Graham Priest and Neil Thomason

Graham Priest &  Neil Thomason (both from University of Melbourne), “Lakatos, Paradox, and Paraconsistency” with commentary by Stuart Shapiro (Ohio State University).  Both the paper and the commentary can be found here.

Comments

[Justin Fisher]

Graham, thanks for the interesting paper.

I'm curious about the 'formalist' view that you hastily dismiss in section 3.3. This was the view that it is misguided to seek *true* axioms, and that we should instead view mathematics as the study of what follows from certain chosen axioms, where many different sets of chosen axioms might have been equally suitable starting points.

I would have thought that this view was not that far from Lakatos, at least if we don't read too much into the word 'axiom'. For Lakatos thought that what mathematicians do is opportunistically choose starting points -- it's only a slight stretch to call these starting points 'axioms' -- and explore what follows from them. If we're somehow dissatisfied with the results we discover, we may then go back and do a 'proof analysis' to find the source of these unsatisfactory results. Sometimes we decide to keep our old starting points and live with the surprising consequences. But sometimes we instead conclude that we should shift our interest to what follows from a different set of starting points -- a different set of 'axioms', in my perhaps strained usage of the term -- one that hopefully will yield more satisfactory results.

You call the 'formalist' view sketched above "wildly implausible", and offer two quick arguments against it.

First, you say that Godel has shown us that (suitably complex) mathematical systems are such that not all their theorems may be drawn out by first-order deductive reasoning from axioms. However, it's not clear to me why the advocate of the general line described above would need to be restricted to first-order reasoning. Insofar as Godel's "informal reasoning outside the system" helped to show interesting features that, in some important sense, follow from the axioms of a system, why isn't it available to a (quasi-)formalist to say that mathematicians are interested not just in the first-order consequences of their starting points, but also the consequences that follow in this other important sense, even if tracking all this requires considerations that are "informal" or "outside the system"?

The question (a) of what counts as 'following from' a given starting point seems orthogonal to the question (b) of whether our starting points are *true* or instead merely the ones we've chosen to work with. Godel has shown us that traditional formalists have given too narrow of an answer to (a), but it still seems to me that they were spot-on regarding (b), and it seems like that's highly relevant to your paper.

Second, and "more importantly", you suggest that not just any set of axiomatic starting points counts as mathematical. I'm not sure this is true, and regardless, it seems irrelevant to the present topic.

I'm not sure it's true because I can easily get myself in the mood where I think, e.g., that pure evolutionary theory, pure computational theory, pure game theory, and pure theoretical economics *really*are* branches of mathematics.

But regardless, the (quasi)formalist view above was -- so far as I could tell -- being proposed as a mere statement of some relevant truths about mathematics, *not* as a set of jointly sufficient conditions for a system to be mathematical. So, even if you're right that these conditions aren't, by themselves, sufficient for a system to be mathematical, that doesn't seem relevant here.