Seminar on Mathematical Logic and Category Theory

On 4 February 1939, Jean Cavaillès and Albert Lautman presented a joint paper on mathematical thought to the Société française de philosophie. Their agreement was more apparent than real, yet their disagreement never became explicit. This talk locates it precisely.

I first distinguish six senses of the word “logic” (normative, mathematical, metamathematical, transcendental, dialectical, and Cavaillès’s “logic of the concept”) and show that only one of them, Hilbertian metamathematics, provides common ground on which the two authors can actually be compared. The familiar opposition between a logic internal to mathematics (Cavaillès) and a logic external to it (Lautman) does not survive this analysis, since Lautman explicitly rejects a Dialectic self-sufficient apart from mathematics. I therefore reformulate the question: where does each author place the instance that makes mathematical necessity and unity intelligible?

The divergence then appears in their treatment of the regress of metatheories. Cavaillès refuses any absolute starting point and presents this refusal as a consequence of Gödel’s second incompleteness theorem; I argue that it rests on an unstated premise, namely that any guarantee of a mathematical theory must itself be mathematical. Lautman, by contrast, halts the regress at the level where the mind grasps the schemas of the Dialectic. Both positions are decisions, though only one presents itself as such.

Finally, I test Cavaillès’s notion of thematization against Turing-Feferman progressions of theories by iterated reflection, reading the autonomy constraint as a formal counterpart of his demand for immanence. This gives a precise criterion for the “necessity” of passing from one theory to the next (a statement unprovable below and provable above, as in Goodstein’s theorem), but also shows that autonomy imposes a bound. The upshot is a dilemma: an instance able to norm validity must exclude, while one able to account for novelty must not. Cavaillès and Lautman pay the same debt on opposite sides.

References:

1. Cavaillès, J., Méthode axiomatique et formalisme, Paris, Hermann, 1938.
2. Cavaillès, J., Sur la logique et la théorie de la science, Paris, PUF, 1947.
3. Cavaillès, J. and Lautman, A., “La pensée mathématique”, Bulletin de la Société française de philosophie, 40, 1946; English trans. G. Crocco and F. Jaëck, “Mathematical Thinking”, M×Φ Annals of Mathematics and Philosophy, 2(1), 2024.
4. Lautman, A., Les Mathématiques, les idées et le réel physique, Paris, Vrin, 2006; English trans. S. Duffy, Mathematics, Ideas and the Physical Real, London, Continuum, 2011.
5. Hilbert, D., “Über das Unendliche”, Mathematische Annalen, 95, 1926, pp. 161–190.
6. Gödel, K., “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik, 38, 1931, pp. 173–198.
7. Turing, A. M., “Systems of Logic Based on Ordinals”, Proceedings of the London Mathematical Society, s. 2, 45, 1939, pp. 161–228.
8. Feferman, S., “Transfinite Recursive Progressions of Axiomatic Theories”, Journal of Symbolic Logic, 27(3), 1962, pp. 259–316.
9. Kirby, L. and Paris, J., “Accessible Independence Results for Peano Arithmetic”, Bulletin of the London Mathematical Society, 14, 1982, pp. 285–293.
10. Bernays, P., review of A. Lautman, Essai sur les notions de structure et d'existence en mathématiques, Journal of Symbolic Logic, 5(1), 1940, pp. 20–22.
11. Detlefsen, M., Hilbert’s Program: An Essay on Mathematical Instrumentalism, Dordrecht, Reidel, 1986.
12. Benis-Sinaceur, H., Jean Cavaillès. Philosophie mathématique, Paris, PUF, 1994; 2nd ed., Vrin, 2019.
13. Halimi, B., “Logic, Cavaillès’s sought-after science”, Revue de métaphysique et de morale, 106(2), 2020, pp. 145–164.
14. Eckes, C., Jaëck, F., Mélès, B. and Szczeciniarz, J.-J. (eds.), Albert Lautman philosophe. Des mathématiques à la Résistance, Paris, Éditions Rue d’Ulm, 2025.

In this talk I will describe a view of mathematical reality to the effect that:

• there is only a finite number of things
• that there is an infinite number of things is, in a sense to be made clear, an illusion
• paradox is an integral aspect of the illusion

The view is made possible by an important technical result in the model-theory of paraconsistent logic: the Collapsing Lemma.

By putting into practice a vision of mathematics as a human cultural product, many innovations and improvements are achieved with respect to both the classical and constructive visions. The main change is to view mathematics as the careful management of information rather than the pursuit of truth, thus retaining all significant conceptual distinctions. In particular, the ideal is distinct from the abstract; exists is distinct from not-for-all-not; logic is distinct from computation; and finally, truth is distinct from consistency.

In this lecture, we will focus on the continuum and illustrate the general principles and results in this example of crucial importance. A comprehensive treatment can be found in G. Sambin, Positive Topology. A New Practice in Constructive Mathematics, Oxford University Press, 2025.

In 1895 Cantor gave a definitive formulation of the concept of set, namely,

A collection to a whole of definite, well-differentiated objects of our intuition or thought.

Let us call such a collection a concrete set. More than a decade earlier, Cantor had introduced the notion of cardinal number by appeal to a process of abstraction:

If we abstract not only from the nature of the elements of a set, and also from the order in which they are given, then there arises in us a definite general concept which I call the cardinal number of the set .

Lawvere has suggested calling Cantor’s cardinal numbers abstract sets. An abstract set may be considered as arising from a concrete set when each element has been purged of all intrinsic qualities aside from the quality which distinguishes that element from all the others. An abstract set is thus an embodiment of pure discreetness.

In my talk I will describe how category theory provides the natural framework for understanding abstract sets, conceived of as static or as undergoing change.

Poster

The aim of the presentation is to give a general overview of the application of a result by Ernst Zermelo to three very different areas of investigation: abstraction principles in neologicism, the axiom of choice in second-order logic, and regularity properties in probability theory. The talk is based on three articles that have recently appeared (see bibliography).

Bibliography:

1. 2019, (with Benjamin Siskind), “Neologicist Foundations: Inconsistent abstraction principles and part-whole” , in Mras, Gabriele M.; Weingartner, Paul; Ritter, Bernhard (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Wittgenstein Symposium. De Gruyter, Berlin, Munich, Boston, 2019, pp. 215–248.
2. 2023, (with Benjamin Siskind and Stewart Shapiro), “A note on choice principles in second-order logic” , The Review of Symbolic Logic, 16(2), pp. 339-350.
3. 2024, (with Guillaume Massas), “Totality, Regularity and Cardinality in Probability Theory” , Philosophy of Science, 91, 721–740.

Poster

What does it mean that an event C "actually caused" event E? The problem of defining actual causation goes beyond mere philosophical speculation.

For example, in many legal arguments, it is precisely what needs to be established in order to determine responsibility. (What exactly was the actual cause of the car accident or the medical problem?) The philosophy literature has been struggling with the problem of defining causality since the days of Hume, in the 1700s. Many of the definitions have been couched in terms of counterfactuals. (C is a cause of E if, had C not happened, then E would not have happened.) In 2001, Judea Pearl and I introduced a new definition of actual cause, using Pearl's notion of structural equations to model counterfactuals. The definition has been revised twice since then, extended to deal with notions like "responsibility" and "blame", and applied in databases and program verification. I survey the last 15 years of work here, including joint work with Judea Pearl, Hana Chockler, and Chris Hitchcock. The talk will be completely self-contained.

Poster

We show that the famous consistency formula Con(PA) for Peano Arithmetic PA, "no x is a code of a derivation of (0=1)," is strictly stronger in PA than the statement "PA is consistent." Hence, despite the widespread belief, the unprovability of Con(PA) in PA does not yield the unprovability of consistency. Furthermore, we demonstrate that "PA is consistent" is provable in PA. These findings apply to a broad class of formal theories, including ZF set theory.

We also discuss the potential impact of these findings on the foundations of mathematics and the theory of cognition.

References:

1. Sergei Artemov, Consistency formula is strictly stronger in PA than PA-consistency .
2. Sergei Artemov, Serial properties, selector proofs and the provability of consistency , Journal of Logic and Computation, Volume 35, Issue 3, April 2025, exae034.

Poster

In our email correspondence from 2016, Vladimir Voevodsky proposed an original concept of mathematical structure. This concept was motivated by his work in homotopy type theory on the one hand, and his reading of Proclus's commentary on Euclid's definition of a plane angle (Definition 1.8 of the "Elements") on the other. In my presentation, I will introduce Vladimir's conception of mathematical structure, compare it with standard conceptions, and touch upon some of the questions Vladimir raised in our correspondence.

This presentation is based on the article arXiv:2409.02935.

Over the past 150 years, some of the most profound and influential theorems in mathematics and computer science have emerged from self-referential paradoxes. These theorems deal with systems that exhibit self-reference, challenging our understanding of fundamental concepts. This presentation will explore:

1. Georg Cantor's Theorem: Demonstrating the existence of different levels of infinity.
2. Bertrand Russell's Paradox: Exposing inconsistencies in naive set theory.
3. Kurt Gödel's Incompleteness Theorems: Revealing inherent limitations in formal systems and the notion of mathematical proof.
4. Alan Turing's Halting Problem: Proving the existence of unsolvable computational problems.

Remarkably, these diverse theorems, along with several others, can be understood as manifestations of a single, elegant theorem from basic category theory. This presentation will elucidate this theorem and demonstrate its various instances.