The a priori of mathematics

The claim (not mine) that mathematics is not a priori, or at least not in part. I understand that you think he is wrong?
He tried to demonstrate this by the fact that non-Euclidean geometries were discovered

What is the connection between non-Euclidean geometries and the a priori of mathematics? If at all I would see an anti-connection, and even that is not certain.

So you agree with me that mathematics is a priori?

In itself, yes. But regardless of that, I don't see why other geometries are an argument against it.

I think that gentleman thought that if a priori knowledge could exist at all - according to his method it probably cannot exist - it must be immune to any revision. Based on this mistaken assumption, the historical revision that classical geometry underwent (in fact, its replacement) was interpreted by him as evidence that it was not a priori in the first place. Thus he sought to "prove" that the distinction that I myself made between a priori and empirical is a failed distinction even in the arena of a serious deductive science like mathematics.
You understand - and then we seek to defeat Hamas - may God have mercy and save

Now I understand what he meant. But there is a mix-up here. Mathematics does not say that between two points there is only one straight line or that parallelograms do not meet (the axioms of Euclidean geometry). It says that if you adopt these assumptions, or in a straight space where these assumptions apply, different theorems hold (for example, that the sum of the angles in a triangle is 180). This is an a priori necessity, not the assumptions themselves. Therefore, different geometries do not touch on this in any way. They simply describe different spaces and in any case have different properties.

Okay, and how would you describe the status of those basic assumptions (between two points there is only one straight line, etc.) that hold those theorems? A priori or empirical?

They have no status, because they are not claims. These are hypothetical assumptions, and what interests the mathematician is whether they hold and what the results (theorems) will be. When you come to claim that the world is Euclidean or not, this is not a claim in mathematics but in physics, and this is of course an empirical claim that is subject to refutation.

So if we apply the principle of grace to that gentleman, perhaps we can say that he actually meant those hypothetical assumptions on which geometry is founded..? Because according to you, these are empirical. But of course, if we do that, we have already gone outside of mathematics, and then his claim that “mathematics (or at least geometry) is not a priori” falls anyway, right?

By the way, do you think it is possible to argue that these basic assumptions (on the basis of which the mathematician builds theorems) are not part of mathematics, but they are also not part of physics? Or can they be considered a logical or metaphysical foundation for mathematics?

No. If it is a claim about the world, it is empirical, and if it is a mathematical assumption, it is not a claim but a hypothetical assumption.