Mathematical Truth (Benacerraf 1973), introduces a problem for mathematical knowledge that I find incredibly compelling. Benacerraf thinks that there are two requirements of a theory of mathematical truth: semantic uniformity, such that what we mean when we talk about math is similar to what we mean when we talk about everything else; and epistemic uniformity, such that the epistemology of math is similar to the epistemology of everything else. Benacerraf thinks that a good theory of mathematical truth has to meet both goals, and that the two goals are incompatible, which leaves mathematical truth in a rough spot.
On the topic of semantic uniformity, Benacerraf says that "A theory of truth for the language we speak, argue in, theorize in, mathematize in, etc., should ... provide similar truth conditions for similar sentences. The truth conditions as- signed to two sentences containing quantifiers should reflect in relevantly similar ways the contribution made by the quantifiers". For examples, he looks at these two sentences:
(1) There are at least three large cities older than New York. and
(2) There are at least three perfect numbers greater than 17.
Then, he asks if these sentences have the same logical form:
(3) There are at least three F's that bear R to a.
Sentence (1) has the same logical form as sentence (3). This means that sentence (1) is true if there are at least three elements in the set of things that exist which satisfy the predicates “large city” and “older than New York”. For sentence (2), can we do the same thing? Benacerraf replies “That sounds like a silly question to which the obvious answer is “Of course.””. However, he continues, that requires numbers to be in the set of things that exist! A Platonic realist has no problem with this, because they posit the real existence of numbers. Benacerraf calls this the “standard view” – sentence (2) has the logical form of sentence (3), and numbers really exist so this is fine.
However, in intuitionist mathematics, sentence (2) would have a very different logical form than sentence (3). Benacerraf then turns to views where mathematical truth comes from being derivable from a set of axioms. Those theories give a truth-predicate for sentences that is strictly syntactic, allowing us to determine the truth or falsity of a sentence in a system of axioms without referencing anything in the real world. So, under syntactic (what Benacerraf calls “combinatorial”) views of mathematical truth, sentence (2)’s logical form is again unlike sentence (3). This is all just to say that it is controversial that sentence (2)’s logical form looks like sentence (3).
Benacerraf thinks “we shouldn’t be satisfied with an account that fails to treat (1) and (2) [the same]”.
The second condition on a theory of mathematical truth, epistemic uniformity, requires that “we have mathematical knowledge, and that such knowledge is no less knowledge for being mathematical”. In other words, we need to be able to know whether at least some mathematical sentences are true or false. Mathematical knowledge “must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have”.
Epistemic uniformity is hard! If we model sentence (2) in the form of sentence (3), it is a claim about properties of numbers. The “standard view”‘ says that numbers exist, so the sentence at least has a truth-value. Numbers are acausal. The existence or non-existence of the number 3 is not observable by humans. This is a problem for causal accounts of epistemology, where knowledge requires some causal link between the state of the world and your knowledge. Numbers are acausal, so we can’t have knowledge of them, and we can’t have mathematical knowledge more generally.
(Some people say that noticing that putting two items next to two items and seeing that you now have four items gives you some evidence that 2 + 2 = 4. However, “adding two items to two items gives you four items” being true or false has a causal effect on your belief after running the text, so you can have knowledge of it even if you can’t have knowledge of “2 + 2 = 4”.)
Here is the crux of the paper: those two properties can’t both be satisfied. If mathematical sentences are like normal sentences, they refer to objects. If mathematical sentences refer to objects, we can’t have causal knowledge of mathematical truth. Eek! This is a big problem for humans ever gaining knowledge of mathematical truth.