This post – which I’ve split up into parts – is a bit of a departure from talking about the subject matter of mathematical ideas, and more about mathematics in general. In particular, a while ago I was asked a question by a philosopher friend about topology and topos theory as he was trying to understand Alain Badiou’s writings about ontology. That eventually led to my reading a bit more about what recent philosophers have to say about mathematics, or to use it for. This eventually led me to look at Fernando Zalamea’s book “The Synthetic Philosophy of Contemporary Mathematics”. It’s not a new book, unless 2009 counts as new at this point 8 years later. But that’s okay: this isn’t a book review either (though I did find one here). However, it’s the book which was the main jumping off point for the thoughts I’m putting down here. It’s also an interesting book, which speaks to a lot of the same concerns that I’ve been interested in for a while, and while it has some flaws (which I’ll speak to briefly in part II), mostly I want to treat it as a starting point.

I suppose the first issue in talking about the philosophy of mathematics, if your usual audience is for talking about mathematics itself, is justifying why philosophy of mathematics in general ought to be of interest to mathematicians. I’m not sure if this is more, or less, true because I’m not a philosopher, but a mathematician, so my perspective isn’t a very sophisticated reader of the subject, but as someone seeing what it has to say about the field I practice. We mathematicians aren’t the only ones to be skeptical about philosophy and its utility, of course, but there are some particular issues there’s a lot of skepticism about – or at least which lead to a lack of interest.

Why Philosophy Then

My take is that “doing philosophy” is most relevant when you’re trying to carefully and systematically talk about subjects where the concepts that apply are open to doubt, and the principles of reasoning aren’t yet finally defined. This is why philosophers tend to make arguments, challenge each others’ terms, get accused of opacity (in some cases) and so on. It’s also one reason mathematicians tend to be wary of the process, since that’s not a situation we like. The subject matter could be anything, insofar as there are conceptual issues that need clarifying. One result is that, to the extent the project succeeds at pinning down particular concepts and methods, whole areas of philosophy have tended to get reframed under new names: “science”, a more systematic and stable version of the older “natural philosophy”, would be one example. To simplify the history a lot, we could say that by systematically describing something called the “scientific method”, or variations on the theme, science was distinguished from natural philosophy in general. But the thinking that came before this method was described explicitly, which led to its description, was philosophical thinking. The fact that what’s left is necessarily contentious and subject to debate is probably part of why academics in other fields are often dubious about philosophy.

Similarly, there’s the case of logic, which began its life in philosophy as an effort to set down systematic ways of being sure one is thinking rigorously (think Aristotle’s exposition of how syllogisms work). But later on it becomes a topic within mathematics (particularly following Boole turning it into a branch of algebra, which now bears his name). When it comes to philosophy of mathematics in particular, we could say that something similar happened as certain aspects of the topic got formalized and become the field now called “metamathematics” (which studies things such as whether given theorems are provable within specified axiom systems). So one reason philosophy might be important to mathematicians is that the boundary between the two is rather porous. Yet maybe the most common complaint you hear about philosophy is that it seems to have become stuck at just the period when this occurred – around 1900-1940 or so, motivated by things like Hilbert’s program, Cantorian set theory, Whitehead and Russell’s Principia, and Gödel’s theorem. So that boundary seems to have become less permeable.

On the other hand, one of the big takeaways from Zalamea’s book is that the philosophy of mathematics needs to pick up on some of the themes which have appeared in mathematics itself in the “contemporary” period (roughly, since about 1950). So the two fields have a history of exchanging ideas, and potential to keep doing so.

One of these is the sort of thing you see in the context of toposes of sheaves on a site (let’s say it’s a topological space, for definiteness). A sheaf is a kind of object which is defined by what it looks like locally in each open set of the space, which is constrained by having to fit together nicely by gluing on overlaps – with the sheaf condition describing how to pass from local to global. Part of Zalamea’s program is that philosophy of mathematics should embrace this view: it’s the meaning of the word “Synthetic” in the title, as contrasted with “Analytic” philosophy, which is more in the spirit of the foundational approach of breaking down structures into their simplest components. Instead, the position is that lots of different perspectives, each useful for understanding one theme, some part or aspect of mathematics, can be developed, and integrated together by being careful to account for how to reconcile them in areas where they both have something to say. This is another take on the same sort of notion I was inspired by when I chose the name of this blog, so naturally, I’m predisposed to be interested.

Now, maybe it’s not surprising that the boundary between the two areas of thought has been less permeable of late: the part of the 20th century when this seems to have started was also when many fields of academia started to become more specialized as the body of knowledge in each became so huge that it was all anyone could do to master one special discipline. (One might also point to things like the way science became a much bigger group enterprise, as witness things like the Manhattan Project, which led into big government-funded agencies doing “big science” in an institutional setting where specialization and the division of labour was the norm. But that’s a big digression and probably needs more data to support it than I’ve actually got.)

Anyway, Whitehead and Russell’s work seems to have been the last time a couple of philosophers famously made a direct contribution to mathematics. There, the point was to nail down a definite answer to the question of how we know truth, what mathematical entities are, how logic functions and gives rise to more complex mathematics, and so on. When Gödel, working as a mathematician, showed how it was incomplete, and was construed as doing mathematics (and if you read his paper, it’s hard to construe it as much else), that probably contributed a lot to mathematicians drifting away from philosophers, many of whom continued to be interested in the same questions.

Big and Small Scales

Still, even if we just take set-theoretic foundations for granted (which is no longer quite as universal as it used to be), there’s a distinction here. Just because mathematics can be reduced to set theory and logic, this doesn’t mean that the philosophy of mathematics has to reduce to the philosophy of set theory and logic. Whatever the underlying foundations, an account of what happens at large scales might have very different features. Someone with physics inclinations might describe it as characterizing the “effective theory” of mathematics – the way fluid dynamics is an effective theory of particular kinds of big statistical ensembles of atoms, and there are interesting things to say about each level in its own right.

Another analogy that occurs to me is with biology. Suppose we accept that biology ultimately reduces to chemistry, in the sense that all living things are made of chemicals, which behave exactly as a thorough understanding of chemistry would say they do. This doesn’t imply that, in thinking about biology, there’s nothing to think about except chemistry: the philosophy of biology would have its own issues, because biology entails new concepts, regardless of whether there happens to be some non-chemical “vital fluid” that makes living things different from non-living things. To say that there is no such vital fluid is an early, foundational, part of the philosophy of biology, in this analogy. It doesn’t exhaust what there is to say about living things, though, or imply that one should just fall back on the philosophy of chemistry. A big-picture consideration of biology would have to take into account all sorts of knowledge and ideas that get used in the field.

The mechanism of evolution, for example, doesn’t depend on the thermodynamic foundations of life: it can be applied to processes based on all sorts of different substrates – which is how it could inspire the concept of genetic algorithms. Similarly, the understanding of ecosystems in terms of complex systems – starting with simple situations like predator-prey models and building up from there – doesn’t depend at all on what kind of organisms are involved, or even that they are living things. Both of these are bodies of knowledge, concepts, and methods of analysis, that play a big role in studying living things, but that aren’t related at all to the foundational questions of what kind of physical implementation they have. Someone thinking through the concepts in the field would have to take them into account and take into account their own internal logic.

The situation with mathematics is similar: high-level accounts of what kinds of ideas have an influence on mathematical practice could be meaningful no matter what context they appear in. In fact, one of the most valuable things a non-rigorous approach – that of philosophy rather than, say, metamathematics as such – has to offer is that it can comment when the same themes show up even in formally very different sub-disciplines within mathematics. Recognizing these sorts of themes, before they can be formalized and made completely precise, is part of describing what mathematicians are up to, and what the significant features of that practice may be. Discovering those features, and hopefully pinning them down enough to get one or more ways to formalize them that are rigorous to use, is one of the jobs philosophy ought to be able to do. Zalamea suggests a few such broad patterns, which I’ll try to unpack and comment on a little in Part II of this post.


Historical Change

Even granted the foundational questions about mathematics, there are still distinctive features of what people researching it today are doing which are part of the broader picture. This leads into the distinction which Zalamea makes between the different characteristics of the particular mathematics current in different periods. Part of the claim in the book is exactly that this distinction isn’t only an arbitrary division of the timeline. Rather, the claim is that what mathematicians generally are doing at any given time has some recognizable broad features, and these have changed over time, depending on what were seen as the interesting problems and productive methods.

One reason mathematicians may have tended to be skeptical of philosophy (beyond the generic reasons) is that by focusing on the typical problems and methods of work of the “Modern” period, it hasn’t had much to say about the general features that actually come up in contemporary work. David Corfield made a similar argument in “Toward a Philosophy of Real Mathematics”, where “real” meant something like what Zalamea calls “contemporary”: namely, what mathematicians are actually doing these days.

This outlook suggests that, just as art has evolved as new ideas are created and brought into the common practice, so has mathematics. It contrasts with the usual way mathematicians think of themselves as discovering and exploring truths rather than creating the way artists do. It probably doesn’t have to be: the continents are effectively eternal in comparison to human time, but different people have come across them and explored them at different times. Since the landscape of possible mathematics is huge, and merely choosing a direction in which to explore and by what methods (in the analogy, perhaps the difference between boating on a river and walking overland) has a creative aspect to it, the distinction is a bit hazier. It does put the emphasis on the human side of that historical process rather than the eternal part (already a philosophical stance, but a reasonable one). Even if the periodization is a bit arbitrary, it’s a way of highlighting some changes over time, and making clear why there might be new trends that need some specific attention.

Thus, we start with “Elementary” mathematics – the kind practiced in antiquity, up through about the time of invention of Calculus. The mathematics in this period was closely connected to the familar world: geometry as a way to talk about space, arithmetic and algebra as tools for manipulating numbers, and so forth. There were plenty of links to applications that could easily be understood as being about the everyday world – solving polynomial equations, for example, amounts to finding quantities that have special properties with respect to some fairly straightforward computation that can be done with them. Classical straightedge-and-compass constructions in geometry give a formal, idealized way to talk about what can be more-or-less well approximated by literal physical operations. “Elementary” doesn’t necessarily mean simple: there are very complex bits of mathematics in these areas, but the building blocks are fairly simple elements. Still, in this period, it was possible to think of mathematics itself as a kind of philosophy of real things – abstracting out ideal properties and analyzing them, devising rules of logic and calculation, and so on. The sort of latent Platonism of a lot of mathematical thinking – that these abstract forms are an underlying reality of particular physical things –

Then “Classical” (the period when Leibniz, Euler, Gauss, et. al.) when mathematics was still a fairly unified field, but with new methods, like the rigorous use of infinite processes. It’s also a period when mathematics itself begins to generate more conceptual issues that needed to be talked about in external language.Think of the controversy over the invention of Calculus, and the use of infinitesimals in derivatives and integrals, the notion that an infinite series might converge, and so on. Purely mathematical solutions were the answer, but arose only after there’d been an outside-the-formalism discussion about the conceptual problems with infinitesimals. This move away from elements that directly and obviously corresponded to real things was fruitful, though, and not only because it led to useful tools like Calculus. Also because that very fact of thinking about idealized or abstract entities opened up many of the areas of mathematics that came later, and because trying to justify these methods against objections led to refinements like the concept of a limit, which led into analytical arguments with “epsilons” and “deltas”, and more sophisticated use of quantifiers like the “for all” and “there exists”. Refining this language opened up combinatorial building-blocks of all sorts of abstract concepts.

This leads into the “Modern” period (about 1850-1950), where people became concerned with structure, axiomatization, foundational questions. This move was in part a response to the explosion of general concepts which those same combinatorial building blocks encouraged. Particular examples of, for instance, groups may very well have lots of practical applications, but here we start to see the study of the abstract concept of a group as such, proof of formal theorems about it, and so on. In algebra, Jordan and Cayley formally set out the axioms of rings, groups, fields, etc. which people had been studying for some time in a less explicit way (as, for instance, in Galois theory). The systematization of geometry by Klein, Riemann, Cartan, and so forth, was similar: particular geometries may well have physical relevance, or be interesting as examples, but by systematizing and proving general theorems, it’s the abstractions themselves that become the real objects of study for mathematicians as such.

As a repertoire of such concepts started to accumulate, the foundational questions became important: if the actual entities mathematicians were paying attention to were not the elementary ones, but these new abstractions, people were questioning what, in ontological terms, those things actually were. This is where the investigation topics like the relation of set theory to logic, the existence of set-theoretic models of formal theories, the relation between provability of theorems and the existence of models with particular properties, consistency of axioms, and so on, came to the forefront.

Zalamea’s book starts with an outline of Lautman’s description of five big themes in mathematics that became prominent in the Modern period, and then extends them into the “Contemporary” period (roughly, after 1950) by saying that all the same trends continue, but a bunch of new ones appear.  One of these is precisely a move away from a focus on the specific foundational description of a structure – in categorical language, we’ve tended to focus less on the set-theoretic details of a structure than on features that are invariant under isomorphisms that change all of that. But this gets into a discussion I’ll save for the second part of this post.

For now, I’ll say just a couple of things about this approach to wrap up this part. I have some doubts about the notion that the particular historical evolution of which themes mathematics is exploring represent truly different “kinds” of math, but that’s not really the claim. What seems true to me is that, even in what I described above, you can see how spending time exploring one issue generates subject matter that becomes a concern for the next. Mathematicians on another planet, or even here if we could somehow run through history again, might have taken different paths and developed different themes – or then again, this sequence might have been as necessary as any logical deduction. This is a lot harder to see, though the former seems more natural to my mind. Highlighting the historical process of how it happened does, at least, help to throw some light on some of the big features of what we’ve been discovering. Zalamea’s book (which, again, I’ll come back to) makes a particular attempt to do so by suggesting three main kinds of contemporary math (with their own neologisms describing them). Whatever you think of the details of this, I think it makes a strong case that looking at these changes over time can reveal something important.