Hacking wades around the edges of the swamps of jargon and academic debate, giving us a feel for them from the outside, letting us get a little muddy without getting us bogged down. He is keen on Wittgenstein, for example, but, after looking at übersichtlichkeit ("surveyability") as an illustration of the problems involved, he writes "you may see why I prefer to evade the thickets of translation, interpretations, commentary, and hermeneutics". And the narrow concerns of modern analytic philosophy are largely ignored:
"issues of nominalism, which at present preoccupy academic philosophers, do not much matter to mathematicians or mathematics. They have very little to do with mathematical activity. That is why my glance at these topics will be fleeting."
Hacking also uses oppositions and typologies which he admits up front are overly simplistic, but which help provide some structure.
The material is organised in titled sub-sections, typically one or at most two pages in length, covering individual topics in a fairly self-contained fashion. There are some significant digressions in these: a description of the engineering feats of ancient Samos, for example, and a look at dictionary definitions of arsenic. The overall effect could easily have been disconnected, but the broad thematic chapters within which these topics are chained provide both a broader context and something of a narrative flow.
The first chapter, "A Cartesian Introduction", looks at two strands that go back to Descartes. The first is the application of mathematics, not to physics and science but to mathematics itself, where Hacking asks whether we should be surprised at connections (or analogies or correspondences) between geometry and algebra. The second is the contrast between leibnizian (step-by-step) and cartesian (understood "all at once") proofs, where Hacking touches on extremes exemplified by the mathematicians Voevodsky and Grothendieck, Wittgensteinian notions of "perspicuity" and "surveyability", the proof that it is impossible to divide a cube into unequal smaller cubes, and computer-verified proofs, among other topics.
What makes mathematics mathematics? Hacking considers a range of dictionary definitions, institutional and neuro-historical answers, logicism as a programmatic answer, Bourbaki's focus on structure, Wittgenstein's "a motley of techniques of proof", the implications of experimental mathematics, Thurston's suggestion of a recursive definition, Hilbert's problems and the Millennium prizes, the contingency or inevitability of mathematics as we know it, and play and ludic proof.
Why have so many philosophers been so fascinated by mathematics? Hacking sees two strands here, one "ancient" and one "modern". The first is driven by surprise at proof results, for example at the discovery of the Monster (the largest of the finite simple sporadic groups). While this intense response may be prominent among mathematicians, in contrast "most thoughtful people ... find that story simply unintelligible". The second strand is a product of the Enlightenment, and specifically of Kant and concepts of the necessary, analytic, and a priori. "Even when spurned by philosophers, the feeling of necessity, of logical compulsion, is one of the reasons that there is philosophy of mathematics at all."
Returning to contingency, Hacking considers two models: a "Latin" model, where mathematics could have evolved in different ways just as the Romance languages did, and a "butterfly" model — invoking developmental biology, not chaos theory! — in which development unfolds in a basically predictable fashion. A possible example of the former is Cantor's approach to infinite numbers, where alternative frameworks seem possible; an example of the second could be complex numbers, which appear more inevitable.
Turning to broader historical trends, Hacking focuses on two areas, on our concept of proof and the divide between applied and pure mathematics. With the first he argues that "the very idea of demonstrative proof seeming to convey apodictic certainty is something of a fluke". He traces its Greek origins, in Thales, Pythagoras, Plato, the Eleatics and the stories Kant and others have told about them ("history-as-parable is often much more useful in philosophy than history-as-fact"). And he touches on ideas that undermine the consensus story — other Greek mathematical traditions, Chinese mathematics, different kinds of proof — even asking "did the ideal of proof impede the growth of knowledge?".
The notion of a divide in mathematics goes back to Plato ("philosophical" and "practical"), Bacon ("pure" and "mixed"), and late eighteenth century Germany (rein and angewandt). Hacking also touches on the French tradition and military history, Hamilton's ideas and Cambridge "pure mathematics", and the definition used by Society for Industrial and Applied Mathematics, while his own typology contains eight different kinds of "Apps". One example is rigidity, with a history running from Maxwell through Buckminster Fuller to current mathematics. Another is the aerofoil, where institutional differences between British and German mathematics help explain why F.W. Lanchester's ideas were ignored in his own country and Britain "fought the First World War with the wrong theory". These show the limitations of the most popular view: "applied mathematics is seldom 'deduction all the way down' between representation and derepresentation".
They have featured extensively already, but Hacking now turns his focus to what he labels Platonism, reserved for "what is plausibly connected to the historical Plato", and "lower-case p platonism, for recent ideas that often have a highly semantic component". One chapter describes and contrasts the "attitudes" of two mathematicians, Alain Connes ("our spokesperson for structuralism", with "a deep commitment to the existence of an archaic reality constituted by the series of numbers") and Timothy Gowers (who is "discovering what can be proven").
The last chapter considers alternative platonisms and their opponents. Paul Bernays introduced the label "platonism" to mathematics in a 1934 lecture, opposing to it a kind of intuitionism. Another contrast can be found in the attitudes of the mathematicians Dedekind and Kronecker. The structuralism of contemporary philosophers differs from that of mathematicians, however. Contemporary philosophical debates oppose platonism with nominalism ("denying the existence of any such objects as numbers, functions, sets, and their ilk"). Here Hacking touches — fleetingly — on denotational semantics and concepts of ontology, commitment, indispensability, presupposition, intuition, and suchlike.
Hacking's approach in all of this is personal and idiosyncratic, but never indulgent or aggrandizing. He is honest about his limitations — "glimpses of work I do not fully understand, but which cannot be ignored" — and open about following his own interests. And his own involvement with many of the people whose ideas he describes is disclosed in a kind of appendix.
Why Is There Philosophy of Mathematics At All? is discursive and readable, presented in easily digestible chunks, clearly explained, and just a lot of fun. Little technical background in philosophy or mathematics is assumed, with involved details either passed over (the content of the Langlands program) or given brief explanations (Saul Kripke's idea of a rigid designator). Full appreciation, however, probably requires both a general familiarity with the history and philosophy of science and some experience doing mathematics.
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