Invention vs Discovery

A lurking question, old as Greece, does ask:
Is math invention or discovery?
The answer’s both, but here the harder task
is knowing how it switches.  So you see,
most truths are found with constructs, which in turn
are built from truths like chickens and their eggs.
One needs to know, should this fact raise concern,
that constructs can be daft; just logic’s dregs.
The truths discovered tell how to define
the structures that make math and world align.

Pythagoras’s theorem about lengths,
is one such truth that he (and others) found.
Observed, discovered, and through mental strength,
it was then proved with pictures that astound.
Much later mathematicians, minds alit,
would formalize, as sets, both “length” and “space”.
These constructs were defined so that by writ
Pythagoras’s truth remains the case.
That is, this “theorem” now seems just defined.
As such, one might then lose their peace of mind.

So in this world where "space" is formalized,
do all those pretty proofs now lose their charm?
Of course not!  We all ought to realize
how “length” could change its meaning without harm
to mathematical consistency
in all the theory giving facts of space.
However, then these “facts” would hardly be
related to the nature we embrace.
And hence, a truth has told how to define
a structure that makes math and world align.