Zambezia (1981), IX (i).MATHEMATICS; APPLICATIONS = PROGRESS?*A. G. R. STEWARTDepartment of Mathematics, University of ZimbabweMATHEMATICS IN ONE form or another is being applied to an increasing number ofdisciplines, spreading from its traditional areas of application in the technologicalsubjects into the business and economic spheres and even into political science andcliometric history. Thus it is essential that every educated person should learn toappreciate mathematics, what it can, and more importantly what it cannot, do. Therecent catastrophe of the misapplication of catastrophe theory (a branch ofqualitative topology) in the social sciences is a case in point. To quote the title of aninteresting article on this point: 'You cannot be a twentieth century man withoutmaths.*1This applies especially to Zimbabwe in 1980 with its vast problems withconflicting priorities. So it is essential that there be an increasing number of well-motivated mathematics graduates, from this University and from the teacher-training colleges, to enter teaching and the private and public sectors in thescientific, industrial and business spheres. In the non-teaching sphere, the need isfor graduates who can apply their mathematics correctly to the problems that theyencounter, whether they are routine or new and unusual problems. An interestingexample of an unusual application is one experienced by the late Professor HannaNeumann (one of the best mathematics teachers that I encountered during mytraining) while she was at the Manchester College of Science and Technology. Abraid manufacturer, unable to work out how to use his new machine for makinganother type of braid, brought his problem to the Department of TextileTechnology. There, one member, recognizing that the problem was a mathematicalone, referred it to a member of the Mathematics Department working in 'abstruse*pure group theory. An application of group theory provided the manufacturer withhis required solution.2In the teaching sphere, the need is for graduates who can teach, and teachsoundly, mathematics for the understanding of basic ideas and not for theaccumulation of facts. The task of producing such graduates poses two majorproblems for the Mathematics Department of the University. Firstly, whatmathematics should be taught to achieve the useful mathematician? Here twoschools of thought prevail.*An inaugural lecture delivered before the University of Zimbabwe on 17 July 1980.1 The Economist, 27 Oct. 1979, 107-14.2 The problem is posed in M, Gardner,' Mathematical games', Scientific American (1962), CCVI,i, 141, and solved in M. Gardner, 'Mathematical games', ibid., ii, 158,2 MATHEMATICS: APPLICATIONS = PROGRESS?The first, the one in which I was brought up, is that it is most efficient to givestudents a thorough grounding in pure mathematics courses. This is in the hope thatthe training will enable them to know how to find out what they need to know whenthey are faced with and motivated by a specific problem. This approach is based onthe notions that the applications are so diverse that it is impossible to cover allpossibilities which might arise and that an out-of-context real life problem, beyondthe scope of experience of both lecturer and student, taught in a mathematicsclassroom, is regarded as pure mathematics and kills incentive much moreeffectively than an equally well-presented piece of mathematics.The second is that an integrated approach of pure and applied mathematics isbetter. Sir James Lighthill, in his Presidential address of 197O3 to the MathematicalAssociation of Great Britain, claimed with true chauvinism that such an approachhas made Britain the leader in mathematical education. He went on to give anexample, taken from traditional applied mathematics, of how to teach appliedmathematics so as to dispel its then current reputation as a dull and boring subjectHanna Neumann, in her 1968 address4 to the Australian equivalent of ProfessorLighthill's audience, took the more 'in vogue' approach by giving examples ofapplications of pure mathematics using the mathematical modelling approach.Mathematical modelling tries to replace the traditional approach, of well-presentedmathematical theory based on physical theories supplied by the scientists, with aproblem-based approach reaching back, with the scientist, to the data.The fields of application of mathematics and the breadth of subject matterwithin mathematics itself precludes anyone from being an expert in more than asmall portion of it Gauss (1777-1855) is often regarded as the last universalist inmathematics and its applications, though Poincare (1854-1912), by not stayinglong enough in any field to round out his work there, came close to mastering thewhole province of mathematics. However, the current trend, even in the researchfield, is towards more interaction between the layers of mathematics that build fromthe core of logical foundations, through pure layers to applied layers to the layerswhich are rightly described by labels such as 'mathematical physics' or 'bio-mathematics'.Every mathematician is compelled by internal or external pressures to accountfor the relevance of his studies, more often than not pursued for enjoyment Thefeeling Is that, if he can find at least one point outside his layer of working to whichMs field of mathematics can be applied, he has satisfied his critics.At this point 1 should like to quote two statements from Alfred NorthWhitehead on the usefulness of pure mathematics. The first is a comment on Plato's'Lecture on The Good* in which Plato propounded an equation between 'TheGood' and the study of the natural numbers:3 M, J, Lighthill, 'The art of teaching applied mathematics', Mathematical Gazette {1971) LV249-70. '4 H. Neumann. 'Who wants pure mathematics?', Australian Mathematical Gazette (1974) I79-84. ' ;A. G. R, STEWART 3The notion of the importance of pattern is as old as civilization. Everyart is founded on the study of pattern. The cohesion of social systemsdepends on the maintenance of patterns of behavior and advances incivilization depend on the fortunate modification of such behaviorpatterns. Thus the infusion of pattern into natural occurrences and thestability of such patterns and the modification of such patterns is thenecessary condition for the realization of the Good. Mathematics is themost powerful technique for the understanding of pattern and for theanalysis of the relation of patterns. Here we reach the fundamentaljustification for the topic of Plato's lecture. Having regard to theimmensity of its subject matter, mathematics, even modern mathe-matics, is a science in its babyhood. If civilization continues toadvance in the next two thousand years, the overwhelming novelty inhuman thought will be the dominance of mathematical understanding.5The paradox is now fully established that the utmost abstractions arethe true weapons with which to control our thoughts on concrete fact6In short, all mathematics is applicable.The second major problem in the production of well-motivated graduatescapable of being useful mathematicians is that of motivation. It faces all teachers ina world where mathophobia is, at best, prevalent or, at worst, regarded as a virtue.No person can be a successful mathematician unless he enjoys the subject wellenough to do the work required to overcome Ms mathophobia. Fortunately, historyabounds with reports of people in whom the enjoyment of mathematics appearsinherentI would like to make a slight digression at this point to single out Pierre deFermat (1601-65), the mathematician from history who epitomizes, for me, thespirit of the mathematician. That spirit is seen to a greater or lesser degree in manyof the men who have advanced mathematics through the centuries, but Fermat is amathematician whose mathematics is not beset by illness, tragedy or overridingphilosophical considerations. By profession he was a lawyer; Ms mathematics washis all-consuming hobby for he felt that, because of his position as a magistrate andjurist, he had to hold himself aloof from all social contact to avoid any hint ofcorruption. He was fortunate to live at a time when it was possible to be an expert inall fields of mathematics and to apply that mathematicsŠa time when mostmathematicians did not need to answer for their mathematical behaviour andworked only with those of similar tastes. He is regarded as a co-founder of thebranches of co-ordinate geometry (with Descartes) and probability theory (withPascal). It is felt by some that, if he had not been too modest to publish his work, hewould have pre-empted Newton and Leibniz in the discovery of the calculus. Hepreferred to communicate his ideas to friends through letters or through Marin'Quoted inF. E, Browder and S. MacLane,' The relevance of mathematics', in L. A. Steen(ed),Mathematics Today; Twelve Informal Essays (New York, Springer-Verlag, 1978), 348-9.* Quoted in L. A. Steen, 'Mathematics today', in Steen, Mathematics Today: Twelve InformalEssays, 5.4 MATHEMATICS: APPLICATIONS = PROGRESS?Mersenne, the mathematical clearing-house of the time. His newly discoveredresults were often posed as problems so that his friends could obtain enjoymentfrom solving them for themselves. He applied his results on maxima and minimaand his principle of least time to a systematic study of optics. His greatestcontribution was to number theory where he contributed many ideas, usuallywithout proof, which led to great development in the subject as his successors triedto justify Ms results. One such idea, Ms famous Last Theorem, states that for everynatural number n greater than 2, there do not exist three natural numbers a, b, c forwMch an + bn = cn. TMs well-known conjecture appeared in the margin of hisLatin translation of Diophantus' Arithmetica opposite a discussion on Pythago-rean triples, that is, natural numbers a, b, c satisfying a1 + b2 = c2 so that they canbe the lengths of the sides of a right-angled triangle. Fermat added that he had abeautiful proof of the result w Wch space did not allow him to write out in the margin.TMs problem is typical of many problems in number theory: so easy to state thatanyone with a Mgh-school mathematics education can understand it, but verydifficult to solve. Many people have contributed proofs showing that the result istrue for an extremely large number of values of {«, a, by c), but no one has yet beenable, even with the aid of computers, to come up with a proof covering all cases.However, for those who do not have an inherent love of mathematics it isessential that motivation be supplied for them by their parents or teachers.In an attempt to find out what else motivates the study of mathematics I decidedto seek an Mstorical answer to the question, 'Mathematics: Applications = Pro-gress?', the title of this lecture. Put another way: does the usefulness of a branch ofmathematics help to stimulate the study and further development of that branch ofmathematics? Several other Mstorical pointers will be considered as well.Most written records that survive are formalized and polished accounts ofresults containing little or no record of the ideas that stimulated them; so it is oftenimpossible to decide the motivation for the study of particular topics. For example,our knowledge of ancient mathematics is dependent on two early Mstories. One byHerodotus claims that the geometry of the Egyptians was that of6 rope stretchers*whose sole interest was the re-surveying of property boundaries after the floodingof the Nile. The other by Aristotle claims that geometry stemmed from the priestswho had the leisure to pursue its study and who used their results as a form of ritualrope stretcMng for laying out their temples.The Mesopotamian and Egyptian ages appear as ones where a fair amount ofprogress was made in arithmetic, algebra and geometry for purely practicalreasons. Some problems found on a Babylonian tablet have the ring of modern-dayfirst-form practical problems; for example, cover a road 100 km long by 1 mm widewith asphalt and compute how many days' wages it costs. Freudenthal,7 commentingon this and similar problems, wonders if Babylonian schoolboys queried the use of7 H. Freudenthal, Mathematics as an Educational Task (Dordrecht, Reichel, 1973), 2,A. G. R. STEWART 5solving such problems and whether they received the same spurious replies fromparents and teachers as their modem-day counterparts.The next period is one of great development in mathematics based in Greeceand Alexandria In Greece mathematics was divided very firmly into two parts,that of the business place and that studied by the thinkers of the day. It is the latterwhose history is remembered. Their mathematics was pure, their results obtainedby deductive reasoning from self-evident truths called axioms, and patterns weresought However, the axioms and arguments were based on the known physicalworld and the mathematicians depended for inspiration, unwittingly maybe, ontheir environment Euclidian geometry, long admired as the model deductivesystem and used for many applications to physical problems, was the geometry of afinite flat-earth society.A similar approach basing the design of the world on the natural numbers,f 1,2, 3...}, soon required modification following the discovery that, no matter howsmall the unit of measure used, it is impossible to express the ratio of the length of aside of a square to the length of its diagonal in terms of natural numbersŠthediscovery of what are called the irrational numbers.Greeks in the first part of this period studied geometry and numbers for theirown sake, but the period was a long and non-homogeneous one. The Alexandrianperiod, of about 300 years, was one of applied mathematics where, inter alia, theknown results of geometry and arithmetic were applied to astronomy, geography,optics and mechanics. The applications seem to have come after the mathematicsand, contrary to the hypothesis that mathematics advances most effectively whenin touch with reality, there was little advance in mathematics at this stage.However, the applications, although very good, were limited by the mysticismgiven to mathematics by the Greeks. The world was assumed to be a spherebecause a sphere is a perfect form and the planets' motions were described by ahighly complex system of circles, circles also being of pure form.In 200 B.C. the Greek geometer, Appolonius of Perga, produced a systematicstudy of the conic sections, the curves being obtained by cutting a double-sidedcone with various planes. This work, in translated form, allowed Kepler in 1604 toreplace the highly complicated description of the motion of planets, given in termsof circles by Ptolemy and Copernicus, by a very simple one based on ellipticalorbits about a sun placed at one focus of the ellipse. Kepler also replaced the notionthat the planets travelled at a constant speed (God is constant so the motion of theplanets, which must be perfect, should be constant) with one in which the speedvaried but the area swept out by the planet in a unit of time was constantThis time delay of 1,800 years between pure mathematical discovery andapplication has been greatly reduced in modem times. Browder and MacLane8 givethe following examples of the reduced time between major pure mathematicaldiscoveries and their applications:8 Browder and MacLane, 'The relevance of mathematics', 327.6 MATHEMATICS: APPLICATIONS = PROGRESS?(a) Cayley invented matrix theory in 1860 and subsequently appliedit as a part of pure mathematics to describe linear geometrictransformations. Sixty-five years later Werner Heisenberg usedCayley's ideas as a tool in quantum mechanics.(b) Einstein applied tensor calculus as a tool in his theory of relativitythirty years after its development by Italian geometers in the1870s.(c) Eigenfunction expansion of differential and integral operators wasdeveloped between 1906 and 1910 and applied twenty years laterin wave mechanics.During the Alexandrian period Euclid published his Elements, the influence ofwhich on the teaching and direction of mathematics lasts till this day. Euclid'sbooks were well written for their time though Ms teachings were difficult to followas instanced by two quotations, the first from Proclus Diadochus and the other fromSamuel Taylor Coleridge's introduction to Ms translation of Book I of Euclid'sElements into verse:Ptolemy once asked Euclid whether there were any shorter way toknowledge than by the Elements whereupon Euclid answered thatthere was no Royal Road to Geometry.9I have often been surprised that Mathematics, the quintessence ofTruth, should have found admirers so few and so languid. Frequentconsiderations and minute scrutiny have at length unravelled the causevizŠthat though reason is feasted, imagination is starvedŠwhilstreason is luxuriating in its proper paradise, imagination is wearilytravelling on a dreary desert. To assist reason by stimulus ofimagination is the design of the following production.10Unfortunately, Coleridge's criticism applies not only to Euclid's Elements but tomany books and lectures since produced His lament about the paucity of admirersof mathematics is put in another way by Paul R. Halmos: 'It saddens me thateducated people don't even know that my subject exists.'11 It is interesting to notethat Halmos' very well written book, Naive Set Theory, was parodied by a lecturerin philosophy at the Australian National University as a form of protest against theintroduction of Modern Mathematics into the Australian High School syllabuses.The decline of Greek mathematics seems to be attributable to several causesamong which was a move away from the advancement of the subject matter to thewriting of commentaries on the books of the day, though this was to be extremelybeneficial to the later development of mathematics. Another was the failure toapply algebra to geometry, although the theory of conies devised by Appoloniuscame close to the successful seventeenth-century algebraic geometry of Descartes.At the end of the era of Greek mathematics, Diophantus wrote an excellent book,9 Quoted in C. B. Boyer, A History of Mathematics (New York, J. Wiley, 1968), 111.10Quoted in Mathematical Digest\l980), XXXVIII, title page.II Quoted in Steen, Mathematics Today, 1.A. G. R. STEWARTArithmetical on algebra, containing Ideas far in advance of many later developments.However, Diophantus was no geometer, and Ms contemporary geometer, Pappus,was no algebraistŠa definite indication of lack of application causing lack ofprogress.European civilization developed through the Roman Empire, while math-ematical development took place in India, China and the Arab countries. TheRoman Empire and its resultant civilization did little to advance mathematics inEurope. It was only the Renaissance translations of the great Greek and Arabiantexts into Latin and their distribution to the centres of Renaissance learning that ledto the development of mathematics in Europe.Consideration of the negative effect of the Roman Empire and pre-RenaissanceEurope on the progress of mathematics may help towards the solution of theproblem of motivation in mathematics by indicating some pitfalls to avoid. Threelarge negative contributions of the Romans and the Church centred on Rome were:(a) the burning of the library at Alexandria with its large collection ofbooks as a result of a Caesar's attempt to burn the Egyptian fleetriding at harbour there;(b) the closure in 5 27 of the pagan philosophical schools by Justinianwho felt they were a threat to Christianity in the area;(c) the inadvertent killing of Archimedes while he was deep incontemplation of a mathematical drawing.Archimedes was one of the greatest Greek applied mathematicians, having deviseda number system that could express a number large enough to count the grains ofsand in the universe, in addition to Ms well known results on buoyancy and levers.He developed military machines and catapults to protect his native Syracuse fromRoman attack and used the reflective powers of parabolic mirrors to set fire to theRoman fleet One of Archimedes' methods of raising water from rivers inspired thedevelopment in Zimbabwe of a new water wheel illustrated in Professor Harlen'sinaugural lecture. However, Archimedes sought to be remembered by what heconsidered his greatest achievement, a pure mathematical result:If a sphere is inscribed in a cylinder, the ratio of the volume of thesphere to the volume of the cylinder is as two is to three and that of thesurface areas of the two solids Is also as two is to three.Marcellus, the Roman commander whose orders not to kill Archimedes weredisobeyed, complied with Archimedes' request and constructed a tomb forArchimedes on which was inscribed a diagram of a sphere inscribed in a cylinderabove the figures 2:3. This allowed one of the leading mathematical commentatorsof our time to make the crack that Rome's greatest contribution to mathematics wasthe construction of Archimedes' tomb. (Gauss in 1850 similarly requested that aseventeengon be inscribed on his tombstone to commemorate his greatestachievement, the proof that a regular seventeen-sided plane igure can be8 MATHEMATICS: APPLICATIONS = PROGRESS?constructed using straight edge and compasses only; but his request was notcomplied with as the mason felt that he could not carve the figure in any way thatwould make it distinguishable from a circle.)The Romans were practical men in accord with Lord Beaconsfield's definitionof practical men as those that repeat the errors of their forefathers. Alfred NorthWhitehead describes them as 'being cursed by the sterility that waits onpracticability'.12 True, they had great engineering works; but their engineering wasdone by rale-of-thumb methods using only sufficient mathematics, borrowed fromthe Greeks, to achieve their aims. Cicero bragged that Romans were not dreamersas the Greeks were, but applied their study of mathematics to the useful: 'We haveestablished as the limit of this art its usefulness in measuring and counting'.13The Romans preferred to foster development in medicine and agricultureŠanomen for the present?It is probably unfair to single out the Romans for criticism, since mathematicalhistory seems to imply that mathematics was being developed almost nowhere atany particular time. This of course is because of the lack of sources and the fact thatthe interest of the historians tended to be European-centred,The Church spread Roman learning, with its paucity of mathematics, to Europeso that the best Europeans at the time of the Renaissance had a mathematicalability below that of the earliest Greek mathematicians. Nor did the Church help tofoster mathematics with comments such as St Augustine's 'The good Christianshould beware of mathematicians'14Ša warning based on the fact that Roman andmedieval mathematicians were often pagans and astrologers.There is an undercurrent through Kline's book, Mathematics in WesternCulture, from which the above quotation was taken, that religion and mathematicsare incompatible. This is an unfounded proposition in my opinion. At most,religious and mathematical thought belong to mutually exclusive sets. As Pascalfound out, you cannot decide on the existence or non-existence of God by purelyrational arguments based on natureŠGod's existence is an independent axiom.The Renaissance period and the introduction of printing heralded the develop-ment of mathematics in Europe. For various reasons scholars began to collect andtranslate the mathematical texts of other cultures, their interest in mathematicstranscending social and ideological barriers. One of the prime movers of this periodwas Regiomontanus, the man from 'King's Mountain', or Konigsberg. A seeker ofknowledge with a love for classical education from Greek, Latin and Arabiansources, with a bent towards science, equally interested in practical and theoreticalstudies and, most importantly for mathematics and science, the owner of a printingpress, he collected the mathematics books of Greece and Arabia and had themprinted. As a result of the endeavour of such people, the existent mathematical' Quoted in M, Kline, Mathematics in Western Culture (New York, Pelican, 1953), 28.'Quoted ibid, 108.1 Quoted ibid., on cover.A, G. R. STEWARTknowledge was brought to many centres, and more importantly, Greek geometrywas brought into contact with Arabian and Greek algebra. A leading book of thisperiod was the Summa of Inca Pacioli, which had four parts: arithmetic, algebra,elementary Euclidian geometry and double-entry bookkeepingŠone of the fewindications in mathematical history of an integration of pure mathematics withcommercial mathematics.The period from the Renaissance to the early nineteenth century is one in whichmost contributors to mathematics were both pure and applied mathematicians.Moreover they were scientists, philosophers, medical men or engineers as well, andthis makes it difficult to decide how much influence each part had on theirmathematical developmentI shall now follow the development of particular topics within the subject ratherthan of mathematics as a whole.One of the leading ideas in the development of Art was that of perspective, orhow to project a three-dimensional scene onto a two-dimensional canvas mostaccurately. Many famous artists worked on the project, applying essentiallygeometric ideas. Despite the interest of great men such as Leonardo da Vinci, thepossibly fruitful alliance of mathematics and Art did not achieve all it might havebecause the interested parties were artists applying mathematics with no profes-sional mathematician to guide them. The study, therefore, died out for more than acentury, before the geometers developed a suitable geometry.The early Renaissance period saw an upsurge in the interest in algebra, and inparticular in the solution of polynomial equations. The formula for solution of thequadratic equation was known in Babylonian times. The Arabs developed amethod of successive approximations which gave engineers and mathematicalpractitioners solutions for cubic equation to any degree of accuracy required.However, exact solutions for cubics were sought for their logical significance aswere exact solutions of higher-degree polynomial equations. The higher degreepolynomial equations were erroneously supposed to have no practical significanceas it was believed that the real world could be described in terms of linear, quadraticand cubic equations which described length, area and volume respectively. Thework was hampered by a lack of suitable notation and the non-acceptance ofnegative numbers. However, formulae built up, using the coefficients of theequation and the operations of addition, subtraction, multiplication, division andthe taking of roots for the solution of the general cubic and quartic (degree four)equations, were published by Cardano in his Ars Magnet of 1545. Such solutionswere called 'solutions by radicals*.Attempts to ind a similar solution to the quintic (degree five) equationcontinued, on and off, till the nineteenth century, when Abel and Galois showedthat no general solution was possible. Galois also developed a theory givingconditions for determining when a given polynomial equation could be solved byradicals. The methods used by Galois, Lagrange and Abel utilized ideas that arenow the basis of the abstract theory of groups. Lagrange, of an earlier generation10MATHEMATICS: APPLICATIONS = PROGRESS?than Abel and Galois, used what is essentially the theorem now known by his namerelating the number of elements in a subgroup to the number of elements in thegroup before Galois coined the term 'group'. Galois talked of groups about fiftyyears before the definition of a group was formalized. Today group theory isintroduced by its formal definition; Lagrange's Theorem proved early on and thenGalois' theory derived as a beautiful example of the application of a large numberof results from the theory of groups and field theoryŠalmost the reverse of thehistorical developmentAbel and Galois changed the emphasis in the theory of equations from solvingparticular problems to consideration of the existence of solutions, a change thatsignalled great progress in algebra. (They also shared a tragic history. Abel wasdogged by poverty and resultant ill-health, dying when he was only twenty-nine. Heand Galois were unfortunate to have important papers lost by the Academy ofSciences in Paris before they could be published To Galois it happened more thanonce and he is the epitome of the misunderstood genius. He rebelled at every turnand lost. After refusing to pay attention to school lessons which he regarded asdreary, he was denied admission to the Ecole Polytechnique when he failed hisentrance examinations after insulting the examiner by describing the examinationquestions as showing a lack of understanding of mathematics. His style of writingwas such that his papers were rejected as unintelligible. He rebelled against thestate and eventually died, aged only twenty, as the result of a duelŠleaving amathematical legacy that took many brilliant mathematicians a long time to sortout) The solution of polynomial equations is one example of an impracticalproblem that led to a great development in mathematics.The needs of astronomy and engineering in the Renaissance period inspiredvast improvements in the sixteenth and seventeenth centuries in trigonometricalnotation and tables and in the decimal representation of numbers, especiallyfractions. An engineer, Stevin, and a Scottish laird, Napier, were largelyresponsible for the current decimal representation of fractions. Napier introducedthe idea of logarithms, for the purely practical reason of speeding calculations. Hislogarithms were based on geometrical ideas and everything was multiplied by tenmillion to give seven-figure log tables without using decimal fractions. At about thesame time a Swiss, Jobst Burgi, developed a similar system of logarithms. Theutility of logarithms was obvious and led to their immediate acceptance andsuccessful application.The development, respectively, of geometry and analysis (the study of calculus,infinite series and infinite processes in general) were complementary and at timesso successful that interest was drawn away from other topics to the detriment of abalanced development of mathematics.In the early Renaissance period the interest was in elementary geometry as thegeometry of the Greeks was too sophisticated for the people of the time, whosemathematics was based on the equally stultifying extremes of the practical Romanapproach and the mystical astrological approach. One of the leading algebraists ofA, G. R. STEWART11the time, Cardano, whose scientific researches were modern in approach, was stillheld by the mysticism of the past and regarded himself as an expert astrologer. He isreputed to have prognosticated the date of his own death and to have committedsuicide on the day to maintain his reputation as an astrologer.Interest in the more advanced geometric theories was aroused after Galileoand Kepler had applied Appolonius' Conies to their studies of the motion of bodieson earth and in the heavens. These applications of mathematics to physics renewedinterest in the harmony of the universe and stimulated the use of the rationalmathematical approach in the study of other subjects. Some people took the idea toextremes; Kepler, for example, is reputed to have applied mathematics to theselection of a new wife, after the wealthy heiress whom he married had died.Unfortunately, the girl selected, with true feminine disregard of mathematics,refused to marry Mm and he had to settle for one of lower rating.Galileo, Stevin and Kepler also re-established interest in the study of infiniteprocesses and infinitesimals by their interest in Archimedean physics. Eudoxus inthe fourth century B.C. had applied a method, similar to the modern theory of limits,called the method of exhaustion, to the problem of finding the length of curves. Ashis work led through a long maze to the calculus of Newton and Liebniz, he canprobably be regarded as the initiator of analysis. Archimedes in the third centuryB.C. used the method of exhaustion to resolve arc-length, area and volumeproblems, and his use of the method inspired the trio mentioned above. Stevinfound centres of gravity by dividing unusual shapes up into an infinite number ofinfinitely small regular shapes which 'filled' the irregular shape. He used the notedfact that the more regular divisions that were fitted into the irregular shape thesmaller was the portion of the irregular shape outside the regular shapes. Keplerused a similar method to study areas inside the elliptical orbits of planets. Galileoapplied the notion of the infinitely small to Ms dynamics. He even had degrees ofinfinity, explaining that objects stayed on the rotating earth because the infinitelysmall distance they had to fall to stay on the earth was infinitely small comparedwith the infinitely small distance they would travel along the tangent to the surface.Therefore, Ms theory of projectiles implies that they would remain on the earth.Another great boost to analysis and its applicability to p hysical problems camefrom the application of algebraic methods to geometry. Algebra replaced thegeometric and visual intuition of synthetic Greek geometry with routine calcula-tions. Despite the very applicable nature of the results, the motives of the initiatorsof the algebraic approach were extremely pure. Descartes' La Geometric was, asAppolonius' Conies had been, a triumph of impractical theory, being part ofDescartes' development of an overall philosophy of life. His co-initiator, Fermat,however, had his feet more firmly on mathematical ground. The new geometry,today synonymous with analytic geometry, was a very powerful and general tool.Its use in solving successfully and easily some difficult problems led to its rapidacceptance by the mathematicians of the day. It found ready applications toproblems involving lengths of paths and areas under curves, a subject that led12MATHEMATICS: APPLICATIONS = PROGRESS?eventually to integral calculus. So although analytic geometry started as a puremathematical theory it very rapidly became an applied mathematical tool.Despite the fact that much of the mathematics involved was pure mathematics,there can be little doubt that the impetus for and the direction of the development ofanalysis came from the needs of applications. Throughout its history, thetheoretical development of analysis lagged behind its uses and the subject is largelytaught in this manner today. The history is too full to give an adequate account of itin the short time available here, but I want to select a few examples to show howmathematicians derived results without the fall theoretical backing required fortheir justification and the effect that this had on the later development ofmathematics.Wallis, one of the greatest English mathematicians before Newton, gainedmany useful results by a non-rigorous method of incomplete induction. In thismethod, mathematical inductionŠa finite processŠwas extrapolated to applyintuitively to infinite processes, a result that had no rigorous foundation. Worsestill, after proving his result for positive whole numbers he assumed them to be truefor negative, fractional and even irrational numbers, coming up with correct resultsmore often than notNewton and Liebeiz developed much of their theory by ignoring second orderinfinitesimals, and by carrying over without justification the ideas and methods offinite polynomial theory to infinite series. At no time did either grasp thefundamentals of the mathematics that they used. However, they had a faith in theirintuition and method. Their methods were often criticized. For instance a Dutchphysician and geometer, Bemhard Nieuwentijt, did not deny the correctness oftheir results but objected to the vagueness of Newton's methods and the lack of aclear definition of Liebniz' s differentials of higher order. Rolle and Varignon soughtto eliminate some of the problems by showing that the methods were reconcilablewith the geometry of Euclid which was at that time held in high regard: to thereligious it was represented as expressing the laws of God; to the non-religious itwas held up as the source of all nature's immutable lawsŠobjections wereoverruled as old-fashioned and outmoded.One of Newton's strongest critics was Bishop George Berkeley, who made ascathing attack on the basis of calculus in a tract, The Analyst, published in 1734.The attack was not prompted by dissatisfaction with the results but by theshattering of a sick friend's Christian faith by arguments put forward by Halley (ofComet fame), a leading proponent of calculus. However, Berkeley hit the theory atits weakest spot and subsequent attempts by British mathematicians to rigorizetheir mathematics, admittedly in the wrong direction, led to a stagnant period inBritish mathematics over the next century. This put Britain a century behindcontinental Europe in analysis and probably accounted for the rise in algebra inBritain in the nineteenth century.Although the attempts of the algebraists to find solutions of polynomialequations led to the introduction of negative numbers and complex (imaginary)A, G. R. STEWART15Lobatchevsky and Bolyai had the courage to publish their results, which wereignored by the mathematics community as a whole, although Lobatchevsky wassacked as Professor of Mathematics at Kazan as a result. Their results gainedpartial acceptance only after the posthumous publication of Gauss' similar results,Gauss having a much greater reputation. Fuller acceptance of the results followedthe construction of a model, called a pseudosphere, in which all the axioms ofGauss, Lobatchevsky and Bolyai held and in which several of their results weremeaningfully illustrated. In the model, 'lines' were defined as geodesies, curves ofshortest distance between points on the surface.Bemhard Riemann (1826-66) took a different approach to non-Euclideangeometry. He distinguished between 'infinite' and 'unending' (a circle is finite butunending) replacing the axiom OD infinite extension of lines by an axiom onunending extension of lines. His ideas led Mm to replace the parallel axiom by onewhich assumed all lines eventually met; that is, he replaced 'one line parallel to thegiven line' by 'no lines parallel to the given line'. Curiously, his ideas led to ageometry with more than one straight line through two given points, and allperpendiculars to a given line meeting in a single point; and with triangles having anangle sum greater than 180°, and increasing with area. His geometry was modelledby using a sphereŠthe real worldŠwith straight lines interpreted as geodesies, inthis case, great circles centred at the centre of the sphere. The thorough acceptanceof his theory and of the possibility of non-Euclidean geometries came whenEinstein used a Riemannian-type geometry to produce a new theory of themovement of the planets that was simpler and more fundamental than Newton's,Einstein did for Riemannian geometry what Kepler did for Appolonius' theory ofconies.Does this mean that Euclidean geometry is false and its application invalid? No!Euclidean geometry is valid as it is based on a set of independent non- contradictoryaxioms and is logically deduced from those axioms. Equally its applications are validin any model that can be shown to satisfy these axioms. The new geometries merelygive alternative ways of interpreting the physical world.Mathematics was now free of the constraint that its axioms were naturallyoccurring and inviolate, that it was tied to the physical world. The price of this new-found freedom had to be paid If the axioms and the resultant theory were to beindependent of the physical world, the terms used and theorems proved had to beprecisely stated. No longer could half the definition be left to a mathematical'youknow', or the proof to an intuitive notion.New geometries emerged rapidly and this proliferation led to a new applicationof group theory. Klein became Professor of Mathematics at Erlangen in 1872 andin his inaugural address outlined what has become known as the EriangenProgramme which resulted in the various geometries being thought of as branchesof one overall theory and not as many separate theories. He described geometry asthe study of properties of figures that remained invariant under particular groups oftransformations; for instance Euclidean geometry is the study of properties that16 MATHEMATICS: APPLICATIONS = PROGRESS?remain invariant under rigid transformations that move figures without disturbingtheir size or shape, the idea most of us learned in Form I for superimposing twotriangles on one another to determine whether they were congruent or notEuclidean geometry has fallen into disrepute and lost its importance in schoolsyllabuses more, I feel, because it is difficult to teach and leam than because it isnow old-fashioned and no longer the force it once was in applications. However, itslong tradition in mathematics means that it still implicitly pervades manymathematical theories and its absence from school curricula is making it harder formodern students to cope with their university studies.In the nineteenth century algebra underwent a similar axiomatic revolution.Attempts were made to extend the notions of complex numbers. The system ofnumbers had been enriched through the centuries by extension; from the naturalnumbers {1, 2, 3 ...}; to the rationals {? \a, b natural numbers!; to the irrationals,the Greek incommensurables; to the positive and negative numbers; to the complexnumbers required to solve all polynomial equations. Was this a continuing processor had the end been reached? Hamilton discovered that no extension was possibleunless the commutative rale ab = ba was discarded. Cayley's matrices were alsofound to violate this law and algebra was freed of the need to restrict its ideas to old-fashioned 'natural5 laws. Once again, and this time in the United States more thananywhere, mathematicians sought to find out what happened if the various laws ofalgebra were denied, giving rise to a spate of new algebras. As with geometry,abstract algebra with its underlying group theory was brought in to unify thetheories into a cohesive ordered whole.Mathematicians in the nineteenth century became increasingly aware of theneed to put mathematics onto a sound basis that unified its many theories; theoriesdifferent on the surface but with many underlying common themes. The notion of aset and the age-old axiomatic-deductive method were chosen as the basis of thenew abstract theories. But the lesson was still not learned. Close scrutiny ofthe underlying mathematical logic came up with paradoxes in the initial free andeasy notion of set An example of such a paradox is:If the barber in the town shaves those and only those men who do notshave themselves, who shaves the barber? An assumption that thebarber shaves himself leads to a contradiction of the fact that the barbershaves only those who do not shave themselves, whereas the otherhypothesis, that the barber does not shave himself, leads to theconclusion that he must shave himself.This paradox is a male chauvinist one as it assumes that the barber is a man. Thecontradiction inherent in the paradox would lead a non-discriminatory person tothe conclusion that the barber is a woman. Still, the point was taken from theseand other paradoxes, and the twentieth century has seen attempts to lay firm logicalfoundations for mathematics, though most mathematicians have accepted that thelogicians have or will come up with a sound basis for their field and press onregardless.A. G. R. STEWART13numbers of the form a + ib (i being the imaginary square route of-1), it was theanalysts who found these numbers most useful. D'Alembert, in 1752, used them ina paper on the resistance of fluids treating them exactly as if they were ordinarynumbers without justification. On the other hand, d'Alembert introduced the'limit' concept into calculus in an attempt to shore up its shaky foundations.However, his presentation of his results lacked a clear-cut phraseology and theywere ignored by mathematicians. Fortunately, Cauchy came along later to presentthe correct approach to the limit concept and used it to put analysis on a sounderlogical footing.It is unfortunately a theme recurring throughout mathematical history that goodresults are often handicapped by almost unintelligible presentation. I have alreadycited the example of Galois. The geometry of Descartes almost failed to gainrecognition because his presentation omitted many points of detail that he foundelementary but which were essential for the understanding of the argument byordinary mortals. Fortunately others came along who amplified Ms works and itwas these amplified versions that were snapped up and used, not the original.Newton was reputedly very poor at communicating his ideas. Here was a man whohad a profound influence on the direction of science and mathematics yet whosestudents often absented themselves from his lectures because they derived nothingfrom them.Fortunately there were others whose teaching, writings and notations wereclear. Standing head and shoulders above all other mathematicians as far asvolume of publication is concerned is Leonard Euler (1707-83)). One of his booksis the basis for most twentieth-century calculus texts. Another, which he dictatedafter going blind to one of Ms domestic workers, had a particularly clear exposition.There are many instances of poor notation delaying the progress of mathematics,but Euler, a great innovator of notations, did much to improve the clarity andrepresentation of mathematical problems. He was also not above taking short cutsto applications of Ms mathematical ideas, often preferring intuition to rigorousproof. His use of divergent series where convergent ones were needed gave muchcause for concern among Ms successors.So the story continues, culminating with Fourier whose brilliant application ofFourier Series to physical problems was based on' a feeling' that all functions couldbe expanded as Fourier Series. A growing feeling that the analysts were skating onthin ice was accelerated by the discovery of naturally occurring pathologicalfunctions which could not be fitted into Fourier's general theory using methodsdeduced before that time. (Mathematicians now know that Fourier's Theorem isnot true for all functions, and have spent over a century developing new theories toprove its truth for increasingly more functions. The boundary between thefunctions for which it holds and those for which it does not has still to be found.)The problems with the foundations of analysis led to an increasing use of rigour,to increasing questioning of the foundations of mathematics, and to more abstractand pure forms of mathematics. One of the results of increasing questioning of14 MATHEMATICS: APPLICATIONS = PROGRESS?fundamentals was to have a profound effect on mathematics as a whole. That was Inthe field of geometry. Euclid's Ten Axioms were regarded as sound when hepostulated them. Two thousand years of working with them to obtain a vastquantity of results which had been applied to solve problems on the very complexpatterns of nature had served to reinforce belief in their soundness. Mathematics,because of the indisputability of Euclidian geometry, was almost equatable with'Truth', However, two axioms gave cause for concern, even to Euclid. They bothinvolved infinite extension of linesŠthe axiom that said a straight line could beextended indefinitely in either direction; and the parallel or fifth axiom, thatthrough a given point not on a given line, one and only one line (in the plane of thegiven point and given line) can be drawn that does not meet the given line, no matterhow far either is extended (alternatively there is one and only one line parallel to thegiven line through a given point). But man's experience is limited to a finite part ofthe universe so infinite extensions of lines is beyond his experience and theseaxioms cannot be regarded as self-evident truths. Consider parallel railway lines,which appear to converge. Mathematicians therefore decided to justify the fifthaxiom by:(a) deducing it from the others; or(b) assuming a new axiom, contrary to the parallel axiom, and thenusing the other nine axioms to arrive at a contradiction. Logicallythis would show that the contradictions of the fifth axiom werefalse and so the fifth axiom had to be true.Saccheri followed the reductio ad absurdum idea of the second approach, and in1773, after having failed to arrive at a contradiction, could not overcome 2,000years of tradition. He gave up his researches claiming Euclid had been vindicated.Gauss followed the same line of reasoning in the earlier part of the nineteenthcentury when mathematics was more susceptible to the questioning of itsfoundations. He came up with the same results but made the correct conclusion:'Other geometries could be as valid as Euclid's'. However, he did not have thecourage to publish.Two Eastern Europeans, Lobatchevsky and Boly ai, assuming that through anypoint it was possible to draw at least two lines parallel to a given line and thatEuclid's other nine axioms held, derived result after result by pure deductivereasoning. They continued even though some of their results were surprising,seemed ridiculous and contradicted visual representation. For instance:(a) The angle sum of a triangle was less than 180° and became smallerthe larger the area of the triangle. Gauss even experimented withthis idea by placing men on peaks of adjacent mountains usingmeasuring devices to determine the angle sizes of the triangle theyformed, but his results were inconclusive.(b) Similar triangles (ones of the same shape but possibly differentsizes) were always congruent (equal in all respects), contrary toEuclidean geometry.A. G, R. STEWART17The increasing abstraction of mathematics has meant an increasingly universaltheory, raising a hope expressed in a comment by Bourbaki (a group ofmathematicians, not a single mathematician): 'mathematic not mathematics'.However, many penalties have had to be paid. The greater care required at eachstage has meant that no one can hope to be an expert in all fields. To derive benefitfrom the abstraction:(a) Intuition must be played down.(b) The terminology must be precise and esoteric so that someonenew to the subject has to learn a vast list of new terms before he canget down to the interesting and enjoyable manipulation of thoseterms to produce results. This is often sufficient to deter peoplebefore they start(c) The translation of an applied problem into a mathematical one andthe translation of the mathematical answer into an applied answeroften takes longer than the solution of the mathematical problem. Irecently came across a paper which consisted of a couple of pagesof translation of a problem into its equivalent in another fieldwhere the answer was immediately obvious.I shall draw no conclusion from the above discussion as there are so many otherfactors involved.Mathematics is not only a utilitarian subject; it is also an absorbing andinteresting hobby. For instance, Pascal was a dilettante mathematician whoabandoned his mathematics for theology after a religious experience. Yet one nightafter this conversion when unable to sleep because of toothache he set himself amathematics problem and became so absorbed in solving it that his pain wasforgotten. He also constructed and sold several calculating machines. When afriend sought his help in solving a problem on the equitable distribution of the stakesafter an interrupted game of dice, Pascal got together with Ferniat to establish themodern theory of probability.There are many unusual and fascinating non-examinable results in mathe-matics that time and the demands of other subjects force out of mathematicscurricula. These can, and for some enterprising people do, make an interesting,stimulating and enjoyable leisure activity. There are many instances throughouthistory, a few of which I have pointed out, where a piece of very pure mathematicshas turned out to be highly applicableŠtoo many for any piece of mathematics tobe safely regarded as useless. To quote-Kline: 'To insist that each step in a chain ofeven geometric reasoning be meaningful, is to rob mathematics and science of twothousand years of development'.15 Mathematical problems are rarely dreamed upby mathematicians; they seem to present themselves for solution in both pure andapplied mathematics.15 Kline, Mathematics in Western Culture, 482.18 MATHEMATICS: APPLICATIONS == PROGRESS?I am biased, I enjoy pure mathematics and regret that I cannot persuade morepeople to try it long enough to enjoy it and discover that it is a worthwhile study. Letme close by quoting C, J. Keyset: 'The golden age of mathematicsŠthat was notthe age of EuclidŠit is ours'.1616 Quoted in Boyer, A History of Mathematics, 649.