Thursday, July 23, 2009

What Follows From the Explanatory Power of the Honeycomb Conjecture?

Following up the intense discussion of an earlier post on Colyvan and mathematical explanation, I would like to discuss in more detail another example that has cropped up in two recent papers (Lyon and Colyvan 2008, Baker 2009). This is the Honeycomb Conjecture:
Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling (Hales 2000, 449).
The tiling in question is just (Hales 2001, 1)

The Honeycomb Conjecture can be used to explain the way in which bees construct the honeycombs that they use to store honey. The basic idea of this explanation is that the bees which waste the minimum amount of material on the perimeters of the cells which cover a maximum surface area will be favored by natural selection. As Lyon and Colyvan put it:
Start with the question of why hive-bee honeycomb has a hexagonal structure. What needs explaining here is why the honeycomb is always divided up into hexagons and not some other polygon (such as triangles or squares), or any combination of different (concave or convex) polygons. Biologists assume that hivebees minimise the amount of wax they use to build their combs, since there is an evolutionary advantage in doing so. ... the biological part of the explanation is that those bees which minimise the amount of wax they use to build their combs tend to be selected over bees that waste energy by building combs with excessive amounts of wax. The mathematical part of the explanation then comes from what is known as the honeycomb conjecture: a hexagonal grid represents the best way to divide a surface into regions of equal area with the least total perimeter. … So the honeycomb conjecture (now the honeycomb theorem), coupled with the evolutionary part of the explanation, explains why the hive-bee divides the honeycomb up into hexagons rather than some other shape, and it is arguably our best explanation for this phenomenon (Lyon and Colyvan 2008, 228-229).
Lyon and Colyvan do not offer an account of how this conjecture explains, but we can see its explanatory power as deriving from its ability to link the biological goal of minimizing the use of wax with the mathematical feature of tiling a given surface area. It is thus very similar to Baker's periodic cicada case where the biological goal of minimizing encounters with predators and competing species is linked to the mathematical feature of being prime.

Baker uses the example to undermine Steiner’s account of mathematical explanation. For Steiner, a mathematical explanation of a physical phenomenon must become a mathematical explanation of a mathematical theorem when the physical interpretation is removed. But Baker notes that the Honeycome Conjecture wasn’t proven until 1999 and this failed to undermine the explanation of the structure of the bees’ hive (Baker 2009, 14).

So far, so good. But there are two interpretations of this case, only one of which fits with the use of this case in the service of an explanatory indispensability argument for mathematical platonism.
Scenario A: the biologists believe that the Honeycomb Conjecture is true and this is why it can appear as part of a biological explanation.
Scenario B: the biologists are uncertain if the Honeycomb Conjecture is true, but they nevertheless deploy it as part of a biological explanation.
It seems to me that advocates of explanatory indispensability arguments must settle on Scenario B. To see why, suppose that Scenario A is true. Then the truth of the Conjecture is presupposed when we give the explanation, and so the explanation cannot give us a reason to believe that the Conjecture is true. A related point concerns the evidence that the existence of the explanation is supposed to confer on the Conjecture according to Scenario B. Does anybody really think that the place of this conjecture in this explanation gave biologists or mathematicians a new reason to believe that the Conjecture is true? The worry seems even more pressing if we put the issue in terms of the existence of entities: who would conclude from the existence of this explanation that hexagons exist?

Hales, T. C. (2000). "Cannonballs and Honeycombs." Notices Amer. Math. Soc. 47: 440-449.

Hales, T. C. (2001). "The Honeycomb Conjecture." Disc. Comp. Geom. 25: 1-22.

Anonymous said...

What about using the fact that bees use a hexagonal arrangement as evidence that this is indeed minimal? That is, one could say "Well, if there were a more efficient method, the bees would have evolved it. So the honeycomb conjecture must be true."

Kenny said...

I do like these cases of mathematical explanations of physical phenomena. But I suspect that the actual mathematical fact that is doing the work is a bit weaker than the fact that is mentioned in several cases.

For instance, with the cicadas, I suspect that it's sufficient that the length of the cycle have no small divisors (where "small" is some sort of vague term). It turns out that every number with no small divisors is either prime or quite large (because it is a power of some non-small prime, or product of non-small primes). Counterpossibly, if 7x7 were smaller than 17, we would find cicadas with cycles of that length.

Similarly with the Honeycomb conjecture. It doesn't matter that this hexagonal lattice is the absolute minimum perimeter for the packing. All that matters is that it has smaller perimeter than any nearby packing - if the only better packings are extremely complicated, then we can't expect evolution to produce them, especially if they are only similar to extremely inefficient packings. Presumably the reason it took until 1999 to prove the conjecture is that no one could rule out this possibility - but we don't need to rule out that possibility to give the evolutionary explanation.

Chris Pincock said...

Anonymous and Kenny, thanks for your suggestions. See the next post for some related discussion.