L. E. J. BROUWER AND KARL POPPER: TWO PERSPECTIVES ON MATHEMATICS

Studies in History and Philosophy of Science, 2024

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism.

Synthese, 2022

Brouwer's intuitionistic program was an intriguing attempt to reform the foundations of mathematics that eventually did not prevail. The current paper offers a new perspective on the scientific community's lack of reception to Brouwer's intuitionism by considering it in light of Michael Friedman's model of parallel transitions in philosophy and science, specifically focusing on Friedman's story of Einstein's theory of relativity. Such a juxtaposition raises onto the surface the differences between Brouwer's and Einstein's stories and suggests that contrary to Einstein's story, the philosophical roots of Brouwer's intuitionism cannot be traced to any previously established philosophical traditions. The paper concludes by showing how the intuitionistic inclinations of Hermann Weyl and Abraham Fraenkel serve as telling cases of how individuals are involved in setting in motion, adopting, and resisting framework transitions during periods of disagreement within a discipline.

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: What does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics.

Publications des Archives Henri-Poincaré -Publications of the Henri Poincaré Archives).

Theoria, Revista de teoria, historia y fundamentos de la ciencia, 2006

This paper is concerned with Cavaillès' account of "intuition" in mathematics. Cavaillès starts from Kant's theory of constructions in intuition and then relies on various remarks by Hilbert to apply it to modern mathematics. In this context, "intuition" includes the drawing of geometrical figures, the use of algebraic or logical signs and the generation of numbers as, for example, described by Brouwer. Cavaillès argues that mathematical practice can indeed be described as "constructions in intuition" but that these constructions are not imbedded in the space and in the time of our Sensibility, as Kant believed: They take place in other structures which are engendered in the history of mathematics. This leads Cavaillès to a critical discussion of both Hilbert's and Brouwer's foundational programs.

2002

I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for the kind of knowledge that such an account of itself would amount to.

Presented at 'Technology, Knowledge, Truth' Conference held by the Melbourne Society for Continental Philosophy, Dec. 13-15th 2017

Karl Popper's falsificationist demarcation of science from non-science rests on a specific ontological commitment to what he calls his 'naive realism'. The self-conscious 'naivete' here lies in the undefined - and for Popper largely undefinable - nature of this reality. All that his falsificationist protocol requires of it is that it be objective - that is to say, that neither our thoughts, nor beliefs or justifications bear on it or its ‘contents’. It is what it is, and nothing we have to say on the matter can alter it one jot. He doesn't, I think, seek to claim that this objective reality exhausts the possibilities of the real - he does not, for example, deny the existence of subjective states. But he does claim an independence of objective reality from those states (though the reverse does not hold true). However, I think it is possible, on the basis of Popper's own arguments, to say more about the character of this objective reality in ways that neither invalidate nor contradict Popper's arguments, but do enrich them in useful ways - philosophically at least. The paper will argue that the terms of what Popper calls his 'evolutionary epistemology' require us to understand the objectivity of this external reality to consist not of actual or material terms', but rather of relations that are external to the terms they relate. More precisely, it consists of relations between relations, which is to say, differential relations. In this light, Popper's external reality, the guarantor of his falisificationist demarcation, appears as not as a realm of actual things, but of virtual problems. What I would like to do, then, is to read Popper in and on his own terms, but 'behind his back' as a philosopher of difference in a Deleuzian mode. In doing so, I seek to follow through on Bergson's (and perhaps Deleuze's) aim of adding to science the metaphysics that it lacks (and does not want) in terms derived from (Popperian) sciences own methodological assumptions in ways that both complicate and shed light on the very demarcation those assumptions seek to defend.

Revista Pesquisa Qualitativa

This paper aims to contribute to the clarification of the role of mathematical intuition and imagination in the constitution of mathematical knowledge, evidencing its epistemological and procedural characteristics. For that, an "epistemology of intuition and imagination" in the field of mathematics is outlined (suggested) emphasizing the need to adopt a dynamic conception of mathematics. In this context, intuition and imagination present themselves as forms of mathematical experience that give access, through paths that are not purely logical, to mathematical knowledge. Its epistemological and rationality characteristics, a rational of a non-logical nature, are highlighted by several examples, resources for moving the ideas involved. The epistemological study of intuition and imagination also allows highlighting its ontology, constituted of more relations than objects. From the pedagogical point of view, we discuss the formative character of philosophical studies involving...

Ruch Filozoficzny, 2020