This is an attempt to think through some topics from the philosophy of math seminar I’m attending.
One could characterize Frege’s view about the epistemology of mathematics as anti-agentive in the following sense. Frege thinks that the status of a mathematical claim as justified, at least within the areas of logic and arithmetic, do not require the involvement of any agents at all. The argument for this is his recasting of mathematical claims in terms of his Begriffsschrift. Such a recasting is possible in a slightly different form with proof theoretic concepts that were developed following Frege. One starts with a set of axioms and primitive rules of inference. A proof then consists of a sequence (or a tree) generated by a simple inductive definition based on the primitives. An axiom is a proof as is any substitution instance of an axiom. A sequence (or tree) generated by the application of an inference rule to the an existing proof is a proof, with a similar condition on substitution. Closing under these conditions generates a set of proofs. A mathematical claim is then justified if it features as the final sentence in one of the sequences in the set of proofs.
It is important for the argument that the inferences mentioned above are not understood in psychological terms. That would undermine the thesis entirely. Rather, the rules of inferences can be seen as sets of collections of symbols. Additionally it is important that rule of inference be formally specified. That is to say that the applicability of a rule should be determinable in an entirely mechanical way, e.g. based on the geometry of the symbols in the specification of the rule.
As long as there is a proof in an appropriate formal system of a proposition it is justified by Frege’s lights. Frege takes his point a bit farther. He thinks that for any mathematical argument, given in a somewhat informal manner, there is already a purely formal proof, entirely agent-free, in the background corresponding to it or underwriting it.
To make things a little sharper, the justification of a mathematical claim is given entirely by its proof. This point is made by several philosophers, but I think Benacerraf puts it nicely in “Mathematical Truth.” Proof, following Frege, is formal object, a sequence of symbols or abstract objects, or some such depending on one’s other epistemological and ontological proclivities. (Frege would want to view it not as a sequence of symbols, but as an analog in terms of abstract objects since he thought formalism was a non-starter.) Proof is, consequently, entirely agent-free. Therefore, the justification of a mathematical claim is entirely agent-free. There is simply no space for an agentive contribution to the justification of a mathematical claim on the Fregean view of mathematical justification. This idea, or at least techniques for developing it, has been developed somewhat in Hilbert and proof theoretic researchers following him.
One of the achievements of Frege and the proof theorists consists in turning proof into a mathematical object in its own right. (Although, I suppose Frege didn’t investigate the mathematical properties of proofs qua objects much, if at all.) This makes salient the distinction made by, I believe, Kreisel between proof theory and the theory of proofs. The former is a completely mathematized approach to proof. The latter is a philosophical investigation into the notion of proof that is found in mathematical practice. Proofs in the sense of the former are rarely given while proofs in the sense attended to by the latter are wide-spread. (I’m not entirely sure I have the distinction or the attribution correct. I’m having some difficulty tracking it down.) Indeed, Frege says that hardly any proof meeting his level of rigor has been offered. The distinction offered seems like one that must be made if one is interested in getting the role of the agent back into the epistemology of math.
There are two connected issues that I think I need to get clearer on for a better grip on Frege’s views about the epistemology of math. One is how exactly this relates to his anti-psychologism. The other is spelling out how this connects to his project of showing that arithmetic is analytic.
Also, the above considerations, at least for Frege, only apply to arithmetic and logic. If one holds roughly this view, what does one say about geometry? We can recast geometric questions in terms of Tarski’s axiomatization, but this wasn’t open to Frege.
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February 3, 2009 at 5:00 pm
Greg
Just for clarification: are justifications in natural science also entirely ‘anti-agentive’? If no, why? If yes, then what’s special about mathematical justification?
February 4, 2009 at 4:54 am
Shawn
Greg,
Good question. I’m of two minds on this. I suspect that a similar line of thought could be applied to natural scientific justification, although such a conception of justification would be rooted earlier than the late 19th century. I don’t want to press that idea though. There is a way in which natural scientific justification is anti-agentive (pardon the terminology) is that it doesn’t (or shouldn’t) depend on any particulars about the investigators, or even that there are investigators. The latter bit is part of what Frege was claiming.
I think one point of divergence is that with natural scientific justification is that there’s a way in which the physical world supplies the justification. It confirms or disconfirms a prediction, yields experimental results, etc. In the case of math, that isn’t the case, at least on many of the views of math floating in Frege’s time. There was much more of a view that mathematics was the creation of the human mind (or at least couched in those terms) or that mathematical objects and claims about them depended on the minds and intuitive faculties of the mathematicians.
There was certainly an abundance of psychological vocabulary used in the framing of proofs in the 19th century. One might think that was just a mode of speech, not really involving agents in any way. The hard point, I take it, for that view is putting things in an entirely agent-free way, i.e. what the notion of proof was that was being invoked. With Frege you get the specification of a formal language and proof given entirely in its terms; same with Hilbert.
March 13, 2009 at 6:05 pm
Ansten
I must submit I find this a very wrong characterization of Frege’s logic. To my eyes, what you are describing is the meta-mathematical reconstruction of the notion of demonstration; but Frege’s ideography is not such a reconstruction — it is a language to talk and think with, not an object to talk and think about. And in this language, an occurrence of the judgement stroke indicates a judgement, of the “inference-stroke” an act of inference — and judgements and inferences are truly acts, not sets of signs or the like.
I also think one should be careful with talking about the existence of demonstrations in some third realm, which you seem to be indicating (“there is already a formal proof…”). This is in effect making Frege into an extreme realist (justification comes down to the possibly “subject-transcendent” existence of proof-objects), and as far as I know there are no indications, at least not in the early and middle period Frege, that such was his view.
Have you looked at the first four paragraphs in Grundlagen der Arithmetik? I think one can extract from those passages quite a clear picture of the relation of Frege’s epistemology to his logicistic and anti-psychologistic programs.