Is Set Theory THE foundation?

What is the foundation of mathematics? A couple of years ago, being totally naïve, I would have said that numbers are the most basic kind of object in mathematics, but I would have been at a loss to give a clear definition of the concept of number. At the time, I knew very little math; just basic arithmetic and what some call “baby logic”. Even so, I’d wondered for a long time about the basis for “the queen of the sciences”, that one field of knowledge where complete certainty could be achieved. Therefore, I decided to jump into the philosophy of mathematics to try to answer that vague question.

I was fortunate enough to find Peter Smith’s website, which has a lot of material & resources for the autodidact. There I could trace a learning trajectory from baby logic to set theory, which is where I suspected I would get, if not the answer, at least a better idea of how the answer might go. (Philosophy is renowned for not providing any definitive answers to anything!) Well, it’s been a tough 2 and a half years, and not only do I not feel close to reaching my goal, I recently discovered something that seems to completely upset the whole plan!

You see, set theory is traditionally considered to be the foundational framework on which the whole of mathematics can be built. The idea is that every mathematical concept can in the end be reduced to the concept of a set. In set theory, numbers can be defined as sets, e.g. \(0\) is the empty set \(\{\}\), \(1\) is the set containing the empty set \(\{\{\}\}\), \(2\) is the set containing the set that contains the empty set \(\{\{\{\}\}\}\), etc.¹ Of course, there is the question of what a set itself is, meaning that the question of what a number is gets pushed back again. Still, there is a great tradition backing set theory as the foundation for mathematics. And all this time until a few weeks ago, my impression was that there was no other candidate for the role. And then I found out about category theory…

The problem with functions

It happened in the early hours of the night. I was feeling a little bored with the study of lambda calculus, which I thought was leading me astray from my main goal. So I decided to poke around other areas in mathematical logic, that branch of mathematics/philosophy covering the foundational aspects of mathematics. And, to my surprise, I found that category theory, while not part of mathematical logic, is nevertheless an active area of research into said foundations. Well, I just had to find out more about it!

It turns out that category theory gives a more general, more abstract conception of a mathematical object than even set theory. This took me by surprise! Not only that, but researchers in this area actually have arguments against the reduction of some mathematical objects to sets. The most prominent among those objects are functions which, in set theory, are reduced to sets of ordered pairs.

But, first, what is a function? The simplest answer, and the most intuitive one, is usually given by way of a simile: a function is like a box into which you put something and out of which you get a result. So, for example, if we are speaking of the successor function, then we can picture a box into which we input a number and we pull out its successor. That is, if we put in a \(1\), we pull out a \(2\). If we put in a \(2\), we pull out a \(3\), and so forth. If we are speaking of the addition function, we imagine a box that takes two inputs, and gives one output. For example, if we input a \(2\) and a \(3\), we get a \(5\). If the box is the multiplication function, and we input a \(2\) and a \(3\) we get, not a \(5\), but a \(6\), since the function multiplies them.

This is how I first learned the concept, and it is, like I said, very intuitive and easy to grasp. However, using this simile is just a heuristic device. A function is not really a box! Besides, isn’t the realm of mathematics a static realm? The world of math is not like the physical world, so there shouldn’t be any kind of motion or change.

Enter set theory. Instead of a magic converter box, set theory tells us that a function is a set of ordered pairs (of objects). Graphically, we can represent a function as a list of ordered pairs. For instance, the successor function can be represented by the list \((1,2)(2,3)(3,4)(4,5)\dots\) and so on.

As you can see, a list (really, a set) implies no motion—there’s no putting anything in and pulling anything out. So, I was pretty satisfied by this definition. It’s nice and “static”, with the kind of permanence that befits that ethereal world of mathematics. But then category theory had to ruin it!

The objections

The first objection has to do with the assumption that mathematical objects have to be static. In geometry, for instance, we have functions that do imply motion, e.g. translation, rotation, etc. And no one would say that geometry is not part of mathematics. In fact, the very word “function” implies a role for a certain action! Now, it may be said that such a dynamic conception of geometry is flawed, but I take the main thrust of the objection to be that, whether math is static or dynamic, its nature must not be assumed one way or another without argument, which is precisely what I had done.

Another objection has to do with the underlying concept of an ordered pair. When studying set theory, the usual definition given for an ordered pair is that by Kuratowski, i.e. that an ordered pair \((a,b)\) is the set \(\{a, \{a,b\}\}\). However, it is not the only definition! (I knew this before reading about category theory, but it just did not register in my mind that it might be a problem.) There are other definitions of ordered pair, all of them valid. But if that is the case, then a function cannot be reduced to a set of ordered pairs, as there is simply “no fact of the matter” about what an ordered pair is in terms of sets. Worse! Many set theorists believe the various definitions to be artificial, proxies for the genuine mathematical object. That is to say, while ordered pairs can be represented as sets, it is quite possible that they are not sets. So if a function is defined in terms of an ordered pair, and the latter is not a set but something mysterious yet to be accurately defined, then we haven’t really defined what a function is!

Aftermath

There are many results in set theory that are said to be counter-intuitive or surprising, e.g. the Löwenheim-Skolem and Burali-Forti theorems. However, this discovery of category theory as a potential rival to set theory for the role of foundation for mathematics came as quite the shocker, even more so than any particular theorem. It felt like the carpet was being pulled from under me. Fortunately, the perplexity eventually subsided, and I now see a new area of discovery awaiting me. It goes to show that even in mathematics not everything has been said and done, and that perhaps what we are witnessing here is the abandonment of a “theory of everything”.