The Unreasonable Effectiveness of Intuitionism in Mathematics

Written on January 28, 2020

Tags: philosophy, math

Platonism is a philosophical position that originated with Plato’s theory of forms. It is the belief that abstract entities such as mathematical objects are “real”, and independent of human thought, and that they “actually exist” outside of time and space. This philosophical position is hotly debated among philosophers, and that’s largely because they’re free to argue without having any real implications for mathematics. Mathematicians will continue Mathing, whatever the philosophers say they’re Really Doing.

But perhaps not. There is actually another tradition within mathematics, based on another philosophical position: Intuitionism. Intuitionism asserts that mathematical objects *are* constructs of our minds. That they exist just as mental representations, and we construct them as needed. When we’re not thinking about them, they’re not somehow “out there somewhere”. I’m not going to argue here that intuitionism is correct; it will play an ancillary role in the argument that platonism is not.

Mathematical intuitionism is coupled with a different form of logic called *constructivism*. It is this different logic which results in a different mathematical tradition, and a different set of theorems. Intuitionism is based around the idea that we construct mathematical objects (in our heads) as needed. The logic, therefore, requires us to construct mathematical objects before we can use them. In classical mathematics, we can show that an object exists without actually being able to construct it. For instance: classically, it is obviously true that there is some digit (0-9) that occurs infinitely often in the decimal expansion of \(\pi\). But which one is it?

Constructively, we must be able to construct the particular digit that occurs infinitely often, and be able to point to it in order to make an existence claim. In order to say that an object exists, we must have the actual object in front of us.

For most people, this logic is *weird*, and I’ll argue later that this weirdness is tied to our attachment to platonism. For now I think an explanation is in order. The way that we arrive at this weird logic, is by taking away a law of inference; in constructive mathematics, the law of excluded middle doesn’t hold. The law of excluded middle states that a proposition is true or it’s negation is true. That is, either the ball is blue, or it is not blue, any “middle” option is “excluded”. In mathematical notation, \(P \vee \neg P\) holds for every proposition \(P\). In classical logic, this is true. If you are unfamiliar with constructivist logic, this will probably seem *obviously* true; how can it be false! Constructivism, however, requires that you actually construct a proof of something in order for it to be true, and sometimes you can’t prove either \(P\) or \(\neg P\), so constructively, you can have situations where neither are true.

So why *is* it that the law of excluded middle is so intuitive? I would argue that if you start with a platonist worldview, then constructivist logic makes little sense. These mathematical objects exist, whether I’ve constructed them or not. These propositions are either true or false, whether I can prove them or not. When objects exist in math-space, their properties already determined, then it only makes sense to claim that any given property is either true or false. However, in the intuitionistic view, the objects live only in our minds, and we have to determine their properties. If there is a property that we can’t determine in finite time, it is just indeterminate.

So far I’ve described platonism, intuitionism, constructivism, and classical logic, and I’ve argued that if you accept platonism, then classical logic naturally falls out. At best, constructivism is *deeply weird*, and at worst it’s nonsense. If I could prove that classical logic is “false” (whatever that would mean), then we could conclude that platonism is a bad philosophical position, since it gives rise to bad mathematical assumptions. But we don’t actually have to go so far as “classical logic is false”. If platonism were true, we would expect the world to function in a particular way: according to platonic rules. In a platonist world, classical logic would be the right description of truth. I don’t have to argue that classical logic is the *wrong* description of truth, simply that it is not the *only* good description of truth.

And it’s not! Constructive logic is an incredibly good foundation of mathematical. It produces a rich mathematical universe that has equivalents to the interesting and useful structures in classical logic. Platonists should be shaken to their core when they see how effective intuitionism is at mathematics. From their perspective, it’s unreasonable! To them constructivism cannot be describing reality; it can describe some small subsection of reality, but shouldn’t provide a basis for understanding reality itself. Platonists should predict that constructivism would be about as effective as any other specific mathematical structure, like a group or a vector space, at describing the whole of mathematics — not very effective. And yet, we see that it does work as a foundation for mathematics. So we can actually ground these philosophies in claims they make about the structure of reality (or at least, the structure of mathematics), and platonism doesn’t pass the test. It’s a failed philosophy. At best, it’s on probation, with a lot of explaining to do.

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Of course, the effectiveness of classical logic also calls into question intuitionism. Perhaps this post would have been better titled *Death of Mathematical Philosophy: The Unreasonable Variety of Foundations in Mathematics*.

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