Perfecting the symmetry: a variety of inference to the best explanation in mathematics

Although a given mathematical theorem can typically be proved in many ways, only some of those proofs are regarded by mathematicians as explaining why that theorem holds. Furthermore, mathematicians routinely use non-deductive arguments to confirm conjectures before they have found proofs (or disproofs) of them. These two features of mathematical practice suggest that some confirmation in mathematics proceeds by “inference to the best explanation” (IBE). Furthermore, a mathematical theorem may exhibit a striking symmetry, and a proof of such a theorem would have to exploit a similar symmetry in order to explain why that theorem holds. This aspect of mathematical explanation reveals an important, heretofore unrecognized form of IBE in mathematics. In such a case, a conjecture is confirmed by some mathematical discovery partly because the conjecture “perfects the symmetry” in that discovery. This paper examines one historical example of such an IBE: Brahmagupta and Snell (independently) regarded Heron’s formula for the area of triangles as confirming a conjecture not merely because that conjecture generalizes Heron’s formula, but also because that conjecture perfects the formula’s symmetry.