Philosophy of Mathematics Seminar

Abstract: According to semantical completeness for first-order logic, given a first-order theory formalized in a language L(T), for any sentence A, T derives A iff all models of T are models of A. Notice that proving the right hand side of the equivalence might appeal to “transcendental techniques” (for instance, the models might be models of synthetic geometry and yet the techniques used to establish the truth of a certain sentence in all models are algebraic). In other words, we might be able to know that a derivation of A exists without however having the slightest clue as to how to display one. In this case I will say that the formalized derivation within T is optional (at least with respect to knowing “derivability” in principle). Second-order theories are less commonly employed in contemporary foundations. However, the program of reverse mathematics rests on second-order formalizations of fragments of full second-order arithmetic and work in the area of neo-logicism has required the use of second-order theories to capture the Fregean distinction between concepts and objects and to formulate abstraction principles (Hume’s principle etc.) mapping concepts into objects. In my presentation I restrict attention to the neologicist arena. As in this context one works with standard semantics, the completeness theorem fails. We might thus be able to prove something about all models of a certain theory T without being able to conclude anything about the (in principle) derivability or refutability of a certain sentence. The interesting situation here is that the “transcendental” (usually set-theoretic) machinery used to establish the semantic claim cannot be easily mimicked to yield derivability results (even in principle) within T and that a careful, and creative, process of formalization is required if one wants to gain a formal derivation within T of a sentence whose truth or falsity in all models is proved by “transcendental” techniques. In this case formal derivations are not optional. In my talk I will contrast the situation in first-order logic (using algebraic and geometrical theories as examples) with that in second-order logic. In the latter case, I will refer to the recent proofs of formal refutability of the conjunction of Hume’s principle and the Nuisance principle (Ebels-Duggan 2021), and provide a more detailed discussion of the refutability of abstraction principles satisfying the “part-whole” constraint (Mancosu & Siskind 2019). 

Registration: If you do not hold a University card which gives access to the upper floors of the Schwarzman Centre, please write to admin@philosophy.ox.ac.uk copied to daniel.isaacson@philosophy.ox.ac.uk at least two working days before a seminar to register your attendance.  

Please note: A notice will be sent out for each talk with the Zoom link.

_{Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Beau Mount}