Seduced by Mathematics: The enduring fascination of mathematics by James D Stein

Author:James D Stein
Language: eng
Format: epub
ISBN: 9789811255489
Publisher: World Scientific Publishing Co. Pte. Ltd
Published: 2023-02-15T00:00:00+00:00

The Work of Man

Leopold Kronecker was a 19th-century German mathematician who famously declared, “The integers are the work of God, all else is the work of Man” [3]. So how do we get to a point where we can actually talk about something called “the square root of minus 1”?

Yes, they knew the rules for computing with expressions of the form a + bi in the 16th century — but how do we know we’re actually talking about something? Mathematics had faced this same dilemma before — first with zero, and then with negative numbers — but you can put real-world interpretations on both of them. In fact, money is a really good model for these concepts, as zero corresponds to having no money, and negative numbers correspond to being in debt.

And that’s probably akin to what Descartes had in mind. But there’s a more serious issue than just “mathematical fiction” associated with the square root of −1. What if it’s more than “mathematical fiction” — it’s total fiction in the sense that no such mathematical entity exists?

One of my former professors encountered this particular stumbling block. Without going into gory details, we can think of the real numbers as a number system of dimension one. It turns out that the complex numbers are a similar system of dimension two. In the 19th century, mathematicians came up with quaternions, which were a similar system of dimension four, and Cayley numbers, which were a system of dimension eight.

As you can probably guess, the next number system up the line would be of dimension sixteen, and my professor spent some time working out theorems that would be true of such a system. Just before he was ready to submit his paper, another mathematician managed to demonstrate that there were no such systems of dimension sixteen — or higher.

Here’s how mathematicians got around that problem for the complex numbers. They defined a system of ordered pairs (a, b) of real numbers which obeyed the following rules of addition and multiplication.

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