The empirical approach tries to put math on the same footing as science. Axioms aren’t absolute, heaven-sent truths. They are empirical facts that have always been true in our experience, to the point that we’re willing to draw conclusions based on them. Axioms then are scientific observations like “the weight of a piece of metal doesn’t change over time”. To put a fine point on this, despite all the hoopla about deductive reasoning, we use *inductive* reasoning (generalizations based on our experience) to justify our axioms! Hypothetically we could run into a situation where we can’t draw a straight line through a point, in the same way that we could find a piece of lead that weighs more on Fridays than Mondays. In such situations we would have to revise the canons of science/geometry, but these scenarios seem pretty far-fetched. It looks like, by and large, the ancient Greeks took this attitude toward mathematical axioms; they saw geometry as a science on par with physics and biology (or what we would call physics and biology - these categories didn’t really exist in peoples’ minds). If Euclid had been able to get into a space ship and visit a black hole, he would have discovered that his axioms didn’t always hold, and he may have revised The Elements.
The formal school of thought dodges the whole question. It holds that, in doing a mathematical proof, we aren’t studying the geometry of a specific universe, a particular system of numbers, etc.; we aren’t studying any specific entity (even a purely abstract one) at all, so it’s not meaningful to ask if the axioms *are* true. When we’re doing a proof, we’re drawing conclusions that apply to anything for which the axioms *happen* to be true. For example, if you re-define “straight line” to mean a great arc on a globe, it turns out that four of Euclid’s postulates are “true” of this new definition, and lo-and-behold his proofs (at least, the ones that don't use the Parallel Postulate) carry over. This school of thought isn’t particularly concerned with what is true or false: it focuses on the mechanics of deductive reasoning, and what things can be concluded from what premises.