The Fundamentals of Mathematics


A toot on Mathstodon made me aware of this. It’s a listing, and brief description, of 243 theorems, as compiled by Oliver Knill. As the title implies they’re all intended to be fundamental theorems of some area of mathematics.

Many areas of mathematics have something called their Fundamental Theorem. The one that comes first to my mind is always the Fundamental Theorem of Calculus. That one connects derivatives and indefinite integrals in a way that saves a lot of work. But also commonly in my mind are the Fundamental Theorem of Algebra, which assures one of how many roots a polynomial should have, and the Fundamental Theorem of Arithmetic, about factoring counting numbers into primes.

The list does not stop there. And it gets into areas where “Fundamental Theorem Of ___ ” is not the common phrasing. They are, where I know something about the area, certainly core, fundamental theorems as promised, though. Or important mathematical principles, such as the pigeon-hole principle. It’s worth skimming around; even if you don’t know anything about the area, Knill provides some context, so you can understand why this might be of interest.

And then after the many theorems Knill provides some thoughts about why these theorems. What makes a theorem “fundamental”. This is something which shows off how culturally dependent and human the construction of mathematics is. And then, from page 147, a set of short lecture notes about the history of mathematics. Even if your eyes glaze over at torsion groups, it’s worth going into those notes at the end.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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