Is mathematical knowledge defeasible by experience, If so, in this way?

Question

Kitcher’s basis for the indefeasibility requirement on a priori knowledge is that empirical indefeasibility is required for independence, in the sense Kant intended, of empirical evidence. I am inclined to agree with Casullo (2003, §2.2) that Kant’s remarks on ‘independence’ are not sufficient to determine whether what he had in mind entailed such indefeasibility or not. On the other hand, a good reason for doubting the interest of the indefeasibility-involving conception is that it is just too easy to show that there is no a priori knowledge on this conception. Even core putative cases of a priori knowledge – mathematical knowledge, for instance – are, familiarly, defeasible by empirical evidence. However good your intuitions, conceptual analyses, proofs or deductions are, if all the mathematical experts tell you that you have made a subtle but crucial mistake, your justification will (and should) be defeated. Thus if there is anything interesting to debate about the claim that mathematics is a priori, the notion of a prioricity in play cannot involve an empirical indefeasibility requirement.

Carrie Jenkins, 2008, "A Priori Knowledge: Debates and Developments", p.2 (this paper is not a published version but a final draft to be found her website.)

I don't understand why mathematical justification will(and should) be defeated simply by mathematicians 'telling that there's been a mistake'. For the justification to be defeated, I think it is necessary to point out what exactly the mistake is.

Then, it seems that the proofs, or whatever, were actually bad and false, thus not knowledge at the outset. it seems to me that the justification in question is defeated in the way that isn't empirical.

Answer 1

Mathematical knowledge is documented in mathematical theorems. To challenge a mathematical theorem means

• to challenge the theorem by a counter example

• or to challenge its proof by indicating an error or a gap.

Answer 2

You are right. A sound mathematical proof remains sound regardless of how many people disagree with it or what their qualifications are. We may think of Cantor, who was ridiculed by mathematicians of his time for his work on infinities, now widely understood to be correct.

However, a mistake is always possible in any calculation or proof. If many mathematicians say there is a problem with a proof, then it's more likely that there is a problem than not. Merely their saying so doesn't prove there is a mistake, but it does provide Bayesian evidence in that direction.

There are, however, empirical ways to falsify mathematical knowledge. Suppose you are solving a Sudoku puzzle, and realize, looking down a column, that you have written a definite 7 twice in that column. In Sudoku you are not allowed to have a number appear more than once in a column. Thus, this observation shows you made a mistake, assuming the puzzle itself wasn't flawed. The observation of the two 7s is a real, physical observation, the 7s being pencil marks on paper, and it falsified the deductive reasoning you'd been doing up until that point.

Speaking generally, mathematics revolves around the production of proofs, which are physical objects, or at least physically instantiated. We can consider mathematics to be an engineering discipline, where the object is to manufacture a piece of paper with ink on it that satisfies a list of properties. For example, that the arrangement of ink is in accordance with the inference rules of ZFC, and the bottom line of ink is the ZFC formula for the Reimann conjecture. Engineers are always trying to make physical objects satisfying some list of properties.

Answer 3

Mathematicians will not just "tell" you that you've made a mistake, they usually point out a counterexample to show you that it doesn't work, leaving scientists of all branches love hating them for ruining years of work, yet preventing them from ruining even more, while providing no concise reason why that is the case.

The thing is usually mathematicians compile a list of axioms that are thought to be true and then a whole universe disentangles out of those. Now if you can show that there are "in-universe" contradictions that would hint at at least one of the axioms not being true, hence the whole think is rubbish. Now from that you don't know what it is or whether it's fixable so yeah... people love/hate math.

Now despite all of that happening in a fictional universe created through these axioms, that is still somewhat of an "empirical" approach in that you use experiments in that universe to contradict the existence of that universe.

Answer 4

For the justification to be defeated, I think it is necessary to point out what exactly the mistake is.

I think it's implied in the scenario, that the expert mathematicians who reviewed your proposed proof have pointed out to you what they say the error is. The argument being made by Jenkins doesn't depend on them not pointing out the mistake.

The issue is that although they might point out the mistake, the nature of mathematics is that one can be utterly convinced by a mistaken proof, while lacking the understanding necessary to make sense of experts who point out the mistake. (People in this situation who assert that the experts are wrong, are referred to as cranks.) Jenkins says the mathematical experts might identify a "subtle but crucial mistake" in a proof you propose.

So suppose you are in a situation where you are convinced by the logic of a proof, but all the experts say the proof is incorrect and point out a subtle mistake. In this situation you can listen to their explanations of your mistake, and try to understand why it is a mistake. But empirical evidence shows that:

• It is common for non-experts to be fooled by a subtle mistake in a purported proof;
• It is common for non-experts to not understand the expert explanations as to why the proof is flawed, and even common for non-experts to believe that the experts are wrong;
• In fact, experts are very rarely unanimously wrong.

So the empirical evidence tells us that as a non-expert it is wise to accept the expert consensus over one's own conviction that the proof is correct, and empirical evidence shows that as a non-expert we should weigh the expert consensus over the proof's apparently inescapable logic. This applies both before we try to understand the experts' explanation of why the proof is flawed, and after we have listened to that explanation but found it logically uncompelling.

Therefore, even if we find a proof compelling, our knowledge that the result is true is still defeasible by empirical evidence. That is Jenkins' argument as I understand it.

Answer 5

Mathematics came out of human experience. But part of it, as Kant argued, is inherent in how our minds function. But this is nothing special, after all music is part of human culture but we did not invent the capability of being able to sense sound. It is already part of us.

Mathematics often uses ordinary intuition to develop new ideas. But because they are expressed in a complex language of its own, this is hard to see.

Take for example, topology. This is a large part of current mathematical thinking. It was invented in the early part of the 20C. It investigates notions of continuity. Had they proven that a line was not continuous, then they would have changed the theoretical apparatus. This is because ordinary intuition tells us lines are continuous.