Math Is the Art of Stating Things Clearly

Why do we use math?

In physics we describe everything, from the smallest of particles to the largest of galaxies, with the language of mathematics. Why should that one field be able to describe so much? And why don’t we use something else?

The truth is, this is a trick question. Mathematics isn’t a language like English or French, where we can choose whichever translation we want. We use mathematics because it is, almost by definition, the best choice. That is because mathematics is the art of stating things clearly.

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter. The bartender stops them, pours two beers, and says “You guys should know your limits.”

That was an (old) joke about infinite series of numbers. You probably learned in high school that if you add up one plus a half plus a quarter…you eventually get two. To be a bit more precise:

\sum_{i=0}^\infty \frac{1}{2^i} = 1+\frac{1}{2}+\frac{1}{4}+\ldots=2

We say that this infinite sum limits to two.

But what does it actually mean for an infinite sum to limit to a number? What does it mean to sum infinitely many numbers, let alone infinitely many beers ordered by infinitely many mathematicians?

You’re asking these questions because I haven’t yet stated the problem clearly. Those of you who’ve learned a bit more mathematics (maybe in high school, maybe in college) will know another way of stating it.

You know how to sum a finite set of beers. You start with one beer, then one and a half, then one and three-quarters. Sum N beers, and you get

\sum_{i=0}^N \frac{1}{2^i}

What does it mean for the sum to limit to two?

Let’s say you just wanted to get close to two. You want to get \epsilon close, where epsilon is the Greek letter we use for really small numbers.

For every \epsilon>0 you choose, no matter how small, I can pick a (finite!) N and get at least that close. That means that, with higher and higher N, I can get as close to two as a I want.

As it turns out, that’s what it means for a sum to limit to two. It’s saying the same thing, but more clearly, without sneaking in confusing claims about infinity.

These sort of proofs, with \epsilon (and usually another variable, \delta) form what mathematicians view as the foundations of calculus. They’re immortalized in story and song.

And they’re not even the clearest way of stating things! Go down that road, and you find more mathematics: definitions of numbers, foundations of logic, rabbit holes upon rabbit holes, all from the effort to state things clearly.

That’s why I’m not surprised that physicists use mathematics. We have to. We need clarity, if we want to understand the world. And mathematicians, they’re the people who spend their lives trying to state things clearly.

5 thoughts on “Math Is the Art of Stating Things Clearly

  1. Madeleine Birchfield

    Really, the limit is defined of a Cauchy sequence, that 2 is the limit of the sequence 1, 3/2, 7/4, 15/8, 31/16…

    Or one could speak of the inverse limit of a series of abelian groups, in this case Z/2Z, Z/4Z, Z/8Z, Z/16Z, whose group homomorphisms are by way of multiplication of 2, yielding the real interval [0, 2]. Or one could speak of the categorical limit in the category of abelian groups. It depends upon how abstract one wishes to define mathematical terms.

    Limits are not the only way to form the foundations of real numbers and mathematical analysis, there are also infinitesimals and non-standard analysis, by way of hyperreal numbers, dual numbers, or smooth toposes. It is the mathematician’s choice, and quite honestly, the physicist does not really care which foundation is used, or even if the mathematical tools have proper foundations, only that the tools work. (cf. quantum field theory and the foundational issues with the maths behind QFT)

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    1. Andrew Oh-Willeke

      “the physicist does not really care which foundation is used, or even if the mathematical tools have proper foundations, only that the tools work. (cf. quantum field theory and the foundational issues with the maths behind QFT)”

      As a native of mathland who spent an extended amount of time time metaphorically abroad in physicsland this is the philosophical difference between the disciplines that is most striking to me.

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