# Tony’s take August 2023

## This month’s topics:

### Math, discovered or invented?

The perennial question is brought up again by Robert J. Marks II on the Evolution News website, July 9, 2023. The evidence, Marks tells us, points towards discovery. I believe most mathematicians think that way too. The way we speak about “searching for a solution” to a problem implies that the solution exists, out there somewhere, and that we just have to find it. The evidence Marks refers to is the occurrence of simultaneous discoveries. “Many mathematical breakthroughs are sometimes independently reported by two or more mathematicians at roughly the same time.” He gives several examples, the most famous being the discovery/invention of differential and integral calculus by Newton and Leibniz in the 17th century and the discovery/invention of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky in the early 19th.

Here is another example that I think should be better known. In 1758 Euler’s publication of his observation that the numbers $S$ of vertices, $A$ of edges and $H$ of faces of a convex polygonal solid must satisfy $S-A+H=2$ (Propositio IV) included, as he described it, the completely equivalent statement: in such a solid, the sum of all the face angles is equal to $(4S-8)$ right angles, i.e. $2\pi S – 4\pi$ (Propositio IX). In his introduction he remarked how surprising it was that in all the years geometry had been studied, these most basic elements of solid geometry had remained undiscovered. But there he was wrong. Back in 1621 Descartes had formulated “Propositio IX” in almost the same terms. This work was never published and in fact was lost, but not before Leibniz had transcribed it (around 1675) from the papers Descartes left behind. It still remained unknown until it was discovered among Leibniz’s papers towards the middle of the 19th century. More details here.

That two mathematicians, over a hundred years separated in time, should write down the same theorem in almost exactly the same words seems to me strong evidence that mathematical truth exists independently of us, waiting to be discovered.