The Need For Non-Euclidean Geometry

I have argued against using whimsy to introduce complex numbers. You know what I mean by whimsy: "No square roots of negative? Let's invent them!" If you don't, go check out John and Betty.

One counter-argument, voiced in the comments and on twitter by MathCurmudgeon, is the Argument From Vi:

@mpershan "The way of mathematics is to make stuff up and see what happens." @vihartvihart
— Curmudgeon (@MathCurmudgeon) March 30, 2014

Historically, this is just not the way complex numbers were invented. They were invented out of need, in particular a need for real solutions to cubic equations.

But, no matter, maybe this sort of whimsical creation is just a regular practice of math, one that we could reasonably expect to resonate with our students. But where else is this sort of casual invention at play in the history of math? James Cleveland tentatively suggests that whimsy was responsible for the discovery of non-Euclidean Geometries.

But check out this 1824  letter from Prof. Gauss to a Taurinus who sent our cranky mathematician some of his geometric work:

"In regard to your attempt, I have nothing (or not much) to say except that it is incomplete. It is true that your demonstration of the proof that the sum of the three angles of a plane triangle cannot be greater than 180 degrees is somewhat lacking in geometrical rigor. But that in itself can easily be remedied, and there is no doubt that the impossibility can be proved most rigorously. But the situation is quite different in the second part, that the sum of the angles cannot be less than 180 degrees; this is the critical point, the reef on which all the wrecks occur. I imagine that this problem has not engaged you very long. I have pondered it for over thirty years, and I do not believe that anyone can have given more thought to this second part than I, though I have never published anything on it."

And now Lobachevsky:

"The fruitlessness of the attempts made since Euclid's time...aroused in me the suspicion that the truth...was not contained in the data themselves; that to establish it the aid of experiment would be needed, for example, of astronomical observations, as in the case of other laws of nature..."

Euclid's Postulates had been explicit for ages, and anyone could have just casually wondered what the world would look like if the fifth was false. That would be in the true spirit of mathematical whimsy -- "What if Euclid's Fifth Postulate was false?" -- but that's not what happened.

Instead, the existence of non-Euclidean geometries emerged from the need to explain why Euclid's parallel postulate couldn't be derived from the other four. Non-Euclidean geometries were posited out of the need to explain the failure of centuries of effort.

[Source: Euclidean and Non-Euclidean Geometries, Marvin Jay Greenberg]