New Book: Mary Leng's Mathematics and Reality

Mary Leng's book is now out! From the back cover:

Mary Leng offers a defence of mathematical fictionalism, arguing that we have no reason to believe that there are any mathematical objects. In mounting this defence, she responds to the indispensability argument for the existence of mathematical objects ... In response to this argument, Leng offers an account of the role of mathematics in empirical science that does not assume that the mathematical hypotheses used in formulating our scientific hypotheses are true.

It's great to see an extended defence of fictionalism out there. In my book manuscript I argue that fictionalism can't work, so I will be reading this quickly to see what options the fictionalist has available! There will eventually be a symposium on this book in the journal Metascience and hopefully other reviews will appear soon.

Gettier Cases for Mathematical Concepts

Although I noted the appearance of this book back when it came out, it is only recently that I have had the chance to read Jenkins' Grounding Concepts. I agree with the conclusion of Schechter's review that "Anyone interested in the epistemology of arithmetic or the nature of a priori knowledge would profit from reading it." But at one point Schechter asks

In supporting Jenkins' view, it would be helpful to have an intuitive case of a thinker who forms a justified true belief that is not knowledge on the basis of a competent conceptual examination of an ungrounded concept. That is to say, it would be helpful to have a clear example of a Gettier case for concepts.

Jenkins uses these sorts of cases to undermine Peacocke's account of a priori mathematical knowledge. But she often resorts to comparisons between maps and concepts in a way that makes the point less convincing.

In the last chapter of my book I discuss the issue. While I do not ultimately agree with Jenkins on all points, I think she has a very good objection to Peacocke's views. Here is how I summarize the issue in the manuscript. I have adapted Jenkins map cases to the sort of mathematical concept cases that I believe Schechter is after: Jenkins'

basic concern is that even a perfectly reliable concept is not conducive to knowledge if our possession of that concept came about in the wrong way. She compares two cases where we wind up with what is in fact a highly accurate map. In the first case, which is analogous to a Gettier case, a trustworthy friend gives you a map which, as originally drawn up by some third party, was deliberately inaccurate. The fact that the map has become highly accurate and that you have some justification to trust it is not sufficient to conclude that your true, justified beliefs based on the map are cases of knowledge. In the second case, simply finding a map which happens to be accurate and trusting it "blindly and irrationally all the same" will block any beliefs which you form from being cases of knowledge. Jenkins extends these points about maps to our mathematical concepts:

A concept could be such that its possession conditions are tied to the very conditions which individuate the reference of that concept (...), but not in the way that is conducive to our obtaining knowledge by examining the concept (Jenkins 2008, p. 61).

The first sort of problem could arise if the concept came to us via some defective chain of testimony. For example, a "crank" mathematician develops a new mathematical theory with his own foolishly devised concepts and passes them off to our credulous high school mathematics teacher, who teaches the theory to us. The mere fact that the crank mathematician has happened to pick out the right features of some mathematical domain is insufficient to confer knowledge on us. The second kind of problem would come up if I was, based on a failure of self-knowledge, like the crank mathematician myself, coming up with new mathematical concepts based on my peculiar reactions to Sudoku puzzles. Again, this approach to mathematics would not lead to knowledge even if my concepts happened to reflect genuine features of the mathematical world.

I agree with Jenkins that examples of this sort show the need for either some grounding for our concepts or some non-conceptual sources of evidence.