Proofs and Catastrophes
Why Is There Philosophy of Mathematics At All?
Cambridge University Press, 2014
Mathematics: (A novel)
Translated by Ian Monk
Originally published in French as Mathématique by √âditions du Seuil, 1997
Dalkey Archive Press, 2012
“In the history of art”, Adorno wrote, “late works are the catastrophes.” Adorno was reflecting on Beethoven’s late style: its “sudden discontinuities”, its episodic, fragmented feeling and its refusal of harmony. The late Beethoven refuses to gather his “fractured landscape” and his splinters of history into a “harmonious synthesis”. Rather, “he tears them apart in time.” The history 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 the geologists 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 the origin myths of the discovery of proof.
In Kant’s heroic version of this origin myth, a “new light” flashed upon the mind of the first man to demonstrate the properties of the isosceles triangle from a priori principles. Legend, Kant wrote, has preserved for us the “memory of the revolution” sparked by this new light. During his telling of Kant’s tale, Hacking remarks that Kant’s word, “revolution”, “is almost worn out with over-use.” He suggests instead Netz’s “catastrophe” or his own metaphor, “crystallization”. These metaphors conjure a geological history of mathematics—catastrophes, crystallisations and chalk formations—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 has been producing it”, Lakatos wrote in Proofs and Refutations: “It becomes a living, growing organism, that acquires a certain autonomy from the activity which has produced it; it develops its own autonomous laws of growth, its own dialectic.”
Lakatos was writing against “formalism” as a kind of forgetfulness that “disconnects the history of 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 in Cambridge in the Michaelmas term of 1956. Lakatos received his doctorate in June 1961 for the work that would become Proofs and Refutations (“Essays in the Logic of Mathematical Discovery”). A year later, in March 1962, Hacking’s doctoral thesis, “Part I: Proof; Part II: Strict implication 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 are philosophies of mathematics at all.”
Why Is There Philosophy of Mathematics At All? returns, therefore, to a question that Hacking first asked over half a century ago. He offers two answers to it. The first answer lies in the experience of being compelled by proof. The philosophy of mathematics endures because “a certain 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 manifold applicability and its importance.”
Hacking’s book is steeped in Wittgenstein’s way of thinking about mathematics and in the strange mathematical vernacular—the glitter and the alchemy—of Wittgenstein’s Remarks on the Foundations of Mathematics. Hacking is perfectly aware that he has been breathing the haunted air: “I bought my copy of the Remarks on the Foundations of Mathematics on 6 April 1959, and have been infatuated ever since.” Wittgenstein’s image of “motley” mathematics is given form in Hacking’s miscellany: in the motley of his digressive, episodic book. The book’s table of contents is six pages long and its list of subsections includes: “Descartes’ Geometry“, “The Langlands programme”, “Eternal truths”, “Leibnizian proof”, “Arsenic”, “Exhaustive classification”, “The experience 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 history of 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 by his Cambridge education (“I was brought up in logicism”) and remarks on the escape that Euclid offered to a thirteen year old boy at a mediocre state school: “I learned about proofs, and delighted 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 in any of the argument’s individual details.”
Early on in Why Is There Philosophy of Mathematics At All?, Hacking retracts one of his youthful pronouncements about proof: “Some decades ago I had the gall to open a lecture with the words: ‘Leibniz knew what a proof is. Descartes did not.'” In that 1973 lecture Hacking argued that 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 its content. It is a sequence of sentences beginning with identities and proceeding by a finite number of steps of logic and rules of definitional substitution to the theorem proved.” Where Descartes 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 conducted and through which truth would, he wrote, be rendered “stable, visible and irresistible, so to speak, 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: “We have forgotten those events, but they are responsible for the concepts in which we perform our pantomime philosophy.”
In Why Is There Philosophy of Mathematics At All? Hacking keeps his pseudo-couple, Leibniz and Descartes, as allegorical figures in a seventeenth-century branching of the ways, but where once those branches were “proofs” and “anti-proofs”, now there are “leibnizian proofs” and “cartesian proofs”: two cultures of proof. “It is astonishing”, he writes, “that we have not yet confessed to the duality of proof, cartesian and leibnizian. These are two ideals, which pull in different directions.” The experience of proof on which the philosophy of mathematics is founded and to which it obsessively returns is the experience of cartesian proof: “seeing as a whole, with clear conviction.” Hacking is writing of the lateness of the cartesian proof: of the ways in which it is out of 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 are hourly becoming more leibnizian.”
In his doctoral thesis, Hacking noted “the uncanny resemblance between trying to recall and trying to prove; between recollecting successfully after some effort, and hitting on a proof.” This aside was prompted by the story of the slave boy in Plato’s Meno who comes, with Socrates’s help, to see the truth of a geometrical theorem about squares. Since this knowledge comes neither from teaching nor experiment the boy must have already known it somewhere within himself: he must be recollecting it. Plato, Hacking wrote, “tried to reduce the puzzling to the familiar, 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’s book is a sustained investigation of the workings of memory: of, in particular, his memories of the culture of proof in Paris in the 1950s. Roubaud’s story begins in the lecture hall of the Institut Henri 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 image or handful of images: the board, benches, heads, chalk drawings, charged with meaning. I remove it from hell, or its limbo. I remove it from my memory so as to erase it, as I do with all the memories that I fix by writing them down, like the chalk “potatoids” drawn by “Choquet” on the 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 the Bourbaki revolution: “The knowledge of sets was within us. It is the most fundamental mathematical 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 mathematicians who began collectively writing a new treatise on analysis. The projected modern analysis textbook evolved into a multi-volume treatise, √âléments de mathématique, which was planned, as Leo Corry writes, to be “the ultimate mathematics textbook.” The Treatise, self-contained and highly formalised, was to express a unified, modern conception of mathematics, with each volume a comprehensive account of a branch of mathematics. By the time Roubaud was writing his memoir, in the 1990s, Bourbaki had become, he says, a “museum piece”. Yet, the branches and interpolations of Roubaud’s book mimic Bourbaki’s axiomatic presentation, his book’s many beginnings 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 been made of the past of mathematics,” Roubaud writes of the rumours that circulated around the lecture theatres of the IHP. Bourbaki seemed to be tearing down the entire edifice of existing mathematics in order to built it anew: they were a kind of catastrophe, a revolution that seemed closer and more plausible than political revolution. Roubaud remembers being “gripped by the vertigo 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 the illusion of an absolute beginning”, Roubaud writes, but this absolute beginning proved impossible: how were you to know that the real beginning had been reached without first examining 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 “Summary of Results” appeared in 1939, but the four chapters on set theory were only published in final form between 1954 and 1957).
Roubaud didn’t know how to read Bourbaki’s General Topology when he first sat down to it: he could 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 the Bourbakian ideal.” The Bourbaki members, Galison writes, “aimed their story of mathematics to be the non-narrative narrative, the account outside time, a structure, an architecture to be contemplated as it ordered ‘mathematic’ from set theory on out.” Roubaud decides to read the Treatise like poetry, which he would re-read and commit to memory until he had “repositioned all of its elements in the present, in the simultaneity of inward time.” In the end, however, Roubaud reads the Treatise against the grain. He extrapolates a theory of memory from the book on topology and uses this theory to express the intertwining of points in inner time: the neighbourhoods of memory in which Bourbaki’s catastrophe is tangled with motley experience.
Alice Bamford is a PhD student at the University of Cambridge.