Does Philosophy of Math Need a Practical Turn?


When we dig beneath the neatly composed surface we find a great buzzing, blooming confusion of ideas, and we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty.

That’s Jeremy Avigad (Carnegie Mellon), writing at Aeoncalling for philosophy of math to be more attentive to the actual development of mathematics and to the actual practices of mathematicians. It could be seen as an argument for doing philosophy of math as a kind of philosophy of science, which to varying degrees is about the real history and methods of science.

I’ve put a longer excerpt from the piece below. I’m curious to hear from others, especially those working in philosophy of math, if they think Professor Avigad is correct in his diagnosis and prescription for the subject.

In part, the philosophy of mathematics was a victim of its own success. For a subject traditionally concerned with determining the proper grounds for mathematical knowledge, modern logic offered such a neat account of mathematical proof that there was almost nothing left to do. Except, perhaps, one little thing: if mathematics amounts to deductive reasoning using the axioms and rules of set theory, then to ground the subject all we need to do is to figure out what sort of entities sets are, how we can know things about them, and why that particular kind of knowledge tells us anything useful about the world. Such questions about the nature of abstract objects have therefore been the central focus of the philosophy of mathematics from the middle of the 20th century to the present day.

In other branches of philosophy, where no neat story was available, philosophers had to deal with the inherently messy nature of language, science and thought. This required them to grapple with serious methodological issues. From the 1950s on, philosophers of language engaged with linguists to make sense of the Chomskyean revolution in thinking about the structure of language and human capacities for understanding and generating speech. Philosophers of mind interacted with psychologists and computer scientists to forge cognitive science, the new science of the mind. Philosophers of biology struggled with methodological issues related to evolution and the burgeoning field of genetics, and philosophers of physics worried about the coherence of the fundamental assumptions of quantum mechanics and general relativity. Meanwhile, philosophers of mathematics were chiefly concerned with the question as to whether numbers and other abstract objects really exist.

This fixation was not healthy. It has almost nothing to do with everyday mathematical practice, since mathematicians generally do not harbour doubts whether what they are doing is meaningful and useful—and, regardless, philosophy has had little reassurance to offer in that respect. It turns out that there simply aren’t that many interesting things to say about abstract mathematical objects in and of themselves. Insofar as it is possible to provide compelling justification for doing mathematics the way we do, it will not come from making general pronouncements but, rather, undertaking a careful study of the goals and methods of the subject and exploring the extent to which the methods are suited to the goals. When we begin to ask why mathematics looks the way it does and how it provides us with such powerful means of solving problems and explaining scientific phenomena, we find that the story is rich and complex. 

The problem is that set-theoretic idealisation idealises too much. Mathematical thought is messy. When we dig beneath the neatly composed surface we find a great buzzing, blooming confusion of ideas, and we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty. And so the call of the sirens is pleasant and enticing: mathematics is set theory! Just tell us a really good story about abstract objects, and the secrets of the Universe will be unlocked. This siren song has held the philosophy of mathematics in thrall, leaving it to drift into the rocky shores.

You can read the whole essay here.

Still from Chuck Jones’ animated adaptation of “The Dot and the Line” by Norton Juster

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