The PMA held a session in the Group Program at the 2020 Eastern Division Meeting of the American Philosophical Association. The session was organized by Juliette Kennedy, and consisted of a book symposium on John Baldwin’s Model Theory and the Philosophy of Mathematical Practice. The participants were John Baldwin (University of Illinois at Chicago), Timothy Bays (Notre Dame University), Colin McLarty (Case Western) and Scott Weinstein (University of Pennsylvania)..
This book can advance philosopher’s understanding of both structure and mathematicians. As to structure, John gives a great look at current working methods, which differ markedly from the usual philosophic ideas for an obvious reason: Mathematicians ideas cannot just be defended in philosophic debate. They must produce results. And they do. John describes how a lot of these developed and what they achieve. This leads to the point about mathematicians. Philosophy of mathematics, and philosophy of science more generally, too often neglect the raw experience of desperately desiring scientific answers. Mathematical proof is so different from philosophic argument that we sometimes treat mathematical knowledge as something that just follows from the axioms. As if you just identify which axioms are relevant, put them in the right order, and then the answer is in front of you. John shows the living experience of the model theory community seeking, sometimes finding, and always reacting to, new concepts and methods to answer various standing questions. He shows how this goes at stages before anyone knows whether it will work. Philosophy of mathematics can benefit immensely from absorbing these insights. I will say, though, that textbooks and research talks and papers all show mathematicians use model theory, per se, much less often than they use categories and functors to describe structure.
John Baldwin heralds the significance of the “dividing lines” at the very beginning of Model Theory and the Philosophy of Mathematical Practice.
After the paradigm shift there is a systematic search for a finite set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. In this framework one can compare different areas of mathematics by checking where theories formalizing them lie in the classification.
For John, the interest of the “syntactic” character of the dividing lines consists primarily in the fact that they are absolute for transitive models of set theory, and in consequence, that many of the central results of contemporary model theory do not rely on theories stronger than ZF. But, a number of the dividing lines, for example, NOP, are syntactic in a more robust sense – they are Π02 properties familiar from combinatorics. We explore whether this aspect of the dividing lines may have any interest from the point of view of the philosophy of mathematical practice. On the one hand, we observe that for every formal system F , there is an NOP theory T , such that T cannot be established to be NOP in F . On the other hand, all “naturally occurring” NOP theories (of which we are aware) can be proven to be NOP in primitive recursive arithmetic. We suggest investigating whether there is a recursively axiomatizable theory T that arises naturally in the course of mathematical practice such that T is NOP, but T is not provably NOP in primitive recursive arithmetic. If not, why? Might this tell us something interesting about mathematical practice? Or about the nature of mathematics itself? If so, might the additional strength required to establish that such a theory is NOP reflect some interesting phenomenon that helps to clarify our understanding of the mathematical topic it formalizes?
Slides of John Baldwin’s Responses.