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.

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March 13, 2009 at 6:56 pm

AnstenYou describe Fregean epistemology as concerned with justification, and then you suggest a problem for this epistemology; but your problem, as far as I can see, does not concern justification, but methods of justification. For the justification of a theorem, certainly, one demonstration suffices — another justification using different methods doesn’t add anything to the status of the theorem as a theorem. Henkin’s proof didn’t make the completeness theorem more justified; what it did was to present novel and fruitful methods, and a theory of justification need not say anything about the fruitfulness of methods.

To make this complaint Fregean one could perhaps refer to the distinction Frege makes in GLA paragraph 3 between the content and correctness conditions of a judgement. Epistemology is only concerned with the latter (Frege bases his account of analytic, synthetic, a priori, and a posteriori judgements on it), whereas the fruitfulness of methods, one might say, falls on the side of content.

March 13, 2009 at 8:04 pm

GregWhy do information and justification need to be linked? What’s so bad about a Fregean accepting that proof 1 alone or proof 2 alone each completely justify a given theorem T, but that having both gives us qua epistemological (and perhaps even psychological) agents more information?

(Another, distinct thought just occurred to me: could the different proofs be somehow analogous to different Fregean ‘modes of presentation’/ senses? That probably wouldn’t work, but maybe…)

March 14, 2009 at 4:42 am

JustinI’m afraid I’ve never understood the agentive vs. anti-agentive business, so perhaps I should just stay quiet…

However, one route would be to note the importance of subsidiary steps in the proof. One proves a number of lemmas in the course of proving the main theorem. Of course, some lemmas are good enough to be named and taught to undergraduates, others less so, all the way down to individual steps of the proof that don’t even earn the ‘lemma’ designation. So even in terms of Fregean epistemology, one has a potential difference. (I don’t know if this would hold up in case studies of alternate proofs).

March 19, 2009 at 10:38 am

RafalWell, you can simply bite the bullet, no? This could consist of the following claims:

1. Strictly speaking, new proofs of a theorem T do not provide us with new insight or better reasons to accept T.

2. However, having interestingly different proofs of T provides us with new nowledge of metalogical facts of the sort: T can be proven in such-and-such system in such-and-such a manner.

3. So, giving a proof of T (epistemologically speaking) is also a proof that T can be proved this way. And this is a new claim that our proof justifies.

4. When we say, `new proofs provide us with better insight into T’ what we mean is only that we get a better grip on its place in the system(s) involved in those proofs. That is: we learn new metalogical facts.