Dynamics in the Eremenko-Lyubich class

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Is maths imaginary?

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.

This star really is imaginary. Or so I am told.

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.

The famous Mandelbrot set, as drawn in the late 1970s.



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).

We can now draw the Mandelbrot set in much more detail. But it is still the same set.

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.


On checking a proof

In many respects the most difficult part of mathematics lies in communicating your work. That is not to say that proving results is easy; but the proof is just the tip of the iceberg (even though it often feels like the principal goal). The proof is of no value or interest unless you can communicate it. And to communicate it, you have to be able to write it down accurately. I’ve not found any notes on how best to do this, so here are a few thoughts of my own.

When I look back at some of the proofs I wrote when I started work on my PhD, I realise how much I have learned. My supervisors – who were very gracious, very helpful, and very dedicated – used to cover my early work in red ink. I then learned how to write a proof through an iterative (and very painful) process, in which I would write something, receive the red ink, fix those problems, receive further red ink, and so on. I became very familiar with red ink. Very, very familiar.

An early example of the great value of red ink

In this note I’d like to comment on how one might spot problems oneself, rather than depend on one’s supervisors in this way. This is not a trivial task, but a really important one. Perhaps I can offer a few pointers which might be of help.

Let’s suppose you have proved a result. You’ve written it all up to your own satisfaction, and wish to share your  achievement with your fellows. I began to make a list of the things you should do, but it was very long, exceedingly tedious, and all boiled down to the word check. Which is a bit boring. So let’s try the following, which is less prescriptive if possibly less all-encompassing. It’s just three words. How hard could that be?

First forget. In developing your proof you, no doubt, came up with all sorts of ideas and intuitions and implications and pictures. You have to (somehow) now lay these all to one side. Your reader will not have any of this in front of them, so you have to be sure that none of your work now depends or uses anything other than the words in front of you. (Incidentally, the best way to do this is to put your proof to one side for a few months, and then come back to it. You’ll be astonished how terrible it will look).

Second focus. Focus on the words in front of you, and what they say. This is easier said than done; because you expect your words to say one thing, you will tend to interpret them in that way. Try not to. Look at what is written and nothing else.

Third check. Read what you have written, word by word, sentence by sentence, and ask yourself the question “why on earth does that follow?” Notice the negation; if you expect things to be wrong you are more likely to spot mistakes than if you expect them to be correct. In my personal experience they are probably incorrect.

I could probably make a list of common mistakes, but it really is hard to make that interesting. So I will highlight just three (three is a useful number here):

The word “clearly”: It is very easy to make the mistake of writing “clearly XYZ” when what you mean is “XYZ seems pretty darned obvious to me but I can’t quite work out why”. If you can’t work out why XYZ is true, chance is that is isn’t.

Things that are true but don’t actually follow: This is a very easy mistake to make; you write something like “Since X, then Y” and assume it is OK because Y really is true. But you are not asserting here that Y is true, and that is not what you need to check. You need to check that Y follows from X and nothing else!

Failure to satisfy all necessary conditions: If you use another result (maybe a book result, or a lemma of your own from earlier) you need to be sure that all the conditions are checked. This is especially true of a book result – if that says something like “If A, B, C, D and E, then F”, then there is no chance to use this result if only A, B, C and D are true.

Yes, this is all amazingly tedious. Yes, this is a very lengthy process. No, there is no alternative (apart from asking a friend to check). Yes, you will be a better mathematician when you can do all this. No, I do not claim to be able to do this all the time myself. Yes, I welcome feedback and other suggestions.