Mathematics Learning Is Not Special
We may treat mathematics too specially, as being unnecessarily privileged compared to other academic disciplines or ways of knowing.
This excerpt, from A Critical Introduction to Testimony, unearths an important issue in my view: we may treat mathematics too specially, as being unnecessarily privileged compared to other academic disciplines or ways of knowing.
There are . . . types of knowledge where we really do have to know things off our own bat, or so many people have argued. Although it may be possible to testify to such knowledge in a purely generic way, a hearer could not properly be said to acquire knowledge on the basis of such testimony.
Williams argues that someone who believes a mathematical proposition, p, on the basis of someone else's authoritative testimony, but cannot mathematically demonstrate its truth, cannot be said to know it. On Williams's account, even a perfectly reliable informant could not succeed in communicating mathematical knowledge by mere say-so, since 'access to mathematical truth must necessarily lie through proof, and [. . .] therefore the notion of non-accidental true belief in mathematics essentially involves the notion of mathematical proof'. Williams is otherwise sympathetic towards testimony as a source of knowledge, and so it is plausible that his position is motivated by the assumption that mathematical knowledge has to meet higher standards than 'ordinary' knowledge, and in particular requires understanding of the proof rather than just of the content of the theorem.
But of course mathematics is not special in this way. This is especially clear from the author's own example:
Imagine that you are a librarian cataloguing the collection of a renowned mathematician who died long ago and who made important contributions to a number of subfields in mathematics, often displaying considerable ingenuity in deriving complex mathematical theorems and rarely making a mistake. In a dusty tome on arithmetic you come across the following statement, neatly recorded in the great man's distinctive handwriting:
[MATHEMATICAL TESTIMONY.] It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.
One can come to know, via a teacher's "testimony," that, for n > 2, there are no natural numbers a, b, and c, such that aⁿ + bⁿ = cⁿ. This can be done without understanding the proof at all. In fact, almost everyone will know Fermat's Last Theorem in just this way and no other.
