Sun 4 Nov 2007

Deriving Mathematics, part 1

Another multi-part series of posts (along with storytelling) I want to try here is laying out my thinking about how we can try to understand (and teach?) formal mathematics. Nothing too revolutionary. Everything I'm going to write about has been done before in various places. I'd like to have a shot at putting my spin on the exercise.

My idea is to motivate everything. Start from what we commonly accept as "true" today — and by "we" here, I mean, say, your basic early high-school student. What do we understand about numbers right now? How can we motivate the leaps that mathematicians make to get to abstract things like sets, rings, fields, formal logic, analytic number theory, ... the list is endless.

Given my finite time here on earth, I'm not going to try and cover everything (surprise!). However, I will claim that a lot of what is taught in senior school classes, even in university courses in Pure Mathematics, really isn't that far away from what "everybody" knows. It's partly a matter of notation, but also, if you learn by building mental models for yourself, a matter of seeing the motivation behind particular idea so that you can put it in perspective.

Let's begin with some thought experiments and on Tuesday I'll start working towards answers.

Think about what you know about numbers. Usually, you'll just think of "them" as numbers, not bothering to group them any further. So let's consider a few thought experiments.

1. What is really different about integers (1, 2, 3, -7, -63, ..) compared to things that aren't integers? What makes the integers a special collection?
2. Amongst the numbers that aren't integers, we're used to dealing with fractions (1/2, 1/3) and decimals (1.23, 3.45, 5.6789). Somewhere along the line, you've probably picked up that numbers like π, the square root of 2, the fourth root of 17, etc, are somehow different from the fractions and decimals. What makes them special?
3. In everyday life, we often do arithmetic on a clock. We add 17 hours to 3:30 in the morning to get 8:30 in the evening and things like that. We subtract 2 hours from 1:00 and get 11:00 — a bigger number than we started with. What are the differences between doing arithmetic on a clock, compared to doing arithmetic with normal numbers?
4. For that matter (following on from the previous question), are there differences between doing arithmetic with fractions than when doing arithmetic with integers?

We're not going to just stick to numbers. As soon as you start generalising, pictures (diagrams) become important both for illustration and as a tool for describing and proving things. Consider these:

1. Take three lines, say, one unit, two units and five units in length. How many different triangles can you make from those lines?
2. An alien somehow telephones you (skip over how he worked out this piece of technology) and asks a simple question: "What is a circle?" Since you're on the telephone, you can't wave your arms or draw a diagram and you have very few technical terms in common. How can you describe a circle to the alien? He's not stupid, but he doesn't have the same knowledge as you.

These are all "establish your assumptions" problems. A lot of mathematics, particularly the ill-defined branch we call Pure Mathematics, works with establishing the assumptions — the pre-conditions — and then seeing which ones can be removed and what the effects are.

My approach in the following articles is going to be along these lines. Start with a well understood example. Consider some of the things it relies (or maybe doesn't rely on but are present) and then move in one of two different directions: look at generalisations of the example and look at other special cases that have similar or overlapping assumptions. Just in passing, I'll note that the integers and the clock arithmetic I mentioned in question 3, above, have a lot of the same underlying assumptions (which you can sort of intuit, since their behaviour is similar).

Once we have a few examples, we start to wonder what makes them different. If two things are similar, but not identical, how can you tell them apart? Why is a circle different from a square? If you were an ant crawling around the surface of each shape, could you tell them apart? If you saw them drawn on paper, presumably you can tell them apart, but why? In other words, why is "circleness" so different from "squaredness"?

Topics: mathematics/series: deriving