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The two greatest mathematicians of the early twentieth century, David Hilbert and Henri Poincaré transformed the mathematics of their time. Their personal interaction was infrequent, until Hilbert invited Poincaré to deliver the first Wolfskehl Lectures in Göttingen in the spring of 1909. A correspondence ensued, which fixed the content and timing of the lecture series. A close reading of the exchange throws light on what Hilbert wanted Poincaré to talk about, and on what Poincaré wanted to present to Hilbert and (...) |
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This paper sets out to show how Eddington's early twenties case for variational derivatives significantly bears witness to a steady and consistent shift in focus from a resolute striving for objectivity towards “selective subjectivism” and structuralism. While framing his so-called “Hamiltonian derivatives” along the lines of previously available variational methods allowing to derive gravitational field equations from an action principle, Eddington assigned them a theoretical function of his own devising in The Mathematical Theory of Relativity (1923). I make clear that (...) |
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While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as “written (...) |
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The arguments are exhibited in favour of the necessity to modify the history of the genesis and advancement of general relativity (GR). I demonstrate that the dynamic creation of GR had been continually governed by internal tensions between two research traditions, that of special relativity and Newton’s gravity. The encounter of the traditions and their interpenetration entailed construction of the hybrid domain at first with an irregular set of theoretical models. Step by step, on eliminating the contradictions between the models (...) |
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This paper explores the main philosophical approaches of David Hilbert’s theory of proof. Specifically, it is focuses on his ideas regarding logic, the concept of proof, the axiomatic, the concept of truth, metamathematics, the a priori knowledge and the general nature of scientific knowledge. The aim is to show and characterize his epistemological approach on the foundation of knowledge, where logic appears as a guarantee of that foundation. Hilbert supposes that the propositional apriorism, proposed by him to support mathematics, sustains (...) No categories |
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The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building on a (...) |
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This paper attempts to show how the logical empiricists’ interpretation of the relation between geometry and reality emerges from a “collision” of mathematical traditions. Considering Riemann’s work as the initiator of a 19th century geometrical tradition, whose main protagonists were Helmholtz and Poincaré, the logical empiricists neglected the fact that Riemann’s revolutionary insight flourished instead in a non-geometrical tradition dominated by the works of Christoffel and Ricci-Curbastro roughly in the same years. I will argue that, in the attempt to interpret (...) |
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El artículo presenta y analiza un conjunto de notas manuscritas de clases para cursos sobre geometría, dictados por David Hilbert entre 1891 y 1905. Se argumenta que en estos cursos el autor elabora la concepción de la geometría que subyace a sus investigaciones axiomáticas en Fundamentos de la geometría . Por un lado, afirmo que lo que caracteriza esta concepción de la geometría es: i) una posición axiomática abstracta o formal; ii) una posición empirista respecto del origen de la geometría (...) No categories |
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Karl Pearson is the leading figure of XX century statistics. He and his co-workers crafted the core of the theory, methods and language of frequentist or classical statistics – the prevalent inductive logic of contemporary science. However, before working in statistics, K. Pearson had other interests in life, namely, in this order, philosophy, physics, and biological heredity. Key concepts of his philosophical and epistemological system of anti-Spinozism (a form of transcendental idealism) are carried over to his subsequent works on the (...) |
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Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) |
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In this paper we present a schema for describing dualities between physical theories, and illustrate it in detail with the example of bosonization: a boson-fermion duality in two-dimensional quantum field theory. The schema develops proposals in De Haro : these proposals include construals of notions related to duality, like representation, model, symmetry and interpretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or boson-fermion duality, has (...) No categories |
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We develop a new understanding of the historical horizon of event ontology. Within the general area of the philosophy of nature, event ontology is a still emerging field of investigation in search for the ultimate materialist ontology of the world. While event ontology itself will not be explicated in full mathematical details here, our focus is on its conceptual interrelation with the dominant current of Idealism in Western thought approached by us as a problem in the history of ideas. Our (...) |
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