Domain Extension and Ideal Elements in Mathematics

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].

show abstract

On Certainty, Change, and “Mathematical Hinges”

Martin

2022

Topoi

View full text Add to dashboard Cite

Coliva (2020a) asks, "Are there mathematical hinges?" I argue here, against Coliva's own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein's well-known riverbed analogy enables us to see how his way of thinking about mathematics can account nicely for a "dynamics of change" within mathematical research-something his philosophy of mathematics has been accused of missing (e.g., by Ackermann (1988) and Wilson (2006)). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained).

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Domain Extension and Ideal Elements in Mathematics

Cited by 3 publications

References 15 publications

Deep Disagreement in Mathematics

Deep Disagreement in Mathematics

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

On Certainty, Change, and “Mathematical Hinges”

Contact Info

Product

Resources

About