Kant (vs. Leibniz, Wolff and Lambert) on real definitions in geometry

Abstract

This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid’s definitions. I argue that Kant accepted Euclid’s definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.

Kant e o método matemático

“… KrV, B759 7 Neste sentido, concordamos com a descrição das "definições reais" da matemática fornecida porHeis (2014). Uma análise completa das definições matemáticas, sua relação com o procedimento de construção e os esquemas matemáticos pode ser encontrada emCapozzi (2021).8 Examinei esta questão mais detalhadamente emMartínez (2022).…”

Dedekind and Wolffian Deductive Method

“… Kant's first Critique is quoted following the usual convention, designating the first and second editions with A and B. Translations of Kant's works are cited as usual, by year and page.11 Kant 1992a, 634: "A definition is genetic if it yields a concept through which the object can be exhibited a priori in concreto; all mathematical definitions are of this sort." 12 On real definitions of circle and parallel in Kant and the modern tradition, seeHeis (2014) and De Risi (2015).13 Friedman (1992, 83) throws doubt on the usual identification of fundamental propositions (Grundsätze) and axioms in Kant, since I.20 is "not an axiom in Euclid, but a basic (and therefore fundamental) theorem." However, Kant's examples of axioms are taken from modern reconstructions of the Elements.…”