A popular view in the philosophy of set theory is that of *potentialism*: the position that the set-theoretic universe unfolds as more sets come into existence. A difficult question for the potentialist is to explain how *classes* (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we'll see how category-theoretic resources can be brought to bear on this issue. I'll first give a brief introduction to topos theory, and then I'll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I'll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intentional classes should be intuitionistic rather than classical, and that the strength of the intuitionistic logic is dependent upon the partial order on the worlds.
SEMINAR - Finished
In this talk I present work in progress by my FNRS MIS funded research team (Pilar Terres, Pierre Saint-Germier, Joao Daniel Dantas) working on explanatory inference. We came up with a criterion for relevance of the entailment relation, relative to a given logic. One of the weak criteria of relevance presented in the literature is the principle of variable sharing: if a (multiple conclusion) sequent is relevantly valid then every formula in the sequent needs to have at least one variable in common with the other formulas in the sequent. I present a couple of cases from which it should be clear that this criterion (while being necessary) certainly is not sufficient for relevance. We solve these problems by analyzing relevance in terms of connectivity. The idea is to say that a sequent is relevantly valid iff a connected graph (of a specific nature) can be established that contains all of the formulas of the sequent. The basis of this idea is the concept of a constitution of a logic. This is a set of sequents that express full logical grounds of all formulas of the language of the logic (the grounded formula of each sequent is underlined, the non-underlined formulas are the partial grounds--examples are “A,B>A&B”, “A&B>A” and “A, A->B > B” in the case of classical logic). The partial grounds (the non-underlined formulas) of each formula determine the way in which formulas of the potentially relevant sequents can be connected to other formulas of the sequent. We will present and motivate the criterion, give a couple of examples, and present some graph-theoretical results concerning this criterion.
Building on the work done by Cobreros, Egre, Ripley, and van Rooij on a non-transitive version of CL denoted by them as ST, we present a generalization of this phenomenon, consisting of providing technical means to find a non-transitive analog for every Tarskian logic. We do this by building on Wojcicki's and Frankowski's results on the characterization of Tarskian logics and p-logics in terms of classes of matrices and p-matrices, respectively. The key to our result is the extension of certain algebras and matrices with so-called infectious values.
Mathematical reasoning is known to be rigidly structured in premises and conclusions that, to avoid infinite regress, lay down a set of axioms and postulates as starting points. In this talk, we propose to analyze the function that axioms and postulates play in structuring mathematics, using the fruitful perspective of the philosophy of language. Building on the preliminary work of Ruffino, San Mauro and Venturi (2020), we extend the application of Speech Act Theory to Mathematics, explaining the type of illocutionary forces in play with axioms and postulates in the mathematical language.
We start historically, by pointing out the different developments of axiomatic theories from such perspective. One can find substantial differences in the axiomatic methods by reading the different illocutionary forces in play, from Euclid's directive postulates to Frege's assertive axioms, and Hilbert's later fusion between axioms and definitions. We argue that this reading can also highlight the nineteenth-century triumph of mathematical ontology, as mathematics became objectual, rather than procedural.
After the historical excursion, we aim at a more recent proposal: Kit Fine's Postulationism. Fine argues that one can lay down axioms by taking postulates as imperatival, as means for introducing new entities. We evaluate Fine's proposal from the Speech Act Theory perspective, arguing that his postulates are not imperatives, but declaratives in disguise. In other terms, we claim that Fine is Hilbert's descendant, rather than Euclid's.
We conclude by proposing our own reading, that in laying down axioms and postulates one can find both assertive and declarative illocutionary forces. This not only shows a hybrid type of Speech Acts related to them but also explains their current linguistic use as consistent with the historical, mainly Hilbertian, picture of modern mathematics. (Joint work with Giorgio Venturi)