This is mostly a post for myself, so that I remember the existence of something I mean to read. I have tried downloading and putting into scattered files stuff I mean to read. I’ve also tried stuffing links of stuff I mean to read into Yojimbo. Maybe putting it here will at least let someone read the things.
Anyway, this is a short essay by Joel Abraham that’s on arxiv.org. It’s Introduction to the p-adic Space. p-adics are a method of thinking about what the real numbers are. Why we need ways to think about what the real numbers are turn up when you think carefully about where our idea of them comes from.
It’s easy to see where the counting numbers like ‘1’ and ‘2’ and ‘3’ come from. They’re part of our evolutionary heritage, the part of mathematics that we know is understood also by apes and crows and raccoons. We understand some of it before we even have language.
With some thinking, and many people helping, we can go from these counting numbers to the idea of ‘0’. And even into the negative counting numbers like ‘-4’. And by thinking about multiplication, and how to reverse multiplication, we get fractions. Rational numbers. Positive and negative, given the chance. But then what are the irrational numbers? We can work out easily there have to be irrational numbers. We can name some of them. But how to give a clear definition of the whole mass of them? It should be more than just “also the other numbers”.
The p-adic numbers are one of ways to go about this. They start with thinking what we mean for two numbers to be “close to” one another. And thinking hard about how to write numbers. and this gets to interesting insights I don’t know as well as I’d like.
For this deficiency I blame Usenet. I first noticed p-adics in the voluminous and not particularly wise rantings of a crank poster to sci.math, back in the day. I forget what point, if any, he was trying to prove. But to first notice a subject as someone’s apparently idiosyncratic scheme of rewriting numbers so that everything we were already used to was useless, and in the service of some clearly nonsense goal (I think he was maybe trying to show how the number meant by 0.99999… was somehow different from the number meant by 1), is to badly hobble it. And I followed strongly mathematical-physics classes as an undergraduate and a graduate student. It’s easy to just miss problems of number representation. (This although p-adics could offer some advantages in numerical computing. They could make more numerically stable representations of irrational numbers.)
As I say, I want to fix that, and a friend linked to this arxiv post. And now that I’ve said stuff about it in public maybe it’ll coax me into going back and reading and understanding it all. We’ll see.