# Critical views of Infinity

The focus of the workshop are views of infinity that are critical of actual infinity. The literature presents us with a heterogeneous constellation of criticism of what is often called “Cantorian infinity”. This workshop will explore and compare a number of approaches to potentialism stemming from the history and contemporary mathematical practice. By comparing a number of approaches, we hope to better understand what are the perceived difficulties with actual infinity and which proposals have been put forth to overcome them. These include, for example, predicativist and constructivist approaches to mathematics and potentialist views of the set-theoretic universe in philosophy of mathematics.

The workshop is part of the project *Infinity in Mathematics: a Philosophical Analysis of Critical Views of Infinity. *This project receives funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 838445. * ** *

**Speakers:**

**Schedule**

*Tuesday 15 of June 2021 (Oslo time)*

13:55 *Opening*

14:00 -- 15:10 **Mirja Hartimo**: *Husserl on foundations: Top-down and bottom-up*

10 minutes break

15:20 -- 16:30 **Øystein Linnebo**: *Potentialism and Critical Plural Logic*

15 minutes Break

16:45 -- 17:55 **Dag Normann** and **Sam Sanders**: *Countability - the simplest kind of infinity?*

10 minutes break

18:05 -- 19:15** Wilfried Sieg**: *Methodological Frames: Mathematical structuralism and proof theory*

*Wednesday 16 of June 2021*

14:00 -- 15:10 **Thierry Coquand**: *Point-Free Topology and Sheaf Models*

10 minutes break

15:20 -- 16:30 **Michael Rathjen**: *Predicativity and the nature of function spaces*

15 minutes Break

16:45 -- 17:55 **Reinhard Kahle**: *Cantor’s Paradise*

10 minutes break

18:05 -- 19:15 **Laura Crosilla**:* Cantor’s Paradise and the Forbidden Fruit*

**Registration:**The workshop will take place via Zoom.

Contact Laura.Crosilla @ ifikk.uio.no for a Zoom-invite.

**Abstracts**

**Thierry Coquand**

**Title: ***Point-Free Topology and Sheaf Models*

**Abstract: **The question whether or not a given collection of mathematical objects should be considered as an actual, completed totality, or should only be considered as a potential, open collection, occurs often in constructive mathematics. One can for instance compare the collection of real numbers, presented as a set of Cauchy sequences in Bishop's book, to the notion of choice sequences in Brouwer.

In algebra, a similar question arises for the collection of all algebraic numbers (or, more generally, for the algebraic closure of a given field).

H. Edwards opposes in this way Kronecker and Dedekind's approach of the theory of algebraic curves:

*The necessity of using an algebraically closed ground field introduced –and has perpetuated for 110 years- a fundamentally transcendental construction at the foundation of the theory of algebraic curves. Kronecker's approach, which calls for adjoining new constants algebraically as they are needed, is much more consonant with the nature of the subject* ("Mathematical ideas, ideals, and ideology", 1992).

We will try to revisit in this talk this question, in the light of recent works on point-free topology and sheaf models.

**Laura Crosilla**

**Title:** *Cantor’s Paradise and the Forbidden Fruit*

**Abstract: **In this talk, I discuss potentialist views of infinity that highlight a key foundational role for the natural numbers within mathematics. My focus are aspects of the foundational reflections by mathematicians Henri Poincaré, Hermann Weyl and Errett Bishop. One of Poincaré’s characterizations of predicativity focuses on invariant definitions and is bound up with his potentialist view of infinity. For Weyl (1918) the sequence of the natural numbers is an ultimate foundation of mathematical thought over which ``a realm of new ideal objects, of sets and functional connections is erected by means of the mathematical process’’. Bishop highlights the pivotal role of the set of integers for uncovering the computational content of mathematics. I single out affinities between these views and argue for the fruitfulness of an analysis of critical views of infinity.

**Mirja Hartimo**

**Title: ***Husserl on foundations: Top-down and bottom-up*

**Abstract: **In his writings on theoretical approach to the world, Husserl distinguishes two approaches, both of which can be given a ”precise mathematical sense.” These approaches are i) a top-down approach, which reveals the ”apriori sctructure of the infinite world” and ii) a bottom-up approach, which builds on judgments about intuited ”morphological realities” and their ”morphological types” and is finitely verifiable (esp. Husserl 2012, 286-290). In her talk, after having elaborated on these two approaches and their relationship to each other, Hartimo will suggest relating them to an extensional, structuralist approach and an intensional approach. The paper will then explain what Husserl has to say about the exact nature of the latter approach and how Husserl seems to think these two approaches can be unified.

**Reinhard Kahle**

**Title: ***Cantor's Paradise*

**Abstract: **We review some historical examples which convinced the mathematical community that Cantorian Set Theory is a paradise. We also discuss how these examples resist philosophical onslaughts.

**Øystein Linnebo**

**Title:** *Potentialism and Critical Plural Logic*

**Abstract: **Potentialism is the view that certain types of entity are successively generated, in such a way that it is impossible to complete the process of generation. What is the correct logic for reasoning about all entities of some such type? Under some plausible assumptions, classical first-order logic has been shown to remain valid, whereas the traditional logic of plurals needs to be restricted. Here I seek to answer the open question of what is the correct plural logic for reasoning about such domains. The answer takes the form of a *critical plural logic*. An unexpected benefit of this new logic is that it paves the way for an alternative analysis of potentialism, which is simpler and more user-friendly than the extant modal analysis.

**Michael Rathjen**

**Title:** *Predicativity and the nature of function spaces*

**Abstract: **Function sets are crucial to the development of constructive mathematics. Classically, however, they entail the full force of impredicativity: the Power Set Axiom. Their acceptance and rationale in constructive mathematics has been an important topic and concern for central constructivist thinkers such as Brouwer, Myhill, Bishop, Goodman, Dummett, Feferman, Martin-Löf.

In the talk, I shall survey some of their views and struggles, and then try to use tools and results from mathematical logic to shed more light on the issue.

**Dag Normann and Sam Sanders**

**Title:** *Countability - the simplest kind of infinity?*

**Abstract:** A set is enumerable if it is the range of a function defined on the set of positive integers. A set is countable if there is an injection (or bijection) from the set to the set of positive integers. In set theory, as developed since the late 19th century, these concepts are equivalent.

However, in rather strong formal theories suitable for Reverse Mathematics the concepts are not equivalent. Enumerable subsets of the reals can be treated as a part of Second-Order Arithmetic (SOA), while countable sets can only be analysed properly in third-order arithmetic, e.g. in Kohlenbach’s higher order Reverse Mathematics.

In this talk we will discuss the complexity of the statement that all countable subsets of the reals are enumerable and survey some classical theorems where the two concepts cannot be interchanged without the introduction of axioms that genuinely goes beyond SOA. We will also touch upon the necessary computational complexity of a functional witnessing the fact that all countable sets of reals are enumerable.

We discuss the foundational and philosophical implications, including a new twist on predicative mathematics.

**Wilfried Sieg**

**Title: **Methodological Frames: Mathematical structuralism and proof theory

**Abstract: **Mathematical structuralism is deeply connected with Hilbert and Bernays’ proof theory and their programmatic aim to ensure the consistency of mathematics. That goal was to be reached on the sole basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from *absolute* *finitist* to *relative constructivist* proof theoretic reductions. The mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frames. Against this background, I examine Bernays’ reflections on proof theoretic reductions – from the mid-1930s to the late 1950s and beyond – that are focused on narrowly arithmetic features of frames.

I propose a more general characterization of frames that has ontological and epistemological significance. It is rooted in the internal structure of mathematical objects and is given in terms of accessibility: *domains of objects* are accessible if their elements are inductively generated; *principles for such domains* are accessible if they are grounded in our understanding of the generating processes. The accessible principles of inductive proof and recursive definition determine the generated domains uniquely up to a canonical isomorphism. The determinism of the inductive generation allows us to refer to the objects of an accessible domain; at the same time, the canonicity of the isomorphism justifies an “indifference to identification”.