“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” – G. H. Hardy
So at the end of your letter on time you asked for a letter about space. Conveniently, space – in its broadest sense – is what Measurement by Paul Lockhart is about. I will come back to that. First allow me a series of digressions.
(The Greeks treated all of mathematics spatially. Three was a line three times longer than a line of length one. To add you put the lines end to end, to subtract you cut away a stick of the relevant length. They did all their maths in this way, shapes were vital.)
As may have come up in these letters before: I was a religious kid. I don’t think that my dislike of mathematics has featured; so cliff notes are that I found maths to be too algorithmic and formulaic – in all of the ways – to be of interest at school. Another thing unlikely to have come up is that I find the song Amazing Grace very sad, and somewhat discomforting, especially the final verse.
Consider all this to be foreshadowing.
(Geometry is ye olde Greeke for measuring the Earth. Plato’s Acadamy had a sign behind the bar that read ‘Let No One* Ignorant of Geometry Enter’, but in ye olde Greeke.)
When I was but knee high to a woodlouse I had the disconcerting feeling of coming upon infinity in a deeply disturbing way. I was small and so this memory is probably a composite of several moments but my brain has been somewhat fried by my travels in time (and indeed space, especially the space enclosed by the pint glass). The way I remember it is as a conversation with my mother in church regarding Heaven and the unending nature of it. What started as an innocent question ended in tears as my mother explained that there was no end, that every day would be followed by another; that at the edge of space there would have to be something else beyond, some sort of nothingness.
(They also believed the universe was infinite in all directions, and that ideas existed in a very real but very abstract way. They also believed that their complex and varied beliefs spanning several centuries and most of the Eastern Mediterranean would be over simplified and bled of all nuance so I could make word count.)
The result was physical discomfort.
(In their maths however, they avoided infinity. Largely thanks to the confusions introduced by Zeno and his tortoises. They also didn’t have a numeral for zero. No one did until the Indian’s started using it. Nothing and infinity are tied up by the idea of the infinitesimal – a fragment that takes up no space but has a value. Something we only really started to get to grips with in Newton / Leibniz’ work.)
I sympathise. The idea that like that extra day after infinite others, there was another mile of space after the infinite miles behind it. It was an emotion I haven’t really had about anything else. It was a real horror, but wasn’t fear. It wasn’t about scale or feeling small. It was about something more abstract. Something about existing in the infinite something about having something definite but completely unknowable.
(Of all the scientific disciples, I’ve been told that mathematicians are the most likely to be religious. This seems weird when their field is about such refined and precise logic which seems kind of incompatable with the non-rigourous thinking implied in the word ‘faith’.)
It hurt my gut, but not in the way sadness does. It was unpleasant, and I sat for a long time wanting to cry. Then I did cry for a bit. Then the whole congregation rose to sing Amazing Grace which ends with the verse:
When we’ve been there ten thousand years,
Bright Shining as the sun,
We’ve no less days to sing God’s praise,
Than when we first begun.
This was all a bit much for me and I had to be taken outside.
(Then again, there is clearly something in the abstraction of the subject that overlaps with the spiritual. Especially in the Abrahamic tradition where God is not so much a personality as a concept.)
Anyways. I was reading Measurement. So the book is Lockhart’s way of teaching maths that isn’t everything that turned me off maths when I was at school. His approach is to teach methods of proof, then to lay down questions or problems that are to do with applying mathematical argument. There are no answers.
He urges the reader to treat maths as a creative endeavour. Find a proof, work on it, polish it like you would a poem. Make it shine. Then try and find a better one, or an alternative one.
The emphasis is on getting you to discover maths. You won’t be the first to prove there’s only so many platonic solids. But the discovery can be totally original to you. It is genuinely thrilling. This maths is out there, existent in a perfectly objective way. Nothing comes closer to Plato’s world of ideals than mathematics.
Some of it feels kind of arbitrary. It is logic so you take a bunch of assumptions and you run them as far as they will go, see what happens. It doesn’t matter how long your first stick is, as long as you can work out how long it is compared to something else.
But somehow it produces specificity. (Pi emerges, e emerges, sine and tangents, the primes.) These things have a realer than real existence. They are necessarily true. They are capital-T Truth. And that is easy enough to understand, but understanding is not feeling and when I finished Measurement I got a hint why so many mathematicians are religious.
In his discussion of (how e emerges), I started to have that infinity feels again. The discomfort. Here was a glimpse of the sun, of real objects, after watching the procession of shadows on the wall. It was also the closest thing I’ve had to a genuine religious experience since being slain in the spirit as a kid.
It was – appropriately – (transcendent).
I didn’t cry this time. Much. But I would still recommend the book.
This has all got rather highfalutin, why don’t you do something about something low next week: Low-life, low-brow, low-culture, low-lands – Whatever, I leave it up to you.
P.S. *’No one is not one word… Don’t quote me on that.‘ – J. S. Loveard