Monday, August 31, 2009

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.

Friday, August 14, 2009

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?