By Nikhil Handa

Mathematics. To many of you, all this word will do will is bring back memories (or maybe even nightmares) of sitting in a stuffy classroom, crunching through questions and following a closely defined method to produce an answer, using a process that can only be described as ‘thinking inside the box’, where the box also happens to have many layers of insulation and an industrial metal exterior. Outside that box is an infinite space of unproved conjectures, undiscovered hypotheses and questions like the one you saw which made you click this article.

Think back to your first maths lesson of primary/high school. Now, please raise your hand if your teacher explained to you what maths actually is. Alright, I see a few hands, but not nearly enough (don’t worry, I can’t actually see you, I think I’ll leave the cyber-spying to Microsoft). It’s possible that those who have raised their hands are thinking of the definition ‘a framework to describe our universe’. In other words, mathematics is a language.

Just like English, just like French and just like Gobbledegook, maths is a language. And similarly to other languages, you can’t make sense of it without some sort of knowledge of the basics, but as you start to learn the more complex concepts, the more you can start to use them yourself. You can also translate maths into English (for example 2+3=5, which you just translated into English in your head in order to read it), like you can translate German into Spanish. Exactly like other translations, there are some ambiguities in the process of doing so. For example, the symbol for ‘implies’ in maths does not mean the same in English (‘implies’ in maths is only concerned with truth, not causality). And finally, identical to all other languages, maths would not make sense without a real world for it to be describing. You can describe there as being ‘1 apple’, but ‘1’ cannot exist by itself in the real world.

Hopefully, that convinces you that maths is a language, and if it doesn't, don’t be a pain and just listen to me. But how does maths being a language relate to our question at hand? Well firstly, let’s say that in order for maths to be real, it can exist in only our universe, and no other theoretical universe, whereas if it is an abstraction, as we created it in our minds, we can obviously have it in any sort of universe, ours or theoretical. Well, this is when the definition I mentioned earlier comes in handy: ‘Maths is a framework used to describe our universe’.

If we study this definition closely, we can notice it says ‘our universe’ (I am aware this didn’t take much close studying, but I just wanted to sound like an intellectual, so let me have my moment). So if we ventured into a theoretical universe, maths shouldn't exist, as this would go against the definition. This, by the aforementioned conditions for maths to be real, should present us with the conclusion that maths is indeed real.

However, what if the definition is wrong?

Boring but accurate Banach-Tarski paradox representation

Boring but accurate Banach-Tarski paradox representation

There exists a theorem in maths called the Banach-Tarski paradox, and it's as messed up as pineapples on pizza. It suggests (and can back up with mathematical evidence) that a sphere can be cut to form another sphere of exactly the same density, mass and volume. In other words, you've just created a copy of the sphere seemingly out of thin air.

The proof for this theorem is weird and wacky, but the point is that it makes logical sense, in mathematics at least, but it doesn't make sense in the real world. Well, we don't think.

As you can imagine, this paradox took the world of mathematics, physics and philosophy by storm to the extent that mathematicians, physicists and philosophers began to work with a blunt pencil and miss their afternoon tea (it's practically unheard of in such professions), and people became obsessed over whether this paradox can be applied to the real world.

If the answer is no, this can only mean that maths is indeed an abstraction, because it's describing something that isn't real. If it was real, how can something real prove something that doesn't exist?

Tasty Banach-Tarski paradox representation

Tasty Banach-Tarski paradox representation

Now, many of you might assume the answer is no, because you cannot imagine starting with a sphere and creating two spheres with no stretching or distortion, and that is fair. If it was possible, there would be no starving people, as they could create more food out of less food, and more importantly, you don't have to go over to the neighbour’s garden if you accidentally kick the football over the fence, as you would have an infinite number of footballs. Think of the possibilities.

However, as always, there exists a group of people (who I can only assume create conspiracy theories in their spare time) who suggest that the Banach-Tarski paradox does exist in real life, but on a microscopic scale.

It's time to do some more thinking back to secondary school science unfortunately, and this time we have to think back to Chemistry class. Yes, yes, I'm sorry, I didn't want to make you do it, but I must. If you remember, your teacher would have babbled on about particles colliding a whole lot. Now, when these tiny sub-atomic particles collide at high energies, it seems that more particles are created, so we have more than we started off with. That sounds rather familiar, doesn't it? The conspiracy theorists thought so as well, so they claimed that this is evidence of the Banach-Tarski paradox in real life, telling us that the paradox is not definitive proof that maths is an abstraction.

Even if this argument of sub-atomic particles turns out to be untrue, that doesn't mean the paradox won't ever be true in real life. Countless times have people encountered theorems and branches of mathematics which they thought had no application to real life, but centuries later, the application to real life was discovered. So, it is very possible that our mathematical way of thinking is just not at the stage yet to prove the paradox as true in real life, but it might be soon.

And so, I now reach a very common conclusion in philosophy. Given the evidence we have now, the most likely conclusion is that maths is an abstraction, on the grounds that the Banach-Tarski paradox cannot apply to our universe and hence cannot be real. However, there is still some possibility that the Banach-Tarski paradox can be applied to our universe, but we just don't know how yet (or cannot prove how yet), making the most likely conclusion be that maths is indeed real.

Unfortunately for us, until that day, we will simply have to endure more awkward conversations with our neighbours asking for our football back. Sorry to leave you on such a sour note.