Would transcendental numbers even be considered to exist if not for significant ones like $\pi$ and $e$?

Question

I mean, even if $\pi$ and $e$ were rational, those values would still exist — actually, I guess even that can be questioned, if you get deep into the philosophy, but I'm not sure what the consensus is among mathematicians. Regardless, what I'm asking is, would we even know that transcendental numbers exist, if we never ran into any that were significant?

Answer 1

It is easy to see that transcendental numbers exist even without producing one. This is because $\mathbb{R}$ is uncountable and the set of algebraic numbers of countable, so in fact most real numbers are transcendental.

Many other specific transcendental numbers have been described than $\pi$ and $e$ as well. I don't know the history of it, but it's possible these other transcendental numbers were proved to be transcendental first. Likely so, in fact, because it is much easier to prove these other numbers are transcendental than it is for $\pi$ and $e$.

Answer 2

Yes, since mathematicians (Liouville, to be more precise) proved that transcendental numbers exist long before it was proved that $\pi$ and $e$ are among them.

Answer 3

"Would we even know that transcendental numbers exists, if we never ran into one?". Yes, we would. Since the field $\overline{\Bbb Q}$ of algebraic numbers in the field extension $\Bbb C \mid \Bbb Q$ is countable, there must exist uncoutably many transcendental numbers - even if we never ran into one.

Answer 4

If nobody had ever tried and failed to write a polynomial equation for $\pi$ or $e$ then it is quite likely that we would not know transcendental numbers exist, because they were a sort of "unknown unknown".

The question doesn't seem to concern mathematics so much as the philosophy of knowledge. The question of writing an equation for $\pi$ in terms of other numbers surely occurred to the ancient Greeks, for example, but the fact that no finite equation actually exists, and hence the notion of transcendental numbers, did not occur to them. See squaring the circle and so on, where it took thousands of years to discover that some problems were insoluable, eventually leading to the notion of transcendental numbers.