How may the terms "a priori" and "a posteriori" be used in(side) of mathematics?

Question

This question seems either trivial or somewhat vague; let me explain further.

I apologize if I am misunderstanding the concepts or missing the point entirely; I am a mathematics student and I certainly lack any serious knowledge of philosophy. Still, I am interested if there is an opinion on this:

The terms "a priori" and "a posteriori" may roughly be identified with knowledge (or justification, rather) that is free from experience, in the first case, and contingent on experience, in the second case.

In mathematics one often finds the "a priori" term used in proof-writing, which I suppose may often just be for the sake of exclamation (somewhat like the use of "a fortiori") or even just for style points. I wonder whether there is merit to using the term in a setting (the mathematical one) in which all knowledge and justification is "obviously" a priori. Am I missing the point here?

On the other hand one find also the term "a posteriori" used in mathematical proofs. Let me give two examples:

(i) Suppose one wants to give a proof that a class of objects C has property P. It is not obvious that every object belonging to C has P, but after some pages of deductions we arrive at the conclusion that, indeed, any such object from C has P. We have only after close observation have found this, but may this justification be considered "a posteriori"? (I have seen the term used in this way.)

(ii) Assume now there is a property P that is satisfied by some, but not all objects of C. One sometimes finds statements like "It is not a priori true that A in C has P, but after fixing A we find, a posteriori, that A has P."

(As an example for those comfortable with some linear algebra: "It is not a apriori true that a k-vectorspace is finite-dimensional. The k-vectorspace of polynomials of degree n is a posteriori finite dimensional.")

I guess the two cases are similar. Would someone explain to me, whether or not this is a gross misuse of language or whether one might consider observations inside of mathematics as some kind of a posteriori knowledge?

Thanks a lot, I am happy to try to clarify further if this was unclear!

Edit: J.-P. Serre, one of the most renowned mathematicians ever and someone who is generally considered a great expositor, is a frequent user of the term "a priori". To give another example, consider this example from his "Local Fields" [p.79, Prop. 17]:

"[...]We know a priori that G(K_n/K) can be identified with a subgroup of G(n) (cf. Bourbaki, Alg., Chap. V, §11);[...]"

The statement is that some object (a Galois group) maybe identified with another, then he cites a source for a proof of the statement.

Another Edit: This is from Hartshorne's Algebraic Geometry:

"[...]We will begin our study in an oblique manner by defining the notion of an "abstract nonsingular curve" associated with a given function field. It will not be clear a priori that this is a variety. However, we will see in retrospect that we have defined nothing new.[...]"

Answer 1

In most areas of math a priori and a posteriori are usually borrowed as an academic jargon simply for "up until now" and "afterwards", respectively.

But some statistics topics such as Frequentist inference with its null hypothesis significance testing (NHST, developed by Fisher, Neyman and Pearson in the early and mid-1900s) and Bayesian inference, one needs to seriously consider some a priori assumptions of one's sampling methods as suggested here.

The problem with frequentist approaches is that you can only quantify evidence against a hypothesis, but never for it. If you want to test whether two groups are equal, then Bayesian statistics are the most appropriate tool.

So by design the NHST framework a priori assumes statistical evidence against a certain hypothesis which limits its applicable use cases. Of course the other Bayesian framework also assumes or needs certain a priori probability distribution(s).

Another famous similar situation is discussed in contemporary philosopher Bostrom's book Anthropic Bias: Observation Selection Effects in Science and Philosophy. There's a puzzle in decision theory called Sleeping Beauty problem. Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.

If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

It turns out that if we a priori assumes self-sampling assumption (SSA), then according to its reference:

For instance, if there is a coin flip that on heads will create one observer, while on tails it will create two, then we have two possible worlds, the first with one observer, the second with two. These worlds are equally probable, hence the SSA probability of being the first (and only) observer in the heads world is 1/2, that of being the first observer in the tails world is 1/2 × 1/2 = 1/4, and the probability of being the second observer in the tails world is also 1/4.

This is why SSA gives an answer of 1/2 probability of heads in the Sleeping Beauty problem.

If we a priori assumes self-indication assumption (SIA) based on all possible observers due to a posteriori evidence, then according to same reference:

For instance, if there is a coin flip that on heads will create one observer, while on tails it will create two, then we have three possible observers (1st observer on heads, 1st on tails, 2nd on tails). Each of these observers have an equal probability for existence, so SIA assigns 1/3 probability to each. Alternatively, this could be interpreted as saying there are two possible observers (1st observer on either heads or tails, 2nd observer on tails), the first existing with probability one and the second existing with probability 1/2, so SIA assigns 2/3 to being the first observer and 1/3 to being the second - which is the same as the first interpretation.

This is why SIA gives an answer of 1/3 probability of heads in the Sleeping Beauty Problem.

Although this anthropic principle was originally designed as a rebuttal to the Doomsday argument (by Dennis Dieks in 1992) it has general applications in the philosophy of anthropic reasoning, and Ken Olum has suggested it is important to the analysis of quantum cosmology... Ken Olum has written in defense of the SIA. Nick Bostrom and Milan Ćirković have critiqued this defense.