[FOM] Re: Three Types of Foundational Inquiry

[sean.stidd@juno.com]

Professor Friedman: thank you for your charitable response to my post. In talking about different types of foundational inquiry I meant to distinguish between different kinds of 'foundational' questions one might ask about mathematics: 

1. What is the best overall theory of mathematics as a whole?

2. What are the basic concepts of mathematics? (Perhaps: at various different levels of mathematical practice.)

3. How should we understand the semantics, epistemology, and metaphysics of mathematics?

I think (FWIW) that you are right to see 1 and 2 (and I would, at least, hold out hope for 3) as a unified undertaking, which makes my original assertion false. What I think was right about what I was trying to say is that sometimes, when 'head butting' arguments break out on the list (each side repeating the same points to one another without seeming progress being made), it is because people hold incompatible commitments either in general philosophy or philosophy of mathematics, that prevent them from acknowledging what seem like straightforward points being made on the other side. (For example, the Dummettian intuitionist has semantic views which prevent him or her from accepting all of 'core' mathematics, and so prefers a slenderer foundation. Someone who thinks that all concepts need to be articulated in something like everyday English in order to be coherent or comprehensible may object to fairly obvious mathematical points because, well, the theoretical set-concept, while connected to several set-concepts in closer proximity to a typical pre-mathematical understanding, is actually a (genaeologically anyway) hybrid concept which happens to get us progress in theoretical foundations if we adopt it.) I don't really know if it would be helpful if people were clearer about their philosophical commitments up front when that type of head-butting dispute breaks out, but it seemed to me when I sent my original post that it might be.

Your approach to these kind of disputes, which seems to me to be to try to turn the conceptual or semantic/epistemic/metaphysical disagreements into disagreements that have some kind of fom-theoretical (F1) content, seems to me to be a good one. But some people will resist that kind of approach because they think the answers to questions of type 2 or 3 need somehow to be settled before we can have 'real convinction' in our mathematical theories, or that certain positions about conceptual or philosophical problems are so obviously the right ones that they rule out certain kinds of talk about mathematics from the very beginning. On the other hand, demon-world skepticism is still with us and (at least to some thoughtful minds) unresolved despite Descartes' efforts, whereas analytic geometry is part of the secondary school curriculum.