The Metaphysics of Mathematics; Against Platonism

Numbers have long fascinated the human mind. They are essential to our lives and practically universal – children across the globe grasp mathematical concepts at a young age. But as important as they are, few people sit down and ask: what is a number?

It might seem like a useless question, but the answer has big implications. For this article, I am going to break a rule of writing and occasionally reference numbers in their numeric form.

Imagine you’re creating a list of “things which exist” in the following scene: two horses grazing in a pasture.

Well, there’s at least four things on our list from the start – 2 horses, 1 pasture, and lots of grass. But that can’t be all. Each horse has 2 eyes, 2 ears, 4 legs, 1 set of teeth, 1 tail, etc. All of that needs to go on our list, too.

But what about “the number 2”? It keeps showing up – the exact same number, over and over. Do we add “2” to the list of “things which exist in our scene”?

We only have two options: yes or no. I’d like to explore the former.

On the List

Let’s say that, in addition to every other object in the scene, the number 2 also exists. This is called “mathematical Platonism” – which I will simply reference as “Platonism.” The number 2 is treated as an “abstract object.” Meaning, the number doesn’t physically exist anywhere – it isn’t located in space or time – but it exists independently as an abstract entity.

If this is true, it implies existence doesn’t necessarily require physicality. At least one other realm of existence is out there – and it’s inhabited by an infinite amount of abstract objects, including all numbers. These objects do not play by regular rules; they have a kind of necessary existence. You can imagine a universe without Earth, or perhaps even one without three-dimensional space. But numbers would still exist. The truth “2 + 2 = 4” would still hold.

Regardless of the construction of the universe, mathematical truths remain the same; they are necessarily universal.

This is why many mathematicians (and philosophers) believe numbers have an almost divine quality to them. They transcend physical space and time; they are “eternal”, in a meaningful way.  Influential German mathematician Georg Cantor believed there existed “different sizes of infinity”, and that God was identical with the “absolute infinity.”

(Fun fact: Cantor also believed God directly communicated these ideas to him, and that he was chosen to reveal them to the world. More on that later.)

The central idea of Platonism is this: mathematical objects exist independent of our conception of them. Without minds, numbers would still be “out there”.

Contrary to what you might think, Platonism is the standard, accepted metaphysical theory by the majority of mathematicians – with big philosophic implications. If Platonism is true, it means that you can have a meaningful non-spatial, non-temporal existence; that the mind has access to this realm; that “real infinities” exist; and that the realm of abstract objects has a close connection with the regular universe – numbers are, in a sense, omnipresent throughout our world.

I am not a Platonist.

Off the List

As I see it, the Platonic error is a common one. It’s a mistake about the relationship between our concepts and the external world.

Numbers are concepts, and without the human mind, they wouldn’t exist. Numbers require units – you have 2 hands, 2 yards, 2 pounds, 2 minutes, 2 miles per hour – but not a disembodied “2”. You need 2 of something.

It’s like the word “several”. You would never say “several exists.” Several what? “Several” is not an object – it’s a quantifier. It’s a way of saying “this many”. The words “this many” do not exist separate from our conception of them.

Quantifiers need a unit to quantify.  To think otherwise is an abstraction error. Take the Pythagorean theorem: A squared + B squared = C squared. What is this saying? Well, without reference to units, it’s not saying anything at all. The Pythagorean theorem becomes meaningful when the formula is attached to distance units. It’s saying, “This distance squared plus this distance squared will equal that distance squared.”

Abstractions are beautiful and profound, but we mustn’t forget they are abstractions from something concrete. Their metaphysical existence is never independent from their conception. Without minds, no abstraction would exist.

But what about mathematical truths? Wouldn’t two plus two still equal four, without minds?

The answer is yes, but only when we carefully explain what we mean. The equation “2 + 2 = 4” is not saying “add the object 2 to the object 2, and you’ll get the object 4.” It’s saying, “take two of any unit, add two more units, and you will end up with four units.” This is true, without reference to abstract objects.

This is true, even without a human mind, because mathematics is an extension of logic. It is necessarily the case that two units added to two units will result in four units. The concepts of “two” and “four” necessitate it. This is no different than other logical truths. Even in a universe without minds, no “square circle” will ever exist, by virtue of what we mean by the terms.

This is why mathematics is universal. It isn’t because numbers exist in a Platonic realm; it’s because mathematical truths are grounded in logical necessity.

Now, that doesn’t mean the symbolic representation of mathematical truths would exist separate from minds. In a mindless universe, you wouldn’t ever see “2 + 2 = 4” written down anywhere. But that formula points to a logical truth that exists without being symbolically represented.

So it’s true: numbers do not have spatial existence. They don’t weigh anything. But that isn’t because they exist independently in some ethereal realm. It’s because numbers are concepts. They exist when we conceive of them. They are descriptors, quantifiers, adjectives – they do not stand alone without a unit.

The Platonic error is not unique to mathematics. Once you see the mistake, it shows up all over the world of ideas. Abstraction is essential to understanding anything – perhaps even synonymous with it. But if we aren’t careful, we’ll end up committing the same metaphysical error as the mathematicians: giving independent existence to our concepts.

These ideas are not without broader implications. Platonism permeates all of mathematics – down to its very foundations. I will write more on this in the future.