This is a follow up to the previous post on Fregean epistemology of math. I want to rehearse a problem for the view, and a tentative response. I’m doubtful that the objection is original to me and would welcome a pointer to a discussion of the issue. The post went on for too long, so I decided to break it up into two parts. The second will follow soon.

There are problems with a broadly Fregean epistemology of mathematics. I’ll sketch the epistemology again. Frege gave a formal proof system and a mathematized notion of proof. A proof is a sequence of sentences, each of which is an axiom or follows from previous entries in the sequence by the rules of inference. As said before, this is anti-agentive in the sense that mathematical justification comes down to whether there are proofs in the formal system, and this is an entirely agent-free notion. The justificatory status of mathematical claims, at least those that can be expressed in the formalism, is established independent of any agents. 

The problem I want to sketch is that this makes it mysterious why multiple proofs of a proposition are informative or desirable. If a proposition is provable, then it is justified. If we’ve found a proof of it, then we’ve recognized its status as justified. Yet, multiple, distinct proofs of such a proposition are informative. They do not add anything to or change the justificatory status of a proposition. On the Fregean story, there does not seem to be space for a contribution from multiple proofs. 

Stepping out of the Fregean picture, we can ask what contribution multiple proofs of a single proposition could make. There are probably a lot, but I’ll try to give only a couple of possibilities. One is that can supply new methods for proof. New methods can result in helping us prove new things as well as being interesting in their own right. The best example of this that I know of is Henkin’s proof of the completeness of FOL. Goedel had proved this before, but Henkin’s method was different from Goedel’s. It simplified the proof greatly and turned out to lead to other developments. This suggests a response on behalf of the Fregean that I’ll return to shortly. Another benefit of multiple proofs is that they could show conceptual dependencies and relations. These do not directly reveal anything about justificatory status. They do improve our understanding of the claims and concepts involved, but understanding is not a notion that fits anywhere in the Fregean account. An example of this kind would probably be Gauss’s proofs of the quadratic reciprocity theorem. Although, I’m unsure of the details, so there could be new methods involved there too. (The notion of method needs to be firmed up to make this sharper.) Another possibility is the preference for constructive proofs over non-constructive proofs. This is more than simply an aesthetic thing. A constructive proof of something previously only given a non-constructive proof is usually regarded as informative.

I mentioned that we could offer a response for the anti-agentive epistemologist basd on the methods example. While the new proofs based on new methods might not reveal anything justificatory on her picture, the further theorems they facilitate proving allow us to verify the justificatory status of new propositions. This fits into her picture, slightly indirectly, in helping with the recognition of more propositions as justified. The Fregean epistemology can accommodate an aspect of proofs of old theorems using new methods, that is it can supply a reason for valuing them. This reason, however, misses one of the main reasons these new methods are useful, namely the light they shed on the old theorems.