ESPM

Potentialism in the Philosophy and Foundations of Mathematics

Abstract: Aristotle famously claimed that the only coherent form of infinity is potential, not actual. However many objects there are, it is possible for there to be yet more; but it is impossible for there in fact to be infinitely many objects. Although this view was superseded by Cantor’s transfinite set theory, even Cantor regarded the collection of all sets as “unfinished” or incapable of “being together”. In recent years, there has been a revival of interest in potentialist approaches to the philosophy and foundations of mathematics. The lecture provides a survey of such approaches, covering both technical results and associated philosophical views, as these emerge both in published work and in work in progress.

Plural Level Theory 

Abstract: The aim of the talk is to outline a plural regimentation of the Level Theory (LT) originally introduced by Button (2021) to frame the cumulative hierarchy of sets in purely set-theoretic terms, with no appeal to stages as primitives. First, after a brief introduction to LT and to the framework within which it moves, I justify the turn towards plural logic and present the language grounding Plural Level Theory (PLT). Second, mirroring the methodology adopted by Button (2021), I sketch Plural Stage Theory (PST), a first intuitive version of the theory later to be refined by PLT. Third, I outline PLT by stating its core components, its axioms and some theoretical results. Finally, I present the next steps for the development of the theory. 

References 

Button, T. (2021). Level Theory, Part 1: Axiomatizing the Bare Idea of a Cumulative Hierarchy of Sets. Bulletin of Symbolic Logic, 27(4), 436–460.

Set-Theoretic Bicontextualism

Abstract: In this talk, I argue for an intermediate position between generality absolutism and generality relativism in set theory. Furthermore, I develop a formal framework that can implement this position, allowing us to differentiate, sentence by sentence, between inherently absolutistic and inherently relativistic set-theoretical sentences.

In a slogan, some set-theoretical sentences express facts about all sets, while others express facts only about some sets. To achieve this, the framework employs a supervaluation-like semantics incorporating both an all-inclusive domain containing all sets (absolutistic domain), and the strongly inaccessible ranks of the cumulative hierarchy (relativistic domain).

In this framework, we can isolate a class of sentences that is true in all models, irrespective of whether they are absolutistic or relativistic. Using this class of sentences, I suggest a way to identify fundamental principles of the set-theoretical ontology.

My approach draws some parallels to Steel’s Multiverse with a core (Steel, 2014), as one of my aims is to identify core features of the set concept. However, unlike Steel, my approach adopts a semantic perspective, defining core features in terms of quantifier domains.

Inference in constructive set theory

Abstract: Constructive set theories have been developed with the aim to provide a foundation for constructive mathematics that, unlike type-theoretic foundations, is close to the standard set-theoric foundation for classical mathematics. As is well-known, the choice of axioms for constructive set theory is not straightforward, as a single principle may have several classical equivalents that are constructively far apart, and therefore it may not be clear which of the equivalents should serve as an alternative to the classical principle. Moreover, some set-theoretic principles, when added to intuitionistic logic, result in a classical theory. Thus in set theory, the logic one starts with may not be the logic one ends up with, and the question what exactly is the logic of a given constructive set theory, or any other constructive theory for that matter, becomes relevant. Such questions are in general hard to answer, but over the last two decades some progress has been made. In this talk constructive set theory is introduced, it is explained what the logical tautologies and logical inferences of a constructive set theory are, and some results are presented for the most well-known constructive set theories IKP, IZF, and CZF. 

Choice Principles and Existential Quantifiers in Constructive Mathematics

Abstract: The Axiom of Choice has been one of the most debated axioms in the history of mathematics. In particular, it has long been discussed whether the axiom is acceptable in constructive contexts. While results like the Goodman-Myhill theorem in constructive set theory and the Diaconescu theorem in topos theory have been considered as evidence for the non-constructivene nature of the axiom, on the other hand, influential figures such as Bishop and Dummett have argued that the axiom’s validity is grounded on the intuitionistic reading of logical constants. Furthermore, Martin-Löf showed that the type-theoretic version of the Axiom of Choice could be derived in the context of his intuitionistic type theory.

Later, Martin-Löf provided a thorough analysis of the relation between the type-theoretic Axiom of Choice and Zermelo’s axiom. In particular, he showed that the latter cannot be derived from the former unless the choice function is assumed to be extensional. However, this extensional form of the axiom cannot be derived in intuitionistic type theory. Rather, it can be proved that it entails the law of excluded middle, as it happens in the set-theoretic context. This analysis reveals that the non-constructive aspect of the Axiom of Choice does not lie in the existence of a choice function, as argued by the early critics of Zermelo’s axiom, but rather in its extensionality.

The aim of this talk is to go one level deeper than Martin-Lof’s analysis by carefully distinguishing among different formulations of choice axioms and rules within a type-theoretic framework, while also examining their logical relationships. Furthermore, this analysis highlights the strong connection between the status of choice principles and the interpretation of existential quantification in constructive contexts.
The study is conducted within the framework of the Minimalist Foundation (MF). MF is a two-level foundation, where the first level (mTT) corresponds to an intensional type theory, while the second level (emTT) is an extensional type theory with a set-theoretic flavor, which makes it more convenient for everyday mathematical reasoning. The two levels are connected by a suitable model construction which interprets emTT into mTT, hence the extensional theory into the intensional one. MF features a primitive notion of proposition and hence a primitive notion of existential quantification, which is not identified with the dependent Σ-type as in Martin-Löf Type Theory. The identification of the existential quantifier with Σ plays a fundamental role in the proof of the Axiom of Choice in type theory. The formulation of the rules for ∃ in MF prevents the derivation of the type-theoretical Axiom of Choice. Furthermore, it turns out that even the weaker Axiom of Unique Choice cannot be derived in MF. However, they are both consistent with the theory. Dropping the internal validity of unique choice is crucial for distinguishing between functional relations and operations, i.e. type-theoretic functions, which are instead identified in Martin-Löf type theory.

The main results I want to discuss are the following:

• In the intensional level of MF, it is possible to provide four non-equivalent formulations of the Axiom of Choice: the type-theoretical Axiom of Choice, the extensional Axiom of Choice, the Axiom of Choice for functional relations and the Axiom of Choice for extensional functional relations. Further, I will also formulate the corresponding choice rules. Then, I will analyze the logical relationships between all these choice principles.
• The extensional level emTT features rules for forming quotient sets. I will show that such rules entail that the four axioms given above reduce to two. Further, I will explain how we can recover within this context the familiar proofs that extensional choice implies excluded middle.
• Finally, I will explain the exact relationship between existential quantification in type theory and choice principles. The results can be summarized as follows:
  – theories featuring a pure existential quantifier do not validate internally any choice principle. Examples are MF and the Calculus of Inductive Constructions.
  – theories featuring a truncated or regular existential quantifier internally validate unique choice. Examples are homotopy type theory and elementary toposes.
  – theories with a strong existential quantifier, which is the Σ-type, internally validate the type-theoretical Axiom of Choice. Examples are intensional and extensional Martin-Löf type theory.

In conclusion, these results provide a more detailed analysis of choice principles in constructive contexts and highlight the importance of distinguishing between operations and functional relations. Furthermore, they show that the notion of constructive existence is not uniquely determined and is closely connected to the validity of specific choice principles. This latter aspect has not been explored in other analyses of the Axiom of Choice in constructive theories. The talk is based on joint work with Milly Maietti.

Logical Constants

Abstract: I discuss criteria for distinguishing between logical and non-logical vocabulary. The primitive logical expressions are the logical constants of a language. A logical constant can be extensionally equivalent with a non-logical expression. Thus, criteria that take only the extension of an expression into account such as the usual invariance criteria fail to be adequate. Metasemantic factors have to be used in the formulation of a criterion: I argue that it must be possible to introduce a logical constant by suitable introduction and elimination rules. Suitable rules have to determine the behaviour of the logical constant uniquely in a sense to be specified. Moreover,the rules have to be truth preserving, again in a sense to be specified. Roughly, these are uniqueness and existence conditions. Although introduction and elimination rules play an important role, my criterion for logicality is not inferentialist. “Tonk” and similar expressions are ruled as logical constants, simply because they are not truth-preserving; there is no need for a harmony and other syntactic constraints on rules, because I employ semantic notions freely. Also, one can introduce logical constants via truth conditions, as is the case for generalized quantifiers; only few of them will qualify as constants by my criterion; but I do not rule them out trivially. Therefore, I can give a unified account of logicality that applies to expressions given by semantic methods and usually ignored by inferentialists as well as to expressions given by inference rules such as “tonk”, which are usually ignored by semantically oriented logicians on their invariance accounts.

IMPLICIT COMMITMENTS, BELIEVABILITY, AND REFLECTION

Abstract: By definition, the implicit commitments of a formal theory Th are sentences that are independent from the axioms of Th, but their acceptance is implicit in the acceptance of Th. In (2017, 2018) the phenomenon of implicit commitments was studied from the epistemological perspective through the lenses of the formal theory of believability. In this talk, we provide a proof-theoretic analysis of this approach and compare it to other expansionistic theories of implicit commitments. We argue that the formal results presented in the paper favor the believability theory over its main competitors. They show that the believability theory is conceptually light and can be meaningfully applied to itself, and to natural theories containing the believability predicate. However, it has its problems: we argue that in its current formulation, the theory cannot deliver all the goods for which it was defined. In particular, being amenable to a generalized conservativeness argument, it does not support the view that the notion of truth is epistemically light. In the end, we discuss some ways out of the problem.

References

Cieśliński, C. (2017). The epistemic lightness of truth: Deflationism and its logic. Cambridge University Press.
Cieśliński, C. (2018). Believability theories: Corrigendum to the epistemic lightness of truth. Deflationism and its logic. http://cieslinski.filozofia.uw.edu.pl/ Corrigendum (accessed: 03.01.2023)

Qualifying the Received View on Urelements

Abstract: In the philosophy of set theory, the received view on urelements is that urelements are necessary to account for the applicability of set theory outside of mathematics, but dispensable for theoretical purposes. In this talk, both components of this view will be qualified. As groundwork, there first is a brief recounting of the role of urelements in 20th-century mathematics, showing that the received view goes back to Zermelo.

Thereafter, three arguments for the theoretical dispensability of urelements are discussed.

The first, Reinhardt’s structuralist argument, argues that all structural questions about (impure) sets can be reduced to questions about pure ordinals. However, this says little about the structural features of models of set theory, and the representational role of set theory requires consideration of non-structural features of sets.

The second argument uses bi-interpretability results between pure set theory and set theory with urelements to argue that one can omit urelements. However, the most general results with less assumptions about the number of urelements obtained by Hamkins and Yao require the philosophically problematic addition of parameters to the set-theoretic language.

The quasi-empirical third argument is based on the hitherto expressive adequacy of set theory without urelements, and can be resisted by noticing that set theory with urelements achieves other Maddian foundational goals, such as a more faithful representation of structures, as shown by Barton and Friedman. The first qualification is thus that urelements are dispensable for some foundational goals, but not for others.

Finally, the last section of the paper shows that on popular accounts of mathematical representation, namely the mapping and the DEKI account, the absence of urelements is an idealisation. Thus, the second qualification is that the purported necessity of urelements for accounts of application hinges on one’s view on idealisations in scientific models.

Deep disagreement in set theory

Abstract: Against the intriguing debate in the philosophy of set theory about the independence problem, this paper studies the situation in the set-theoretic community. Are set-theoretic practitioners indifferent to these philosophical issues or do they take a stance, and how does that effect set-theoretic research? I provide a thorough study of the disagreement between set theorists holding either an absolutist* or pluralist* view. In such a case, the conciliationist challenge is raised. In the literature, some argue by means of benefits for scientific progress in favour of the steadfast view ([Lougheed(2018)] and [De Cruz and De Smedt(2013)]), while others argue that the parties of the disagreement should suspend judgement [Matheson(2018)]. In the set theory case, it turns out that the answer changes with changing the epistemic goal. I argue that if the parties of the disagreement aim at set-theoretic progress, then they are required to remain steadfast, but if they aim at true beliefs about the propositions on which they disagree, then they are required to suspend judgement.

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.

Mathematical Fictionalism by Halves

Abstract: The idea that, while we should be realist about some mathematical objects, there are some mathematical theories that we should treat as mere fictions, can be found in a number of places. Donald Gillies suggests such an approach to deal with uncountable infinities, and while Quine is realist about some uncountably infinite sets, his view that work in the higher reaches of set theory should be viewed as ‘mathematical recreation… without ontological rights’ can also be read as a form of partial fictionalism, or ‘fictionalism by halves’ as I will call it. Even Godelian Platonists who are realists about the full iterative hierarchy of sets may resist treating all of the alleged set theoretic universes in the set theoretic multiverse as equally real. Hamkins suggests that such single-universe Platonist will have to ‘explain away as imaginary’ all the other set theoretic universes that set theoreticians work on, and in doing so they might be seen as supporting a form of fictionalism by halves. This paper notes that such fictionalisms ‘by halves’ go against a developing ‘pluralist’ orthodoxy in the philosophy of mathematics (shared by many Platonists and fictionalists alike), and argues that Hilary Putnam’s indispensability argument might be best thought of as an argument for an anti-pluralist form of fictionalism by halves.

Considering the thesis that Peano Arithmetic is sound and complete with respect to arithmetical truth

Abstact: In “Arithmetical truth and hidden higher-order concepts”, published in 1987 and republished with some emendations in 1996, I raised the question whether first-order Peano Arithmetic has a special place among axiomatizations of arithmetic, to which, on the basis that it is sound and complete with respect to a notion of arithmetical truth articulated in that paper, I answered yes (a view which has been referred to as “Isaacson’s thesis” in publications since 2008). I still hold to this view, but in my talk I will present some reformulations and emendations of the grounds I originally offered for it.

Exploring Mathematical Understanding through Cognitive Capacities, Actions and Object

Abstract: This presentation introduces a process-based investigation of the constitution of one’s understanding of mathematical truths. Using three distinct proofs of the Pythagorean Theorem as case studies, it demonstrates that different mathematical agents can have different understandings of the same proposition. This differentiation arises in two ways: 1) Mathematical agents constitute their understanding of the  proposition using different cognitive capacities and 2) the ways they understand the proposition are constrained by the types of objects that they introduce in their inquiries. To illustrate these points, the three proving processes of the Pythagorean Theorem are analysed in terms of objects, actions, and cognitive capacities. As an agent’s actions in her proving processes necessitate the use of certain cognitive capacities, these cognitive capacities justify why she understands a proposition in a specific way. As an agent performs actions with objects, the types of objects constrain the way the proposition is understood. Based on these case studies, three epistemological payoffs of such a process-based investigation of mathematical understanding are proposed: First, this approach entails a pluralistic stance on mathematical understanding. Second, it provides a method for conducting epistemological studies from the first-person perspective. Lastly, it allows us to argue for the limits of one’s understanding.