The Form and Function of Duality in Modern Mathematics

Krömer, Ralf; Corfield, David

doi:10.4000/philosophiascientiae.976

Cited by 3 publications

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“…In this article we have tried to contribute to what we could call the "philosophy of (mathematical) duality" [27,30,53,69,71] by analyzing a major event in the history of this notion, namely the construction of a geometric object (an affine scheme) associated to any (commutative) ring. Let's stress the main argumentative thread of the proposed analysis:…”

Section: Discussionmentioning

confidence: 99%

“…[65,Chapt.6,Sect.4,[253][254][255][256][257][258][259][260][261][262][263][264][265][266]] and [105] for a mathematical approach, Refs. [9,27,30,53,69,71,83,98] for philosophical-oriented analyses of this notion in modern mathematics). 3 Among the different instantiations of the mathematical notion of duality, the duality between algebra and geometry occupies a privileged place.…”

Section: Introductionmentioning

confidence: 99%

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Grothendieck’s theory of schemes and the algebra–geometry duality

Catren

Cukierman

2022

Synthese

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We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck's theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations A → B into other similar structures B. This vantage point will allow us to analyze the relationship between the algebrageometry duality and (what we shall call) the structure-semiotics duality (of which the syntax-semantics duality for propositional and predicate logic are particular cases). Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck's theory of schemes permits to reconstruct general (commutative) rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized-by means of what we shall generally call Gelfand transform-as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics (e.g. Stone duality, Gelfand duality, Pontryagin duality, Galois-Grothendieck duality, etc.) can be understood as particular cases of this general pattern. In the wake of Majid's analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck's notion of functor of points can be understood as a "relativization of the a priori" (Friedman) that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding (co)domains.1 According to J.-P. Serre, "[t] his first approximation to the theory of affine schemes was 'in the air ' in 1954-1955, at least with some restrictions on the rings in question. One of Grothendieck's contributions was to remove these restrictions: the best category of commutative rings is the category of all commutative rings! " [51, 256]. For a discussion of the genesis of the notion of scheme see Refs. [36,35,92].

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Grothendieck’s theory of schemes and the algebra–geometry duality

Catren

Cukierman

2022

Synthese

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“… 13 The terminology of “functional” and “axiomatic” conception of duality has been coined by Saunders MacLane in his [49]. Kromer and Corfield [47] also mention the labels “formal” or “Gergonne-type” conception for the axiomatic conception, and “concrete” or “Poncelet-type” conception for the functional conception. The reason for the labels “Poncelet-type conception” and “Gergonne-Type conception” will become clearer in the historical Section 4. …”

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confidence: 99%

“…Modern standard textbooks on projective geometry like [16, 16] and [15] have essentially followed their lead. For a discussion of this point see [47, 101].…”

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confidence: 99%

Projective Duality and the Rise of Modern Logic

Eder¹

2021

Bull. symb. log

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The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels "duality" or "reciprocity". The aims of this article are, irst, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Speci ically, we want to see in which ways geometer's preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been in luenced by re lections on duality. This is a ``preproof'' accepted article for The Bulletin of Symbolic Logic. This version may be subject to change during the production process.

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Duality as a category-theoretic concept

Corfield

2017

Studies in History and Philosophy of Science Part B: Studies in

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The Form and Function of Duality in Modern Mathematics

Cited by 3 publications

References 11 publications

Grothendieck’s theory of schemes and the algebra–geometry duality

Grothendieck’s theory of schemes and the algebra–geometry duality

Projective Duality and the Rise of Modern Logic

Duality as a category-theoretic concept

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