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Of Apes and Algebraists Main Feature 1 of Apes and Algebraists John Heard Explores the World of 19Th-Century Pure Mathematics

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Of Apes and Algebraists Main Feature 1 of Apes and Algebraists John Heard Explores the World of 19Th-Century Pure Mathematics NO. 83: JUNE 2007 ISSN: 1751-8261 Contents Of Apes and Algebraists Main Feature 1 Of Apes and Algebraists John Heard explores the world of 19th-century pure mathematics. BSHS Image Competition 3 “I believe that mathematical reality lies English mathematicians had experienced Reports of Meetings 4 outside us, that our function is to discover something like a century of intellectual BSHS Postgraduate Conference or observe it, and that the theorems that isolation from their European neighbours. A Exploring and Being Explored we prove, and which we describe grandilo- long-running feud between the supporters of ‘Outreach Day’ Bursaries 5 quently as our ‘creations’, are simply notes of Newton and those of Leibniz concerning the our observations.” origins of the differential calculus had melded BSHS 1947-2002 6 G H Hardy, A Mathematician’s Apology, 1940 seamlessly into a general English antipa- The Questionnaire 1947 6 thy, encouraged by the French Revolution, It may seem curious that pure mathemat- towards European rationalism. Consequently Past Presidents Reflect 7 ics, which has such a long and honourable throughout the eighteenth century the tradi- history, should have found considerable dif- tional English view of mathematics continued The Origins of BSHS 8 ficulty in justifying its existence less than two to exist largely unsullied by new trains of Replica Instruments 10 hundred years ago. Yet the opening decades thought, to the extent that much of European Readers’ Letters 11 of the nineteenth century, when natural phi- mathematics would have been unintelligible losophy was finally partitioned into distinct to Englishmen – even had it been written Reviews 12 News 14 Listings, Viewpoint info. 15 BSHS Conferences, BJHS 16 Editorial In this issue our celebration of the Society’s 60th anniversary continues. There are reflections by five of our former presi- dents and two scientists who have been members of BSHS since its foundation in 1947. This issue also features an essay by the Society’s first secretary on the origins of the BSHS. The main feature, by John Heard, examines the justifications that 19th-cen- tury pure mathematicians gave for their Fig 1: First Catch Your Invariant: J. J. Sylvester and Arthur Cayley likened pure research, which was likened to geographi- mathematics to geographical exploration and discovery (Paul Du Chaillu, Explora- cal exploration and discovery. tions & Adventures in Equatorial Africa, 1861). © Zoological Society of London. For those interested in the Society’s and specialist sciences, posed awkward ques- in English. This traditional English view was, Outreach and Education activities, see the tions for English pure mathematicians, who broadly speaking, that mathematics was details of the 2007 competition, searching found that their objectives were increasingly subsumed under natural philosophy, and for an image for use in history of science at odds with the prevailing Victorian ethos of so gained its interest and importance from communication, and two considerations of utility. Various strategies were invoked so as to being a necessary tool for studying the Book the education uses of replica instruments. commend pure mathematics to an increas- of Nature: it was part of the great enterprise I now have an Assistant Editor, Rosemary ingly uncomprehending public, and of these of glorifying God by trying to understand His Wall, and my thanks go to her for her help probably none has been so long-lasting, or handiwork. with this and future issues. Contribu- so useful to a hard-pressed pure mathemati- This comfortable insularity began to be tions to Viewpoint 84 should be sent to cian in a tight corner, than that of presenting breached early in the nineteenth century, newsletter[a]bshs.org.uk by 17 Aug 2007. pure mathematics as the exploration of a real when English mathematicians at last became mathematical world. acquainted with European developments, Rebekah Higgitt, Editor At the dawn of the nineteenth century a principle feature of which was a level of 2 Viewpoint No. 83 abstraction that effectively divorced math- most colourful mathematicians of the nine- ematics from its roots in natural philosophy. teenth century, but also had a good claim to Mathematics, from this point of view, was no be second only to his great friend Cayley in longer the science of natural quantities, and the Victorian pantheon, such as it was. Many this had an important effect on the criteria by stories circulated about Sylvester; some are which mathematicians judged truth in math- greatly exaggerated, if not downright false, ematics. Under the banner of natural phi- but are perfectly believable nonetheless losophy, mathematics was true to the extent – he was a character who could have found that it successfully mirrored and explained himself on the wrong end of a manslaughter natural processes and this requirement was charge. He also wrote verse, of which he held the touchstone that guaranteed the validity an unaccountably high opinion, and although of the mathematical processes. For example, once he started speaking he could prove in Newton’s differential calculus the exact impossible to stop, he was the ideal person nature of fluxions was hotly disputed, but to convey to the members of the British the success of his theory justified their use, Association a memorable apologia for the notwithstanding that what was going on was mathematical enterprise. hidden in a fog of impenetrable metaphysics. He decided to take the attack to Huxley, However, in the new dispensation math- and asserted that mathematics was quite ematical truth needed a new touchstone, the opposite of being “purely deductive”. which not surprisingly turned out to be an Indeed, Sylvester said that mathematical appeal to purely rational thought. Mathemat- analysis advances “from continually renewed ics first required premises the truth of which introspection of that inner world of thought were apparent to all right-thinking people, of which the phenomena are as varied and and on those premises were to be built logi- require as close attention to discern as cal constructs in which each step was derived Fig 2: Arthur Cayley those of the outer physical world ... , that it from earlier ones by the application of logical is unceasingly calling forth the faculties of principles that were as evidently correct as Poe’s short story The Purloined Letter; clearly observation and comparison, that one of its the original premises. Whatever propositions he believed that the theme would be familiar principle weapons is induction, that it has resulted from this purely rational procedure to his readers. And during the next two dec- frequent recourse to experimental trial and were true by definition, however counter- ades George Boole, Sir George Airy and Wil- verification, and that it affords a boundless intuitive they might appear. liam Thompson all expressed concern at what scope for the exercise of the highest efforts Allied with this new view of mathematical they saw as a waste of talent, with able math- of imagination and invention.” He used truth was an extension of the legitimate field ematicians such as Arthur Cayley throwing language to describe pure mathematics that of algebra. The eighteenth century had hap- themselves into a life of abstract ratiocination mirrored the language that Huxley used to pily embraced the labour-saving concept of when there were so many unsolved practical describe science, and drew parallels between the formula, in which an unknown quantity is questions to which their attention would be the activities of natural historians – of whom, expressed in terms of other unknown quanti- better directed; indeed, Airy characterised of course, Huxley was one of the most pre- ties; but now it was proposed that algebra modern geometry as “puerile”. The critics were eminent – and those of pure mathematicians, could properly ignore underlying quanti- especially alarmed at the effect on student who roamed the mathematical world seeking ties completely, becoming simply a formal mathematicians, whose studies should have new and interesting specimens which they calculus for the manipulation of symbols. been fitting them for occupations that could then observed, recorded and captured for the This new algebra, when prosecuted with be put at the service of industry and Empire. benefit of future generations. unchecked rationality, yielded curious results: Amongst those taking a tilt at the pure So as to bring home this parallel most for example, when equating a generating mathematicians was Thomas Henry Hux- forcefully, Sylvester referred to “the acciden- function to its infinite series it was impermis- ley, “Darwin’s Bulldog”, who in 1869, during tal observation by Eisenstein, some score sible to assert that the equation was true an attack on Comtian positivism, declared or more years ago, of a single invariant [an for some values of x but not others, since that pure mathematics “knows nothing of algebraic construct] ... which he met with in algebraic rules could not refer to quantities; observation, nothing of experiment, noth- the course of certain researches just as acci- hence algebraists succeeded in proving, to ing of induction, nothing of causation!”, and dentally and unexpectedly as M. Du Chaillu their satisfaction at least, that: said elsewhere that “mathematical training might meet a Gorilla in the country of the is almost purely deductive”. This was all too Fantees … Fortunately [Eisenstein] pounced much for one of his readers, James Joseph down upon his prey and preserved it for the 1 2 3 = 1− 2 + 2 − 2 + ... Sylvester, who thought that pure mathemati- contemplation and study of future mathema- 3 cians had been supine long enough, and that ticians.” This arresting imagery would have Some English mathematicians adopted he was no longer prepared to take impre- been very familiar to most of Sylvester’s audi- these ideas with all the enthusiasm of the cations against his life’s work lying down. ence, because Paul Du Chaillu had for many convert, but there were also notes of alarm His chosen venue for a reply was the 1869 years explored the equatorial regions on the sounded in various quarters.
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