(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.This theorem is much easier to prove and was known for a long time.
(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.
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.We could imagine supporting this claim using experiments with bees and careful measurements.
(2’) Any honeycomb built using regular polygons has perimeter at least that of the regular hexagonal honeycomb tiling.
(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.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).
(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.
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.