# 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 *Aeon**, *calling 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.

It’s a nice essay, and I’m glad to see philosophy of maths being discussed, but (and I would say this, wouldn’t I?) I don’t agree that phil of maths has lost its way by retaining a concern with the central metaphysical and epistemological questions of what are mathematical objects and how do we know them.

There’s a very long history of philosophers of maths complaining that philosophers of maths have done the subject a disservice by focussing on these abstract questions and ignoring the messiness of mathematical practice. Eg I was prompted today to pick up my copy of the 1983 “New Directions in the Philosophy of Mathematics” volume to find an article for a student, and this volume is devoted almost entirely to articles that make the same complaint. Tymockzo (the editor) opens the volume by saying that its origin was a seminar held in 1979 where philosophers, mathematicians, and logicians met “to discuss common concerns about the nature of mathematics” and “were frustrated by the inability of traditional philosophical formulations to articulate the actual experience of mathematicians”. Indeed, under the influence of Lakatos in particular, my own PhD dissertation started out as an attempt to get back to the realities of mathematical practice, and to approach philosophy of maths from there. I’d say since Lakatos at least (and perhaps even before) there’s been a steady rumble of philosophers of mathematics saying to other philosophers of mathematics ‘look, forget all that boring detached ontology, take a look at real life maths as it is practiced’).

But, while it is obviously important for philosophers of mathematics to work with actual mathematics in mind (and not just their idealized image of how mathematics should be), and while there are interesting philosophical questions to consider about how mathematics develops that may not have much obvious to do with the traditional ontological questions (I’m thinking eg of Maddy’s call to consider the contours of mathematical depth, though in fact there is also a metaphysical angle to her interest in this), the metaphysical and epistemological questions don’t just go away if we stop looking at them. And if one looks at how those philosophers of maths who still do care about the ontology are discussing those questions, it doesn’t look to me at all like these questions are being approached without due sensitivity to mathematics as it is actually practiced. One of the key questions in the ontological debate is how mathematics gets used in empirical science, and whether accounting for that use requires a realist understanding of mathematical objects. And for that philosophers of maths whose central concern is with the ontological question have had to get their hands dirty in looking at how mathematics becomes applied. So I think it does philosophy of mathematics a disservice to present those whose concerns are with metaphysics and epistemology in the mathematical context as ignoring the realities of mathematics as practiced (albeit given the direction the debate has gone, the recent concern has tended to be with mathematics in applications rather than pure mathematics).

I also worry sometimes about some of the regular calls to pay more attention to mathematical practice that, while this is all well and good, this doesn’t get us far if those who are looking at mathematical practice are not able then to use that attention to articulate clear *philosophical* questions about mathematics for which we might hope (also by looking at mathematical practice) to find answers. There are some nice cases where attention to mathematical practice does raise interesting philosophical issues aside from the standard ones (e.g. aside from articulating the notion of mathematical depth I mentioned earlier, there’s been some nice work on mathematicians’ use of apparent aesthetic notions in mathematics, and interesting questions to look at about the notion of explanatoriness at work within mathematics, and whether there is a clear distinction between (e.g.) explanatory and non-explanatory proofs that may have implications for philosophical accounts of explanation elsewhere). But just looking at mathematics as practiced doesn’t guarantee that you’ll find philosophically interesting aspects to remark on. One of my own frustrations with my own early case study work in the philosophy of mathematics is that I had to put a lot of work in (over a year observing a postdoctoral maths research seminar at the Fields Institute in Toronto, where, incidentally, I was not only the only non-mathematician in the room, but also the only woman too!) to come out with observations on mathematical development that, on reflection, seemed to me to be pretty obvious and not particularly philosophically exciting. So in response to Jeremy I’d probably say, yes it may well be a good thing for more philosophers of mathematics to get their hands dirty and pay a bit more attention to actual mathematical practice, but it will only really pay off if they have a good sense of what, from a philosophical perspective, it is that they’re looking for. And that bit might require them to spend at least a bit of time in the armchair!

I think Mary is about right here, though ultimately I’m not convinced that general questions about the existence of mathematical entities are well grounded. That aside, her central point is right. What does it mean to ‘attend to what mathematicians actually do’? Early in my career, I had a similar experience to her, only with historical mathematics rather than the seminars on C-star algebras that she attended. I found I had to spend an awful long time in the archives to extract even a tiny quantity of philosophical juice. I couldn’t write historically-based philosophy papers fast enough to sustain a career. I still wouldn’t be able to tell a graduate student what is the correct method in philosophy of mathematical practice. So far I have supervised precisely one PhD student to completion in this area. He is now going round giving talks about the urgency of having a methodological debate.

Also! If Lakatos is the ur-text for complaining about philosophers not attending to mathematical practice then there is some pre-history, e.g. Raymond Wilder _Mathematics as a Cultural System_

Loved this article when I read it yesterday, and I think Avigad nails a lot of things on the head, such as the rather incredible achievements that have been made in the last 150 years or so (mathematical logic, Godel, etc.). I will say (and I’m speaking only as someone with an undergraduate degree in philosophy but with a deep love for the subject, especially phil of math) that there does seem to be some of this going on with philosophers like Christopher Pincock and Sorin Bangu (I just downloaded a book of his that delves into the intersection of cognitive science, psychology, and neuroscience with mathematics: “Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science”), and more would definitely be welcome.

That said, “philosophy of mathematics” is always going to be concerned with questions of what mathematical objects/truths are, or how it is so powerful at describing the physical world (my own thoughts tend towards some sort of Pythagoreanism whereby the universe is in some deep sense mathematical because it is structured, and as Mike Resnik would say, mathematics is the “science of patterns”). Those questions seem to me like they will always be somewhat orthogonal to the question of mathematical intuitions potentially residing in the brain, or mathematics being solely something whose concepts we acquire from environmental and cultural backgrounds, etc.

I may have a bit of a bias towards mathematics and phil of math since I think it to be the most incredible edifice of human knowledge ever constructed (think for a moment: we’ve launched members of our own species to that bright thing in the sky we call the moon; we’re capable of knowing where the planet Uranus will be in exactly 1000 years to the day; we can compute the value of physical constants to 12 degrees of significance and then experimentally confirm that the prediction is correct…all directly because of mathematics). So yea, in general I’d be a fan of more quality, interesting work in that specialty.

“hink for a moment: we’ve launched members of our own species to that bright thing in the sky we call the moon; we’re capable of knowing where the planet Uranus will be in exactly 1000 years to the day; we can compute the value of physical constants to 12 degrees of significance and then experimentally confirm that the prediction is correct…all directly because of mathematics”

Physics played a certain role in those too…

David,

I definitely recognize that. Other things even: engineering, the general curiosity and adventurousness of the human species, etc.

My point is that “physics” wasn’t anywhere near capable of doing any of those things until it’s mathematization by individuals like Galileo, Newton, etc.

If there is a problem here, I doubt it is a special problem for the philosophy of mathematics. Some philosophers think about what’s happening in relevant disciplines, most of us just talk about things philosophers like to talk about. (I’m not making a call as to what we should be doing.)

And I feel Avigad overstates the degree of engagement with relevant disciplines in other areas of philosophy.

“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.”

Meditations on Generative Grammar seems to me to be a fairly rare topic in a phil language intro course or seminar. Debates about generative linguistics have been nowhere near as central to phil lang as debates about the reality of mathematical objects in phil math. If there was an analogous fixation in phil language it has to be the debates surrounding Millianism.

Most of our philosophical fixations in pretty much any area are probably not “healthy” in Avigad’s sense.

I think part of this is terminological. In philosophy of science there’s a moderately clean divide between general philosophy of science and the philosophy of particular sciences. The former tends to be concerned with quite broad epistemological and metaphysical issues: do we have knowledge of unobservable entities? to what extent is science a rational process? what are laws of nature? Examples in the former are often quite stylized and thin on technical details, and you can read the literature without advanced scientific knowledge. The latter is concerned with more specific questions that are much more embedded in a particular theory: what are the units of selection in evolution? Are symmetry-related solutions to physics equations redescriptions of the same underlying reality? How do we understand quantum mechanics? The examples are highly technical and you can’t really read the literature without a reasonably in-depth understanding of the theory.

This being philosophy, this divide is neither clean nor uncontested; indeed, count me among those who think general philosophy of science sometimes goes astray because its picture is too detached from the details. But still, the distinction is useful and tracks something important.

Pretty much all of philosophy of math is the equivalent of general philosophy of science. The questions it asks are very general (“what is mathematical knowledge?’ “do mathematical objects exist?”) and the examples mostly don’t go much beyond arithmetic and real analysis. There is logical space for a different kind of philosophy of math that is much more engaged with details and specifics, and would require practitioners to know a lot more math; we don’t really have a name for that other sort of philosophy of math, and very few people do it. I would strongly expect insights from it would helpfully inform “general philosophy of math”, and indeed my lay suspicion is that the (very set-theoretic) picture of math and mathematical reasoning embedded in general philosophy of math has tensions with practice that would be good to explicate. But it needn’t be seen as in competition with general philosophy of math, any more than philosophy of physics is in competition with general philosophy of science.

David, you say, “pretty much all of philosophy of math” is on the generalist side of the divide you mention, but a strong counterpoint to that would be the extremely active work currently happening in the philosophy of set theory—think of work of Penelopy Maddy, Peter Koellner, Hugh Woodin, myself, Carolin Antos-Kuby, Neil Barton, Toby Meadows, Sy Friedman and many others, who are doing philosophical work that is tightly connected with sometimes extremely technical work in set theory on the topics of forcing, large cardinals, inner model theory, the model theory of set theory, and so on. This is perhaps one of the currently most active subareas of the philosophy of mathematics, and it seems to be quite closely connected with the mathematical work in this area.

That bit I did know – but set theory is an interestingly special case because (at least as I understand it) it is itself very out of the mainstream of modern pure mathematics. There’s a tradition in philosophy (going back at least to Quine, I guess) of seeing set theory as a paradigm of pure math, but it’s actually very unusual.

I don’t find this to be accurate.

Set theorists have huge interaction with other parts of pure mathematics, and they are often well placed to answer foundational questions arising in other parts of the subject, such as questions about the axiom of choice or infinite cardinalities or foundational questions concerning Grothendieck universes in category theory or measure theory. Such kind of interaction can be seen, for example, on http://MathOverflow.net, where the set-theory tag is one of the most popular. In addition, descriptive set theory and particularly the theory of Borel equivalence relations is deeply connected with virtually all other parts of pure mathematics, for research in this area is focused on the set-theoretic complexity of the classification problems arising in those other fields.

Meanwhile, it is true that Quine’s version of set theory, New Foundations, is indeed outside the main stream, not only of mathematics, but of set theory itself.

I’m mostly basing the comment on conversations with mathematicians; the comment was sociological and I don’t know the sociology of mathematics that well. On a cursory look, the last Fields medal awarded for work in set theory seems to have been Cohen in 1966, which seems to support the hypothesis, but it’s scarcely conclusive (not least because it relies on my attempt to classify Fields medalists’ results on the basis of pretty brief descriptors).

My conversations with mathematicians very much agree with David’s. Most mathematicians view set theory as a strange topic that is not at all central to their interests.

The only way this statement makes sense is if by “set theory” you mean only the topics currently being researched in this area. But if you follow any branch of mathematics to it’s bleeding edge you will find topics that only a handful of people in the world care anything about. Hence, your statement does have an interpretation that is correct (i.e., most mathematicians don’t really care too much about the topics currently being researched in set theory), but I really doubt that any mathematician would agree with this post as written. Most mathematicians understand the foundations of mathematics in terms of set theory. The few who don’t prefer category theory.

Well to be fair that second kind of philosophy of Maths does happen. It tends to get called ‘Philosophy of mathematical practice’, and there’s even an Association for the Philosophy of Mathematical Practice. So it does exist and there’s plenty of people working in that area.

But I think where your comment is right is in directing us to think of that part of philosophy of mathematics as not necessarily in conflict with the part that focuses on the metaphysics and epistemology of mathematics, but just a specialism with a somewhat different focus.

(Reply was to David – x-posted with Joel.)

Thanks, that’s helpful; I gadn’t quite realized that was what philosophy of math practice addressed.

The Association for the Philosophy of Mathematical Practice that Mary mentioned has been around for almost 10 years now. For more information, including links to related events, a list of people working in the area, and a sample of recent publications in the area, see here:

http://www.philmathpractice.org

I’ve definitely thought of this divide between “traditional philosophy of math” and “philosophy of mathematical practice” as having some connection between the divide between “general philosophy of science” and “philosophy of particular sciences”. However, I think one notable difference between the divides is that “philosophy of mathematical practice” often still aims at mathematics as a whole rather than specific sub-fields within mathematics. This is how I see my work on the nature and role of mathematical proof, and much other work on mathematical explanation. In some ways, philosophy of mathematical practice is more like “philosophy of particular sciences” with “mathematics” as the particular science, rather than being related to “traditional philosophy of mathematics”.

There could be interesting and important work on philosophy of particular mathematical subfield. For instance, philosophy of category theory, or philosophical questions about the duality between algebra and geometry, or following the informal discussion among mathematicians about what it would mean for there to be a field of 1 element. This would be a lot more closely analogous to work in philosophy of biology about species concepts, or work in philosophy of physics about the interpretation of quantum mechanics. (Philosophy of set theory probably fits here, but is much more of a bridge to traditional philosophy of mathematics as well.)

Dichotomies make for punchy thought pieces and fruitful discussions. But a less coarse taxonomy might too yield dividends. Sure, there’s phil-math-as-metaphysics-and-epistemology and phil-math-as-phil-scientific-practice. But expand your ideas of what counts as philosophy and of what counts as relevant in our quest to understand mathematics and you’ll find at least two other possible (and actual) manifestations of the philosophy of math: the semiotics of math and the phenomenology of math.

I could certainly see someone arguing that these latter two approaches fit in one or the other of the former two categories, and that’s fine. But if they do so belong, they’re species of their genus and thus importantly differentiated from other species.

Representative examples of both the M&E and the practice approaches tend to artfully ignore the first-personal point of view, singular or plural. The practice approach is where you’d perhaps expect to find accounts of mathematical experience. And you’ll find some marvelous examples. But much of that approach takes an anthropological, from-the-outside perspective on the practice. A necessary perspective, for sure, but not exhaustive.

In his suggestive little essay, “Meditation in a Toolshed,” C. S. Lewis argued, in his way, that philosophers and others who prize the acquisition of understanding should not only “look at” things from the outside but “look along” things from the inside as well. He also argued that the cognitive fruits of “looking along” something are conceptually prior to the cognitive fruits of “looking at” something. I think his lessons, such as they are, bear application to the philosophy of mathematics.

It is certainly my impression that very little philosophical attention is devoted to understanding many of the things going on in mathematics, especially in the last 75 years, apart from work in foundations and logic, and in this way the philosophy of mathematics differs strikingly from the philosophy of physics. This is not just due to a relative lack of attention to issues about mathematical practice. A great deal of work has been devoted to understanding the content and meaning of some important results in mathematics, but very little work of that sort has been done on results in, say, algebraic geometry or number theory. Similarly, there are good questions to ask about how a branch of mathematics developed, what its central concepts are and how best to think about them, and how it is related to other branches of mathematics. There does not seem to be a lot of development of questions like these about most of the central work going on in mathematics in the past half-century.

I think this is partly due to inaccessibility. Very few philosophers are likely equipped to have much to say about, for example, Michael Atiya’s ideas, all over the news yesterday, about the Riemann Hypothesis, simply because we lack the relevant training. In typical cases I think that philosophers are too far removed from what is going on in mathematics even to know what ideas and results and developments might be rewarding to study. Philosophical attention to these things pays off when one is able to situate a result in conceptual space or historical context in some way that exposes more meaning or significance than the mathematicians themselves have been able to articulate. That is a tall order when one is not readily able even to appreciate everything that the mathematicians have articulated.

One worries, with Jeremy, that if the image of mathematics put forward by philosophers is sufficiently cut off from what mathematicians have actually been doing for many decades now, that image might be more a projection of how some careful thinkers image mathematics must be rather than even a partial depiction of what it is.

Mary is right that the call to arms over this alleged problem is as old as the problem itself. But it is also true that if there is a problem of this sort it is time-sensitive. A philosophy of math might be less attractive if it has been developed by a group of thinkers who are, as I am, largely unaware of or inattentive to what mathematicians have done in their own lifetimes.

People interested in these debates might like to read chapter 6 of Danielle Macbeth’s excellent recent book “Realizing Reason” (Oxford, 2014). There she contests standard mathematical logic’s understanding of mathematical proof as a deductive chain of reasoning which is as far as possible gap-free. She argues that in fact such a chain is not necessary or sufficient for proof in mathematics. It is not necessary because working mathematicians often leave large sections of their working undescribed (as ‘obvious’). It is not sufficient as rendering any significant proof gap-free often lengthens it to the point of incomprehensibility. Proofs are not meant to convey deductive entailment so much as understanding, she suggests – thus the mathematical logicians’ formalizations are no use in actual mathematical practice.

For a philosopher who has engaged knowledgeably and enthusiastically with the vast new intellectual horizons opened up by C20th mathematicians such as Grothendieck, see the incomparable Fernando Zalamea – especially his book “Synthetic Philosophy of Contemporary Mathematics” (Sequence 2012).