Benacerraf Mathematical Truth

This article is as influential as the other Benacerraf article we read. Together, they set the agenda for contemporary philosophy of mathematics. Both are somewhat dated. I'll focus on the morals that still influence the philosophy of mathematics, and only comment briefly on the rest. Benacerraf wrote, relatively recently, an article about how he sees the situation today and what he still thinks is interesting about these articles called "What mathematical truth could not be."

Benacerraf, in MT, emphasizes that philosophy of mathematics is part of philosopy. A satisfactory theory of mathematical truth and knowledge must fit with a satisfactory general account of mathematical truth and knowledge and, conversely, a general account of mathematical truth and knowledge cannot be satisfactory unless it coheres with some account of mathematical truth and knowledge since, after all, we do have some knowledge of mathematical truths. That is not an ambitious claim: 2+2=4 is already a substantial enough mathematical truth to cause trouble for most general accounts of truth and knowledge, and we do know that 2+2=4.

What is the problem? We need an account of mathematical truth that not only gets the truth values right, but one that shows that mathematical truth is truth in relevantly the same sense in which other kinds of truths are true, that is, that it isn't a mere coincidence or pun that we use the same word in both cases.

Similarly, we know that 2+2=4, and any satisfactory account of that fact and of knowledge generally must make it possible to see that this is knowledge in the ordinary sense of the term, or, if it isn't, admit as much, stop pretending that it is knowledge, and account for the usual mistaken belief that it is knowledge in some convincing way. That doesn't seem to be a promising alternative, and so, by and large, I'll ignore it. But Benacerraf is not ruling it out, merely indicating that it looks really hard, and there is not serious attempt at it.

These are already important points, but Benacerraf goes on to point out that all philosophies of mathematics that we actually have are of two types:

1. Those that get truth "right"
2. Those that get knowledge "right"

The central problem is getting both right at the same time.

The two clearest examples are, on the truth side, the usual simplified version of Gödelian Platonism, and, on the knowledge side, naïive formalism.

The Platonist view says that mathematical truths are truths about mathematical objects and their relations, just as all truths are truths about objects and their relations. This kind of view makes it obvious in what sense mathematical and mundane truths are all truths: A truth bearer is true just if the relations it asserts obtain between the objects that are its subject matter actually do obtain, and that works for cats, on, mats, 2, +, =, and 4 in exactly the same way, with no special variations.

The problem arises when we get to knowledge: We know the cat is on the mat because we can see it. We don't know that 2+2=4 because we can see it.

In fact, Gödel does try to offer an account of mathematical intuition that plays a role analogous to perception. His theory is nontrivial, despite what is usually said, but it is Kantian and heavily influenced by Husserl, and so the likes of Benacerraf find it completely unhelpful.

On the other side, the formalist easily accounts for how we know when various theorems are true: we can see proofs and verify relations (that is, check that a proof is correct according to some system) in much the same way that we can see a cat on a mat. The shoe is on the other foot: What does that have to do with the numbers 2 and 4 (which are not the numerals) and their relations?

One can explain that it is true that the expression "2+2=4" is derivable in some system. What is missing is what that has to do with arithmetic in any ordinary sense.

Set theory provides us with a notion of a model and with a completeness theorem: We can show that what can be derived is "true of" a model. This is very like what we want, but there is a chicken and egg problem: we only get the nice fit on the basis of a prior set theory, and how do we take that? Nonetheless, the picture has been very important, because most philosophers (of mathematics and not) have used it as a sort of guide toward what we hope to achieve.

Various "disquotational" or "minimalist" theories of truth are in fashion now. The basic idea is that truth is not a substantial notion, merely a linguistic device for removing quotes:
"Snow is white" is true if and only if snow is white.

Whatever you may think about such views, they don't have much to do with our current problem. In stating our problem, we used the word "truth" a lot, but it can be read throughout as merely a convenient device for removing quotes. What we are concerned about is how it is that 2+2=4 and the cat is on the mat, and how we can have good reasons to know that.

-- ProfessorShaughanLavine? - 21 Feb 2005