# Is maths real?

A friend has just started a Bachelor of Arts degree here at Waikato. As part of her first year study, she’s chosen to do a Philosophy paper. Apparently, one of the questions that has been posed, is "Is maths real?".

Well, what is real? You certainly can’t put ‘maths’ in a box and give it to someone. like you could with a chair or a chicken. But does it have more substance than just some made-up statements about how to add things or describing how large angles are? I’ve often wondered, for example, whether it would be possible to have a universe in which the value of pi was four. In our universe, it isn’t. But is pi, the ratio of the circumference of a circle to its diameter, necessarily 3.14159265…in all universes? I don’t know. I guess it depends on what a circle is.

So is physics real? I would think that it is more real than maths. I mean, physics is supposed to describe reality. Gravity is gravity. Objects attract each other proportional to their masses and inversely proportional to the distance between them. That’s what happens. It’s hard to be a physicist if you’re not a positivist, or at least have strong positivist leanings. In other words, if you don’t believe that there is a **real **world out there that we can know about, and that we can find out about this world **objectively** (e.g. by doing experiments), you are going to struggle to be a physicist. (It is true that quantum theory throws a spanner in the works at this point. The quantum world is weird indeed -an example here – and raises big issues about what is real.)

Recently, I was at a teaching seminar that seemed to be populated mostly be social scientists. In social science, a common paradigm is social constructivism, or namby-pamby waffle as it is known by positivists. In social constructivism, what the world is is constructed in one’s mind, and that what you can find out about it inevitably depends upon how go about finding out about it. In other words, everything is relative.

So, back to the point. Where does maths sit in all of this? I’m not sure it does. It’s hard to believe that maths depends on your point of view. It doesn’t matter how you look at it, 1 + 2 doesn’t equal 4, and it never will. But neither does it fit well with reality, either. 1 + 2 would be 4 in any universe, wouldn’t it? Maybe maths sits in some strange space of its own, separate from ties with this universe but not ‘made up’ in anyone’s head. So what is it? Er, that’s getting too close to philosophy for my liking. I await my friend’s response with interest.

Perhaps the best answer is to say that maths is as real as philosophy.

## One Response to “Is maths real?”

The usual philisophical response is to speculate at length about what the question is. It’s been a while and I’m not sure how much I’m adding. But here’s one to get you started.

It’s kind of your department, but I’m pretty sure the ratio of a circle to its diameter depends on the curvature of the space. Euclidean geometry is not the geometry of actual space and [even if we accept geometrical figures are something that really exist in the physical world] there are in fact approximately zero actual circles whose circumference:diameter is pi.

Does this make Euclidean geometry wrong? Depends. The claim it precisely describes the real world is false. But Euclidean Geometry does follow necessarily for the premises – logicians would call it valid. And if that’s all it claims to be then it’s correct. And in this sense it’s truer than any natural science because the only thing that could make it wrong is if logic didn’t work.

I assume physicists like maths as a tool for modelling the real world. I gather high mathematicians are just as happy working in other modes, or following up the possibilities which they know for a fact they can’t prove.

But I suspect the actual “is real” question would be whether the things of interest to maths – for instance numbers – actually exist. Maths is clearly a great tool for modelling the world but if numbers are real then what manner of thing are they – and if not how come it works as a tool?

Historically this kind of question relates to a lot of ideas about what kinds of ‘thing’ or ‘existence’ – like your seperate mathematical realm (which might be ‘platonic’) – are real or conceivable. I only really did undergrad phil and I think a lot of it is just learning how arguments work by extreme historical examples.

(Somewhere in the Public Address forum someone explained to me a real-world description of numbers that seems sufficient and didn’t seem to be subject to paradox, which is as hard as you may well be imagining. I can only remember a version that’s wrong, but it’s to do with sets of things of certain sizes – which rather explains the strong analogy between numbers and actual groups of objects. I’m told comp sci actually dealt with some of these questions more efficiently than “philosophy”.)

Incidentally, it may have been the phil influence that meant when I was introduced to imaginary numbers I started wondering what made them more imaginary than the other ones. Then I found I couldn’t do my calc homework till I repented.

Oh, and of course: pi is wrong http://www.youtube.com/watch?v=jG7vhMMXagQ