Why Is There Philosophy of Mathematics At All? by Ian Hacking
My rating: 4 of 5 stars

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?

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