosophy. 3) The assumption of Euclidean
parallelism in Whitehead's early works directly influences

the outcome of his view of nature.

 

CHAPTER 1
PROBLEMS WITH THE THEORY OF PERCEPTION

The Fundamental Thesis in Perception
Whitehead has asserted that natural philosophy should

be based upon the immediate data of perception.
He states:

The relatedness which is the subject of
natural knowledge cannot be understood without
reference to the general characteristics of
perception. Our perception of natural events and
natural objects is a perception from within
nature, and is not an awareness contemplating all
of nature impartially from without....[PJerception
involves a percipient object, a percipient event,
the complete event which is all of nature
simultaneous with the percipient event, and the
particular events which are perceived as parts of
the complete event....The point here to be
emphasized is that natural knowledge is knowledge
from within nature, a knowledge "here within
nature" and "now within nature," and is an
awareness of the natural relations of one event in
nature (namely, the percipient event) to the rest
of nature. Also what is known is not barely the
things but the relations of things, and not the
relations in the abstract but specifically those
things as related (12-13).

In this passage Whitehead expresses some of the
fundamental notions which make up his natural philosophy.
An integral part of this thesis is that our knowledge of
nature is, without exception, tied to perception of nature
by means of, in Whiteheadian terms, a "percipient event".
On the surface this thesis appears trite. Without the
involvement of some type of perception, knowledge of the

natural world would be impossible.

66

-:~ --.-_ --“ >—"r'l"‘w1

67
It is noted, however, that the physical sciences deal
routinely with entities which are not perceivable (for

example, the electron or the nucleus of an atom). It is,

however, assumed that these entities underlie the basic
structure of nature. Nor are the notions of space and time,
as they are used by the physical sciences, considered
available for our common perception. Space itself is not
perceived. It is the objects which appear to be located
within space that are studied. Nor are individual instants
of time perceived but time is considered with respect to the
passage of events.

It is Whitehead's contention that the physical sciences
separate natural entities into those things which are
perceived and those things which are not. The physical
science construct knowledge of the natural world through the
use of both types of entities. If Whitehead is to preserve
his thesis that a knowledge of nature must be founded upon
that which is perceived, he must overcome this separation of
types of entities established by the physical sciences. He
must be able to bridge the gap between the perceived and the
non-perceived.

One way he can accomplish this task is to demonstrate

that all of nature is open to perception.‘4 For Whitehead,

 

“The phrase "open to perception" is used by Whitehead to
indicate those events or relations occurring between events
or objects which are directly or indirectly preceivable (by
some precepient event).

68
entities or experiential events of nature can be expressed
in terms of basic relationships. These relations are
consistent throughout all of nature. We can be said to
perceive a non-observable entity if we understand that the
relations involved in this non-observable entity are
consistent in content and structure with the relations of
these observable entities from which they are derived
(abstracted).65 Whitehead believes that he can illustrate
this consistency through the application of the method of
extensive abstraction to abstractive sets of events. He
would say that non-observable entities are available to
perception because the relations which are found in any
abstractive set of events are consistent with the relations

of the observable event from which the abstractive set is

formed.
The Method of Extensive Abstraction and Mathematical
Limiting

An argument could be raised against this theory by
employing the nature of mathematical limiting. By showing

that the method of extensive abstraction is a type of
mathematical limiting, we could contend that this method,

applied to an event, results in something that is not a

 

“The use of the term consistent might be considered
somewhat ambigious. Here, it is to be understood in the

same sense as it is consistent with perception to say that
.molecules have no color or electrons have no determinate

position.

"‘1'"!

_- _. ‘m— ——.——. 1 ~—_

W___.__ -.---

69

natural entity but is an idealized construct of some sort,
i.e., it does not exhibit the same basic relations exhibited
by an entity found in nature. If this were the case,
Whitehead would have violated the requirements of his
natural philosophy. Having started with a perceived event
in nature he would have proceeded through the process of
abstraction to an idealized construction which is not
perceived (does not contain the same basic relations as the
original entity). It would not be an event (a natural
entity). To investigate this line of argument, we must
consider both the process of mathematical limiting and the
process exhibited by the method of extensive abstraction. I
will begin with the notion of mathematical limiting.

One way to illuminate this mathematical method is with
an example. In his book, Beyond Numeracy,°‘ John Allen
Paulos illustrates the principle of mathematical linuting in
the following manner:

Take a circle that is one foot in diameter
and inscribe it in an equilateral triangle. Now
inside this triangle inscribe a circle and then
inside this smaller circle inscribe a square.
Inside the square inscribe a still smaller circle,
inside of which you next inscribe a regular
pentagon. Continue with these nested inscribings,
alternating between a circle and a regular polygon
whose sides increase by one with each iteration.
It's clear that the area of the inscribed figures
decreases with each repetition, but what is the
ultimate area achieved by this sequence of
figures? At first glance it appears that it
should be zero, the process leading only to an

‘

“John Allen Paulos. Beyond Numeracy. Alfred A. Knopf: New
York, 1991 .

70

isolated point. Remember, however, that as the
number of sides of the polygons increases, they
become more circular and after a while the process
becomes, almost at least, one of placing a circle
inside another circle with very little lost in the

area from one step to the next....The limit of
this procedure is a circle concentric with the
original one and having a diameter of

approximately one inch, 1/12th that of the
original (Paulos 129).

The notion of mathematical limiting is not only
applicable to geometrical instances, but also to the sum of
a numerical series, "the area of a curved region, the
asympototic limit of a curve or a sequence of curves...and a
host of other, more esoteric mathematical constructs"
(Paulos 131). For our purposes, we wish to isolate the
basic principles of mathematical limiting in order to
compare these with the basic principles at work in
Whitehead's method of extensive abstraction.

In the quote given above, Paulos refers to the
application of mathematical limdting to idealized
constructs, that is, to idealized representations which are
expressed mathematically. The process begins with a

mathematical representations? of some occurrence and ends

with an idealized approximation in the form of a numerical

 

“’The term ”mathematical representation" is intended to mean
a mthematical expression which exhibits certain qualities
(of some natural occurrence or natural entity. The type of
Imathematical representation will vary with the particular
cease to which it is being applied. In the examples given
here, an arithmetic mathematical representation is being
given in the form. of the numerical expression of area or
Volume.

71
value. This would seem to be true of all instances of
mathematical limiting.

Now let us consider Whitehead’s method of extensive
abstraction. If we can show that the method of extensive
abstraction is an instance of mathematical limiting, we will
have proved that Whitehead has not overcome the problem of
the perception of non-observable entities because the method
of extensive abstraction must begin with some natural entity
and end with some idealized mathematical construct rather
than an event.

We have seen that Whitehead derives the concept of time
from natural events by applying the method of extensive
abstraction to durations. To show that the method of
extensive abstraction is an instance of mathematical
limiting we must prove that a duration can be expressed by a
mathematical representation and that the result of the
process is some idealized construct, in this case, an
extensionless instant of time.

At first glance, it would seem that this is a case of
mathematical limiting. A duration, it would seem, should be
able to be expressed by some arithmetic mathematical
representation and an instant of time is certainly be an
idealized construct which is not found in nature. A closer
examination will show, however, that there are several major

flaws in this presentation. First, Whitehead would not

agree that a duration can be given an arithmetic

 

is

72

mathematical representation. It is admitted that an
arbitrary numerical value could be assigned to some portion
of a duration in such a way that the numerical value would
decrease with the diminution of the series of durations.
But this is not an arithmetic mathematical representation of
a duration, because the numerical values do not express any
property of the duration.

The method of extensive abstraction is being applied to
an abstractive set of durations and an abstractive set has
the following properties: 1) ”of any two of its members one
extends over the other, and [2] there is no event [duration]
which is extended over by every event [duration] of the set”
(104). The method of extensive abstraction differs from the
mathematical process of limiting in this case because there
is no actual numerical limit being expressed. Neither can
we say that there is convergence towards a duration (an
event) as a limit. Such a convergence would violate the
second condition listed above since in an abstractive set of
durations there is no smallest duration. If there were, it
would necessarily be extended over by every event of the set
and this constitutes a violation of the second condition.

Whitehead addresses this problem in the following
passage:

The properties of an abstractive class secure

that its members form a series in which the

predecessors extend over their successors, and

that the extension of the members of the series

(as we pass towards the ”converging end"
comprising the smaller members) dimdnishes without

73

limit; so that there is no end to the series
in this direction along it and the diminution
of the extension finally excludes any
assignable event. Thus any property of the
individual events which survives throughout
members of the series as we pass towards the
converging end is a property belonging to an
ideal simplicity which is beyond that of any

assignable event. There is no one event
which the series marks out, but the series

itself is a route of approximation towards an
ideal simplicity of "content" (104).

While the method of extensive abstraction may seem to
converge toward some ideal limit in nature, in fact no such
limit can be stated. What the method of extensive
abstraction does do is to produce a series of progressively
simpler natural events. In the case of durations this is a
series in which the temporal extension progressively
decreases.

If we were to assign numerical values to the members of
this series, these values would converge to a mathematical
limdt. It is noted, however, that it is not the abstractive
set which converges but the series of assigned numerical
values. An abstractive set approaches or approximates an

instant of time but there is no such entity existing in

nature. This illustration has been concerned with durations
but the same technique is used in establishing an
approximation of a dimensionless point.

Whitehead's theory of perception applies to these

considerations because it would seem. reasonable to assume

that the durations toward the end of an abstractive set are

74

not perceived entities. In our previous discussion of
perception it became obvious that Whitehead's use of the
term "perception" is not that which is commonly accepted.
If we accept his theory of perception, we must also accept
that even the smallest of durations would be perceivable,
not through sense-perception but indirectly, through
awareness of the relatedness of these durations with the
rest of nature (and so with the percipient event).

It can be seen that we have not succeeded to show that
the method of extensive abstraction is an instance of
mathematical limdting. Nor have we proven that Whitehead
fails to show that even infinitely small events are
perceivable. The discussion has, however, illuminated mmch

about the procedure of the method of extensive abstraction

and Whitehead's application of it.

A Review of the thod of Extensiv betraction

Whitehead has intended the method of extensive
abstraction to provide a means of reducing the relations
exhibited by events to their greatest simplicity. In this
way he hopes to illuminate the basic structure or framework
of the events we perceive in nature. These are the same
basic relations upon which the natural sciences depend for
'the expression of the laws of nature. Abstractive sets of
events provide the vehicle for the simplification of
relations exhibited by events when they are subjected to the

unethod of extensive abstraction. An abstractive set is, in

75
essence, a set of nested events, each lying (at least, in
part) within another, each containing smaller and smaller
events, i.e., the events form an infinite non-convergent
series.

Since the larger events toward the beginning of the
series are observable, even those toward the smaller end of
the series are to be considered observable. These latter
events have been simplified in such a way that they retain
only those relations which are common to the whole series.
In this manner, the abstractive set, taken as a whole
series, is considered to be open to perception. If
Whitehead is to maintain the central thesis of his natural
philosophy (stated above) he must be able to prove that

these assertions are true.

CHAPTER 2 .

CRITICISHS: THE METHOD
OF EXTENSIVE ABSTRACTION

There have been a number of criticisms offered against
Whitehead's use of the method of extensive abstraction. I
will consider the criticisms of three authors, C. D.
Broad,64 Adolf Grunbaum,65 and W. Haysfl“s in the following
discussion. These authors present opinions for and against
the applicability of Whitehead's method of extensive
abstraction in the construction of his natural philosophy.
They also comment on each other's Opinions on this subject.
After a discussion of these criticisms, I will offer my own
comments on the applicability of the method of extensive

abstraction to Whitehead's project.

Comments by C. D. Broad
We have been considering the use of Whitehead’s method

of extensive abstraction as it relates to his requirement
that the natural sciences must be based on perceived

entities. C. D. Broad, in Scientific Thought. discusses
this aspect of Whitehead's work. He presents a comparison
between the definition of a point, i.e., a point established

 

“C. D. Broad. §Q$§p§d§dp_lhdpghp. Routledge & Kegan Paul
LTD: London, 1953, cf. Chapter 1, pp. 26-52.

“Adolf Grunbaum. ”Whitehead's Method of Extensive
.Abstraction." in British Journal for thg Philosophy of
Science . Nov. 1953 . pp. 215-226 . .

‘S‘W. Hays. The Philosophy of Whitehead. Collier Books: New
York, 1962. cf. Chapters 6 and 7, pp. 108-125.

 

76

77
through the use of the method of extensive abstraction, and
the development of a definition for irrational numbers
(Broad 38-52).

Broad begins his discussion by raising the following
question: "Have we any right to believe that finite objects
consist Of parts Of pd magnitude, or that such parts, if
they exist at all, will have relations in the least like
those which hold between finite areas and volumes?" (Broad
37-38). He intends to show that a method of re-definition,
applied to the commonly accepted definition of a point, will
solve the problem of establishing volumes of physical
objects from collections of dimensionless points. Broad
contends that Whitehead's method of extensive abstraction
allows this process of re-definition to accomplish its
purpose (Broad 38-39).

Broad presents two criteria for the definition of a
point, stated as follows: ”...(i) that points must have to
each other the kind of relations which geometry demands; and
(ii) that points must have to finite areas and volumes such
a relation that a reasonable sense can be given to the
statement that such areas and volumes can be exhaustively
analyzed into sets of points” (Broad 39). These criteria
can be related to Whitehead’s project" in the following

manner: criterion (1) requires a theory of points adequate

 

7"The project is to present an analysis of the relations of
events in nature through the use of the method of extensive
abstraction.

78
for use by the physical sciences; and criterion (ii) can be
maintained by allowing the use of the method of extensive
abstraction in the construction of points (where abstractive
sets are diminished in such a way that the diminishing
series is infinite and non-converging and the relation of
extension is retained throughout the set). Even the
smallest members of such a set are extended and this

extension can be expressed in terms of area or volume.

Re-definitions: Irrationals and Integers
Broad provides support for his thesis by referring to

the internal structure of a scientific definition.72 He
contends that the internal structure of a definition does
not matter as long as the definition maintains all the
logical relations involved in the entity under
consideration. He uses the definition of irrational numbers
as an example of the method of re-definition (Broad 42-44).
We call irrationals such as J2 or J3 numbers (just as
we call integers such as 2 or 3 numbers) because these
irrationals obey formal relations which are a subset of the

formal relations which all numbers obey, including

integers.7° To illustrate his point, Broad considers the

 

72By the internal structure of a scientific definition,
Broad is speaking of the explicit linguistic formulation of
the definition. Broad refers to this as the inner nature of
the term (Broad 39).

7°It should be noted that integers obey some formal
relations which irrationals do not. Some of the properties
involved with the subset of formal relations which both
integers and irrationals obey include the associative

79
operations of addition and multiplication as they are
applied to both irrationals and integers. He points out
that although these operations produce quite different kinds
of entities when they are applied to irrationals and to
integers, the logical properties expressed by both types of
entities are the same. Broad asks us to accept on faith

that this similarity in logical properties is maintained in

 

property, the commutative property, the existence of an
identity element, and the existence of inverses of elements.
These properties can be defined as follows:

Let a, b, c, e R, where R represents the set of all real
numbers (including integers and irrationals).

Associative Property under addition:
a+(b+c)=(a+b)+c.
Associative Property under multiplication:
a(bc) = (ab)c.
Identity Property under addition:
R contains an element 0 such that
a + 0 = 0 + a = a.
Identity Property under multiplication:
R contains an element 1 such that
a - 1 = 1 - a = a.
Inverse Property under addition:
For each aeR there exists a unique element (-a)eR
so that
a + (-a) = (-a) + a = O.
Inverse Property under multiplication:
For each nonzero aeR there exists a unique element
(1/a)eR so that
a(1/a) = (1/a)a = 1.
Commutative Property under addition:
a + b = b + a.
Commutative Property under multiplication:
ab s be.

These properties, since they apply to the set of all real
numbers, apply to integers and irrationals alike. Other
properties include those of closure, the distributive
property for multiplication over addition and the trichotomy

property. Karl J. Smith. Finitp fldtdgpdtics. Scott,
.Foresman, and Co.: Glenview, Illinois, 1975. pp. 335-336.

80
all operations which can properly be applied to both

irrationals or integers (Broad 40).

There is a striking difference between the internal
structure of what is meant by an irrational and what is
meant by an integer, but we refer to them both as numbers.
This is because they obey the same subset of formal logical
properties, i.e., we assign the term "number" to any entity

which possesses these particular logical properties.

Re-definitions: Points

 

The same type of considerations can be applied to the
definition of a point. Broad indicates that a point is
often defined as the ultimate limit of some diminishing
series of areas or volumes (Broad 46). This concept works
quite well in mathematical considerations but it can appear
paradoxical when we try to visualize how the combination of
a collection of massless point-particles can constitute a
volume or area. It seems counter-intuitive that when a
collection of points are combined they can constitute a
physical object. However, this notion is generally assumed
in the applied physical sciences. It is the project of both
Whitehead and Broad to overcome this apparent paradox.

The difficulty seems to arise from the traditional
definition of a geometrical point. Broad attempts to show
that a point can be defined in a different manner (different

in internal structure from the traditional definition) but,

 

81

at the same time, retain those relations required of a
geometrical point for use in the physical sciences. This
re-definition should make clear how volumes and areas can be
constructed by combining collections of points. He intends
to accomplish this feat through the use of Whitehead’s
method of extensive abstraction.

When the method of extensive abstraction was applied to
a duration the result was a series of progressively
diminishing durations with an increasing simplicity of
relations.74 The end product of this process was the
approximation of a moment in time. This abstractive set of
durations is infinite and results in a simplicity of
relations expressed by an idealized instantaneous moment.

When we consider the spatial side of nature and apply
the method of extensive abstraction to the space associated
with an instantaneous moment, the result is diminishing
series of volumes or areas. These series are also infinite
and approach or approximate dimensionless points.
Traditionally, a connection between such an infinite series
and the point itself is never established.

Broad suggests a re-definition of the point (similar to
the re-definition of irrational numbers discussed above)

which will eliminate this problem. Instead of considering

 

"Refer to the section, The Hethod of Extensive Abstraction
and Hathematical Limiting, for an explanation of what is
:meant by an increasing simplicity of relations. In
jparticular, see the quote from Whitehead, p.72-73, in this

paper-

"...a

”‘33-‘30.” .

 

82

the approximate limit of the series as determining a point,
Broad suggests that we re-define a point so that it consists
of the entire series of diminishing volumes. Although this
seems intuitively incorrect, let us consider it in terms of
the criteria Broad requires for the definition of a point.
If this re-definition can fulfill these criteria, we are
justified in considering it a definition of a point because
it retains all the relations required of a point by the
physical sciences.

The first criterion requires that a point fulfill those
relations required by geometry.7' Broad contends that the
re-definition of a point meets this requirement, as well as
the other geometrical entities constructed through the use
of the method of extensive abstraction (i.e., lines, planes,
etc.). He offers no proof for this assertion but asks us to
take this point on trust (Broad 47). Instead of a proof he
offers an example of how two different series of concentric
circles generate two unique points. These two points can
then be used to define a straight line connecting them.
These constructions illustrate two of the basic properties
required of geometrical points.

The second criterion asks: "Do points, straight lines,

etc., really exist in the same sense as volumes, or are they

 

7“Broad is referring to the logical relations necessary to
construct a coherent physical geometry. It is assumed that
the geometry is Euclidean and the logical relations are the
postulates required by Eucild.

83
merely convenient and perhaps indispensable fictions?”
(Broad 51).?“ Broad asserts that points are not merely
fictions. They have a reality of their own which is
different from that of volumes. He stresses that only
particular entities can be directly perceived and that
points are not particulars but represent classes of series
of volumes. These series of volumes really do exist and the
larger elements of the series can easily be perceived.
Points themselves are to be considered logical functions of
such series and they have existence in the sense that ”they
are determinate functions of real series of actually
existing particulars” (Bread 51). This is to say that
points are expressions of the relations which exist in such

series. This is, Broad, asserts, exactly what Whitehead

 

"It should be noted that Broad's argument is not without
critics. A number of questions could be raised against the
points Borad makes in his argument. First, it might be
questioned whether points really represent classes. It
could be argued that they are the classes of a series of
volumes. Second, it can be questioned whether or not a
class of volumes will have all of the relevant properties of
a point. Two points determine a line, but two sequences of
volumes do not. The question can also be raised as to how a
realism about points is supposed to follow from the
construction given in Broad's argument? Broad also refers
to a "determinate function" without giving us a clear
definition for his use of this term. Finally, it might be
asked, if Broad is a realist about points, then why does he
present his re-definition in the first place? Why does he
go to such lengths to construct them out of volumes? These
points offer a number of reasons why Broad's arguments
should not be accepted out of hand. I am indebted to Dr.
Hark Risjord, Department of Philosophy, Hichigan State
University, for bring these objections to my attention.

84
intends to accomplish through his use of the method of

extensive abstraction.

Criticism: Adolf Grgnbapm
Broad's interpretation of Whitehead's method of

extensive abstraction is not without critics. Adolf
Griinbaum"7 addresses Broad's criteria for the definition of
a point and attempts to show that even with the assumption
of these criteria, Whitehead's method fails. Broad's
criteria for the definition of a point state ”(i)...points
must have to each other the kind of relations which geometry
demands; ~and (ii)...points must have to finite areas and
volumes such a relation that a reasonable sense can be given
to the statement that such areas and volumes can be
exhaustively analyzed into sets of points” (Bread 39).
Grunbaum believes that Whitehead's "...Hethod is vitiated by
Zeno's mathematical paradox of plurality7- because of an
important violation of the two conditions stated by

Broad..." (Grflnbaum 216).

 

77Adolf Grflnbaum. ”Whitehead's Method of Extensive

Abstraction." British Jourpal for pp; Philosophy of Scippce.

Nov. 1953. pp. 215-226.
7'"Zeno's mathematical paradox of plurality...[is intended]

to show that since spatial and temporal intervals are
extended, it is self-contradictory to regard them as
aggregates of respectively unextended points and instants"
(Grflnbaum 216).

85

anumerable and Non-denumerable Infinite Sets

GrUnbaum's argument draws directly from the
mathematical concept which states that, given the standard
notion of additivity customarily assumed in mathematics, an
aggregate of points can only be conceived as an interval of
length if the number of points in the aggregate is non-
denumerably infinite. If we were to consider a finite line
segment as a series of denumerable” point-elements,'° we
find that we cannot define measurement of length of a line
segment in terms of an aggregate of these point-elements.
The measurement of length of such a finite line segment can
be defined in terms of an aggregate of unit point-sets,-1
which are super-denumerably infinite.

Grfinbaum's argument, presented briefly, states that an
aggregate of any unit point-sets of a denumerable infinite
set always results in a length of zero. He illustrates this

point in the following passage:

 

"A denumerably infinite set of points is defined as an
infinite set that can be placed in one-to-one correspondence
with the set of natural numbers. It is said to be countably
infinite. A super-denumerably infinite set of points is
defined as a set of points which cannot be put into a 1-1
correspondence with the set of natural numbers. An example'
of a denumerable infinite set is the set of rational numbers
found on the interval [0,1]. The set of points which make
up the real number line on the same interval is non-
denumerablly infinite.

°°For a definition of "point-element," see the quote from
Grflnbaum below (p. 86 of this paper).

"Grflnbaum discusses this point at length in his article, ”A
Consistent Conception Of The Extended Linear Continuum As An
Aggregate Of Unextended Elements.” iloso of ie .
Oct. 1952. 19. pp. 288-306. p. 295.

86

...that length or extension is also defined
as a property of point-sets rather than of
individual points and that zero length is assigned
to the unit set, i.e., to a §§§ containing only a
single point. While it is both logically correct
and even of central importance to our problem that
we treat a line interval of geometry as a set of
point-elements, the definition of "length” renders
it strictly incorrect to refer to such an interval
as an ”aggregate of unextended points." For the
property of being unextended characterizes unit

point-sets but is not possessed by their
respective individual point-elements, just as

temperature is a property only of aggregates of

molecules and not of individual molecules. The

entities which can therefore be properly said to

be unextended are included in but are not mempprs

of the aggregate of points constituting a line

interval. Accordingly, the line interval is a

union of unextended unit point-sets and not an

"aggregate of unextended points" (Grflnbaum

"Consistent" 295).

This theory asserts that the length of a finite line
segment, for example, [0,1], cannot be defined in terms of
an aggregate of individual points and even though
measurement of length is defined for an aggregate of unit
point-sets, this length will always be zero because this
aggregate is denumerably infinite. We have not escaped
Zeno's paradox. Grflnbaum asserts, however, that a finite
line segment actually consists of a non-denumerably infinite
set of unit point-sets (Grflnbaum. 217). Measurement of
length of an interval (consisting of such a set of non-
denumerably infinite unit point-sets) is established by
considering the aggregate of such unit point-sets and in
practical application is established by the standard formula
for determining the length of any finite line segment (i.e.,

lb-al, on the interval [a,b]).

87
Grunbaum uses this theory against Whitehead's method of
extensive abstraction by showing that any series of points
Whitehead can derive from abstractive sets is necessarily
denumerable.°2
GrUnbaum also asserts that the

...set-theoretical meaning. of super-
denumerability. eludes all logically possible
sensory exemplification, since any collection of
non-overlapping three-dimensional regions of space
is at most denumerably infinite....

Thus the manner in which the super-
denumerable kind of infinity is conceived shows
clearly that sense-awareness cannot suggest the
idea of a collection of perceptible regions whose
cardinality exceeds [aleph zero], and, more
significantly, that a fortiori, sense awareness
cannot exhibit the actual existence of such a
collection in sensed nature. And on Whitehead's
assumptions, it is the latter condition which must
be satisfied, if Cantor's actual super-denumerable
infinities of points, which are required by modern
geometry and physics for metrical consistency, are
to be preserved through epistemological
reconstruction (Grfinbaum 217-218).

Thus, Grfinbaum asserts, Whitehead's use of the method
of extensive abstraction to construct points fails to comply
with Broad's first criterion for points (that such points
must be consistent with the principles of geometry). Since
Whitehead's method cannot derive non-denumerable infinite

sets of points, it does not provide entities applicable to

 

a2Such a collection of abstractive sets is considered
denumerable because the establishment of points by
abstractive sets is an ongoing process and each point, as it
is established, can be put into a 1-1 relationship with the
natural numbers. For this reason any set of points
developed from abstractive sets is always denumerable.

88
geometry in its modern form which requires such sets for its

application (Grunbaum 218).

Differentiation Between Points

Grunbaum also attempts to prove that the method of
extensive abstraction keeps Whitehead from meeting Broad's
second criterion. He points out that "all the properties of
abstractive sets which are essential to the Method are
§22§£2§Q£19 (Grunbaum 223). Having established this point
(which Whitehead himself concedes'a), Grunbaum asks how we
can know from sense perception that there exists two
different abstractive sets which define two unique points.
If we claim that such sets exist, how are we to establish
their particular difference by sense-perception? Whitehead
has given us no reason to believe that abstractive sets do
not fall under Hume’s characterization of the structure of
appearance.04 "Hume has shown that the structure of
appearance is irremediably non-isomorphic with that which
geometry attributes to space in virtue of the infinite
divisibility of intervals" (Grunbaum 219). To say that the
structure of appearance is non—isomorphic to that which

geometry attributes to space is to say that the structure of

 

‘°Alfred North Whitehead. The Aims of Education and Other
Essay . Macmillan: New York, 1929. pp. 176-177. This work

was written in the same period of the other works under
consideration and so I am taking the liberty of assuming
that the views expressed in this work are consistent with
the views expressed in the other works of this period.

"David Hume. A Treatise of Human Eature. 2nd. ed.
Clarendon Press: Oxford, 1978. p. 27.

89

the space of our perception cannot be placed in a one-to-one

relationship with the structure of the space developed by
geometry.
Hume states:

'Tis therefore certain, that the imagination
reaches a minimum, and may raise up to itself an
idea, of which it cannot conceive any subdivision,

and which cannot be diminished without total
annihilation. When you tell me of the thousandth

and ten thousandth part of a grain of sand, I have
a distinct idea of these numbers and of their
different proportions; but the images, which I

form in my mind to represent the things
themselves, are nothing different from each other,
nor inferior to that image, by which I represent

the grain of sand itself, which is suppos'd so

vastly to exceed them. What consists of parts is

distinguishable into them, and what is
distinguishable is separate. But whatever we may
imagine of the thing, the idea of a grain of sand

is not distinguishable, nor separable into twenty,

much less into ten thousand, or an infinite number

of different ideas (27).

Hume is asserting that because our perceptions of space
are at most limited to entities of finite area, we cannot
even consider the possibility of distinguishing between such
infinitesimal entities as two points. Whitehead's derivation
of points is achieved through abstractive sets of
dimdnishing areas. It would seem that sense perception
provides no means for distinguishing between these
abstractive sets which Whitehead needs to confer separate
identity for points lying close to one another. The
difference in identity between classes of sensible volumes

defining a given point and a class defining another. point

separated from the first by a super-denumerable infinity of

90
intervening points seems beyond the powers of the method of
extensive abstraction. For this reason, Grunbaum concludes
that Whitehead's method of extensive abstraction does not
provide for the distinction in identity between points in

the mathematics of physics (Grflnbaum 220).

Continuity of Inclusion
Whitehead attempts to guarantee the existence of

infinite series of subsensory events by simply claiming
”continuity of inclusion" among the members of an
abstractive set: "...every event contains every other event
as parts of itself" (CN 76). Broad adds further support to
Whitehead's position by saying of abstractive sets:
"...ordinary perception makes us acquainted with their
earlier and bigger terms, and the assumption that Space is
continuous guarantees the [existence of the] later ones"
(Broad 44). But this gives us no reason why we should not
accept Hume's characterization of "the structure of
appearance" as correct.

According to Grunbaum the argument from the continuity

of inclusion (referred to above) “derives its semblance of
plausibility wholly from a tacit appeal to the infinite
divisibility which geo-chronometryes attributes to physical
space-time (but not to sensed space-time!)" (Grunbaum 224).

He further asserts that Broad begs the question if his

 

II""Geo-chronometry refers to that branch of the physical
sciences which deals with the measurement of space and time.

its. ’- mmnu'a A: rm'

m—n‘i‘ .7?” u: .;
1

‘ é

91

assertion that space is continuous is based on the strength
of the Cantorean continuity"6 affirmed by geo-chronometry
because that continuity is "defined on the basis of
relations between non-perceptual point-elements which are
postulated ad initio" (Grunbaum 224).

Since neither Whitehead nor Broad has been able to give
a satisfactory defense of the principle of continuity of
inclusion" Grunbaum insists that they both have failed in
their project to show that elements established by

abstractive sets are open to perception.

Assggptions: Grunbaum
It should be noted that Grunbaum bases his arguments

against Whitehead's method of extensive abstraction on two
assumptions. The first assumption, which is explicitly
stated, is that Broad's analysis of the criteria for a
definition of points is correct (Hays 118). The second
assumption, which is implicitly assumed, is that the type of
continuity upon which Whitehead's bases his theory of nature
is that assumed by the classical theory of a finite line
segment, i.e., that of the Cantor-Dedekind continuum (Hays

125).

 

“See footnote 13 on page 16-17 of this paper for an
explanation of Cantorean continuity.

'7The principle of the continuityhof inclusion is required
by condition (ii), proposed by itehead’s criteria for

abstractive sets because "...condition (ii) in the

definition of "abstractive set” rules out a last, smallest
member of the set and thereby requires an infinite
membership for all abstractive sets" (Grunbaum 222,note 8).

 

92

I am willing to concede that Broad's criteria for the
definition of a point seem consistent with Whitehead’s
project. It is also understandable why Grunbaum might
assume that the type of continuity Whitehead attributes to
his theory of nature is that of the classical theory of the
finite line segment. In Whitehead's explanation of the
relation of serial order, he asserts that ”the serial order
among moments of the same time-system has the Cantor-
Dedekind type of continuity" (115). This is the continuity
Grunbaum attributes to a finite line segment. If Whitehead
had intended that the continuity expressed by the extensive
continuum is of the Cantor-Dedekind type, it would be
consistent with that continuity assumed by the physical
sciences. However, if this were the case, it has been seen
that it would be difficult for Whitehead to establish many
of other the properties necessary for this type of
continuity.

There are indications elsewhere in Whitehead's works
that he intends another type of continuity. Hammerschmidt,
in Whitehead's Philoso h of Time,'° sees Whitehead's
purpose as discarding the traditional conception of
continuum ”as a mere closely packed collection of discrete
points" and replacing it with "the conception of the

continuum as an exhibition of the interconnection of

 

“William. W. Hammerschmidt. Ehitehead's zhilosgphy of Time.
King's Crown Press: New York, 1947.

93
regions, derived from their basic extensiveness"

(Hammerschmidt 423)."9 W. Hays, in The Philosophy of
Whitehead, indicates that the continuity which Whitehead

describes can be considered to be close to that of the
mathematical intuitionist (Hays 112-125). I will return
shortly to these questions concerning Whitehead's conception

of continuity and the extensive continuum.

Comments: W. Hays

Hays offers several comments on Grunbaum's criticisms
against Whitehead. A part of Grunbaum's criticisms, Hays
feels, arises because Whitehead is not clear about whether
or not his method is "an exact description of some actual
process of convergence" (Hays 118). The indication that the
method of extensive abstraction arrives at an actpal process
of convergence is important to Grflnbaum's argument that
points established by the method of extensive abstraction do
not represent unique, individually recognizable points. If
such convergence did occur, each abstractive set would
converge to zero and this would offer no help in
distinguishing between different points (Grunbaum 220).

Grunbaum, it will be remembered, maintains that points
established by the method of extensive abstraction are

indistinguishable from any other point derived by the same

 

‘°We can find this notion in Whitehead's work's in several

places, among these are Principlep of Natpral Kppwlggge,
p.4, and Concept of Nature, p.59.

. .-J'..N- W m H
I ' ’ . . .‘flLI I'o. ‘(E‘A:m

0."""'.i‘

94
method. Points established by Whitehead's method would not
display the same properties as points on a finite line
segment. These latter points are considered to be unique
and can be distinguished one from another. Grunbaum denies
that these properties apply to points constructed through
use of the method of extensive abstraction.

If Whitehead is to be successful in his project he
needs to show that the method of extensive abstraction
provides events derived from perceivable entities and that
these are applicable to the methods used by the physical
sciences. Grunbaum contends that Whitehead does not fulfill
these requirements.

He also supposes that the method of extensive
abstraction obliterates precision of meaning made possible
in the statement of natural laws by use of real variables
because the method of extensive abstraction fails to provide
unique distinguishable points (Grunbaum 219). Whitehead
admits that there is no way for his natural philosophy to
discriminate between a space which possesses the compactness
of a series of rational numbers (it is dense) and a space
which possesses the compactness of the series of real
numbers (it is continuoue)'° (Whitehead "Historical” 203).

In this respect, Whiteheadian natural philosophy would have

 

’°Alford North Whitehead. "Historical Changes.” in Essay's
in Science and Philosophy. Philosophical Library: New York,
1947. p. 203.

cm: .1“. ’ 1‘-.-
' 9.

‘ "A ‘L .nl‘ to

Erin"; I~. .

95
difficulty maintaining a strict distinction between real

number variables.

Comments: Arguments Against tpgppggpppg__pg__§§pgp§;yg
Abstraction

Grfinbaum's criticisms have lead us to another point of
consideration, i.e., the problems involved with Whitehead's
conception of the extensive continuum. Before discussing
this point, I would like to comment on the criticisms which
have been offered against Whitehead's method of extensive
abstraction. It would seem that Grunbaum's criticisms raise
a number of issues which Whitehead would have difficulty
dismissing. Whitehead himself maintains that his theory
cannot discriminate between denumerably infinite sets and
non-denumerably infinite sets. This alone seems a crushing
blow to Whitehead's assertion that the use of the method of
extensive abstraction can establish geometrical entities
(points, event-particles, etc.) which are applicable for use
by the physical sciences. The only way, it would seem, for
Whitehead to extract himself from this criticism would be to
show that the type of continuity expressed in nature is not
the continuity of the classical theory of a finite line
segment (as Grunbaum assumes).

The idea that an entity can be considered perceivable
(even when it is admittedly smaller than ordinary perception

would allow), because it expresses the same relations as

those expressed by larger perceivable entities of the same

“fl

WIT." '77" 7"

.Y't‘Z‘ .!

 

98

abstractive set, seems contrary to our notion of perception.
Whitehead must show that the method of extensive abstraction
does, in fact, isolate the relations which are consistent
throughout an abstractive set. It is not obvious that the
mere process of diminishment of a series of events would
produce the required simplification of relations.

Whitehead's project is to express the spatial and
temporal properties (those necessary to construct the
geometry of the natural world) founded solely upon
perceivable data. Even if one accepts that minute entities
are perceivable (in the Whiteheadian sense, i.e., they
express relations similar to those expressed by entities
from which they were abstracted), there still remains the
problem of the establishment of non-denumerably infinite
sets. As stated above, the answer to this problem lies in
the type of continuity Whitehead attributes to nature. I
will table my discussion of this problem until we have
achieved a greater understanding of the continuity expressed

by the extensive continuum.

a

L

I‘-‘.”: “WM-t_a .-

CHAPTER 3

PROBLEM OF THE
EXTENSIVE CONTINUUH

Extensive Continuum of Events

Whitehead offers few direct comments in his early works
on the type of continuity he considers to be exhibited in
nature. Events are seen as the primary entities which
compose nature as a whole and,as such, are considered to be
continuous with each other. This can be seen in Whitehead's
references to the relation of extension as it applies to
events. All events are extended over by other events and
all events also extend over other events. Such extension
enables us to perceive a continuity in nature. In order to
better understand this type of continuity we might wish to
examine those properties exhibited by the relations which
comprise the extensive continuum.

As mentioned above, there has been some controversy as
to the exact nature of Whitehead's extensive continuum. It
has been suggested that Grfinbaum considered the properties
expressed by Whitehead's extensive continuum to be those
properties of continuity expressed by the classical theory
of a finite line segment. Other authors" have suggested

that the properties of the extensive continuum might

 

asW. Hays and G. T. Kneebone deal quite extensively with the
similarities between Whitehead's extensive continuum and the
intuitionist's infinitely proceeding sequence. This will be

explained in greater detail below.
97

 

 

98
resemble those properties of continuity expressed by the
intuitionist's infinitely proceeding sequence.

It should be noted that much of the criticism discussed
in this paper against Whitehead's natural philosophy has
been grounded on the assumption of a particular kind of
continuity which Whitehead was believed to have held. In a
consideration of these criticisms it would be advantageous
to explore those properties exhibited by Whitehead's
extensive continuum as well as the properties of those other

types of continuum mentioned above.

General Properties of a Continuum
Let me begin with a brief description of those

properties generally attributed to a continuum. If we rule
out the possibility of a discrete continuum, it could be
stated that it is the nature of a continuum to be infinitely
divisible and to have limitless extension (Hooper 205).:2
For example, if we consider the continuum of the classical
theory of a finite line segment, the natural numbers can be
divided in infinitely many ways because we can divide
between any two numbers. The property of limitless
extension is illustrated by considering that between any two
points there is another. Indeed, between any two points
there is an infinite number of distinct points. If we

include discrete continua, we must modify these general

 

’ZSidney E. Hooper. ”Whitehead's Philosophy: Space, Time and
Things.” Philosophy, 18, 1943. pp. 204-230.

sifl'mi
,‘fi‘

 

99
properties to that of being "gapless," which gives us the
intuitive notion of being continuous. For the purposes of
this paper, we only need to regard the general properties of

a continuum to be infinitely divisible and extensionless.

Properties of the Extensive Continuum

The extensive continuum proposed by Whitehead arises
out of the general character of the world. It is not to be
conceived as an infinite receptacle into which all matter
can be placed. This is Newton's notion of absolute space.
It also should be noted that the extensive continuum does
not involve shapes, dimensions, or measurability. Whitehead
maintains that these characteristics are additional
determinations which are to be associated with some
particular duration (Hooper 208). Hooper states that the
extensive continuum also has a necessary connection with
"real” potentiality.99 Thus, the extensive continuum can be
seen as "...the primary determination of ‘order' of real
potentiality arising out of the general character of the
world" (Hooper 208).

Hooper continues with the following assertion:

...the extensive continuum is one vast
relational complex providing a niche for all

potential objectifications of actual ...[events],

past, present and future. The future as well as
the past and present are involved, because all

 

9910 speak of "real” potentiality is to speak of some fact
which is real in the sense of arising from the actual world
and which expresses the potentiality of that which is to be
derived through the progressive advance of nature.

 

 

100

possible actual...[events] in the future must
exemplify these determinations in their relations

with the already actual world from which they
arise. Thus the extensive continuum underlies the
whole process of the world, and expresses the
solidarity of all possible standpoints throughout
this process....[TJhe extensive continuum is a
complex system of...£eventsl united by certain

specific relationships, such as ”whole and part,"
"overlapping of parts,” "contact" and other

relationships derived from these primary

types....[T]here is involved in the notion of a

continuum the two properties of indefinite

divisibility and of unbounded extension. This is

so for the reason that "there are

always...[eventsl beyond...[eventsl and because

...nonIeventJ is no boundary”” (Hooper 208-209).

The extensive continuum is regarded as potential in
respect to its potentiality for division (Hooper 209). In
the creative advance of nature, that which was previously
potential in the extensive continuum becomes the initial
stage of the development for new events, each event
culminating in something unique and all its own. Every
actual event can be thought of as being Bu-auhgra in the
extensive continuum in virtue of its relationships with
other events (i.e., it has definite position). In another
sense, each event can be thought of as being every'hgre
throughout the extensive continuum because each event
pervades or is related to all other events in the continuum
(i.e., it has unlimited influence on other events through
the relations it possesses).

The extensive continuum, then, lays claim to the

relations of extension and divisibility. It also exhibits

' :3“

11.5"“ I"! .m:.ns..~.‘n--s

‘ziune An

101
the relations of whole and part, overlapping of parts, and
contact, among others. From these relations it has been
seen that Whitehead derives the properties of lying between
and serial order. It is obvious that the extensive
continuum meets the general requirements for a continuum.

If the extensive continuum also possesses those
properties of the classical theory of a finite line segment,
then the criticisms Grflnbaum launches against Whitehead's
natural philosophy might be considered valid or, at least,
they would gain in plausibility. Before we can attempt to
determine the truth of this assertion, we must examine those
properties of continuity expressed by this view of a finite

line segment.

Continuity of a Finite Line Segment
The continuity expressed by the classical theory of a

finite line segment is the continuity of the real number
line. As mentioned above, Whitehead refers to the
continuity of a serial order as that of the Cantor-Dedekind

type. He offers this proclamation without any attempt at

explanation or proof (Hammerschmidt 48).’4 This is clearly

 

"Hammerschmidt continues as follows:

"...[Whitehead] asserts that the numerical and analytic
aspects of the theory of extensive abstraction can be
validly interpreted in terms of the conventional system of
points in a real continuum. If his assertion is justified,
the continuum deduced from extensive abstraction permits the
use of usual modern mathematical methods, and the metrical
analysis is that of usual geometry. Once the elementry
geometrical conceptions are defined by the abstractive sets,
the new system and the conventional modern system will have

h

)9". -‘ u (“out «ml-4 thing‘s

“Em _i.‘__".;

102
a reference to the continuity represented by the classical
theory of a finite line segment.
Whitehead's project requires him to derive concepts
applicable to entities used by the physical sciences. The
classical notion of continuity applies to series of entities

which are ordered by a relation with the properties of

, #4.:

asymmetry, transitivity, and connectedness (Hays 109).

Connectedness refers to the property that between any two

. .‘l'. I ”Map-tuna. no

members of a set, they are related to one another and the

asymmetrical property refers to a set such that if one thing

is so related to a second, then the second cannot be so
related to the first." Whitehead's definition of serial
order establishes that the relation of ”lying between"
possesses the properties of transitivityO‘ (iv-b), asymmetry

(iv-c), and connectedness (iv-a).97

 

equivalent analytical properties” (Hammerschmidt 48).

"Irving H. Copi. Introduction to Logic. 7th ed. Hacmillan:
New York, 1986. p. 385.
"Transitivity, as a property of the relation of lying
between, is triadic and can be expressed in set notation in
the following way:
Let B be the relation of betweenness and x,y,z,u e A,
where A is the set of all points on some (Euclidean)
straight line, then

(x) (y) (z) (u) ( (Bxyu a Byzu) 9 Bxyz)

Gerald Hassey.Philosophy of Science. Hacmdllan: New York,

p. 191. .

’7The references following the particular properties
mentioned here refer to the sections of Whitehead's
definition of serial order, found on p. 16 of this paper.

103
In "The Nature and Status of Time and Passage,”'0 Paul
Weiss emphasizes that the moments of time also express a
relational ordering which possesses the properties of
connexity, asymmetry, and transitivity (Weiss 155).

The relation of the moments is connexive;

i.e., any pair of them whatsoever, if they are not

in the relation of stretch and moment, are such

that one is before the other (Weiss 155-156).

The relation between the moments of time are
asymmetrical; i.e., if a moment is before another,
that other is not also before it (Weiss 156).

The relation between the moments of time is
transitive; i.e., if any moment precedes a second
which precedes a third, the first precedes the
third as well (Weiss 157).

This is the same type of serial ordering which was
established by Whitehead. Because the moments of time are
derived by the same procedure as points or event-particles
lying along routes or on stations (by the method of
extensive abstraction), it can be assumed that this same
relation of serial ordering holds among points or event-
particles along a route, point-track, or null-track (i.e.,
they express linear order).

Although we have established certain similarities
between properties expressed by the extensive continuum and
properties expressed by the classical theory of a finite

line segment, problems arise if we press this similarity too

far. One problem mentioned above concerns Whitehead’s

 

"Paul Weiss. "The Nature and Status of Time and Passage."
in P 11 so ical Essa s for Alf d ort te ad.
Longmans, Green and Co.: London, 1936. pp. 153-173.

 

 

 

104

inability to produce non-denumerably infinite series of
event-particles by the method of extensive abstraction.9’
The source of this problem lies in the fact that the points
on a finite line segment are considered to be initially
given (i.e., it is assumed to be an axiom of this branch of
classical mathematics that the points of a finite line
segment are not derived or constructed or established by
division of an already existing line segment. They are
assumed to be initially given in the concept of any finite
line segment).

Whitehead's extensive continuum, on the other hand, can
only express potentially infinite series of entities. This
is due to the fact that the extensive continuum reflects the
progressive advance of nature (i.e., all entities of nature
are never completely established and so, as explained above,
the set of all such entities must be considered to be
denumerably infinite).

Another problem arising in a comparison between
properties of a finite line segment and those of the
extensive continuum lies in the fact that the continuum of a
finite line segment consists of unique points or elements.
By unique I mean that they are identifiably different (in

location or order along the line segment). Event-particles

"It will be remembered that event-particles for Whitehead
serve the same function as points in the conventional system

Of Euclidean geometry (33).

1' 5.09.5.1 “an. 11min“) -‘fi
. E ‘.<
' -

tit-Ha -. n -_~..
I‘

105
or points in the extensive continuum seem to lack this
distinctness.‘°°

This point may be illustrated in the following manner.
Event-particles, established by the method of extensive
abstraction, retain their character as events. Whitehead
expresses diverse views about the essential character of
events. On one hand, he states that an event is a distinct
entity. Events never change, they are just what they are
and only that (62). On the other hand, he asserts that
events also have an indistinct side to their nature.
"...[Elvents appear as indefinite entities without clear
demarcations and with mutual relations which are of baffling
complexity" (73).

Grunbaum seems to refer to this ambiguity concerning
events when he insists that event-particles cannot be a
substitute for determinate points on a continuum (Grunbaum
219). He calls to this problem the ambiguity of
convergence. Thus, event-particles ”provide no means for
distinguishing a given point from a continuum of others"
(Grunbaum.219). This is due to the fact that all abstractive
sets converge to entities which are exactly alike and so, in
the case of points, there is no way to differentiate between
two separate points which lie close to each other in an

interval (Grunbaum 220).

'°°See Grflnbaum's argument, given above, for an explanation
of this point.

106

Broad has tried to show that, by constructing
neighboring series of abstractive sets, each of which
defines a different event-particle in definite relation to
one another, this problem is overcome. This procedure is
that which Whitehead uses to establish serial order among
events. This represents a possible solution to this problem
raised by Grunbaum.

Even if there is a way to determine unique points in a
continuum, there still remains the problem that any such
series can never be anything but denumerable in nature. In
this respect, Whitehead's extensive continuum does not
maintain the similarity of its properties to those of a
finite line segment.

In the course of these considerations we seem to have
come to an ippasse. Whitehead's extensive continuum. does
not express the same relational properties as a finite line
segment and so we must question its applicability for use by
the physical sciences. However, this applicability
represents a major part of Whitehead's project. It should
be noted that the greatest difficulty in establishing
similarity in properties with a finite line segment seems to
lie in the fact that the extensive continuum exhibits the
property of infinite potentiality as an integral part of its
character. Such potentiality is not a property of the

classical theory of a finite line segment.

 

*‘flf‘.
. \ ,.v

»*r. . .
I A ._. ~-

~53 .
,-

107

Intuitionist's Notion of Continuum
This property of potentiality expressed by Whitehead’s
extensive continuum has led Hays to suggest a similarity

between the extensive continuum and the intuitionist's
infinitely proceeding sequence. Hays specifically points to

Brouwer'sIO‘ intuitionist mathematics for a concept of

continuity and continuum which might be similar to
Whitehead's theory (Hays 112).‘°= He expresses this in the

following passages:

It has often been pointed out that there is a
radical difference between the continuity of our
experience and mathematical continuity which
applies only to series. By experiential
continuity we refer to the fact that the
perceptual field is given as a connected whole
which cannot be split up into ultimate simple
elements. Mathematical continuity, on the other
hand, deals precisely with such elements, i. e.,
an infinite collection of individuals arranged in
a certain order.

It is clear from Whitehead's conception of
the extensive continuum as exhibiting the
properties of the inclusion, overlap and contact
of regions, that he has not the classical
mathematical continuum in mind. It is in any case
difficult to see how he could conceive it as a
closely packed infinite series of points, since he
regards points as derivative from these extensive

relationships. In this respect his position would
seem in some ways to resemble the mathematical

intuitionist's notion of the continuum, where the
fundamental relationship is taken to be that of
part to whole. In the mathematical continuum of
Brouwer, for example, the individual real number

 

‘°‘Luitzen Egbert Jan Brouwer (1882-1966) founded the system
of mathematical intuitionism.

‘°2The complexities of the intuitionist mathematics is
beyond the scope of this paper. Its importance. in the
present context lies in Hays' assertion of similarity
between Whitehead’s concept of continuum and the
intuitionists' conception of continuum.

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108

is defined as "an infinite sequence of nested
division intervals of increasing level"‘°° (Hays

112).

Also:

Brouwer...has argued that statements which
concern all real numbers, 1. e., all the values of
a real variable, are to be interpreted in terms of
the totality of natural numbers. He defines the
individual place in the continuum not by a set but
by a sequence of natural numbers created by free

acts of choice. Whitehead's conception of the

continuum as a connected system of regions rather

than a series of discrete points brings him closer

to the intuitionist's continuum, where the

fundamental relation is that of part to whole

rather than element to set (Hays 122).

After a brief outline of the intuitionist notion of
continuum, we will be in a position to discuss those
properties expressed in this notion of continuum and seek
any similarities between these properties and those
expressed in Whitehead's extensive continuum. The

information presented concerning intuitionist mathematics

has been drawn chiefly from Stephan Korner's The Philosophy

WW4 and G. 'r. Kneebone's Hathematical Logic
and The Foundations of Hathematics.1°-

 

‘°°H. Weyl, Philoso h of Hathematics and atural Science,
pp. 51-53.

'°‘Stephan Korner. The Ppilosophy of Hathematics. Hutchinson
University Library: London, 1960. cf. Chapters VI and VII.

‘°‘G. T. Kneebone. Hathematical Logic and The Foundations of
Hathematics. D. Van Nostrand Co. Ltd.: London, 1963. cf.
pp. 243-257 and pp. 341 - 352.

we K'.‘ - ‘.fi_ 175?“. . ‘1'7. fin“;

 

109
Basic Properties: Intuitionist Hathematics
In order to understand the intuitionist notion of
continuum it will be necessary to consider a few of the
basic principles upon which this mathematics is formulated.
The primary principle of intuitionist mathematics is that

only those concepts which can actually be constructed are

considered as part of the mathematics. In practice
intuitionist mathematics consists of "creatlingl or
constructtingl mathematical objects....It is not [considered

a part of this program] to show the legitimacy of these
constructions by either logic or formalization. For they
are legitimate in themselves, they are self-validating”
(Korner 124-125).

The mathematics of the intuitionist is considered an
intuitive or mental activity and is not based upon empirical
evidence. The ability to construct a mathematical object in
the mind by intuition alone is considered enough

justification for its validation.

Propertigs of an Ipfinitgly Proceeding Sequence

The aspect of intuitionist mathematics which is most
applicable to our concerns is the notion of an infinitely

proceeding sequence.‘°‘ Such a sequence must be established

 

‘°‘K6rner describes an infinitely proceeding sequence as ”a
sequence which can be continued ad infinitpp no matter how
the components of the sequence are determined, whether by
law, free choice or what you will" (Korner 128).

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1

110

by construction and the intuitionist establishes a real
number generator’°7 as one way to construct such a sequence.

The notion of an infinite set of elements expressed by
a finite line segment is counter to the principles of
intuitionism because the elements of a finite line segment
are not constructed but are considered to be initially
given. The construction of an infinitely proceeding
sequence is never complete. It produces a potential
infinity of entities. The intuitionist, in constructing the
set of all real numbers, does so by a process which can

produce only a potentially infinite set of real numbers.

Similarities of Properties

In this respect it is clear why Hays considered
Whitehead's notion of the extensive continuum to be similar
to that of the mathematical intuitionist. Both can provide
only potential infinite series of entities rather than an
actually existing infinite series. Whitehead's use of the

method of extensive abstraction to derive points represents

 

‘°7The term "real number generator” indicates a mathematical
technique used by the intuitionist to produce an infinite
series of real numbers. The technique was developed by
Brouwer as a way of introducing the continuum into
mathematics, by treating it not as a set of existent
entities but as the potential generation of real numbers.
Incorporated in this generation is sufficient freedom for
the range of possible outcomes of the process of
construction to correspond to what a classical mathematician
would refer to as the totality of all real numbers.
Basically, a real number generator consists of some specific
requirement which limits the range of possibilities of
construction in such a manner that real numbers are produced
(Kneebone 250-253).

111

a type of construction where a continuing series of points
or event-particles is produced. Since this set of entities
is never complete, it can never represent an actual infinite
set. In this way the method of extensive abstraction is
similar to an ”infinitely proceeding sequence," e.g., the
series of numbers produced by a real number generator.
Also, both Whitehead's extensive continuum and an infinitely
proceeding sequence exhibit the relation of part to whole as
an important aspect of their continuity (Hays 112).

Another property the two share is that of overlapping.

This is because the mathematical intuitionist denies the law

of excluded middle.1°e It is impossible to determine if

 

‘°‘The law of excluded middle asserts that any statement is
either true or false (Copi 306). Intuitionist mathematics
denies this law. For the intuitionist, there are only
simple mathematical constructions and mathematical
arguments. If these are valid this can be seen immediately,
by direct intuition. However, for the intuitionist,
negation (sp) is "also a record of a construction, and is
really an affirmation which states: "I have effected in my
mind a construction B which deduces a contradiction from the
supposition that the construction A were brought to an end."
The proposition "I have not effected a construction...." is

of no interest to either the intuitionist or the classical
mathematician. But whereas the classical mathematician

admits "there exists a mathematical construction...", even
if nobody has so far been able to effect it, such a
proposition could from the intuitionist point of view only
be an empty promise...[and] not a piece of mathematics"
(Korner 132).

When we consider the intuitionist meaning of p and sp
we can see that the proposition (p or «p) is not a
universally valid principle of mathematical logic. For the
intuitionist, p or sp are not the only possibilities

available for a proposition, p. There is another
possibility, because, in intuitionist logic, the
proposition, esp p, is also true, where is defined as
not coinciding. On y if a=b is contradictory, i.e., ”only

if we can effect a construction which deduces a

 

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112

pairs of numbers in an infinitely proceeding sequence

overlap. This is because there is always a certain
ambiguity about the discreteness of individual points on an
interval developed by such a sequence<Hays 121). This
possibility of the overlapping of points on an interval is
similar to the properties displayed by points which are
derived from events. Since events are necessarily bound up
in the overlapping property of extension, points abstracted
from them retain some degree of extension (and so, the
property of overlapping also).

We have discussed several important similarities
between the relational properties exhibited by Whitehead's
extensive continuum and the properties exhibited by the
intuitionist's infinitely proceeding sequence. These have
included the property of potential infinities of entities,
the relation of part to whole, and the property of
overlapping. I refer to these as important similarities
because each of these notions of continuum express these

relational properties as a part of their basic constitution.

Differences Between Philosophical Systems

There also exist certain important differences in the
primary objectives of these two systems. These differences
extend beyond the properties displayed by the two types of

continua; they reach well into the basic philosophies upon

 

contradiction from the supposition that a=b", are we
entitled to assert that a and b do not coincide, i.e., a#b.

trim!" ...‘- . o'LTé-‘Tlnie'éilufi

113

which these systems are based. Perhaps the most obvious
difference lies in the fact that Whitehead's natural
philosophy is integrally tied to his concept of perception.
Such a basis in empiricism would be totally inappropriate
for the mathematical intuitionist. Any mathematical
construction of an intuitionist is based only on pure
perception or intuition. It will be remembered that pure
perception does not include perception in our common sense
of the term (Korner 120).

Another difference can be seen in Whitehead's emphasis
on logical principles to express the structure of nature.
While the mathematical intuitionist has developed a logic
consistent with his view of mathematics, it is quite
different from the logic used by Whitehead and Russell in
their work Principia Hathematica‘°’ as well as the logic
expressed by Whitehead in later works (Korner 140).

Whitehead has insisted on the actuality of the
extensive continuum (CN 59, CN 76). Whatever type of
continuum Whitehead intends it to be, it must be is one
which is representative of the reality of nature. The
extensive continuum, through the actual existence of events,
expresses the past, present, and future in the progressive

advance of nature. These differences are related to areas of

 

‘°’Bertrand Russell. Principia Hathematics. 3 vols., written
with A. N. Whitehead. Cambridge, 1910-1913.

 

mu] hear”:-

r. ti 1.

 

 

114
Whitehead's philosophy which are indispensable to his
project as a whole.

A summary of the points we have established can be
illustrated by noting that those areas of dissimilarity
established between the extensive continuum and the
classical theory of a finite line segment are just those
areas where the properties of the extensive continuum are
similar to those of the intuitionist's infinitely proceeding
sequence. Where the infinity of elements which make up a
finite line segment are considered to be initially given,
those entities established by the extensive continuum and an
infinitely proceeding sequence are only potentially
infinite. The points of classical a finite line segment are
unique and clearly individual while the entities of the
extensive continuum and those of an infinitely proceeding
sequence lack a degree of distinctness.

The mathematics based upon the properties of the real
number system (the system represented by a finite line
segment) is the mathematics traditionally and currently used
by the physical sciences. It has been stated that
"[rlegarding convergence and the theory of infinite
series,... Brouwer's...[notion of] convergence leads to two
quite different theories of which only the first is similar
to the classical theory....The classical foundations of

calculus, all the more the modern theory of real

 

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. ‘.

rim-WT 9h. '—"4 . .-.. 27?. L‘fi'Ne‘inm-(a;

 

115
functions..., clearly become meaningless in this light."‘1°
Because the property of potentially infinite series pervades
both Whitehead's natural philosophy and intuitionist
mathematics neither is able to develop the classical system
of mathematics which is generally used in the physical
sciences.111 This distinction between the properties of the
real number system and those of Whitehead's extensive
continuum is devastating to Whitehead's natural philosophy
because a major part of Whitehead's project was to develop a
natural philosophy which could be consistently applied by

the physical sciences.

Conclusions: Extensive Continuum

The preceding considerations have been intended to
illuminate ways in which the use of the method of extensive
abstraction is a method for representing the basic framework
of nature. We have considered similarities and
dissimilarities between the properties exhibited by the
extensive continuum and two types of continua whose
properties are known and which play an important role in the

mathematics.

 

“°Abraham A. Fraenkel and Yehoshua Bar-hillel. Epppggtiggg,

of Set Theory. Amsterdam: North-Holland Publishing Co.,
1958. p. 258.

“‘Because the property of potentiality is related to both
infinities and infin tesimals in the mathematics of the

intuitionist and Whitehead, it is understandable that the
use of such potential infinite series in calculus, which is
integral to the physical sciences, is inappropriate.

 

 

 

 

116

It has been established that the extensive continuum
differs from each of these two types. When we compare the
properties of the extensive continuum to those of a finite
line segment, we find that there are certain essential
properties which cannot be expressed by the extensive
continuum. In the same manner, when we compare the
properties of the extensive continuum to those of an
infinitely proceeding sequence, we find that while many of
the relational properties are similar, certain aspects of
this latter type of continuum are counterproductive to major
aspects of Whitehead’s project. This is especially clear in
Whitehead's emphasis on a natural philosophy which will
provide the mathematical and conceptual tools necessary for
the application of the physical sciences.

The discussion given above has shown that the
properties exhibited by the extensive continuum are not
those same relational properties exhibited by a classical
view of the finite line segment or an infinitely proceeding
sequence. It has also shown that the mathematics derived
from the relational properties of the extensive continuum

are not generally considered applicable for use in the

physical sciences.

Conclusions: Hethod of Extensive Abstraction

Themethod of extensive abstraction enters into this
discussion because this method is fundamental in the

derivation of entities abstracted from the extensive

 

i
:

117

continuum. This method, when applied to abstractive sets,
provides a tool for Whitehead to illuminate the basic
relations among events, which represent the very fabric of
nature itself. If Whitehead's philosophy cannot produce
those entities necessary for its success, the method of
extensive abstraction also fails as a tool for providing
these entities.

It seems obvious that there remains issues of concern
involving the use of the method of extensive abstraction and
doubts as to its effectiveness in providing the type of data
Whitehead intended. We cannot,then, say with surety that
the method of extensive abstraction is the effective tool
needed for Whitehead's project.

With the relations which occur among the events and, in
particular, those events which are durations, Whitehead
intended to reconstruct the geometrical entities necessary
to build a complete geometry of nature. Using the method of
extensive abstraction as a vehicle for this project, he has
tried to develop not only temporal entities but also spatial
geometrical elements, such as levels, rects, puncts, event-
particles, matrices, and finally, planes, lines, and points.

As a result of this discussion I conclude that
Whitehead’s project is not entirely wrong-headed, but
suggest that the method of extensive abstraction needs re-

working. I arrive at this conclusion because of the integral

part the method of extensive abstraction plays in the

 

118
derivation of such entities as points, event-particles,
moments, routes, etc., all of which are necessary for the
development of a mathematics, in particular, a geometry,
applicable for use by the physical sciences. It has been
shown that because of the problems of convergence with
respect to abstractive sets and the development of potential
infinities in the case of the derivation of points and
event-particles, such entities are not fully applicable in

the mathematics of modern geometry and physics.

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r;

CHAPTER 4

PROBLEHS WITH THE
RELATION OF PARALLELISH

In his early works107 Whitehead assumed a four-
dimensional flat space-time and proceeded in his development
of those relations necessary to reconstruct the geometry of
time and space from this point (Hammerschmidt 50). The use
of the method of extensive abstraction on abstractive sets
is not sufficient in itself for Whitehead to complete his
project. The relation of cogredience and the notions of
dimensionality and parallelism are also necessary. None of
these can be developed through the application of the method
of extensive abstraction to events and must to be assumed as
primary elements in Whitehead's natural philosophy.

In Whitehead's later works (Process and Reality, in
particular), he generalized his philosophical system for
generating those elements necessary to produce the geometry
of nature. One could insert particular instances of
dimensionality and parallelism into the system and arrive at
a completed geometry based on those elements inserted as
variables. The resulting geometries would be Euclidean or
non-Euclidean, depending on the types of dimensionality and

parallelism inserted into the system (Hammerschmidt 69).

 

‘°7Those works written before Science In the Hodern World.

Alfred North Whitehead. Science of the Hodern World. The
Free Press: New York, 1967. Original copyright: 1925.

119

 

 

120

This generality is not exhibited, however, in
Whitehead's earlier works. The assumption of a four-
dimensional flat space-time (three spatial dimensions,
assumed to be perpendicular to one another and a temporal
dimension united in such a manner as to create a four-
dimensional space-time) and Euclidean parallelism have
resulted in establishing the geometry of nature as
Euclidean. I

The notion of parallelism (and the type of parallelism
represented) is vital to the development of Whitehead's
natural philosophy. Because the notion of parallelism
cannot be developed through the use of the method of
extensive abstraction, it must be inserted into Whitehead's
philosophical system as a presupposition. The type of
parallelism to be inserted must be determined from our

perceptions of nature.

Parallelism apg Tepporality
Whitehead thought that this derivation could be

accomplished through our perception of durations. In the

exposition Of Principles of Natural Knowledge given above it
was stated that Whitehead begins his development of his

natural philosophy with the application of the method of
extensive abstraction to temporal durations. The result of
this application was a moment in time, itself a duration. A
duration also exhibits a spatial representation of all

events simultaneous to that particular duration. This is

-.,'.'v"1'l

 

121

the space referred to when we say a moment represents all of
nature at that instant of time.

A central characteristic of durations/moments is that
all durations/moments parallel to one another determine a
family. There is also a family of moments associated with
every family of durations. These families constitute a
time-system. The term parallel is assumed to be equivalent
to the term non-intersecting (Kneebone 348). Whitehead
considers the type of parallelism being implied here to be

parabolic or Euclidean (113).

Parallelism and Spatiality
Whitehead develops a natural geometry with spatial

elements which are introduced through a consideration of
moment in time. This geometry of instantaneous space is
developed from puncts, rects, and levels, all of which
exhibit the parallelism assumed in connection with
durations/moments.

From the development of instantaneous space, Whitehead

continued to produce the geometry of four-dimensional space-

time. This geometry is constructed from the elements of
event-particles, point tracks, null tracks, and matrices,
all of which exhibit the same type of parallelism assumed
from the beginning (Euclidean parallelism).

The development of time-less space (that space
necessary for the application of the physical sciences)

proceeds in the same manner, based on the elements of

 

122
points, straight lines, and planes, all of which also

exhibit Euclidean parallelism.

Assupption of Euclidean Parallelism
From the brief synopsis of the exposition of Ppinciplea

of Natural Knowledge given above, it can be seen that the

notion of parallelism permeates almost every aspect of this
work. It is also clear that the assumption of Euclidean
parallelism must necessarily result in the development of
geometries which are Euclidean. In later works Whitehead
generalized his procedure so that a change in the variables
of dimensionality and type of parallelism lead to geometries
which can be other than Euclidean. This does not negate the
fact that in Principles of Natural Knowledge the assumption
of Euclidean parallelism is a major factor in Whitehead's
deduction that the geometry of nature is Euclidean.

This difficulty is perhaps not as vital as those issues
raised against the method of extensive abstraction but it
does lead to the faulty deduction that the geometry of
nature is Euclidean (a notion which at this time is not
accepted in the physical sciences). Since the notion of
parallelism is intrinsic in Whitehead's natural philosophy,
the assumption of Euclidean parallelism and Whitehead's
assertion that the geometry of space is Euclidean is

problematic.

1*

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‘ v

PART IV

CONCLUSION

The thesis of this paper has been to show that certain
notions found in Whitehead's natural philosophy are faulty
and that the application of these notions result in the
apparent failure of Whitehead's entire project. These
notions are the method of extensive abstraction and the
assumption of Euclidean parallelism. I have examined the
method of extensive abstraction with respect to Whitehead's
theory of perception and also with respect to its role in
the development of the extensive continuum. In both cases
the use of the method of extensive abstraction has shown to
produce questionable or faulty results. It has also been
shown that the assumption of Euclidean parallelism in the
development of Whitehead's geometry of nature results in a
geometry that can be only be Euclidean.

It is my contention that Whitehead's natural philosophy
is not entirely wrong-headed but rather that the use of the
method of extensive abstraction and the assumption of
Euclidean parallelism contort his early works so that the
whole attempt appears to be a failure. It seems entirely
reasonable that one could construct a natural philosophy in
which empiricism and the notion of events (as represented by

Whitehead) are primary. The development of such a

philosophy would require methods as yet unformulated.‘

123

 

 

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