# Weyl 2021

A workshop dedicated to Hermann Weyl's Philosophy of Mathematics

Hermann Weyl (1885-1955) was perhaps one of the last mathematicians to produce profound, influential work across a vast territory in mathematics, physics and philosophy. He proposed a highly original, radically new foundation of analysis in his 1918 book “Das Kontinuum”. In 1921, Weyl published “Über die neue Grundlagenkrise der Mathematik”, where he declared:

“So I now abandon my own attempt and join Brouwer.”

To celebrate the 100 years since the publication of Hermann Weyl’s “Über die neue Grundlagenkrise der Mathematik”, we will hold a Workshop dedicated to Hermann Weyl’s Philosophy of Mathematics, with special (but not exclusive) focus on his intuitionistic turn.

*Speakers:*

**Stefania Centrone** (TU Berlin)

**Laura Crosilla** (University of Oslo)

**José Ferreirós** (Universidad de Sevilla)

**Janet Folina** (Macalester College),

**Dagfinn Føllesdal** (University of Oslo)

**Mirja Hartimo** (University of Helsinki)

**Øystein Linnebo** (University of Oslo)

**Henri Lombardi **(Université de Franche-Comté)

**Pierluigi Minari** (Università degli Studi di Firenze)

**Stefan Neuwirth** (Université de Franche-Comté)

**Michael Rathjen** (University of Leeds)

**Wilfried Sieg** (Carnegie Mellon University)

**Iulian Toader** (Universität Wien)

**Mark van Atten** (Husserl Archive (CNRS / ENS), Paris)

The workshop will be online. Registration is required to attend the workshop. Please fill in this form to register.

**Registration deadline: ** **6 December** 12pm (CET).** ** You will receive the Zoom invite on 7 of December by email. We encourage you to **keep your webcam on whenever possible** to help recreate the atmosphere of an in-person meeting.

The workshop is organized within the Infinity and Intensionality project, jointly with ConceptLab.

**Schedule **(all times are CET)

*Wednesday 8 of December*

14:30-15:00: **Dagfinn Føllesdal**, *Opening Lecture*

15:10-16:10: **Stefania Centrone and Pierluigi Minari**, *Oskar Becker's "On the logic of modalities": between modal logic and philosophy of mathematics*

16:20-17:20: **Wilfried Sieg**, *Mathematical structuralism: Weyl’s positions*

17:30-18:30: **Laura Crosilla and Øystein Linnebo**, *Weyl and two kinds of potential domains*

*Thursday 9 of December*

14:30-15:30: **Michael Rathjen**, *On “real” statements*

15:40-16:40: **José Ferreiró****s**, *Second thoughts on set theory*

16:50-17:50: **Iulian Toader**, *Normative Quantifiers*

18:00-19:00: **Janet Folina**, *Generality, Circularity and Foundations*

*Friday 10 December*

14:30-15:30: **Mirja Hartimo**, *From Husserl's definiteness to Weyl’s Über die neue Grundlagenkrise der Mathematik, 1921*

15:40-16:40: **Mark van Atten**, *Intuitionistic induction*

16:50-17:50: **Henri Lombardi and Stefan Neuwirth**, *Hermann Weyl and Paul Lorenzen on predicative mathematics*

17:55-18:50: *General Discussion*

**Titles and Abstracts** (in alphabetical order)

**Stefania Centrone and Pierluigi Minari**

**Oskar Becker's "On the logic of modalities": between modal logic and philosophy of mathematics**

In his 1930 Essay "On the Logic of Modalities" O. Becker pursued two loosely related goals. The first one, more technical in nature, was to find axiomatic conditions capable to reduce to the finite the number of irreducible modalities in Lewis’s Survey system. The second one, more philosophically oriented and, in a sense, much more ambitious, was to treat the logic of modalities from a phenomenological perspective and to understand, from this perspective, the philosophical and logical-ontological problems underlying, and posed by, Intuitionism. In our talk, we will focus on Becker's discussion of Weyl's position in *The Continuum* and on Becker's anticipation of Goedel's modal translation of intuitionistic logic.

**Laura Crosilla and Øystein Linnebo**

**Weyl and two kinds of potential domains**

According to Weyl, “inexhaustibility” is essential to the infinite. However, Weyl distinguishes two kinds of inexhaustible, or merely potential, domains: those that are ``extensionally determinate'' and those that are not. In this talk, we present Weyl's distinction and explain its logical and philosophical significance. In particular, the distinction sheds light on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.

**José Ferreiró****s**

**Second thoughts on set theory **

The talk will describe some of the historical circumstances surrounding Weyl’s foundational contributions. Concerning Weyl it will deal primarily with the period before 1921, both in Göttingen and then Zurich, where Weyl was surrounded by people like Polya, Zermelo, Bernays, etc. In Das Kontinuum Weyl described the way in which he gradually found his way out of the circle of set-theoretic ideas -- reading his 1910 Habilitation lecture, one can indeed confirm that Weyl was still at that time a rather classical follower of set-theoretic mathematics along the lines of Dedekind, Cantor and Hilbert. The pattern of an initial enthusiasm for set theory, followed by subsequent second thoughts, is not at all unusual. I will discuss how it is also present in other experts of the first half of the 20th century, such as Borel, Lebesgue, Lusin, lesser known figures like Rey Pastor, or even van der Waerden.

**Janet Folina**

**Generality, Circularity and Foundations**

Weyl and Poincaré both argued that there was something circular, and thus objectionable, in logicist and set-theoretic foundationalism. Critics have dismissed this type of objection as presupposing constructivism - or worse, as psychologistic. With help from some views from Bernays I aim to show that the objections are substantive, and reveal an important disagreement about what we are looking for in a foundation for mathematics.

Bernays distinguishes two concepts of generality, and argues that neither subsumes the other. After fleshing out this view I’ll argue that it clarifies and supports the circularity objections. Finally, I’ll propose that the different types of generality are aligned with different approaches to the foundations of mathematics: the logical and the (more broadly) epistemological. Bernays’ distinction thus provides another way to understand Weyl’s turn to Brouwer and intuitionism in 1921.

**Mirja Hartimo**

**From Husserl's definiteness to Weyl’s Über die neue Grundlagenkrise der Mathematik, 1921**

Since the turn of the century, Husserl’s philosophy of mathematics combined structuralist and constructivist elements. Thus, in his Doppelvortrag (1901) Husserl complements his merely formal notion of definiteness as categoricity with a more concrete and constructive notion of definiteness, definiteness in “pregnant sense.” In this sense, definiteness requires establishing the existence of the domain of the theory by so-called “existential axioms”. In Ideas I (1913), responding to the debates about the paradoxes, Husserl attempted to do this by “regional axioms,” with which he sought to capture the essences that the objects have apriori, but synthetically. In this talk, after having noted the similarities between Husserl’s and Weyl’s notion of “extensional definiteness” in Das Kontinuum, I will briefly examine, from a Husserlian point of view, the further development of the notion in Weyl’s Brouwerian 1921 paper Über die neue Grundlagenkrise der Mathematik.

**Michael Rathjen**

**On “real” statements**

As late as 1954, referring to Hilbert's program, Weyl writes that *``Intuitive reasoning is required and used merely for establishing the consistency of the game''*. The consistency of a theory T (assumed to contain a modicum of arithmetic) established in this way, however, entails that a certain class of theorems of T has to be accepted as true even by the finitist and a fortiori by the intuitionist, as Kolmogorov pointed out already in 1925.

These formulas are of universal form, i.e., they start with a string of universal quantifiers followed by a quantifier free matrix and are usually referred to as the Pi1 formulas.

They were assigned a privileged status by Hilbert who termed them real statements. The class of Pi1 formulas has some famous denizens, notably Fermat's last theorem, the Riemann hypothesis and Goldbach's conjecture. There is still no satisfactory answer to Kreisel's question ``What more than its truth do we know if we have proved a Pi1 theorem in a theory T?''.

For the larger collection of Pi2 theorems, i.e., those starting with a universal quantifier followed by an existential quantifier, there is an answer in terms of growth rates of functions.

In this talk, I plan to discuss possibilities of classifying Pi1 theorems by unearthing their transfinite content.

**Wilfried Sieg**

**Mathematical structuralism: Weyl’s positions**

The “structuralism” of modern mathematics found one of its earliest and clearest expressions in Dedekind’s work, in particular, in his foundational essay *Was sind und was sollen die Zahlen? *[1]. Dedekind rejected (Kantian) intuitions for the grounding, but also the systematic development of mathematics; that is clear from the introduction to his essay and the contemporaneous letter to Keferstein [2]. Weyl was critical of Dedekind’s as well as Hilbert’s perspective on “structural axiomatics” and the associated “abstract conception of proof”. That criticism emerged also in Weyl’s discussions with Emmy Noether in the early 1930s; see [3]. However, in one of his last papers [4], Weyl sketched a more balanced view of the axiomatic and constructive approaches.

[1] Dedekind, R. 1888, *Was sind und was sollen die Zahlen?*, Vieweg. Braunschweig.

[2] Dedekind, R. 1890, Letter to Keferstein, in: van Heijenoort (ed), *From Frege to Gödel*, Harvard University Press, Cambridge, 1967, pp. 98-103.

[3] Toader, I.D. 2021, Why did Weyl think that Emmy Noether made algebra the Eldorado Of Axiomatics?, *HOPOS *11 (1), pp. 122-142.

[4] Weyl, H. 1985, Axiomatic versus constructive procedures in mathematics (manuscript for a talk around 1953), edited by T. Tonietti, *The Mathematical Intelligencer *7 (4), pp. 10-17 and 38.

**Iulian Toader**

**Normative Quantifiers**

Weyl’s reasons for his move towards intuitionism, in particular his ideas on quantification, were quite novel and really interesting. After presenting some likely historical sources, I argue that he believed the law of excluded middle is meaningless for infinitary mathematics not because quantified statements, themselves, are meaningless, but because they are rules for proper judgment. Infinitary mathematics, according to Weyl, is a normative system.

(Dedicated to the memory of David Charles McCarty)

**Mark van Atten**

**Intuitionistic induction**

In the intuitionistic tradition after Brouwer, one finds two different answers to the question what the evidence of the principle of induction consists in. In this talk, both will be presented in their historical context, and compared with a reconstruction of Brouwer’s view. Finally, Brouwer’s view thus reconstructed will be juxtaposed with Weyl’s.

**Organisers:** Laura Crosilla, Øystein Linnebo and Michael Rathjen

This workshop is organized jointly with ConceptLab and the Infinity and Intensionality project, funded by the Norwegian Research Council (NFR)

See the following link for another workshop on Weyl's Philosophy of Mathematics we organised in 2018: Das Kontinuum - 100 years later.