なぜ数学するのか？
Why Mathematics?

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なぜ数学するのか？

この記事では、ポール・ロックハート著『測定』の抜粋を紹介し、数学を学ぶことについて一緒に考えたいと思います。この本は美しい表紙と親しみのある楽しい語り口調もさることながら、手書きの図形が随所に現れ、読書体験として幸福感を与えてくれるものとなっており、私のお気に入りの書物の一つです。

　二部構成の本著の第一部『SIZE AND SHAPE』、49~50ページからの抜粋を、まずは原文の英語で読んでみましょう。

…. I want to address a serious question. Why are we doing this? What is the point of making up these imaginary shapes and then trying to measure them?

It’s certainly not for any practical purpose. In fact, these imaginary shapes are actually harder to measure than real ones. Measuring the diagonal of a rectangle requires insight and ingenuity; measuring the diagonal of a piece of paper is easy––just get out a ruler. There are no truths, no surprises, no philosophical problems at all. No, the issues we’re going to be dealing with have nothing to do with the real world in any way. For one thing, the patterns we will choose to measure will be chosen because they are beautiful and curious not because they are useful. People don’t do mathematics because it’s useful. They do it because it’s interesting.

But what’s so interesting about a bunch of measurements? Who cares what the length of some diagonal happens to be, or how much space some imaginary shape takes up? Those numbers are what they are. Does it really matter what?

Actually, I don’t think it does. The point of a measurement problem is not what the measurement is; it’s how to figure out what it is. The answer to the question about the diagonal of a square is not  ; it’s the mosaic design. (At least that’s one possible answer!)

The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying.

It’s also a great philosophical exercise. We are capable of creating in our minds perfect imaginary objects, which then have perfect imaginary measurements. But can we get at them? There are truths out there. Do we have access to them? It’s really a question about the limits of the human mind. What can we know? This is the real question at the heart of every mathematics problem.

So the point of making these measurements is to see if we can. We do it because it’s a challenge and an adventure and because it’s fun. We do it because we’re curious, and we want to understand mathematical reality and the minds that can conceive it.

Measurement by Paul Lockhart