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10) famously distinguished those judgments in which the predicate belongs to the subject as something that is (covertly) contained in the latter concepts, which he called analytic judgments or judgments of clariﬁcation, from those judgments in which the predicate lies entirely outside the subject, although it stands in connection with it. He called the latter kind of judgments synthetic or judgments of ampliﬁcation. ” By contrast, all the propositions of mathematics, including numerical formulas and the principles of geometry, are synthetic according to Kant.

Riehl, Alois. 1922. Führende Denker und Forscher. Leipzig: Quelle & Meyer. Ryckman, Thomas A. 2005. The reign of relativity: Philosophy in physics 1915–1925. New York: Oxford University Press. Schlick, Moritz. 1917. Raum und Zeit in der gegenwärtigen Physik: Zur Einführung in das Verständnis der allgemeinen Relativitätstheorie. Berlin: Springer. Schlick, Moritz. 1918. Allgemeine Erkenntnislehre. Berlin: Springer. Schiemann, Gregor. 2009. Hermann von Helmholtz’s mechanism: The loss of certainty.

Helmholtz’s objections to Kant are discussed in Chap. 3. For now, it is noteworthy that Helmholtz raised the question whether necessity is an essential characteristic of a priori knowledge. Kant clearly ruled out logical necessity by considering both sensible and intellectual conditions of knowledge. Arguably, non-Euclidean geometry is consistent with Kant’s view that pure intuition alone can provide mathematical concepts with objective reality. Kant’s argument was directed against Leibniz’s and Wolff’s attempts to infer the existence of mathematical objects from the lack of contradiction in mathematical concepts.