Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).

An obvious counter-example is the law of excluded middle itself, discovered by Aristotle between 400 and 300 B.C.

LEM is a result in the field of propositional logic. However, what about results outside of logic, like number theory or analysis? The simplest such result I know of is the existence of step functions, but I don't know when that was discovered.

I'll leave "not constructively valid" a bit open-ended, but just saying that there is no *known* constructive proof doesn't count. A valid answer could show that the result implies a constructive taboo, or that it's independent of a constructive type theory or set theory, for example. (In the case of step functions, their existence implies the analytic LPO, a constructive taboo.)

very first postulatein Euclid'sElementsis “to draw a straight line from any point to any point” (“ᾘτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν”). The fact that any two points in the plane $\mathbb{R}^2$ are connected by a line is equivalent to the robust division principle, which is not provable constructively. $\endgroup$Elements, it's often hard to decide whether the statement is “constructively valid”, because even thinking classically Euclid is a bit careless about stating when points are supposed to be distinct and such stuff, so we can probably make pretty much everything valid by sprinkling “apart” fairy dust whenever he speaks of two points and such. But is this cheating? $\endgroup$2more comments