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This is almost certainly not the historical justification for the term "smooth", but I have found that the term happily coincides with my analytic intuition of smoothness. Namely, I think of a scalar …

If $p$ is a prime, then every minor of the Fourier matrix $(e^{2\pi i jk/p})_{1 \leq j,k \leq p}$ is non-singular. This fact was proven by Chebotarev in 1927 (answering a question of Ostrowski), Dani …

I believe that Fefferman's disproof in 1971 of the $L^p$ boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner a …

It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of t …

Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that …

In additive combinatorics, one often seeks to count patterns such as an arithmetic progression $a, a+r, \ldots, a+(k-1)r$. When doing so, one is naturally led to expressions such as $$ {\bf E}_{a,r \ …

(Converted from a comment to an answer as requested.) Non-Euclidean geometry was initially developed in hopes of deriving the parallel postulate from the other axioms of Euclidean geometry, as can be …

After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence …

In number theory, I would say that the counterpart of the "Maximise" principle is the "Local to global principle": if there is no local obstruction to solvability of some number-theoretic problem (e.g …

In algebraic geometry, I would say that the counterpart of the "reflection" principle is the Lefschetz principle, as discussed in this previous MathOverflow question: if something is solvable in a "bi …

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes r …

If first-order logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "Godel, Escher, Bach", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of enc …

This isn't an exact analogue to P != NP, in which two large classes exist and it is undecided whether they are equal or not; instead, two large "universes" exist, of which only one is the truth, with …

This inequality is also a corollary of the main result of Fefferman, Charles; Stein, Elias M., Some maximal inequalities, Am. J. Math. 93, 107-115 (1971). ZBL0222.26019. which asserts that $$ \| \s …

Here are some that came to mind: Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two t …