Proofs and Catastrophes

Review of Hacking, Why is There Philosophy of Mathematics at All? and Roubaud, Mathematics: A Novel.
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  26/02/2018The Oxonian Review » Proofs and Catastrophes » Print - The Oxonian Review - - Proofs and Catastrophes Posted By LE Ludtke  On December 8, 2014 @ 9:30 am In Academia,Issue,Issue26.5,Mathematics | Comments DisabledAlice Bamford   Ian Hacking Why Is There Philosophy of Mathematics At All?   Cambridge University Press, 2014 £17.99 (paperback) 304 pages ISBN: 9781107658158  …………   Jacques Roubaud Mathematics: (A novel) Translated by Ian Monk Originally published in French as Mathématique  by Éditions du Seuil, 1997 Dalkey Archive Press, 2012 £10.99 (paperback) 312 pages ISBN: 9781564786838 ………  “In the history of art”, Adorno wrote, “late works are the catastrophes.” Adorno was reflecting onBeethoven’s late style: its “sudden discontinuities”, its episodic, fragmented feeling and itsrefusal of harmony. The late Beethoven refuses to gather his “fractured landscape” and hissplinters of history into a “harmonious synthesis”. Rather, “he tears them apart in time.” Thehistory of mathematics, too, has its catastrophes. The “early history of Greek mathematics”,Reviel Netz suggests, “was catastrophic, not gradual.” Netz borrows his images of time from thegeologists and the biologists and Netz’s catastrophic history of mathematics is invoked, in turn,by Ian Hacking in Why Is There Philosophy of Mathematics At All?   as he digs back through thesrcin myths of the discovery of proof.In Kant’s heroic version of this srcin myth, a “new light” flashed upon the mind of the first manto demonstrate the properties of the isosceles triangle from a priori principles. Legend, Kantwrote, has preserved for us the “memory of the revolution” sparked by this new light. During his  26/02/2018The Oxonian Review » Proofs and Catastrophes » Print telling of Kant’s tale, Hacking remarks that Kant’s word, “revolution”, “is almost worn out withover-use.” He suggests instead Netz’s “catastrophe” or his own metaphor, “crystallization”. Thesemetaphors conjure a geological history of mathematics—catastrophes, crystallisations and chalkformations—and invoke the paradoxical logic of mathematical construction and “discovery”,which Wittgenstein described as “alchemy” and Imre Lakatos described as alienation. “Mathematics, this product of human activity, ‘alienates itself’ from the human activity which hasbeen producing it”, Lakatos wrote in Proofs and Refutations : “It becomes a living, growingorganism, that acquires a certain autonomy   from the activity which has produced it; it developsits own autonomous laws of growth, its own dialectic.” Lakatos was writing against “formalism” as a kind of forgetfulness that “disconnects the historyof mathematics from the philosophy of mathematics, since, according to the formalist concept of mathematics, there is no history of mathematics proper.” Lakatos and Hacking arrived inCambridge in the Michaelmas term of 1956. Lakatos received his doctorate in June 1961 for thework that would become Proofs and Refutations  (“Essays in the Logic of MathematicalDiscovery”). A year later, in March 1962, Hacking’s doctoral thesis, “Part I: Proof; Part II: Strictimplication and natural deduction”, was approved. In the preface to his thesis, Hacking wrote: “We must return to simple instances to see what is surprising, to discover, in fact, why there arephilosophies of mathematics at all.”  Why Is There Philosophy of Mathematics At All?   returns, therefore, to a question that Hackingfirst asked over half a century ago. He offers two answers to it. The first answer lies in theexperience of being compelled by proof. The philosophy of mathematics endures because “acertain type of philosophical mind is deeply impressed by experiencing  a Cartesian proof, of seeing why such-and-such must   be true.” The second answer is in the applicability of mathematics. Proof, which Hacking calls the “Ancient answer”, and use, which he calls the “Enlightenment answer”, meet in passing in Hacking’s discussion of Wittgenstein’s remark: “mathematics is a MOTLEY of techniques of proof—and upon this is based its manifoldapplicability and its importance.” Hacking’s book is steeped in Wittgenstein’s way of thinking about mathematics and in thestrange mathematical vernacular—the glitter and the alchemy—of Wittgenstein’s Remarks on theFoundations of Mathematics . Hacking is perfectly aware that he has been breathing the hauntedair: “I bought my copy of the Remarks on the Foundations of Mathematics on 6 April 1959, andhave been infatuated ever since.” Wittgenstein’s image of “motley” mathematics is given form inHacking’s miscellany: in the motley of his digressive, episodic book. The book’s table of contentsis six pages long and its list of subsections includes: “Descartes’ Geometry   “, “The Langlandsprogramme”, “Eternal truths”, “Leibnizian proof”, “Arsenic”, “Exhaustive classification”, “Theexperience of out-thereness”, “Kant shouts”, “Plato, theoretical physicist”, “Plato, kidnapper”, “Cambridge pure mathematics”, “Aerodynamics”, “Hauntology”, “Some things Dedekind said” and “A brief history of nominalism now”.Hacking’s thoughts on the philosophy of mathematics are woven with vignettes from the historyof mathematics and with memoir. The lived experience of proof, which Hacking offers as the “Ancient answer” to his question, is, in part, his own. Hacking warns us of the shadows cast byhis Cambridge education (“I was brought up in logicism”) and remarks on the escape that Euclid  26/02/2018The Oxonian Review » Proofs and Catastrophes » Print offered to a thirteen year old boy at a mediocre state school: “I learned about proofs, anddelighted in them. Hence I am a gullible victim of Plato’s abduction of mathematics, and also of Kant’s Thalesian myth.” Experience is a vexed and unruly concept, tangled with memory.Experiences of proof are dependent on what Eric Livingston called “cultures of proving”. Like “perceptual gestalts  “, Livingston writes, mathematical proofs articulate an organised “whole” of reasoning, practice and expectation through material detail, though the “whole” “is not present inany of the argument’s individual details.” Early on in Why Is There Philosophy of Mathematics At All?  , Hacking retracts one of his youthfulpronouncements about proof: “Some decades ago I had the gall to open a lecture with thewords: ‘Leibniz knew what a proof is. Descartes did not.'” In that 1973 lecture Hacking arguedthat Leibniz “knew what a proof is” in the sense that his idea of proof anticipated our twentieth-century idea of formal proof: “A proof, thought Leibniz, is valid in virtue of its form, not itscontent. It is a sequence of sentences beginning with identities and proceeding by a finitenumber of steps of logic and rules of definitional substitution to the theorem proved.” WhereDescartes believed “proof irrelevant to truth”, Leibniz thought that truth was constituted by proof and imagined a completely general Universal Characteristic in which proofs could be conductedand through which truth would, he wrote, be rendered “stable, visible and irresistible, so tospeak, as on a mechanical basis.” We are still stuck, Hacking said in 1973, in the seventeenth-century conditions of possibility out of which the concepts of proof and anti-proof emerged: “Wehave forgotten those events, but they are responsible for the concepts in which we perform ourpantomime philosophy.” In Why Is There Philosophy of Mathematics At All? Hacking keeps his pseudo-couple, Leibniz andDescartes, as allegorical figures in a seventeenth-century branching of the ways, but where oncethose branches were “proofs” and “anti-proofs”, now there are “leibnizian proofs” and “cartesianproofs”: two cultures of proof. “It is astonishing”, he writes, “that we have not yet confessed tothe duality of proof, cartesian and leibnizian. These are two ideals, which pull in differentdirections.” The experience of proof on which the philosophy of mathematics is founded and towhich it obsessively returns is the experience of cartesian proof: “seeing as a whole, with clearconviction.” Hacking is writing of the lateness of the cartesian proof: of the ways in which it is outof time. Its ideal—that you must be able to see the proof as a whole, in your mind, all at once—feels belated in our world of mechanized proof: a world in which, as Hacking writes, “we arehourly becoming more leibnizian.” In his doctoral thesis, Hacking noted “the uncanny resemblance between trying to recall andtrying to prove; between recollecting successfully after some effort, and hitting on a proof.” Thisaside was prompted by the story of the slave boy in Plato’s Meno  who comes, with Socrates’shelp, to see the truth of a geometrical theorem about squares. Since this knowledge comesneither from teaching nor experiment the boy must have already known it somewhere withinhimself: he must be recollecting it. Plato, Hacking wrote, “tried to reduce the puzzling to thefamiliar, so proof to recollection. He may have spoken more truly than is currently recognized.” The story of the slave boy is also told by Jacques Roubaud in Mathematics: (a novel) . Roubaud’sbook is a sustained investigation of the workings of memory: of, in particular, his memories of   26/02/2018The Oxonian Review » Proofs and Catastrophes » Print the culture of proof in Paris in the 1950s. Roubaud’s story begins in the lecture hall of the InstitutHenri Poincaré (certificate “Differential and Integral Calculus”) one winter morning in 1954:I have waited over thirty-seven years before daring to stop and stare deliberately at that imageor handful of images: the board, benches, heads, chalk drawings, charged with meaning. Iremove it from hell, or its limbo. I remove it from my memory so as to erase it, as I do with allthe memories that I fix by writing them down, like the chalk “potatoids” drawn by “Choquet” onthe blackboard, long ago. But before erasing it, I charge it with meaning.In mathematics—here, in chalk and later, in topology—Roubaud finds metaphors for the art of memory. He reworks the Platonic parable to describe the confusions and conversions of theBourbaki revolution: “The knowledge of sets was within us. It is the most fundamentalmathematical knowledge. But we had to go and seek it out inside ourselves, just as the Boy,under Socrates’s careful guidance, came across the concealed idea of the ‘diagonal’, by way of anamnesis and recollection.” Nicolas Bourbaki was a pseudonym adopted in the 1930s by a group of French mathematicianswho began collectively writing a new treatise on analysis. The projected modern analysistextbook evolved into a multi-volume treatise, Éléments de mathématique , which was planned,as Leo Corry writes, to be “the ultimate mathematics textbook.” The Treatise, self-contained andhighly formalised, was to express a unified, modern conception of mathematics, with eachvolume a comprehensive account of a branch of mathematics. By the time Roubaud was writinghis memoir, in the 1990s, Bourbaki had become, he says, a “museum piece”. Yet, the branchesand interpolations of Roubaud’s book mimic Bourbaki’s axiomatic presentation, his book’s manybeginnings refract their attempts to erase the past of mathematics and his own abandoned great Project was, he acknowledges, indebted to their Treatise.Bourbaki tried to wrench mathematics out of history. “Apparently, a clean slate had just beenmade of the past of mathematics,” Roubaud writes of the rumours that circulated around thelecture theatres of the IHP. Bourbaki seemed to be tearing down the entire edifice of existingmathematics in order to built it anew: they were a kind of catastrophe, a revolution that seemedcloser and more plausible than political revolution. Roubaud remembers being “gripped by thevertigo of beginning” as he read Bourbaki’s advice to the reader of the Treatise: “This series of volumes […] takes up mathematics at the beginning, and gives complete proofs.” “I needed theillusion of an absolute beginning”, Roubaud writes, but this absolute beginning provedimpossible: how were you to know that the real beginning had been reached without firstexamining the “pre-beginning”? How to begin with the “beginning” of the Treatise when Book I,in which the famous theory of sets was to be presented, had not yet been finished? (A “Summaryof Results” appeared in 1939, but the four chapters on set theory were only published in finalform between 1954 and 1957).Roubaud didn’t know how to read Bourbaki’s General Topology   when he first sat down to it: hecould determine no narrative thread. Indeed, Peter Galison suggests in “Structures of Crystal,Buckets of Dust” that “reading as such, the sequential absorption, seems to pull against theBourbakian ideal.” The Bourbaki members, Galison writes, “aimed their story of mathematics tobe the non-narrative narrative, the account outside time, a structure, an architecture to becontemplated as it ordered ‘mathematic’ from set theory on out.” Roubaud decides to read the  26/02/2018The Oxonian Review » Proofs and Catastrophes » Print Treatise like poetry, which he would re-read and commit to memory until he had “repositioned allof its elements in the present, in the simultaneity of inward time.” In the end, however, Roubaudreads the Treatise against the grain. He extrapolates a theory of memory from the book ontopology and uses this theory to express the intertwining of points in inner time: theneighbourhoods of memory in which Bourbaki’s catastrophe is tangled with motley experience.Alice Bamford  [1]  is a PhD student at the University of Cambridge.Article printed from The Oxonian Review: URL to article: URLs in this post:[1] Alice Bamford: Copyright © 2009 Oxonian Review of Books. All rights reserved.
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