Structures, Apples and Pears, and Why Numbers are Not Objects — A Conversation with Alexander Paseau

In early July, I met with Alexander Paseau, who is Professor of Mathematical Philosophy at Oxford. Professor Paseau specialises in logic and philosophy of mathematics, and is currently—among other things—finishing work on a new book, in which he defends structuralism in the philosophy of mathematics. We had the chance to meet in person in Caffè Nero near Paddington station, in London, to talk about this.

Our conversation was superb—we talked about structures in mathematics, a bit about apples and pears in one’s education, whether J.S. Mill would be accepted into Oxford (yes, he would), and whether structuralism is an intuitive position (yes, but it depends). All of us at Tvērums are delighted and honoured to start our online publishing endeavours with this interview.

Alexander: Some stuff we say will just be chat and won’t make it in to the magazine, I assume. For example, if I asked you how your coffee is.

Emil: It’s quite alright. I wouldn’t mind putting that in the magazine. I might put this very thing in the magazine.

A: I don’t know if you read the FT—Financial Times. On Saturdays, they do an interview with someone of interest. And they always take them out for lunch, and the lunch is very much part of the interview. They start with a few paragraphs about the person and their ideas and then the first course comes along, and they include the menu and the price and everything, and interweave a description of the food with the discussion. So, it’s a very integrated interview experience.

E: Oh, that’s great.

A: I mean, apple juice and iced coffee is less exciting.

E: That’s true. We met two weeks ago in Munich, though. Your flight was cancelled. Did you lose any luggage?

A: I didn’t lose any luggage.  I didn’t even get on the plane. I got to the airport then tried to scan my boarding pass, which didn’t work. They told me to go to desk C1. I went to desk C1, where they informed me that my flight was cancelled. They told me to speak to BA (British Airways). I told them—but you are BA and I want to speak to you (Both laugh), to rearrange my flight. They said—no, you’ve got to call this number. I called the number, and, of course, no one answered. Then they booked me on a flight 24 hours later. So I just accepted that. And then I asked them to compensate me, and they told me that they couldn’t, because of bad weather, which was news to me. Thunderstorms, they said. The weather that afternoon looked completely fine to me, from where I was standing. There you go.

E: But then you came to Munich, and you did a marathon of classes for two days. And thanks for that.

A: Yes. It was a pleasure. Thank you for attending!

E: Well… It was my pleasure! Now, in the masterclass, we covered a lot of philosophy of mathematics and logic. We also talked about your book, the one that you’re finishing working on at the moment, called What is Mathematics About?

A: Right.

E: What is mathematics about? (Alexander laughs)

A: Well, let me first say why that’s an interesting question. One of the great puzzles about today’s mathematics is that it combines two features. First, it seems to be objectively true, and  indisputably true. No one wonders whether Pythagoras’ theorem is true, or whether 1 + 2 = 3; everyone accepts that mathematics is true. So: maths seems to be objective and independent of us, and we don’t get to make it up. But here’s the second feature: maths is not experimental; it’s not  about the physical world. When we learn mathematics at school, at university, we don’t perform experiments. Maths seems to be purely conceptual, and more abstract than the physical sciences, which are about the natural world in some shape or form. That for me is the real puzzle; it’s what, for me, gives the philosophy of mathematics its interest. On the one hand, objectively true, on the other hand not about the physical. That’s the starting point, for me.

Now, to answer your question—my view is that mathematics is about structures. Arithmetic is about any structure of a certain kind satisfying specific axioms. Setting up the  axioms can be quite  explicit, the way logicians do it, or more implicit, the way most  other people do it. When you do school mathematics, you have an implicit conception of what a structure is. And this is not just true for arithmetic, it’s true for the real numbers, for geometry, for graphs, for functions, really for all areas of mathematics—apart from the one area whose job is to provide structures for mathematics. That, in short, would be my answer—mathematics is about any structure that satisfies certain axioms from the relevant branch of mathematics. But then you also need some guarantee that these structures exist, and you need some foundation that furnishes or provides these structures. Mathematics of the last few decades assigns set theory that role.

E: Yes, that is the case. Let’s talk about structures a bit more, and let’s try to get an intuitive feel for what that sort of thing might be. Would you agree with the characterisation that when, for example, I say “one”, “two” and “three” and so on, I’m not referring to objects in any sense, but rather I’m referring to places or empty spots that can be characterized by their relations to one another, and nothing else?

A: Yeah, I would agree with that. They are placeholders in any particular structure. Let me give you an example. It’s a bit like talking about American politics. You can do that by talking about specific people, right? Right now, the President is Joe Biden, the Vice-President is Kamala Harris and so on. Or you can do it in a much more abstract way, say, if you’re thinking about the structure of the government: there’s the President, the Vice President, then there’s the Senate, the House of Representatives and so on; and you say things like: “If the Senate is in deadlock, then the Vice President has a casting vote” and so forth. You can talk about these political offices in an abstract way, without identifying the role with any particular individuals. It’s not a perfect analogy, but it is a pretty good one.

E: Oh, yes, that makes sense. We can even talk about the structure of the government colloquially.

A: Exactly. Without needing to know anything about the particular individuals who hold the offices.

E: So, in mathematics, do we have these particular individuals holding the offices of, e.g., the natural numbers—the individual holding the office of the 1st natural number, the 2nd natural number…

A: No, I do not think that.

E: You do not think that?

A: I don’t.

E: Can you elaborate?

A: Yes. Because mathematics is so very abstract and so very general. Doing mathematics is like talking about any political system. For example, you could come up with your own political system—oh, look, here’s a president, a prime minister, and two legislative chambers and so on. And you could just sort of make it up, according to some principles. That’s sort of how I think that mathematics proceeds—there aren’t any particular entities that play those roles. You’re just talking about an abstract structure of a certain sort that you can make up as you like, subject to consistency.  It’s contingent that in the American case, the specified political structure happens to be instantiated in the physical world. But mathematics doesn’t have to be tied down to actual existing systems. It can conceive of any such system, as long as it’s consistent.

E: Alright, I see. We have the structures, we don’t have mathematical objects in the structures, just the structures.

A: Yes, no objects beyond the structures. But the structures have to come from somewhere. And that’s the second part of my view.

E: Right. And that’s what I’m aiming at with the following question, perhaps. Isn’t it hard to characterise what a structure is?

A: Mathematics has answered that question for itself. It says that they’re sets. Sets of a particular kind. Suppose you have a set of entities that play the role of the natural numbers. A property of those numbers will just be a subset. The even ones will be the subset {0, 2, 4, 6 …}. That subset will be your property. So with sets you can mimic properties. If you take a set theory course as a student, they might teach you how to do that. In fact, it’s a mathematical theorem that you can use sets to build anything you like. Thus, sets can be used as building blocks to build any structure you like; and mathematics has solved that question by saying: just let sets do it.

In the philosophy of mathematics, taking this more or less universal feature of mathematics—that it builds its structures out of sets—at face value was popular a few decades ago. But it’s fallen out of fashion, for various reasons we can go into. Now, I want to try and revive that view. I think this picture, which I call structuralism-within-set-theory (but it doesn’t really matter what you call it), that you use sets to build to structures of mathematics and everything else is just general talk about things that satisfy various axioms – well, I think that’s the correct philosophy of mathematics. There’s a slight twist to that, which I mentioned in Munich, namely, it doesn’t really matter that the structures are sets. It could be anything. But there’s got to be some foundation or other.

E: That also sounds very structuralist—like structuralism about the theory of structure.

A: My point is that there has to be one theory of structure. If you have 52, then 51 of them are redundant. Whether that one foundation that provides the structures is set theory, category theory, or something else altogether doesn’t make a huge difference, as long as it’s more or less mathematically equivalent to contemporary set theory. Each foundation has its own pros and cons, but they don’t matter terribly. What matters is the overall picture: one foundation that provides the structures for mathematics; and any other area of maths is general talk about structures (provided by the foundation) that satisfy that area’s axioms.

E: I think now we have a pretty decent idea of what structuralism within-set-theory or within-a-theory-of-structure is. You’ve taught courses on this at university, I assume?

A: Correct.

E: I used to teach philosophy to high school students a few years ago. When we eventually got around to philosophy of mathematics, I’d always ask them if they though numbers are out there in some sense. What I found interesting was that none of them seemed to be platonists, none of them seemed to be structuralists. Rather, it seemed as though they had some intuitionistic sympathies. They said: well, numbers are in the mind, mathematics is a construction in the mind and so on. My anecdotal data prompts me to think that at least high school students think of mathematics that way. Have your students at university shown such dispositions? Are they generally welcoming when you introduce them to structuralism?

A: Okay, that’s quite an interesting observation. I’ve got a few comments about that.  When you’re doing mathematics, it’s quite intuitive to think of the objects you’re talking about as being independent of us and to think of yourself as discovering facts about them. I mean, it just seems as if there must be an answer to whether the number 2256 + 1 is prime or not. It’s just a fact you can discover. On the other hand, when you take a step back and think philosophically about it, it just seems very odd that non-spatiotemporal objects somehow exist. In the moment,  platonism is intuitive, but when you start reflecting on it, platonism starts to seem contrary to common sense—the objects of mathematics are quite unlike the physical ones we’re used to.

What I think is interesting about the high school students you taught is that the idea of mathematics as structure really kicks in a little bit later in one’s mathematical education. High school mathematics is pre-19th century mathematics. It consists of Euclidean or Cartesian geometry, say plotting curves, or finding derivatives or integrals of functions, or number theory, or solving quadratics—that sort of thing. But as you know from the Munich masterclass, I think—along with other philosophers of mathematics—that the 19th century witnessed a decisive transformation in mathematics. Mathematics before the 19th century could reasonably be conceived as the quantitative study of space and time. But then in the 19th century it changed completely and became something much more abstract, much more structural. Once you start doing abstract algebra, group theory, ring theory, field theory, abstract analysis, topology, functional analysis and so on, the level of abstraction is much higher than in high school mathematics. And then the thought that mathematics is structural becomes  much more obvious and attractive. And this is hidden when you’re doing high school mathematics. Now, I’m speculating, but that might account for the intuitions of your high schoolers.

E: Great! But we did mention platonism, so perhaps we should unpack that for the readers.

A: Platonism is a label that is not particularly faithful to Plato. Plato, I think, was a sort of platonist (both laugh), you might not be surprised about that. Actually, there is a recent book entitled Plato Was Not a Mathematical Platonist, which I haven’t read, but, I have to say, the title seems a bit off, or at least not quite right. I realise of course it’s meant to be provocative. Anyway, platonism is not supposed to be very faithful to Plato’s philosophy. Roughly, it’s the idea that mathematical objects do exist, they’re just not spatiotemporal, nor causal. To use a piece of philosophical jargon, they’re abstract. Not in the sense that mathematicians use the word ‘abstract’, but the way philosophers do. As I said earlier, that’s an unintuitive thought. We’re accustomed to thinking of objects as physical—tables and chairs, and the like, and then science tells us that there are much smaller particles that make up the chairs, you know, molecules and atoms and so on, and that there are very large things like galaxies and the universe. Then the idea that there exist things that are not spatiotemporal comes along, and it’s initially very strange, or at least it was for me, but that’s exactly what platonists believe.

E: Unlike Plato though, who thought that the realm of Ideas, or the realm of Forms, is more real than the world of appearances, what we nowadays call platonists about mathematics do not have this distinction. The mathematical abstract objects are on a par with the rest of the world, they’re just different kinds of things.

A: Exactly. They’re happy egalitarians about existence. (Both laugh lightly.)

E: But is a structure an abstract object?

A: Well, I think a structure can be a set, if it’s implemented in set theory. What you don’t want to do if you take this view is to say: “Well, look—there’s the sets that implement the structure and then there’s the structure itself on top of it.” Once you’ve got sets, they’ll do all the work for you. Why on Earth would you then say: “And on top of that there are all these abstract structures.” There are people who hold this view—they reify structures—but I don’t think that’s the right way to go. Because then there’s all sorts of thorny questions about these abstract objects and abstract structures, as detailed in What is Mathematics About?

E: Alright. Thanks. (Slight pause.) So… I’ll have a look in my notes.

(Pause, notebook pages ruffle)

E: Oh, yes—the apples and pears!

A: (laughs) Okay, apples and pears, yes. I’m having apple juice—that’s relevant.

E: You are! How many apples went in the juice?

A: Possibly zero, by the taste of it. (Both laugh.)

E: When I was in elementary school, my mathematics teacher always used to draw apples on the blackboard, and we did addition and subtraction with the apples. And then there was another level of abstraction, when she drew a couple of apples and pears and asked us how many pieces of fruit are there? What might be the concept of number in the background here?

A: Yeah, I think what’s going on there is that for pedagogical reasons it’s very useful to introduce mathematics through its applications, and clearly number theory applies to the physical world, and you can talk about collections of things and assign them numbers—three apples and two pears make five pieces of fruit. Then you sort of realise it need not be three apples and two pears but could just as well be three tables and two chairs, and then you realise it doesn’t really matter what these objects are. But I think that pedagogically it’s extremely sensible to start with physical objects that young children can handle and can think of in concrete terms.

Now, there is a view that mathematics, and number theory in particular, is about these collections of things—of physical objects. It was advocated by John Stuart Mill, the 19th century philosopher, who made contributions to all sorts of areas of philosophy, and not just philosophy, but also politics and economics. He wrote a book called A System of Logic, in which he defended this view philosophically, that mathematics is the study of physical objects, just like physics, but more high-level and general.  Maths as a sort of super-high-level empirical theory. This doesn’t seem plausible to us, because mathematics has become so abstract, as I said earlier. Mill’s philosophy is at best a philosophy of elementary mathematics, where the examples one uses are concrete objects.

E: Which ties together nicely with your previous thought about the ladder of abstraction in one’s mathematical education.

A: Absolutely.

E: I’m also glad you mentioned Mill because he’s, of course, very well known in Latvia, but more for his work in ethics, than in philosophy of mathematics.

A: He was a universal thinker who contributed to so many areas. At Oxford, we have a degree called Philosophy, Politics and Economics. We often joke that he would be the perfect candidate for this course, because he made such significant contributions to all three subjects.

E: Ah, I had a decent question here, but I think I’ve lost it… (Slight pause) Yes, it’s gone. Wasn’t there a wax sculpture of Mill somewhere, or am I mixing up philosophers?

A: You’re thinking of Bentham. That’s Bentham at UCL (University College London).

E: Oh, wait—that’s here! I can go visit!

A: (Laughs)

E: I think due to time constraints we should start wrapping up. Oh, it’s a pity I lost that question—it was a good one, I think.

A: We could’ve solved the problems of philosophy of mathematics! (Both laugh.)

E: Sounds very Wittgensteinian! (Both laugh.) I’ll ask you about Russell, then. This is a question I plan on asking everybody. Everybody, who agrees to the interview, that is.

A: Oh, I see—you don’t mean to ask people on the street as well?  

E (laughing): No, no, no. Well, actually, that might also be fun. I’ll look into that. (Both laugh.) In any case, there’s this quote from Russell: “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”* Now, Russell wasn’t a structuralist, but still—do you find the quote agreeable?

A: I think what’s going on there is that he’s groping for a kind of structuralist view, which he’s not quite able to formulate at that point in time. It’s the early 20th century, so structuralism hadn’t quite crystalized yet, but he’s sort of seeing through a glass, darkly, that (something I think of as) structuralism is the right answer to the question of what mathematics is about. I would divide the quote into two parts. I think the first part— Mathematics may be defined as the subject in which we never know what we are talking about—is a kind of proto-structuralist idea. What I think he’s saying—charitably interpreted—is something structuralist: it doesn’t matter what the entities of mathematics are. I don’t know what exactly he had in mind, but that’s one way you could interpret him. But I think the second half of the quote—nor whether what we are saying is true—is definitely wrong. That’s just flat-out false, I would say.

E: Yes, I find that bit very puzzling.

A: I mean, Russell was a great writer, but sometimes he let the style win over the thought.

E: You think this is just literary embellishment?

A: Yes, it’s literary flourish, and it does sound great. But I think if he’d been a little more careful, he wouldn’t have written quite that.  Remember, Russell actually didn’t revise most of his texts. At the time he was working on the philosophy of mathematics, he  wrote at an astonishing rate. He averaged several thousand words a day over long periods. All dictated to an amanuensis, I think. He was a very fluent and brilliant writer, but he was less careful than a philosopher writing today would be.

One of the quotes that I particularly like by Russell about mathematics is this: “The 19th century, which prided itself upon the invention of steam and evolution, might have derived more legitimate title to fame from the discovery of pure mathematics.”** This really chimes with my story of how mathematics was transformed over the course of the 19th century. Russell, as usual, put it very slickly.

E: Indeed! I think this is a good time to conclude; thank you very much for meeting me. You know, I just realized I had a blazer on this morning, and I think I’ve accidentally left it at a café near University College London.***

A: Oh, that’s terrible! You should go get it!

E: Yes, I think I should.



* Bertrand Russell, “Recent Work on the Principles of Mathematics,” in The International Monthly 4 (Jul 1901), pp. 83-101.

** Ibidem.

*** Luckily, E found the blazer exactly where he left it.