Q&A: The A Priority of Mathematics
The A Priority of Mathematics
Question
I heard an argument against the a priori status of mathematics. The argument hinges on the invention/discovery of non-Euclidean geometries, which describe actual reality (according to the person making the claim). Seemingly, this innovation undermines the a priori (and deductive) status people seek to grant mathematics.
I wasn’t convinced, because:
1. Even if the old Euclidean system is incomplete or problematic, that does not mean it is not an a priori system.
2. If the “true” geometry of the world was discovered (say, Riemannian geometry), then it itself is also a priori.
3. There are philosophers of science who maintain דווקא that the Euclidean system is the true one and correctly describes physical reality, and therefore it too is a priori (without this detracting from parallel systems)
Do you agree? Disagree? Both…?
Answer
Either I didn’t understand the argument, or it is really stupid. I don’t see any argument here, and in any case I don’t see what there is to answer.
Discussion on Answer
What connection is there between non-Euclidean geometries and the a priority of mathematics? If anything, I would see an inverse connection, and even that is not certain.
So then you agree with me that mathematics is a priori?
Personally, yes. But regardless of that, I don’t see why other geometries would count as an argument against it.
I think that gentleman thought that if a priori knowledge could exist at all — and in his view it probably cannot — then it would have to be immune to any revision. Based on that mistaken assumption, the historical revision that classical geometry underwent (in effect, its replacement) was interpreted by him as evidence that from the outset it had not been a priori. That was how he tried to “prove” that the distinction I myself drew between a priori and empirical is a failed distinction even in the arena of a strict deductive science like mathematics.
You see… and then we expect to defeat Hamas… may God have mercy and save us.
Now I understand what he meant. But there is a confusion here. Mathematics does not say that exactly one straight line passes between two points, or that parallels do not meet (the axioms of Euclidean geometry). It says that if one adopts those assumptions, or in a straight space where those assumptions hold, various theorems follow (for example, that the sum of the angles in a triangle is 180). That is an a priori necessity, not the assumptions themselves. Therefore different geometries do not touch this at all. They simply describe other spaces, and therefore those spaces have different properties.
Okay, and how would you describe the status of those basic assumptions (exactly one straight line passes between two points, etc.) that yield those theorems? A priori or empirical?
They have no status at all, because they are not claims. They are hypothetical assumptions, and what interests the mathematician is: if they hold, what will the results be (the theorems). When you come to make a claim about the world — that it is Euclidean or not — that is not a claim in mathematics but in physics, and that is of course an empirical claim that can be refuted.
So if we apply the principle of charity to that gentleman, maybe we could say that what he really meant was those hypothetical assumptions on which geometry is founded…? Since in your view those are empirical. But of course, if we do that, we have already left mathematics, and then his claim that “mathematics (or at least geometry) is not a priori” collapses anyway… isn’t that so?
By the way, in your view can one argue that these basic assumptions (on the basis of which the mathematician builds theorems) are indeed not part of mathematics, but also not part of physics? Could one place them as 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, then it is not a claim but a hypothetical assumption.
The claim (not mine) is that mathematics is not a priori, or at least that part of it is not. I take it that in your view he is mistaken?
He tried to illustrate this by means of the fact that non-Euclidean geometries were discovered.