The Ghost and the Spirit of Pythagoras

Arkady Plotnitsky ()
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Arkady Plotnitsky: Purdue University, Literature, Theory, and Cultural Studies Program, Philosophy and Literature Program

A chapter in Handbook of the History and Philosophy of Mathematical Practice, 2024, pp 63-89 from Springer

Abstract: Abstract This chapter introduces the concept of radical Pythagorean mathematics, as part of mathematical modernism, which emerged around 1900 and extends to our own time. Arguably, the greatest discovery of Pythagorean mathematics of (sixth-fifth century BC) was that of the incommensurability of the side and the diagonal of the square. This discovery established the complex relationships between geometry and arithmetic and, correlatively, the possibility of a type of logos, a logos containing the alogon within it – a logical thought containing the unthinkable within it. I shall call the mathematics defined by this combination “Pythagorean mathematics.” I shall argue that, as part of mathematical modernism, there emerged a new attitude toward the unthinkable in thought, an attitude that accepts this situation as a positive, enabling condition of thinking and knowledge. I shall call the mathematical thinking defined by this attitude, while keeping the irreducible relationship between geometry and algebra (which subsumes arithmetic within it), radical Pythagorean mathematics. The original Pythagorean attitude, at least in one reconstitution of Pythagorean thought, was, as against the Platonist attitude, tolerant of this condition, but still sought to overcome it by finding a logos without the alogon within it. This attitude, often sliding into a Platonist one, has remained dominant throughout the history of mathematics, physics, and philosophy, including during the modernist period, notable as much for the presence of the radical Pythagorean thinking as for a resistance to it.

Keywords: Algebra; Geometry; Mathematical modernism; Pythagorean mathematics; Radical Pythagorean mathematics; Quantum theory (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/978-3-031-40846-5_7

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