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.

When I say to people that I’m a mathematician, in a lot of cases people misunderstand what I do. Let me explain…

Mathematics, maths or math if you live in the U.S of A. is often confused with something called ‘numeracy’. When I say to people that I am a mathematician they think that I spend my time adding columns of numbers together, doing long multiplications, busying myself with percentages and getting very excited about pie charts. It’s got to the stage now where sometimes I just don’t even bother elaborating on what I, as a mathematician, am really interested in.

What I’ve just mentioned above, things like addition, multiplication, division, percentages, reading bar-charts and all the rest, are indeed part of mathematics, but really a part of mathematics that mathematicians are not particularly interested in – at least not nowadays. Yes, as I mathematician I can do all of these things quite comfortably but I’m not necessarily any better at adding up numbers than the next person. Yes, I know a few tricks that can speed up basic calculations but not necessarily any tricks that the proverbial Joe Bloggs wouldn’t know. Yes, I have to be able to do these things but not necessarily because I find them deeply interesting and they’re not the kinds of things that get me out of bed in the morning.

Just being numerate and able to handle numbers well doesn’t make you a mathematician just like the ability to wield a spanner doesn’t necessarily make you a mechanic. Obviously you need to know how to use a spanner to be a competent mechanic but just knowing how to tighten and loosen bolts is hardly something that is going to give someone the confidence to let you try and fix their car engine. There are plenty of people out there who know maths but who would still find mental arithmetic difficult beyond a certain point. This might seem crazy because surely as a mathematician they should find mental arithmetic a doddle…right? And that right there is where the confusion is coming in – equating mathematics with numeracy. These two things have an overlap but they are a million miles apart from each other.

Unfortunately many people don’t make any distinction between the two; maths is numeracy and numeracy is maths. I guess this is not helped by the fact that at school, most of the time spent in ‘maths’ lessons is spent doing numeracy; so unsurprisingly when people think maths they think times-tables, long multiplication, columns of numbers and boring stuff like that. But mathematics is about logical deduction, studying abstract concepts, precision, analysis. As a mathematician I’m more interested in studying algebraic structures such as groups, rings, fields, modules and Lie algebras; I’m more interesting in things like homology groups, transfinite numbers, set theory, mathematical logic and representation theory. None of these topics really require me to have anything more than an average level of numeracy – it helps to be comfortable with numbers but I can’t multiply six digit numbers in my head any better than the aforementioned Joe Bloggs. And I don’t need to be able to.

Most of the time I don’t bother telling people what I did during my maths degree – it’s not particularly interesting unless you have an interest in mathematics and (I really don’t want to sound like I’m being elitist or patronising when I say this) most people won’t know what you’re talking about because of the language that’s used. Does Random Man on the Street know what I’m talking about if I told hime that I did a course on Algebraic Topology or a course on Partial Differential Equations, or Fourier Analysis – I doubt it. I let people believe what they want to believe about what my degree entailed. If they believe that I spent most of my time looking at pages of numbers and doing big sums then I don’t usually bother to try and correct them – it’s just not worth it (I know that from personal experience by the way).

Just to be clear – no I can’t add a page of numbers up at a glance; no I can’t multiply two twenty digit numbers together effortlessly; and no I can’t interpret any old random statistic that is just given to me. I suppose this is why maths gets a bad reputation for being too boring and that mathematicians are just weird because they’re interested in percentages things like that – when you see maths and numeracy as one and the same thing then it’s like seeing literature and the alphabet as the same thing. There’s not really anything I can do to change this. Which is just too bad as many people will never experience mathematics and how deep the subject goes and how beautiful some of the theories are. I’ll just keep on keeping on…I think that’s the best way forward.

I’ve been asked countless times in the past – “what’s the point of maths?” Faced with this question many mathematicians will probably give a list of reasons about why maths is so important and how it can be used in society for this and that in order to convince the questioner of the point in the existence of mathematics and of the virtues of studying mathematics or numeracy.

I feel differently; I don’t feel at all inclined to do that. But surely as a maths tutor isn’t it my duty to inspire and recruit to the ranks of mathematicians? Yes it is my duty to inspire whenever I can and inform people about mathematics but if someone doesn’t want to be convinced then they won’t be. Some people call football ‘the beautiful game’ – try and convince me of that and I can tell you right now that you’re really wasting your time. I don’t want to be convinced of it and therefore, I won’t be convinced no matter what anyone says and how passionately they might say it.

Why exactly does there need to be a reason for studying, or a point to, maths? Would it be sensible to ask someone who plays tennis what the point of tennis is? Would it be sensible to ask a musician what the point in playing the piano is? How about a Formula 1 driver? A chef? Not really.

Why would anyone choose to play football? Well I suppose if they play Premiership football then the money might have something to do with it but primarily, I would imagine, because they enjoy it I imagine. It doesn’t solve any of the world’s problems as far as I can see, though.

Why would anyone choose to play the piano? Maybe simply because they enjoy it. Music and the piano don’t really solve any major problems facing humanity even when Bob Geldof and Bono get involved.

Why would anyone choose to learn about mathematics? Because they enjoy it? It doesn’t necessarily solve any ….oh wait actually it does solve some big problems facing humanity.

Isn’t it strange that the thing that clearly has applications and benefits that surround us every second of the day is the thing that is constantly challenged about its point. The things that really don’t make that much difference in the big picture are never even questioned.

I don’t do mathematics because it necessarily changes anything; I do it because I enjoy learning about it and for the sheer pleasure of doing it. The applications that there are of mathematics are not my primary reason for doing it; I’m pleased that there are applications and I sometimes take an interest in them but I wouldn’t know half of what I know about the subject if I was motivated only by it’s actual applications in real life.

No-one says that you have to be interested in mathematics. But nevertheless, some people are. But If you really hate something so much, whether maths or anything else for that matter, then don’t learn about it, don’t do it…simple. You don’t have to know about numbers and mathematics in life any more than you need to know about football or how to speak Latin or how to recognise an original Picasso. But for a few exceptional cases, you would probably find life much more difficult if you were completely innumerate than if you didn’t know a single rule of the game of football or a single Latin word or who Picasso was.

There are parts of mathematics that have applications in real life situations and you don’t have to look very far at all for them – in fact you’re probably reading this on one of those applications right now. Then there are parts of maths that don’t have applications. Should we only limit ourselves to those aspects of the subject that have applications right here, right now? Non-Euclidean geometry came about as a theoretical pursuit and it wasn’t until decades later that it found an application in Einstein’s theory of relativity – one of the single most important theories in the history of science. Some of the theory may never find an application other than within mathematics to produce more mathematics – but how do we know which theory will lead to something and which to nothing?

If we are to limit ourselves to stuff that only has applications here and now then should we abandon much of history, linguistics, physics, literature, music or sport even if people love doing these things for the sake of doing them? Why should mathematics, or anything else for that matter, have a point? Why does it need a purpose other than it makes people happy to do it? Mathematics is one of the most beautiful, and in some instances one of the most pointless, of human creations. Why shouldn’t it be enjoyed?

Well that’s my rant just about over – I’m glad to get it off my chest. I hope that you don’t think that all this means that I’m not enthusiastic about maths or that I don’t want to pass on my enthusiasm to other people, though! I continue to learn about mathematics on an almost daily basis and enjoy doing so; I’m proud of the fact that I can usually answer the deep, probing questions that I’m asked by my students about maths and I get excited when I get asked something that I don’t know the answer to because it means I have more to learn. Mathematics is one of the most important parts of my life; I just don’t feel that its existence needs justification or that it has to have a point. I’ll continue to show people the beauty of mathematics and either they will be convinced by its beauty or they won’t. But no hard feelings, though, if not.

Well here it is – I know you’ve all been waiting for this; the Leeds Maths Tuition monthly review for November 2016. It’s been a good month – nothing special but I’ve been having quite a bit of fun. For a start I had to relearn how to solve my Rubik’s Cube – it’s about ten years since I first figured out how to solve the cube and it’s about three years since I last actually bothered to solve it so unsurprisingly I forgot how to do it. Actually it wasn’t all that bad having to re-learn – I really only had to figure out a little bit and then I was away – good old muscle memory!

I’ve also been playing around with my Japanese abacus (soroban) – this also inspired me to learn more about Japanese mathematics so I’ve done a short review of a book called ‘The History of Japanese Mathematics’ by David Eugene Smith; I’ll leave you to guess what it’s about. I mentioned my soroban in last month’s video when I’d dug it out and blown off the cobwebs – I’m really starting to get into it and it’s also got me onto something called Flash Anzan which is a technique for doing mental arithmetic. I’ve started to look into this technique for doing mental arithmetic – it seems to be quite popular in Japan (at least to the point where they have national Flash Anzan championships) but I think it would be really difficult to make it in any way mainstream in the UK.

Well I don’t want to give too much away about what’s in the video – just watch it if you really want to know. There’s nothing in it that will really help you with your studies – that comes in some of my other videos – but it’s just me getting a few things off my chest. I’ll be doing another video next month and who knows what I’ll be talking about. I bet you can’t wait!