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Frobenius Quantales, Serre Quantales and the Riemann–Roch Theorem
The Riemann–Roch theorem for algebraic curves is derived from a theorem for Girard quantales. Serre duality is shown to be a quantalic phenomenon. An...
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Bernhard Riemann, the Ear, and an Atom of Consciousness
Why did Bernhard Riemann (1826–1866), arguably the most original mathematician of his generation, spend the last year of life investigating the...
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“Comprehending the Connection of Things”: Bernhard Riemann and Conceptual Thinking in Mathematics
The subject of this chapter is the conceptual nature of Bernhard Riemann’s thinking and its impact on mathematics, physics, and philosophy. Following...
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From Gauss to Riemann Through Jacobi: Interactions Between the Epistemologies of Geometry and Mechanics?
The aim of this paper is to argue that there existed relevant interactions between mechanics and geometry during the first half of the nineteenth...
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Conceptual Structuralism
This paper defends a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of...
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What Is a Curve?: A Pythagorean Essay
The chapter is an essay on the idea of a curve from the Pythagorean and then radical Pythagorean perspective. The historical scope of this chapter...
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“To Create More Worlds”: Mathematical Practice as Philosophy
The primary aim of this chapter is to consider, by building on the preceding argument of this study, mathematicians’ working philosophy of...
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Cassirer and Klein on the Geometrical Foundations of Relativistic Physics
Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from...
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Introduction
This book belongs to the philosophy rather than the history of mathematics, and the main reasons for considering the history of mathematics in the...
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On the Computational Properties of the Uncountability of the Real Numbers
The uncountability of the real numbers is one of their most basic properties, known (far) outside of mathematics. Cantor’s 1874 proof of the...
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Introduction
I began the Preface saying: Mathematics and physics have an intimate relationship, albeit on-and-off. Mathematics provides both machinery for...
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Primes and Particles
Mathematics and physics have been borrowing notions from each other for a very long time. Some mathematical object or construction provides a superb...
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So Far and in Prospect
Kinship and Particles; Primes and Particles; Effective Field Theory; Packaging Functions Connecting Spectra to Symmetries; Multiples Ways of...
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Confirming Mathematical Conjectures by Analogy
Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less...
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An Approach to Building Quantum Field Theory Based on Non-Diophantine Arithmetics
The problem of infinities in quantum field theory (QFT) is a longstanding problem in particle physics. To solve this problem, different...
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Biologie und Psychologie über Furcht, Ängste und Angst
Dieses Kapitel soll einige Informationen darüber geben, welche typischen Zugangsweisen sich in der biologischen und psychologischen Angstforschung...
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The Methods Behind Poincaré’s Conventions: Structuralism and Hypothetical-Deductivism
Poincaré’s conventionalism has been interpreted in many writings as a philosophical position emerged by reflection on certain scientific problems,...
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The Frege–Hilbert controversy in context
This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In...
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Mathematical Practice, Fictionalism and Social Ontology
From the perspective of mathematical practice, I examine positions claiming that mathematical objects are introduced by human agents. I consider in...
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Continuity in Leibniz and Deleuze: A reading of Difference and Repetition and The Fold
The status of continuity in Deleuze’s metaphysics is a subject of debate. Deleuze calls the virtual, in Difference and Repetition , an Ideal continuum ,...