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.

Review of van Fraassen's Scientific Representation

Here is a short review of van Fraassen's 2008 book Scientific Representation: Paradoxes of Perspective. It will eventually appear in the BJPS. I found the book to be very impressive, although I had trouble understanding the way in which van Fraassen deploys indexical judgment to avoid problems like the Newman problem which have sunk some versions of structuralism. As I put it in the review,

The central unresolved issue with van Fraassen’s empiricist structuralism is what his appeal to context in the solution of his “problems of perspective” comes to. I would distinguish the weaker claim that an ability is not the same as a description of that ability (p. 83) from the stronger claim that a given indexical proposition is distinct from all the propositions expressed by any scientific theory: “Is there something that I could know to be the case, and which is not expressed by a proposition that could be part of some scientific theory? The answer is YES: something expressed only by an indexical proposition” (p. 261). The stronger point seems to link having an ability to knowing an indexical proposition. But here van Fraassen says that even though what these essentially indexical propositions express is a crucial part of any account of representation, this is outside the scope of any scientific theory. As a result, it is not possible to arrive at a scientific theory of representation. This is much less plausible than the mere distinction between an ability and a description of that ability. The weaker claim allows for the possibility of a fully naturalistic theory of how we can think and locate ourselves with respect to our representations, i.e. the abilities which underlie our knowledge of the relevant indexical propositions. While linguistics and cognitive science are not adequate at this stage of science, it is hard to see why what these indexical propositions express would be beyond their scope. It is possible that van Fraassen takes his discussion in Part IV to undermine the demand for this sort of naturalistic completion of a scientific theory of scientific representation, but if this is his intention, then I have failed to follow the argument. It also possible that van Fraassen did not intend to exclude these indexical propositions from the scope of scientific investigation, but then he owes us a clearer account of how our abilities relate to our knowledge of indexical propositions.

Discussion Note of Batterman on Mathematics and Explanation

I have written a short discussion note of Batterman's recent article on mathematical explanation in science. If you have looked at the article, you may recall that he criticizes my "mapping account" as an account of how mathematics helps in explanation, especially the sorts of explanations using asymptotic reasoning which Batterman himself has spent so much time on. The basic point I make in my reply is that I was trying to provide an account of descriptive or representational content in terms of mappings and that I agree that this approach to description is not sufficient to ground explanatory power in all cases. Still, I argue that a theory of explanatory power can build on what I offer for descriptions, and that any account of explanation must say something about how explanations differ from descriptions.

Comments are, of course, appreciated!

Mathematical Explanation in the NYRB

In his recent review of Dawkins' Oxford Book of Modern Science Writing Jeremy Bernstein characterizes one entry as follows:

W.D. Hamilton’s mathematical explanation of the tendency of animals to cluster when attacked by predators.

The article in question is "Geometry for the Selfish Herd", Journal of Theoretical Biology 31 (1971): 295-311. (Online here.) Given the ongoing worries about the existence and nature of mathematical explanations in science, it is worth asking what led Bernstein to characterize this explanation as mathematical?

The article summarizes two models of predation which are used to support the conclusion that the avoidance of predators "is an important factor in the gregarious tendencies of a very wide variety of animals" (p. 298). The first model considers a circular pond where frogs, the prey, are randomly scattered on the edge. The predator, a single snake, comes to the surface of the pond and strikes whichever frog is nearest. Hamilton introduces a notion of the domain of danger of a frog which is the part of the pond edge which would lead to the frog being attacked. Hamilton points out that the frogs can reduce their domains of danger by jumping together. In this diagram the black frog jumps between two other frogs:



So, "selfish avoidance of a predator can lead to aggregation."

In the slightly more realistic two-dimensional case Hamilton generalizes his domains of danger to polygons whose sides result from bisecting the lines which connect the prey:



Hamilton notes that it is not known what the general best strategy is here for a prey organism to minimize its domain of danger, but gives rough estimates to justify the conclusion that moving towards ones nearest neighbor is appropriate. This is motivated in part by the claim that "Since the average number of sides is six and triangles are rare (...), it must be a generally useful rule for a cow to approach its nearest neighbor."

So, we can explain the observed aggregation behavior using the ordinary notion of fitness and an appeal to natural selection. What is the mathematics doing here and why might we have some sort of specifically mathematical explanation? My suggestion is that the mathematical claim that strategy X minimizes (or reliably lowers) the domain of danger is a crucial part of the account. Believing this claim and seeing its relevance to the aggregation behavior is essential to having this explanation. Furthermore, this seems like a very good explanation. What implications this has for our mathematical beliefs remains, of course, a subject for debate.

Southern Journal of Philosophy Relaunched

The Southern Journal of Philosophy has relaunched with a new publishing agreement with Wiley, a new webpage and a new editorial board (including me). As the webpage indicates

The Southern Journal of Philosophy has long provided a forum for the expression of philosophical ideas and welcomes articles written from all philosophical perspectives, including both the analytic and continental traditions, as well as the history of philosophy. This commitment to philosophical pluralism is reflected in the long list of notable figures whose work has appeared in the journal, including Hans-Georg Gadamer, Hubert Dreyfus, George Santayana, Wilfrid Sellars, and Richard Sorabji.

The jewel of each volume is the Spindel Supplement, which features the invited papers and commentaries presented at the annual Spindel Conference. Held each autumn at the University of Memphis and endowed by a generous gift from the Spindel family, each Spindel Conference centers on a philosophical topic of broad interest and provides a venue for discussion by the world's leading figures on that topic.

I hope the philosophers will take advantage of this special venue for pursuing new and exciting directions for research in philosophy.

New Entries in Internet Encyclopedia of Philosophy on the Philosophy of Mathematics

Under the editorial guidance of Roy Cook a number of new entries in philosophy of mathematics have appeared on the Internet Encyclopedia of Philosophy. As I understand it, the aim of this site is to present relatively short summaries which are accessible to a wider audience, esp. undergraduate students, than some other options.

Check out these recent entries:

Bolzano's Philosophy of Mathematical Knowledge (by Sandra Lapointe)

The Applicability of Mathematics (by me -- more shameless self-promotion!)

Mathematical Platonism (by Julian Cole)

Predicative and Impredicative Definitions (by Oystein Linnebo)


A list of the all of the philosophy of mathematics entries can be monitored here.