Perhaps to finish off, let me talk a little about the future of mathematical notation.
If there is any new notation, what should it be, for example?
Well, in books of symbols there are perhaps 2500 symbols listed that are supposedly common in some field or another and arenít letters in languages. And with the right drawing of characters, quite a few of these could be made perfectly to fit in with other mathematical characters.
What would one use them for?
Well, the most obvious possibility is notation for representing programs as well as mathematical operations. In Mathematica, for instance, there are quite a few textual operators that are used in programs. And I've long thought that it would be very nice to be able to use actual special characters for these, rather than combinations of ordinary ASCII characters.
It turns out that there's a very smooth way to do that sometimes. Because we picked the ASCII characters well, one can often get special characters that are visually very similar but more elegant. For example, if I type - > into Mathematica, it automatically gets turned into a nice arrow. And what makes all this work is that the parser for Mathematica can accept both the special character and non-special character forms of these kinds of operators.
Well, I've often wondered about extensions to this. And gradually they're coming. Notice the number sign or pounds sign--or is it called octothorp--that we use for places where parameters go in a pure function. Well, it's bit like a square, just with some tentacles. And in the future there'll probably be a nice square, with tiny little serifs, that is the function parameter thing. And it'll look really smooth, not like a piece of computer language input: more like something iconic.
How far can one go in that direction: making visual or iconic representations of things? It's pretty clear that things like block diagrams in engineering, or commutative diagrams in pure mathematics, and flow charts and things work OK. At least up to a point. But how far can that go?
Well, I'm not sure it can go terribly far. You see, I think one is running into some fundamental limitations in human linguistic processing.
When languages are more or less context free--more or less structured like trees--one can do pretty well with them. Our buffer memory of five chunks, or whatever, seems to do well at allowing us to parse them. Of course, if we have too many subsidiary clauses, even in a context free language, we tend to run out of stack space and get confused. But if the stack doesn't get too deep, we do well.
But what about networks? Can we understand arbitrary networks? I mean, why do we have to have operators that are just prefix, or infix, or overfix, or whatever? Why not operators that get their arguments by just pulling them in over arcs in some arbitrary network?
Well, I've been particularly interested in this because I've been doing some science things with networks. And I'd really like to be able to come up with a kind of language representation of networks. But even though I've tried pretty hard, I don't think that at least my brain can deal with networks the way I deal with things like ordinary language or math that are structured in 1D or 2D in a context free way. So I think this may be a place where, in a sense, notation just can't go.
Well, in general, as I mentioned earlier, it's often the case that a language--or a notation--can limit what one manages to think about.
So what does that mean for mathematics?
Well, in my science project I've ended up developing some major generalizations of what people ordinarily think of as math. And one question is what notation might be used to think abstractly about those kinds of things.
Well, I haven't completely solved the problem. But what I've found, at least in many cases, is that there are pictorial or graphical representations that really work much better than any ordinary language-like notation.
Actually, bringing us almost back to the beginning of this talk, it's a bit like what's happened for thousands of years in geometry. In geometry we know how to say things with diagrams. That's been done since the Babylonians. And a little more than a hundred years ago, it became clear how to formulate geometrical questions in algebraic terms.
But actually we still don't know a clean simple way to represent things like geometrical diagrams in a kind of language-like notation. And my guess is that actually of all the math-like stuff out there, only a comparatively small fraction can actually be represented well with language-like notation.
But we as humans really only grock easily this language-like notation. So the things that can be represented that way are the things we tend to study. Of course, those may not be the things that happen to be relevant in nature and the universe.
But that'd be a whole other talk, or much more. So I'd better stop here.
Thank you very much.