Why Is There Philosophy of Mathematics At All? by Ian Hacking (30-Jan-2014) Paperback

by Ian Hacking

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.

Genres: Philosophy, Mathematics, Nonfiction

Paperback

First published January 31, 2014

About the author

Ian Hacking is Professor Emeritus of Philosophy at the University of Toronto, specialised in the History of Science.

Wikipedia entry

Reviews

[BlackOxford · Rating 4 out of 5]

Mathematical Epistemology

This is a rambling compendium of mathematical wisdom and opinion by a leading philosopher of science. Given its subject matter and the issues it addresses it might appear irrelevant to everyday concerns, particularly politics. If so, appearances are decidedly deceiving. The central theme of the book - the nature of scientific proof - is arguably the most important issue confronting democracies around the world. Controversies about Covid-19 and Donald Trump make the point conclusively.

Epistemology, the study of how we put our experiences into language, that is to say, how we prove what we think we know when we make claims about reality, has been a hot philosophical topic for several centuries. The bad news is that the epistemological project is dead in the water and has been for some time. The connections between language and that which is not-language are too unreliable, too unstable to tell with any certainty what language is false, misleading, incomplete, or biased. There are no agreed upon rules, methods, or algorithms by which any statement whatsoever can be fact-checked, verified, or proven to be absolutely true or false.

The reason for our ultimate inability to distinguish between fact and fiction is not because we have lost contact with something we casually call reality, but because as soon as we bring our experiences of reality into language, those experiences, and the reality to which these experiences are a response, become distorted in ways that are impossible to correct. Language, whatever else it might be, is not reality. But we have nothing but language in which to express reality. Without language, we would not even be able to discuss reality; we would also not have the epistemological problem.

This inadequacy applies even to the most precise and well-studied language we have at our disposal - mathematics. Unlike any other language, mathematics is minutely and invariably defined. Its vocabulary (numbers) is infinite but every element is always related to every other element in a precise way. The way in which these elements are related to each other (their grammar or rules of use) can be stated axiomatically. And the results produced by mathematicians (proofs) remain valid from era to era, culture to culture, and even according to most mathematicians, from galaxy to galaxy.

Mathematics is as close as we are ever likely to get to the perfect language. The reason for this is that it is a language which refers only to itself. It makes no pretence to describing a reality beyond itself. The mathematical world is entirely self-contained. Its potentially infinite expressiveness has been obviously useful but because its expressions are equally obviously only about itself not about the reality which transcends all language. Furthermore, it contains elements that could not exist in any reality outside of the mathematical language in which they are expressed (dimensionless points, negative numbers, etc.). Mathematics is its own reality.

So mathematics shouldn’t have an epistemological problem. Its language doesn’t have to be correlated with anything except itself. Yet despite its apparent linguistic perfection, mathematics has a dirty little secret. What constitutes mathematical proof is not at all a settled matter. Even mathematics suffers the epistemological impasse, although in a somewhat different way than other languages. Non-mathematical languages confront the apparent problem of connecting language with that which is not-language. Mathematics shows that the problem exists in connecting language with itself.

Hacking identifies two dominant schools of thought about mathematical proof - the Cartesian and the Leibnizian. Both are commonly held within the community of mathematicians, often simultaneously by the same people. But, as Hacking explains these are rather different conceptions, not only of proof but also of what constitutes truth.

Cartesian proof involves comprehending a whole, seeing the beginning middle and end of an argument comprehensively in an appreciation of a mathematical problem. The essential element of a Cartesian proof is intuition, that instinctive sense that things fit together in a certain way. It involves an aesthetic of overall rightness, not in the details but in the overall trajectory and elegance of a mathematical argument.

Leibnizian proof is a step by step mechanical process of inference that moves methodically (tediously perhaps from a Cartesian perspective) from minute inference to minute inference. The aesthetic of this sort of proof is one of rigour rather than elegance. It is literally the way that modern digital machines work - in a plodding, uncreative, unimaginative process of logical progression.

According to Hacking, most mathematicians think of themselves as Cartesianists, yet present themselves professionally as Leibnizian. Few, in fact, would deny the importance of both methods and are probably unaware of their methodological transpositions as they go about their work because “What counts is the lived experience of the entire mathematical activity of bringing a proof into existence...”

In other words the two aesthetics sit side by side in mathematics without conflict because the community of mathematicians recognises their joint value and doesn’t attempt to prioritise or differentially value them. In fact, I don’t think it would be offensive to Hacking or his colleagues to say that this acceptance of the two aesthetics is a reasonable definition of the community of mathematicians itself. There is no formal agreement among them because their doesn’t need to be. This is just how the community works.

That the relative importance of these two aesthetics has varied over time, and among important mathematicians is a central part of Hacking’s argument. But this is just a more or less polite way of saying that there are no fixed, permanent, or eternally valid criteria of proof in mathematics. What counts as proof is a matter of the mores of the current community. This is not to say that historical proofs are often rejected (although they may be from time to time); but that the way in which new proofs are generated and approved within the community is.

This process of changing criteria of proof is a subtle one, mainly because it takes place not through the use of established mathematical language but in the creation of variants and dialects which are engaged in, at least for a time, independently. The relative emphasis of elegance and mechanical rigour (or other aesthetic criteria for that matter) may shift substantially yet remain unnoticed by the users of the ‘received’ language. Historically this was the case with geometry and algebra for example; and more recently between geometry and number theory.

There are currently several hundred recognised sub-fields of mathematics, many with their own specialised language and communal methods of proof. The so-called Langlands Project is an effort to effectively translate among these mathematical dialects. As more and more mathematical languages are generated - by the Project itself as well as the course of mathematical research - it is clear that such an attempt at ‘unification’ has what amounts to an infinite horizon. This means that its goal will never be reached.

In short, even in mathematics, the closure of the ‘epistemological gap’ by the identification of a definitive method of proof is really not something on which to bet the farm. Hacking quotes Ludwig Wittgenstein approvingly on the matter: “I should like to say: mathematics is a MOTLEY of techniques of proof - and upon this is based its manifold applicability and its importance.”

In other words mathematical language is powerful just to the extent it is basically uncertain. It might be instructive to remember this when we employ the rather less precise language of everyday life. Politics rules, OK?

[WarpDrive · Rating 3 out of 5]

After getting used to the quality of Shapiro's books on philosophy of mathematics, I must say that I was quite disappointed by the general lack of depth and by the piecemeal approach that is evident throughout this book by Ian Hacking.
The specific philosophical positions are not investigated in detail, and the book is often no more than a long list of bibliographic references, where the position of the individual authors are explained in a couple of sentences. The author himself does not elaborate on his own statement that he is not a "Platonist" - and I got the feeling that he does not elaborate his own position simply because he does not that much to say about it.
On the positive side, a few interesting points are raised:
- the distinct cognitive origins of geometry and numbers (according to modern research, our cognitive apparatus elaborates judgments of numerosity and spacial/geometrical judgments in two distinct systems), but their intimate connection in reality
- the "unity in diversity" of the mathematical world - for example the mathematical breakthrough by Andrew Wiles, who applied specific features of elliptic functions to the demonstration of Fermat's last theorem; and also the fact that the very same mathematical models/techniques originally created in the realm of pure mathematics or to solve a specific problem in a particular branch of the physical world, unexpectedly and repeatedly applied to other unrelated categories of problems. Why is that the "same mathematical structures - models - often turn out to be useful representations of seemingly unrelated phenomena or fields of experience"? Does this tell us something deep about the relationship between mathematics and the physical world?
- the two different perspectives on mathematical proof: the "Cartesian" flavor (get it all in your mind, all at once, grasp the meaning of the whole concept in its entirety), and the "Leibnizian" flavor (a potentially long-winded chain of deductively rigorous atomic steps, each resulting by the application of a single rule of inference)
- the development of "experimental mathematics", utilizing the powerful computational and modelling capability of modern computers
- the sense of "giveness" that emerges strongly when mathematical structures appear to be "discovered" by exploration (for example: the finite group is by itself an elementary notion (group of symmetries of a finite object), but the last finite simple group that was discovered by mathematical reasoning (the "Monster") has a huge number of elements (it is as if "the Monster has been sitting out there, quietly grinning, waiting for us to discover it")
- the fact that, regardless of the individual philosophical position, there is an amazing level of matching between nature and maths; and moreover there is the astonishing fact of the connection between human beings' sense of structure/symmetry that they project on nature, and the deep nature of reality itself. Why is that something merely attractive to the human mind - certain symmetries, for example - turn out to unlock some of the secrets of nature ? Think about the power of symmetry in modern fundamental physics
- there is a very interesting discussion on Godel's first incompleteness theorem, which according to some Platonists supports their view (as there are discoverable truths that lie beyond any consistent axiomatization adequate for the expression of recursive arithmetic - it is so "out there" that it can't be captured by a finite system of axioms). Without going into detail, I agree with the author that this theorem, by itself, is not a definite support for the Platonist perspective.

Overall, it is a reasonably interesting book, with a few interesting points - it is unfortunate that they are submerged in a sea of short bibliographic references and under-developed arguments.

[L. A. · Rating 4 out of 5]

Hacking at his most frustrating. This book was the most interesting when it was about the historical development of mathematical concepts and the practices of mathematicians, and the least interesting when Hacking starts talking about the philosophers and often enough the philosophical opinions of mathematicians. He realizes that the latter is mostly irrelevant and shows this in that he doesn't advance any particularly interested opinion on the more philosophical debates. In light of this, I think he chose a good topic for the book and I wish he would write more about the historical development and social and personal practices in mathematics leaving the irrelevant philosophizing to the wayside. As far as philosophy goes, his bias towards analytic philosophy does him a great disservice. He may have learned from Foucault but he could stand to have learned a thing or two about more dialectical approaches like those of Deleuze (not to mention a dialectical materialist approach) which would be very interesting to see born out against what he sees as the limitations of the terms of the debate in analytic philosophy.

[Michal Paszkiewicz · Rating 5 out of 5]

A very thorough and well researched book. My want-to-read shelf has vastly increased.

Ian Hacking covers history, ideas and unanswered questions in this thought provoking book, leaving me with a much better understanding of the philosophy of mathematics and the various positions one could hold over the reality of maths.

[Tech Nossomy · Rating 2 out of 5]

A treatise on the philosophy of mathematics, with much detail and references on the various philosophies and their relation to each other. The central question however, on whether a philosophy of mathematics is needed is unanswered. For example, which questions can we answer with a philosophy of mathematics that cannot be answered without one? The book lacks in this type of practical approach and rigour. As a matter of evidence, on page 81 it is stated: "the philosophy of mathematics covers a lot of subjects, and different philosophers have different opinions of what is most central or interesting." So even after 80+ pages the reader is still none the wiser.

Deficiencies and omissions:
* There is no dissection of the foundations of mathematics: how axioms, theorems, lemmas, propositions, conjectures, definitions, assumptions, conditions and proofs are related or how and why this ultimately historically developed is not discussed. At the least it would provide a meaningful starting point in identifying potential gaps.
* The point that mathematics developed out of convenience and with its associated nomenclature more than out of principle or dogma is insufficiently argued.
* At the time of publication the University of Tilburg was already renamed Tilburg University.
* Cardano was not the originator of the formula that he is known for, but it is by Tartaglia. This is commensurate with Stigler's law, by the way.
* "Many of the examples of recent mathematics take an idea from one branch of mathematics and use it to create or to solve a fundamental problem in another." but the author gives no description as to why that is. Also, and more interestingly, is the converse true?
* "From egg to infant to teenager to adult to geriatric, as recorded in Shakespeare’s seven ages of man." Those are 5 ages, so that's 2 ages missing. Either way, I would not have thought the musings of an entertainer to be authoritative.
* P versus NP: each time the author opens an opportunity to discuss computability, it is discounted as a conventional wisdom.
* The word 'paradox' barely occurs in the text, but the way paradoxes relate to the limitations of mathematics is omitted.

The book ends with two statements that provide a good starting point:
"Perhaps we are not seeing the wood for the trees, not seeing the big picture because we focus on so many small ones."
(good point, but this is stale news and in fact it spawned neo-Darwinism)
and
"Numbers are abstract conceptions"
(again a good point, just not one to end the book with)

The book offers a wide array of discussions on the meaning of mathematics and does so in a comprehensive albeit not too cogent way. Perhaps the search for the need of a philosophy should be followed along more mundane lines that go beyond "unreasonable effectiveness":
- what is the fundamental reason we find short proofs appealing?
- how do biases, of which an example is discussed in chapter 4-B, still affect mathematical progress?
- what specifically happens when a conjecture that was relied on for being true is proven false?
The point is not to expect that a new philosophy will arise when thinking about these questions, but rather how a philosophy would help to understand the way mathematics imprints upon us a certain rigidity of the world around us.

Perhaps we should go even further:
What if we lived in a polar world or Fourier-analytical world and the cartesian world had not been invented yet? Or what if fractional calculus was invented before integer calculus was, because there was no concept of integers? Or what if our counting system started out as fractional, or perhaps even irrational, or quaternionic, or infinitesimal, instead of with base-10? Describing a world in a comprehensive way in which these systems are the norm would help us understand better the limitations we place on ourselves and henceforth the meaning we place upon our own abilities. I expect this to be a humbling exercise. Unfortunately the book stops short of offering actual practical handles that allows the reader to step outside both the traditional and current viewpoints. Moreover, it does not help that the text is fraught with opinions, diversions, frivolities, caveats, metaphors and tangential discussions, not to mention the liberal but often needless use of parentheses that only break up legibility. Getting to the heart of the matter appears to be a real struggle.

[Liquidlasagna · Rating 1 out of 5]

Amazone

Robert Black
absence of argument
4/10

This is a very self-indulgent book. It's full of references to the author's own works, and when quoting others he always mentions if at all possible his personal connexion to them, so that one has the impression of permanent name-dropping.

He has obviously done a lot of reading about mathematics, both popularizations and obiter dicta of famous mathematicians about their subject, but his grasp of logic and foundations looks a bit shaky:

he says (p. 23) that Gödel thought that by adding new axioms we could get a complete set theory (as though Gödel had forgotten his own incompleteness theorem!), suggests (p. 33) that some might doubt the proof that there is no uneven dissection of the cube because it's proved by reductio (but it's a negative claim and nobody objects to reductio to negative conclusions) and announces that Kripke semantics provides the key to understanding intuitionist logic (apparently unaware that it only provides a *classical* semantics for intuitionist logic, not an intuitionist one).

These slips wouldn't matter if there were philosophical meat in the book, but there isn't.

After Hacking himself the most quoted author is Wittgenstein, but no attempt is made to construct a philosophy of mathematics out of Wittgenstinian ideas (as for example was done by Michael Dummett); on the contrary, all Hacking seems to have taken from Wittgenstein is the idea that you can write a book of philosophy via a motley association of ideas without any structured argument whatever.

He vaguely says that he's a logicist, i.e. one who thinks mathematics reduces to logic, without saying what counts as logic or how the reduction is supposed to go.

He announces that modern philosophy of mathematics makes the mistake of presupposing a semantics in terms of reference and truth, to which he opposes the slogan (and he admits it's no more than that) that meaning is use, without giving any hint of what semantics this might lead to, or how that would affect the philosophy of mathematics.

The moral is, one shouldn't write a book on a subject on which one has nothing to say.

Robert Black