Call for Papers: Mathematical and Scientific Philosophy

Readers of this blog should check out the fall meeting of the Indiana Philosophical Association:

Call for Papers

Mathematical and Scientific Philosophy

with a special session on the Darwin Bicentenary

Indiana Philosophical Association Fall Meeting
Invited Speakers: Colin Allen, Elisabeth Lloyd, Larry Moss

Saturday AND Sunday, 5-6 December 2009
Indiana Memorial Union, IU Bloomington

We invite submissions—from philosophers, logicians, and historians and philosophers of science—on topics that fall under the theme of the meeting. Papers should be 35 minutes reading time, i.e., no more than 17 double-spaced pages. Papers will be blind reviewed; the author’s name and affiliation should therefore appear only on the cover sheet.

Send one copy of your paper and a short, one–paragraph abstract to one of the following.

Peter Murphy
Department of Philosophy
Esch Hall 044U
University of Indianapolis
Indianapolis, IN 46227
murphyp at uindy.edu

Bernd Buldt
Department of Philosophy
CM 21 026
IPFW
Fort Wayne, IN 46805
buldtb at ipfw.edu

Charles McCarty
The Logic Program
Sycamore Hall
Indiana University
Bloomington, IN 47405
dmccarty at indiana.edu

Electronic submissions of papers and abstracts in MSWord or pdf formats are encouraged.
Deadline for Submissions: 15 October 2009
Deadline for Notifications: 9 November 2009
For further information, please email Charles McCarty at dmccarty at indiana.edu

Krugman on Mathematics and the Failure of Economics

Probably anyone who is interested in this article has already seen it, but Paul Krugman put out an article in Sunday's New York Times Magazine called "How Did Economics Get It So Wrong?". The article is very well-written, but a bit unsatisfying as it combines Krugman's more standard worries about macroeconomics with a short attack on financial economics. I am trying to write something right now about the ways in which mathematics can lead scientists astray, and one of my case studies in the celebrated Black-Scholes model for option pricing. Hopefully I can post more on that soon, but here is what Krugman says about it and similar models which are used to price financial derivatives and devise hedging strategies.

My favorite part is where Krugman says "the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth". But he never really follows this up with much discussion of the mathematics or why it might have proven so seductive. Section III attacks "Panglossian Finance", but this is presented as if it assumes "The price of a company's stock, for example, always accurately reflects the company's value given the information available". But, at least as I understand it, this is not the "efficient market hypothesis" which underlies models like Black-Scholes. Instead, this hypothesis makes the much weaker assumption that "successive price changes may be considered as uncorrelated random variables" (Almgren 2002, p. 1). This is the view that prices over time amount to a "random walk". It has serious problems as well, but I wish Krugman had spent an extra paragraph attacking his real target.

Almgren, R. (2002). Financial derivatives and partial differential equations.
American Mathematical Monthly, 109: 1-12, 2002.

New Book: Mathematics and Philosophy

I have just completed a review of the relatively new collection, edited by Bonnie Gold and Roger Simons, called Proof and Other Dilemmas: Mathematics and Philosophy. The review will appear eventually in SIGACT News.

I think everyone who is interested in the interaction between mathematics and philosophy should be encouraged by the volume. The editors have brought together philosophers and mathematicians to try to increase interest in philosophy on the mathematics side. This is a difficult task, and I still have the impression that a philosophy-mathematics collaboration is more difficult than other kinds of interdisciplinary work, e.g. philosophy-physics or philosophy-cognitive science.

From the review:

Hopefully these brief summaries suggest how the editors have sought to link philosophy of mathematics more closely with the interests of mathematicians. There is certainly a need for more engagement between mathematics and the philosophy of mathematics and I believe that this volume marks a productive first step in this direction. It is worth briefly asking, though, what barriers there are to philosophy-mathematics interaction and whether this volume will do much to overcome them. As I have already emphasized, philosophers and mathematicians tend to approach a philosophical topic with different priorities. The mathematicians in this volume often emphasize examples and exciting developments within mathematics, while the philosophers spend most of their energy clarifying concepts and criticizing the arguments of other philosophers. When taken to extremes either approach can frustrate the members of another discipline. Philosophers rightly ask mathematicians to clarify and argue for their positions, while a mathematician may become impatient with endless reflection and debate. A related barrier is the different backgrounds that most philosophers and mathematicians have. Philosophers are typically trained through the careful study of their predecessors and are taught to seek out objections and counterexamples. While most philosophers of mathematics have an excellent understanding of foundational areas of mathematics like logic and set theory, for obvious reasons few have reached a level of specialization in any other area of mathematics. By contrast, most mathematicians will not have much of a background in philosophy and will be tempted to appeal to the most interesting examples from their own mathematics even if they are not accessible to philosophers, let alone many other mathematicians. I am happy to report that most of the philosophical and mathematical discussion in this volume should be fairly accessible to everyone, but this probably happened only because the editors were looking out for complexities that might put off the average reader. Finally, it would be a bit naive to ignore the substantial professional barriers that stand in the way of any substantial philosophy-mathematics collaboration. To put it bluntly, nobody should try to get tenure by publishing for a community outside their home discipline. That said, it is encouraging to see philosophers and mathematicians at least trying to engage each other's interests and I hope these efforts will be continued and expanded in the coming years.

Computer Simulations Support Some New Mathematical Theorems

The current issue of Nature contains an exciting case of the productive interaction of mathematics and physics. As Cohn summarizes here, Torquato and Jiao use computer simulations and theoretical arguments to determine the densest way to pack different sorts of polyhedra together in three-dimensional space:

To find their packings, Torquato and Jiao use a powerful simulation technique. Starting with an initial guess at a dense packing, they gradually modify it in an attempt to increase its density. In addition to trying to rotate or move individual particles, they also perform random collective particle motions by means of deformation and compression or expansion of the lattice's fundamental cell. With time, the simulation becomes increasingly biased towards compression rather than expansion. Allowing the possibility of expansion means that the particles are initially given considerable freedom to explore different possible arrangements, but are eventually squeezed together into a dense packing.

A central kind of case considered is the densest packings of the Platonic solids. These are the five polyhedra formed using only regular polygons of a single sort, where the same number of polygons meet at each vertex: tetrahedron, icosahedron and octahedron (all using triangles), cube (using squares) and dodecahedron (using pentagons). Setting aside the trivial case of the cube, Torquato and Jiao argue that the densest packing for the icosohedron, octahedron and dodecahedron all have a similar feature. This is that the result from a simple lattice structure known as the Bravais lattice. Again, using Cohn's summary:

In such arrangements, all the particles are perfectly aligned with each other, and the packing is made up of lattice cells that each contain only one particle. The densest Bravais lattice packings had been determined previously, but it had seemed implausible that they were truly the densest packings, as Torquato and Jiao's simulations and theoretical analysis now suggest.

The outlier here is the tetrahedron, where the densest packing remains unknown.

Needless to say, there are many intriguing philosophical questions raised by this argument and its prominent placement in a leading scientific journal. To start, how do these arguments using computer simulations compare to other sorts of computer assisted proofs, such as the four color theorem or the more recent Kepler Conjecture? More to the point, does the physical application of these results have any bearing on the acceptability of using computer simulations in this way?

Michael Murray and Jan Cover (Purdue) Take on Evil

My colleague Jan Cover appears in the latest edition of Percontations, a Bloggingheads series which has in the past tackled other philosophical topics like the nature of time. This time the nature of evil is discussed, with special reference to God and Leibniz.

The Honeycomb Conjecture (Cont.)

Following up my earlier post, and in line with Kenny’s perceptive comment, I wanted to raise two sorts of objections to the explanatory power of the Honeycomb Conjecture. I call them the problem of weaker alternatives and the bad company problem (in line with similar objections to neo-Fregeanism).

(i) Weaker alternatives: When a mathematical result is used to explain, there will often be a weaker mathematical result that seems to explain just as well. Often this weaker result will only contribute to the explanation if the non-mathematical assumptions are adjusted as well, but it is hard to know what is wrong with this. If this weaker alternative can be articulated, then it complicates the claim that a given mathematical explanation is the best explanation.

This is not just a vague possibility for the Honeycomb Conjecture case. As Hales relates

It was known to the Pythagoreans that only three regular polygons tile the plane: the triangle, the square, and the hexagon. Pappus states that if the same quantity of material is used for the constructions of these figures, it is the hexagon that will be able to hold more honey (Hales 2000, 448).

This suggests the following explanation of the hexagonal structure of the honeycomb:

(1) Biological constraints require that the bees tile their honeycomb with regular polygons without leaving gaps so that a given area is covered using the least perimeter.

(2) Pappus’ theorem: Any partition of the plane into regions of equal area using regular polygons has perimeter at least that of the regular hexagonal honeycomb tiling.

This theorem is much easier to prove and was known for a long time.

If this is a genuine problem, then it suggests an even weaker alternative which arguably deprives the explanation of its mathematical content:

(1) Biological constraints require that the bees tile their honeycomb with regular polygons without leaving gaps so that a given area is covered using the least perimeter.

(2’) Any honeycomb built using regular polygons has perimeter at least that of the regular hexagonal honeycomb tiling.

We could imagine supporting this claim using experiments with bees and careful measurements.

(ii) Bad company: If we accept the explanatory power of the Honeycomb Conjecture despite our uncertainty about its truth, then we should also accept the following explanation of the three-dimensional structure of the honeycomb. The honeycomb is built on the two-dimensional hexagonal pattern by placing the polyhedron given on the left of the picture both above and below the hexagon. The resulting polyhedron is called a rhombic dodecahedron.



So it seems like we can explain this by a parallel argument to the explanation of the two-dimensional case:

(1*) Biological constraints require that the bees build their honeycomb with polyhedra without leaving gaps so that a given volume is covered using the least surface area.

(2*) Claim: Any partition of a three-dimensional volume into regions of equal volume using polyhedra has surface area at least that of the rhombic dodecahedron pattern.

The problem is that claim (2*) is false. Hales points out that Toth showed that the figure on the right above is a counterexample, although “The most economical form has never been determined” (Hales 2000, 447).

This poses a serious problem to anyone who thinks that the explanatory power of the Honeycomb Conjecture is evidence for its truth. For in the closely analogous three-dimensional case, (2*) plays the same role, and yet is false.

My tentative conclusion is that both problems show that the bar should be set quite high before we either accept the explanatory power of a particular mathematical theorem or take this explanatory power to be evidence for its mathematical truth.

Schupbach Crushes Pincock!

Over at Choice and Inference, Jonah Schupbach has initiated a discussion of my PSA 2008 paper on mathematics, science and confirmation theory. Readers of this blog may be interested in how it is going ...