Formalization in Logic, Language, and Mathematics Conference #2

Andrew Arana
Title: “The geographicity of mathematics as a philosophical problem”

Abstract: In his book Philosophie des mathématiques (Vrin, 2008), Jean-Michel Salanskis raised the “geographicity” of mathematics as a problem for the philosophy of mathematics. The problem is to account for the fact that mathematics is divided into branches, and has been so throughout its history. Such accounting would require the accommodation of this fact within philosophical theorizing on the metaphysics and epistemology of mathematics. I will discuss this problem, the opportunities it provides for the philosophy of mathematics, and some steps toward a response to the problem.

Jeremy Heis (UC Irvine, LPS)
Title: “Russell and Kantianism, 1897-1900"

Abstract: Russell famously announced his break with his early Kantian and idealist commitments in his paper, “Recent work on the Principles of Mathematics,” written in January 1901, during the period where he was completing his first paper on the logic of relations. In criticizing Kantianism, Russell argued that since Kant’s conception of mathematics depended on his recognition that traditional logic was too weak to underwrite mathematical inferences, Kantian philosophy was refuted by the powerful new logic. However, this was a very surprising claim for Russell to make in 1901, since Russell himself had never claimed, during his “Kantian” period from 1897-1900, that mathematics requires pure intuition in order to underwrite inferences that logic is too weak to underwrite on its own. So what was Russell’s Kantianism really like? In this talk, I trace Russell’s engagement with Kantian intuition during 1897-1900 to a conviction — which he gradually modified, deepened, and abandoned — that mathematics concerns magnitudes, and that magnitudes, being everywhere homogeneous, can only be represented using intuition. Along the way, I show how this argument has deep roots in Kant’s own thinking. What emerges from this narrative is a new way of understanding Russell’s criticisms of Kantianism, a new way of looking at the relationship between Kantianism and the rise of modern mathematics, and ultimately a new take on what, for Russell, was the philosophical payoff of logicism.

Cyrille Imbert
Title: "The Cognitive and Social Process of Computing Pseudo-Random Numbers for Scientific Applications: Ingredients for a Reliability Crisis"

Abstract: Various social and cognitive factors may affect the reliability of actual mathematical practices. Here, I investigate this question with respect to the generation of pseudo-random numbers (more precisely, unpredictable sequences of symbols). Gargantuan amounts of them are increasingly used from physics to nuclear medicine. Fortunately, scientists have developed excellent (parallel) random number generators over the decades, like the Mersenne Twister, which has a period of 2¹⁹⁹³⁷-1 and passes most randomness tests. Nevertheless, strong evidence can be found that many computational results may be invalid because of the pseudo-random numbers they actually rely upon. How deep and pervasive this problem is and whether we face a rampant randomness crisis constitute a million-dollar question.

Toby Meadows (UC Irvine, LPS)
Title: “A modest foundational argument for the generic multiverse”

Baptiste Mélès
Title: "Programs without algorithms"

Programs are often defined as "algorithms encoded in a formal language" (viz. programming language). However, describing algorithms is not essential to programs. In order to show this, I will exhibit several source codes, which are by no means exotic, and nevertheless do not describe algorithms. Examples will be taken from SQL, PROLOG and Coq. Programs do not all describe algorithms. Algorithms do not depend only on programs, but on a whole programming environment, which includes the compiler, the machine and some mathematical theorems.

Gregory Scontras (UC Irvine, LPS)
Title: “Formalizing the pragmatics of truth-value judgments”

Abstract: Research into children’s understanding of language commonly employs truth-value judgments: whether a sentence can truthfully describe a given scenario. On the basis of such judgments, researchers have concluded that young children perform quite differently from adults when it comes to understanding ambiguous utterances with multiple potential meanings. However, subtle changes to the truth-value judgment task setup make children more adult-like. I summarize key results from the literature on child ambiguity resolution, noting three factors that affect children’s disambiguation behavior. One of these factors concerns children’s processing ability: how easy it is to access the different grammatical interpretations. The other two factors concern children’s ability to manage the pragmatic context: understanding what the topic of conversation is, and modulating expectations about the world being described. In an attempt to identify the role of each factor in language understanding, I then formally articulate a computational cognitive model within the Bayesian Rational Speech Act framework that specifies the role of each of these three factors in providing truth-value judgments. The results suggest that pragmatic factors may play a larger role than grammatical processing factors in explaining children’s observed non-adult-like behavior, and the computational model offers a hypothesis as to why that’s so: pragmatic factors have a larger impact on informativity, which serves as the ultimate arbiter of utterance endorsement in the truth-value judgment task.

Kai Wehmeier (UC Irvine, LPS)
Title: "A different perspective on modal semantics – ou : La trilogie nancéienne, épisode 3"

Abstract: My previous two talks in Nancy discussed the relationship between quantifiers and modalities (2020) and the question whether variable-binding is extensional (2022). I now want to bring the two topics together by construing the modal operators as bona fide quantifiers and modal logic, accordingly, as a particular quantificational logic. This affords a non-standard perspective on the nature of modal semantics, which I shall explore.