What Is a Curve?: A Pythagorean Essay

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

Chapter Chapter 4 in Logos and Alogon, 2022, pp 137-158 from Springer

Abstract: Abstract 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 follows that of this study in general. While invoking earlier historical junctures, the history of the idea of a curve considered here extends from the mathematics at the rise of modernity, especially analytic geometry and calculus, with main attention given to Fermat and Descartes. Then, it moves to Riemann, who remains a central figure in this chapter, as he is throughout this study, in this case, as the inventor of the concept of a Riemann surface, which is a curve over ℂ. Finally, it moves to algebraic geometry, especially in the work of Weil and Grothendieck. The chapter offers a thesis concerning the idea of curve in mathematics grounded in a defining role of algebra in modern mathematics, nearly making modern mathematics algebra altogether—nearly, but not quite. This chapter places the role of algebra in our thinking about curves in relation to the Pythagorean thesis of this study as a whole, according to which modern mathematics and most mathematics, from the Pythagoreans on, are defined by the interplay between algebra and geometry.

Keywords: Algebraic geometry; Category theory; Curve; Elliptic curve; Étale cohomology; Riemann surface; Topos theory (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/978-3-031-13678-8_4

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