Saturday, July 25, 2009

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.

Kenny said...

I wonder what your suggestions make of the economic argument for a mathematical conclusion that I thought of last year. Especially in this case, neo-classical economics is if anything undermined by a lot of the observations. But there's still somehow something natural about the assumptions involved that gives some motivation to the mathematical claim (which of course is a simple enough one to prove anyway).

Chris Pincock said...

Kenny, thanks for linking to your earlier post. I had read it last year, but had forgotten about it -- this seems to be a symptom of the blogosphere, at least for me ...

I don't see any reason to argue that mathematical claims never receive support from scientific reasoning. I am currently casting suspicion on inference to the best explanation as a support for mathematical truth. It doesn't seem that your example is like that, and I share your conclusion that the sort of reasoning you outline would give us some evidence about what the sum of the infinite series was.

I think what bothers me in this case is that the scope of the mathematical claim is so much wider than the scientific phenomenon in question. That is, we are considering all polygon tilings and singling out the hexagonal one. In your case, by contrast, the sum in question is fully "instantiated" in the economic phenomenon considered.

More needs to be said on this distinction, but this is the sort of line I think I should take. See the bridges case for a potential example.

Alexandria Shearer said...

I really enjoyed your posts on the honeycomb conjecture because it demonstrates how badly mathematicians or scientists want formal concepts to model nature or our reality. There is this tension between trying to formally arrive at truth conclusions and following our ingrained inclinations to make metaphors or analogs using our own experience. For example, when thinking about limits we express them dynamically, as in "k approaches infinity" even when the limit of an expression is static? We tend to add motion in our understanding because it is a useful conceptual metaphor.

And paradoxes like Banach Tarsi are driving mathematicians to create new logics that are more consistent with our intuition. It is like giving in to our own predisposition to opt for analogs, metaphors in nature. Some cognitive scientists even argue that math is an embodied enterprise because some basic fundamentals of math have cognitive foundations.

Where is your "faith" in logic challenged?