The Undecided Nature of Weyl’s Thought: Continuum

Introduction

Weyl was an outspoken critic of classical mathematics who, starting in 1910, became more active in the field’s foundations. Intuitionistic thoughts and practices were heavily influenced by his philosophical convictions. Though he could hardly be termed an idealist, Weyl was greatly influenced by Kant’s thesis of the primacy of intuition. In 1912, he discovered philosophical light in Husserl and Fichte’s metaphysical idealism. As a mathematician, Brouwer was also interested in the philosophy of mathematics, which he addressed in the second chapter of his dissertation. In Brouwer’s philosophy, Gerrit Mannoury was a driving factor. He demonstrated to Brouwer in his lectures that mathematics was not something that existed independently of people but rather was something generated by human beings themselves. These beliefs aligned with Brouwer’s idealistic mindset, which led to his choice of research subject.

Constructivism developed into intuitionism with the publication of Das Kontinuum, and Weyl’s famous book “On the New Foundational Crisis in Mathematics” described this change^[1]. For more than the intrinsic contradictions, Weyl slammed Cantor’s set theory for seeing mathematics’ subject matter as abstract entities that exist apart from the human mind. Although Weyl regarded the normal numeral as a human notion, and insisted on working with precisely specified sets and functions, Brouwer’s intuitionistic agenda, founded on mathematical objects as constructions of the human mind, had piqued Weyl’s attention.

Since intuitionism advocated making the necessary compromises in the pursuit of mathematical stability, Weyl shifted away from intuitionism over time in the 1920s. The intuitionistic method he used was “intolerable” for applied mathematicians since it was “awkward”^[2]. Weyl retreated to Hilbert’s axiomatic approach in the 1920s after abandoning intuitionism^[3]. His later works demonstrate that he never completely adopted Hilbert’s program and continued to vacillate throughout his life in his method to the fundamental problem. Weyl never fully embraced Hilbert’s program.

Researchers and mathematicians know that Weyl often shifted his stance on the central conundrum. Over time, “we all alter our ideas,” said Feferman. “Weyl was definitely no exception,” he said. It’s Feferman’s turn^[4]. He claims that Weyl’s “intellectual gains” in the twenties stemmed from his philosophical thinking, pointing to Fichte’s development of the ideas about space and substance^[5]. According to Norman Sieroka, Weyl’s philosophical movements from formalism to intuitionism to constructivism are closely tied to his philosophical divide between Husserlian phenomenology and Fichte’s constructivism. This is what Norman Sieroka believes^[6]. As a result, such accounts can only give a partial understanding of Weyl’s motivations for changing his mind or the source of his ambivalence.

Aiming to shed light on how professionals develop new ideas, I look at the philosophical roots of normative framework shifts. Weyl’s thinking was influenced by a basic problem: the difficulty of separating intuitive from mathematical continuity. An investigation of this subject will expose Weyl’s continual attempts to solve such fundamental concerns. That Weyl’s repeated heart-shifting over the basis of mathematics reflects a deeper, intra-subjective, self-criticism provoked by normative uncertainty is what I want to illustrate in this article^[7].

The Continuum Idea

Weyl presented a semi-constructive alternative to the R system’s set-theoretic basis in Das Kontinuum. In particular, he was aware of the “unbridgeable gap” among intuition (of space, time, and motion) and mathematics modeling as a “discrete” accurate conception, which would develop as a result of his method. Weierstrass, Cantor, Hilbert, and Dedekind had all offered alternatives to Weyl’s method. “A definite period” was replaced with “every time point inside a given time range” for Weyl, who struggled with the shift from what was intuitively evident to the definition of a time-period. It’s impossible to understand the latter. Last but not least, he said that the essential features of time, as they seem in our experience, are not specific time points.

Two years later, Weyl published a contrasting perspective on the continuum in his New Foundational Crisis in Mathematics piece. He labeled Das Kontinuum’s assessment of the continuum, which Weyl referred to as “atomistic” or “discrete,” a “atomistic” interpretation since it was founded on a false assumption that the continuum is composed of separate points and each point is independent when taken by itself^[8].

To Weyl, the “chasm” between both the intuition and logical continuum can only be crossed by Brouwer’s intuitive construction of the continuous. Thus, it relies on a whole new way of thinking about real numbers. It is more common to talk about “dual intervals” rather than discrete spots when talking about numbers x in the real world.

There can be no discrete sections in a true continuous since the actual nature of a continuum mandates that every segment may be further split without limit. To explain the continuum in classical analysis, points were typically utilized; each point was regarded a fundamental member of the set. The classical continuum could be divided into parts by considering each point to be a member of the set, but this division had its limits since a point is considered a part that includes no other parts. This was the end of the line. Mathematical continuity may be “harmonized” with the intuitive one via the use of intervals rather than points in the production of numbers Much as it occurred in Das Kontinuum as an unbridgeable barrier between intuitive and mathematical continuums, it happens today as the gap between a new mathematical continuum and the collection of discrete components^[9].

Weyl subsequently addressed the continuum question in “Philosophy of mathematics and natural sciences” as an outstanding topic that has not yet been adequately answered. In this book, Weyl’s theory of mathematics was examined. In 1954, he published “Axiomatic and Constructive Procedures in Mathematica,” but he remained dubious that he had found a solution to the “riddle of the continuum”^[10].

Towards A Philosophical Approach of Normative Transitions

Menachem Fisch, a philosopher, has been fascinated for decades by the question of how someone may see his own norms of decency as unreasonable. In Fisch’s newest publications, it is difficult to change our normative convictions just via intra-subjective conversations, according to Fisch^[11]. People who are strongly dedicated to alternative frameworks may destabilize the status quo if they are subjected to normative scrutiny. This is what makes his theory stand out. To borrow Fisch’s word, another way of stating it is to be ambiguous.

For this reason, Fisch has shown that the origins of a certain area are often explained by practitioners who are normatively ambiguous. Observers may observe how Tycho Brahe attempted to unite Earth’s geocentricity with that of the sun, and how Galileo’s theory of free fall and projective motion is an excellent illustration of how Fisch’s theory may be put into effect. Even while their mature works are clear evidence of the ideas to which they were fully devoted, Brahe, Galileo, and Peacock’s more mature work was paved by their unpublished and hence unrecognized transitional works. The fact that Tycho, Galileo, and Peacock were able to negotiate between their own normative frameworks and new concepts implies that they had grown sufficiently ambiguous^[12].

If it is not given the recognition it needs, Fisch believes that ambivalence represents a significant turning point in the history of science, one that is easily overlooked or dismissed as nothing but a condition of confusion or hesitancy. Ambivalence and persuasion are two different things. It is not essential to convince someone of the sordid revelations of one or more of his normative standards to make him ambivalent about these norms. Ambivalence in the present theory might emerge even if practitioners are not convinced that the current theory is erroneous. After being exposed to normative criticism, practitioners may find themselves in a condition of ambivalence for months, years, or even decades.

How to Deal with Weyl’s Ambivalence

For centuries, mathematicians and philosophers have tried to figure out why Weyl’s core attitudes changed over time. In Weyl’s Das Kontinuum, Solomon Feferman, a mathematician and philosopher, saw a “significant development in the pre-dicativist program”^[13]. Weyl’s criticism of the set-theoretical method and his ideas on the irreducibility of positive integers and the need of explicit explanations for mathematical objects like sets and functions influenced this stance to some extent. Pre-dictivism was also attributed to Weyl by him^[14]. While Feferman claimed that Weyl saw the predicative approach as significant, and that he “never fully gave up the achievements of his 1918 work,” Weyl repudiated the predicative method only 2 years following Das Kontinuum^[15].

On the subject of philosophy, Norman Sieroka says that Weyl’s change in philosophical position was driven by his preference for both Fichte and Husserl, but he discriminated between the two in 1920s, relating Fichte’s phenomenology to Brouwer’s intuitionism and Husserl to formalism. On the “right path” for Weyl was the Fichte “formalistic-constructivist” technique of Fichte, after a brief period of time with the “intuitionistic-phenomenological” approach. Wiel’s “interdisciplinary intellectual neighborhoods” set the framework for “a much more comprehensive explanation of the importance of shifts and invariances in light of historiographical concerns.”^[16]

Mathematicians tend to ignore Weyl’s intuitionistic notions and ascribe them to his philosophical interests, which they perceive as entirely unrelated to his mathematical work. However, Michael Atiyah concentrates only on Weyl’s contributions to group theory, completely disregarding Weyl’s longstanding ambiguity on the core problem^[17]. However, while dedicating an entire section of his book to discussing Weyl’s move to intuitionism, Max Newman’s study ends with a functional tone that implies that Weyl’s major goal was to make Brouwer’s mathematics understandable to other practitioners^[18].

Some of Weyl’s irrational conduct might be explained by a larger historical context of cultural revolutions. His work deals with this subject in “Plato’s Ghost: The Modernist Transformation of Mathematics,” which was authored by the mathematician and math historian. Mathematical advancements are seen through the eyes of society at large by Jeremy Gray. Gray asserts that mathematics experienced a “modernist transformation” between the years 1890 and 1930. People who have access to “the right chances” support works that represent actual intellectual concerns and thereby contribute to the greater cultural change process^[19]. The First World War had an impact on the mathematical environment as well. Even while Gray’s method focuses mostly on one direction, namely, how individual works influence the transition process, it would be naive to disregard the way cultural and historical processes influenced Weyl’s ideas on certain theories and concepts^[20]. “The attitude of joyful times – the periods immediately following the First World War” influenced the tone of Weyl’s paper from 1921, as Dirk van Dalen points out^[21]. This perspective on Weyl’s ambiguity, however, may provide fertile ground for further research into Weyl’s ambivalence and its effect on cultural transformations.

Conclusion

Hermann Weyl’s intuitionism is something I’ve attempted to challenge with a different viewpoint in this piece. Weyl’s narrative, according to traditional historical accounts, lacks an understanding of normative framework alterations. Weyl’s ambiguity, I believe, may be unlocked by Menachem Fisch’s idea of ambivalent thought.

In Fisch’s view, it is feasible to have mixed feelings about our standards but not the other way around. We can only modify our whole frame of reference for what is good and wrong when we are faced to objective critique. The ambivalence process has no time limit, and it may continue for the duration of a practitioner’s work. Weyl is a great example of someone whose career was marked by ambiguity, proving that ambivalence does not always lead to a clear conclusion.

Weyl was the first to point out the defects in every mathematical theory he studied, something that other mathematicians may not have been able to accomplish. In his works, he maintained an ambiguous tone that was often attributed to his fluctuating philosophical views. Philosophically speaking, Weyl’s ambivalence may be explained by considering the intra-subjective process of shifting one’s normative framework. You should keep in mind the fact that Weyl’s so-called shifting perspectives really represent just the first step in an extensive, multi-step self-reflection in his quest for a solid basis on which mathematics might be constructed from scratch.

Bibliography

Atiyah, Michael. 2003. “Hermann Weyl.” Biographical Memoirs 82:320-335

Feferman, Solomon. 1988. “The Development of Programs for the Foundations of Mathematics in the First Third of the 20th Century”. doi: https://pdfs.semanticscholar.org/5a3a/112929a5a2a1f1fec4447cc584628918c0fb.pdf

Feferman, Solomon. 2000. “The Significance of Weyl’s Das Kontinuum.” In Proof theory: History and philosophical significance, edited by V. F. Hendricks et al., 179-194. Dordrecht: Kluwer.

Fisch, Menachem. 2017. Creatively Undecided: Toward a History and Philosophy of Scientific Agency. Chicago: The University of Chicago Press.

Fisch, Menachem and Yitzhak Benbaji. 2011. The View from Within: Normativity and the Limits of Self-Criticism. Notre Dame: University of Notre Dame Press.

Gray, Jeremy. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.

Newman, Max. 1957. “Hermann Weyl, 1885-1955.” Biographical Memoirs of Fellows of the Royal Society 3:305-328.

Scholz, Erhard. 2004. “Philosophy as a Cultural Resource and Medium of Reflection for Hermann Weyl.” https://arxiv.org/abs/math/0409596 (last accessed March 20, 2022).

Sieroka, Norman. 2009. “Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism”. Philosophia Scientiae 13(2):85-96.

Sieroka, Norman. 2019. “Neighbourhoods and Intersubjectivity: Analogies Between Weyl’s Analyses of the Continuum and Transcendental-Phenomenological Theories of Subjectivity”. In Weyl and the Problem of Space: From Science to Philosophy, edited by: Bernard, Julien & Lobo, Carlos, 98- 122. Springer, Switzerland.

van Dalen, Dirk. 1995. “Hermann Weyl’s Intuitionistic Mathematics.” The Bulletin of Symbolic Logic, 1(2):145–169.

Weyl, Hermann. 1921. “On the New Foundational Crisis in Mathematics.” In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, edited by Paolo Mancosu, 86-118.Oxford: Oxford University Press.

Weyl, Hermann. 1955. “Insight and Reflection” In The Spirit and Uses of the Mathematical Sciences, edited by T. L. Saaty and F. J. Weyl, 281–301. New York, McGraw-Hill.

[1] Hermann Weyl. 1921. “On the New Foundational Crisis in Mathematics.” In From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press

[2] van Dalen, Dirk. 1995. “Hermann Weyl’s Intuitionistic Mathematics.” The Bulletin of Symbolic Logic, 1(2):145–169.

[3] Ibid

[4] Feferman, Solomon. 2000. “The Significance of Weyl’s Das Kontinuum.” In Proof theory: History and philosophical significance, 179-194.

[5] Scholz, Erhard. 2004. “Philosophy as a Cultural Resource and Medium of Reflection for Hermann Weyl.” https://arxiv.org/abs/math/0409596

[6] Sieroka, Norman. 2009. “Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism”. Philosophia Scientiae 13(2):85-96.

[7] Fisch, Menachem. 2017. Creatively Undecided: Toward a History and Philosophy of Scientific Agency. Chicago: The University of Chicago Press.

[8]Ibid Weyl 1921

[9] Ibid Weyl 1921

[10] Weyl, Hermann. 1955. “Insight and Reflection” In The Spirit and Uses of the Mathematical Sciences, 281–301. New York, McGraw-Hill.

[11] Fisch, Menachem and Yitzhak Benbaji. 2011. The View from Within: Normativity and the Limits of Self-Criticism. Notre Dame: University of Notre Dame Press.

[12] Ibid

[13] Feferman, Solomon. 1988. “The Development of Programs for the Foundations of Mathematics in the First Third of the 20th Century”. doi: https://pdfs.semanticscholar.org/5a3a/112929a5a2a1f1fec4447cc584628918c0fb.pdf

[14] Ibid

[15] Feferman 2000

[16] Sieroka, Norman. 2019. “Neighbourhoods and Intersubjectivity: Analogies Between Weyl’s Analyses of the Continuum and Transcendental-Phenomenological Theories of Subjectivity”. In Weyl and the Problem of Space: From Science to Philosophy, 98- 122. Springer, Switzerland.

[17] Atiyah, Michael. 2003. “Hermann Weyl.” Biographical Memoirs 82:320-335

[18] Newman, Max. 1957. “Hermann Weyl, 1885-1955.” Biographical Memoirs of Fellows of the Royal Society 3:305-328.

[19] Gray, Jeremy. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.

[20] Ibid

[21] Dirk van Dalen. 1995