Mathematics – Created or Discovered?

It’s often said that “mathematics is the language of the universe.” This suggests that mathematics is some kind of universal constant with an existence that is independent of life and certainly humans; how much truth is there in this? I think the answer to that is a quite unsatisfactory “it depends on your point of view.” Well I’m going to give my thoughts on this – I won’t be able to give a definitive answer because I’m not a philosopher and it’s an unsolved problem that has existed for thousands of years and will probably exist for as long as humans exist. This is a very difficult topic to write about, especially in something the length of a blog post – after all, whole books have been written on this subject by some of the greatest thinkers in history; I’m not sure I can really compete with that.

Most mathematicians, from my experience, take a Platonist view of mathematics – that is, that mathematics has an objective existence in some realm, somewhere out there where we can’t get to. Mathematical objects are no less real than the everyday objects that we encounter like cars or books or houses. There’s all sorts of theories; as an example – it could be said that there exists (in one of these realms which we have no access to) a “perfect” right-angled triangle with sides of length 3, 4 and 5 and any triangle that we can have any actual experience of can only ever be an imperfect approximation to this perfect triangle. Moreover this “perfect” triangle has its own objective existence outside of human thought and at some point we discovered the concept of triangle.

And this would be the case for any mathematical object that we can find in any mathematical work of any kind whether it be a linear transformation, quaternions, a vector space, the set of real numbers, projective space…..anything. This objective existence of mathematical objects would mean that mathematics is discovered by us, not created by us.

I used to think this; but I don’t believe this now. I believe that mathematics IS a completely human creation. I do not believe that mathematics has any existence beyond human thought; if humans stopped existing then mathematics would stop existing. I’m not just wanting to be different – I will try to explain.

Over the years that I have studied mathematics I’ve learned that there isn’t just a single mathematics; Cohen’s forcing technique, for example, means that we can rewrite the rules of mathematics and force a mathematical theory to have certain properties. Each person could have, if they so wished, their own mathematical theory where things could be true (don’t even get me started on what true means) in one theory but false in another. Why should we flatter ourselves so much to think that we’ve got the universe’s theory all sussed out?

I don’t believe that mathematics CAN have an existence beyond human thought. I agree that mathematics can be used as a tool to describe many things in nature and to describe the universe in which we appear to inhabit but I don’t believe that the universe is constantly calculating in order for things to happen this way or that way. Things happen, of course – but things happen because they happen and not because there is a numerical or mathematical reason why. The universe appears to exist to us, but I don’t think that it exists, or at least appears to exist, for the purpose of being analysed or mathematically described by humans. The universe just is and the things that happen just happen.

Of course, I could be completely wrong on all of this and just and probably just talking out of my arse. I’m not saying that I’m right – it’s just my own personal set of beliefs. After all, how could we conceive of something like numbers and mathematics if they had no existence in the first place? Does that mean that Santa Claus is real after all? Maybe, then, nothing really exists objectively and everything is just a subjective creation – I think that’s what Kant was getting at. Why bother to study something that doesn’t really exist? Well that’s the subject of another post of mine on the purpose of mathematics. No matter what the reality is, whether created or discovered, mathematics is still something that I love doing. Something doesn’t have to exist for it to be interesting.