Sylvia Pauw

This paper argues that, according to Johann Heinrich Lambert (1728–1777), general concepts of geometrical figures have logical features of what are now called non-propositional functions. In arguing this, I develop observations by Cassirer, and build on Friedman’s work on Kant. I argue that Lambert’s account of the role of postulates and problems in Euclid’s Elements implies that he regards general concepts of geometrical figures as ‘operations’ that have simpler concepts as input, and more complex concepts as output. I show, moreover, that Lambert takes it to be precisely specified which concepts serve as inputs to these operations. In other words, they have what we would call a ‘domain’. My reading supports Dunlop’s recent thesis that, according to Lambert, postulates and constructions primarily serve to make possible geometry’s specifically fruitful type of generality. In addition, my reading sheds new light on Lambert’s notion of a ‘real definition’ (Sacherklärung). I argue, contra Heis, that the real definition of the concept parallel line depends on the controversial ‘parallel axiom’ (i.e. Euclid’s fifth postulate), in Lambert’s eyes.

This paper argues that, for Bernard Nieuwentijt, mathematical reasoning on the basis of ideas is not the same as logical reasoning on the basis of propositions. Noting that the two types of reasoning differ helps make sense of a peculiar-sounding claim Nieuwentijt makes, namely that it is possible to mathematically deduce false propositions from true abstracted ideas. I propose to interpret Nieuwentijt’s abstracted ideas as incomplete mental copies of existing objects. I argue that, according to Nieuwentijt, a proposition is mathematically deducible from an abstracted idea if it can be demonstrated that that proposition makes a true claim about the object that idea forms. This allows me to explain why Nieuwentijt deems it possible to deduce false propositions from true ideas. It also implies that logic and mathematics are not as closely related for Nieuwentijt as has been suggested in the existing secondary literature.

An interpretative question Kant's Critique of Pure Reason raises, is how we should understand the relationship between the categories and the so-called 'logical forms of judgment' Kant deduces them from. In her Kant and the Capacity to Judge, B ́eatrice Longuenesse provides an answer to this question. In this thesis, I evaluate Longuenesse's account by considering its application to a specific group of categories: the categories of Quantity. I argue that for these categories, Longuenesse's account is problematic. The same, however, holds for a possible alternative analysis of the categories of Quantity: Manley Thompson's (1989) analysis. I show that Longuenesse and Thompson have both built their analyses on an untenable conception of the role of the categories of Quantity. The relationship between the categories and the logical forms of Quantity seems to be more arbitrary than Longuenesse, Thompson and others have argued