I was “talking” to someone on twitter recently, who claimed that mathematical objects are ‘imaginary’. I guess this is received wisdom; after all, if mathematical objects are not imaginary, they must exist somewhere. And no-one (at least not anyone I know) has stumbled across an exponential function at the bottom of their garden, or stubbed their toe on a stray modular domain, or been attacked by a swarm of acute angle triangles.
Yet, the fact that mathematical objects are real is the daily experience of mathematicians (though few would ever claim this, because they are much too cautious). I’d like to try to explain this experience. Since I am not a philosopher, there will be no robust philosophical arguments. I will not discuss ontology. Try not to be disappointed.
Imagine you were an astronomer. (No, go on. Give it a go.) You point your telescope up in the air and – lo – a new star appears. You call a friend, and tell her the news. She points her telescope in the same place and – lo – the same star. You write up your discovery, and a team of astronomers in Belgium train their more powerful telescopes on the same spot, and describe the colour and size of the star. You have another look, and see they are correct. An international team in Chile use radioastronomy to discover that your star is actually two stars, orbiting around each other. It is later discovered that there is a large exoplanet orbiting one of these stars.
Now – I guess – it could be argued that there is no star. It could be argued that you invented it, and then let everyone else know how to do the same. The star is some sort of socially constructed illusion. In my view this is a purest nonsense. There is a real star, it is really out there. That, after all, is the belief of (most) astronomers. Otherwise, we might as well give up the whole astronomy thing altogether.
So I am getting to my point. Thanks for being patient.
My point is that this is also the daily experience of mathematicians. Let’s suppose I am studying transcendental dynamics (as I do), and I study a new set which seems of interest (well, you never know). I email a colleague, and they confirm the set looks as I said, and maybe they spot something else; perhaps it has dimension one, or is dense in the plane, or something technical like that. We write a paper. A team of Belgian mathematicians read our paper, and note that, in fact, our set has other interesting properties. They email us and we find that this is indeed the case. More papers follow, and then someone (in Chile, perhaps) observes that our set is actually the union of two interesting sets, and gives some further properties of each. When we look into it, we see that this is indeed the case. This is how (pure) maths is done.
Essentially this story (for it is a story; I have not discovered any sets of interest to Belgians) is no different to the story about the star. And it is very difficult not to believe the punchline is the same; the set exists ‘outside our heads’, just as the star exists ‘outside the heads of the astronomers’. (I’m not trying to claim mathematical proof here; I’m just trying to communicate how it feels to do mathematics).
A real-life example of this story is the famous Mandelbrot set. This was first discovered in the 1970s, when it was very difficult to draw a picture of it. But mathematician talked unto mathematician, and more and more properties were discovered. Technology has moved on, and now highly detailed pictures exist. It is a remarkable object: for example, the set is so intricate that if you try to draw a line around the edge, you will find that your ‘line’ is actually two-dimensional. It is even more intricate than the coast of Norway. Nonetheless, all mathematicians would agree they have been studying ‘the same thing’ all this time.
So it seems undoubtedly true that mathematical objects exist. I am as confident in the existence of the Mandelbrot set, or the sine function, or Riemann surfaces of genus zero as I am in the existence of Belgium. When we study mathematical objects, we discover them – we do not invent them. There are thing that exist that are not material objects.
You may feel that this is silly, because if they exist, then where is their home? (It is probably not Belgium). How do we see them? What are they made of? These are a good questions.