Geometría y conformidad a fin.

Una lectura sobre el § 62 de la Crítica de la Facultad de Juzgar de I. Kant

Abstract

This article provides a detailed analysis of § 62 of Immanuel Kant's Critique of Judgement (KU). This passage is fundamental to establishing Kant's complex typology of purposiveness (Zweckmäßigkeit). The central thesis defended is that, although demonstrations of geometric properties generate a subjective satisfaction analogous to that experienced in judgements of taste (by strengthening the spirit through the concordance between imagination and understanding), they cannot be considered beautiful in the Kantian sense. The basis for this position lies in the fact that the activity of the imagination in a mathematical demonstration is always subordinate to and guided by a specific concept, which excludes the essential condition of beauty, namely: the free and indeterminate operation of the imagination. To defend this thesis, an argument is presented that follows a critical and exegetical sequence. First, the Kantian framework of purposiveness of § 62 of the KU is analysed, differentiating between three types: subjective (associated with beauty), formal objective (characteristic of geometric figures), and material objective. This analysis explains why Kant dismisses the beauty of geometric properties, suggesting that they be called relative perfection because their judgement is intellectual and based on specific concepts. Secondly, Kant's provocative suggestion to call the demonstration beautiful is addressed, and Breitenbach's (2015) interpretation is presented. She supports a non-conceptualist reading, arguing that the imagination makes a contribution free of conceptual determination by drawing new connections during the intermediate moments of the proof. Thirdly, I present a critique of this position. To this end, I argue that Kant's conception of mathematical proof intrinsically requires the construction of a concept in intuition (whether geometric or characteristic, as in algebra). This need for a specific conceptual guide at each step invalidates its potential for aesthetic judgement. Finally, it is concluded that mathematical proofs imply a subjective complacency that presupposes a specific conceptual activity and, therefore, they do not fully identify with the experience of beauty in the terms of KU.