That Damned Number Line

I admit it: I’m a math neanderthal. I just don’t, in so many ways, understand The Number Line, that most basic of mathematical instructional devices.

It’s come up because I’m reading Everything and More: A Compact History of Infinity, by David Foster Wallace. Actually, that’s not reall true. I’ve stopped reading it because I don’t understand, which is one the basic principles explained at the beginning of the book. And I’ve re-read the chapter a number of times, and really, I just don’t get it.

Now, I understand the concept of putting numbers in a line to show their relationship to each other (2 comes after 1, etc). I understand that you can theoretically put and infinite number of numbers between any other two numbers (actually, I start to wig out at that thought, but onwards). What I don’t get is when the idea of ‘space’ is related to the number line. Suddenly, this abstract metaphor that I was quite happy with takes on concrete, measurable properties and muddles it up for me. Why is it important how much space on the number line the set of real numbers occupies, vs, say, the set of irrational numbers (and I may well be mixing terminology here, but onwards). Maybe I don’t get this odd crux of applying physical properties to this completely abstract idea.

And worse, it gets me to thinking about things like Zeno’s (or Xeno? Or someone else entirely?) paradox about how you can never actually go anywhere, because to get anywhere, you have to cross 1/2 that distance, and to cross that 1/2, you have to get 1/2way, etc. and so on and so forth, ad infitum. And I’m generally ok with that, because (and this maybe completely wrong) this paradox assumes that there can be a ‘moment’ in time to worry about travelling within. But I’ve always just kind of assumed our relationship with points of time was also a metaphor, like the number line. There aren’t really any points, we just need to pretend there are, so we can do physics and so on. Which, then, of course, gets me doubting the very nature of physics of which so much seems dependant on the idea of points of time. But because you can always fit an infinite (if infinitely small) amount of time between any two points on the timeline, then… and I run full circle back into that old paradox with points of time.

And there are perfectly reasonably explanations for why we can, in fact, travel anywhere, and I even feel like I understood them at the time, but of course, these have flitted through my head like so much ephemera, and I’m left starting at this abstraction called a number line and worrying about the validity of it’s very existence all over again.

And, if you didn’t have 0 (zero), which, if I’m right in recalling, the Greeks didn’t, how on earth do you have a number line? Like, where do you start it? And how do you do basic arithmetic? Like I want to move 3 spots to the left of 3 on the integer-only number line. Where does that put me? Do I fall off the edge of the metaphor? etc.

I think mathematics would quickly drive me absolutely nutters, where I ever to have studied it. It may, in fact, be why I was never terribly good at it: it’s very foundations clearly bother me. And don’t get me started on the random utlization of base 10 for math. Or why every other base essentially seems to be a corruption of it, the way we use them. Or other meetings of language & math.

Ok. back to work now.