# Generality in explanations

A favorite example in the debate about the applicability of mathematics and mathematical explanation is the one concerning the life cycle of cicades (cf. Ginammi 2014, p. 109; Mancosu SEP): the (biological) fact that cidades have a prime life cycle is to be explained by the (mathematical) fact that prime periodes minimize the intersections with other periodes (and therefore make the cycades better protected from predators).

However, as Ginammi explaines when he discusses this example in his disseration about the applicability of mathematics, it is pointed out by Pincock (2012, ch. 10.2) that in the explanation of the prime life cycles of cicades, we could replace the proposition

(p) prime periodes minimize intersections with other periods

with

(p*) prime periods less than 100 years minimize intersections with other periods.

In this case, the weaker (p*) seems to have as much explanatory value as (p), because both explain the actual life cycles of cicades equally well. This example suggests that in Biology we can adopt propositions of different strenght to explain the same conclusion, and we have no decisive reason to choose one over the other (cf. Ginammi 2014, p. 110).

However, do we really not have a reason to prefer one of (p) and (p*)? According to an ideal of science that originated from Aristotle’s Analytica Posteriora, proper scientific proofs should state their premisses in there most general (hence strongest) form (cf. Betti & De Jong 2010). In accordance with this tradition, in Bolzano’s view for example

(t) Equiangular triangles have angles that together equal two right angles

should never be deployed in a proper scientific proof. The reason for this is that having angles that together equal two right can be truthfully predicated of all triangles, and not only of those that are equiangular. Therefore, as Bolzano argues, in a proper scientific proof we should use the maximally general

(t*) Triangles have angles that together equal two right angles (cf. Bolzano 1837, §447).

Pincock argues, as Ginammi quotes him (2014, p. 110), concerning the different propositions that we can adopt to explain the prime life cycles of cicades:

I do not believe that the ability to explain nonactual instances of these phenomena should heighten the explanatory power of these explanations

At first sight, Pincock seems to have a point here. Why should we adopt premisses that seem way to strong to explain the fact that we are concerned with?

However, I would like to pose the opposite question: why should we not adopt the stronger premise?

Evidently, both (p) and (p*) are truths that are themselves in need of proof. How do we prove these truths? It seems that in both cases, the most straightforward choice would be a proof that appeals to the concept of prime number. Moreover, it seems that the proof of (p*) just is the proof of (p), where the conclusion is narrowed down to prime numbers less than 100. But now suddenly it looks artifical to use (p*) in the proof of our biological fact: the proof of the mathematical fact that it is based on gives us the more general (p) “for free”.

Maybe this points us to an essential difference between proofs in mathematics and proofs in the empirical sciences: whereas the former are proofs about concepts and for this reason are naturally stated in their most general (strongest) form, the latter are proofs about actual things, and can for this reason be narrowed down to cover just the actual instances.

Bolzano was not sure whether his ideal of proper science that he explicated for the deductive sciences should also hold for the natural sciences. And maybe this is an interesting question to raise concerning the problem of applicability of mathematics: given that we appeal to mathematical concepts in our proofs of empirical facts, does this imply that our ideal of generality that we have for proofs in mathematics give us a reason to prefer stronger premises over the weaker in a proof of an empirical fact?