The Powers of Pure Reason by Alfredo Ferrarin;

Author:Alfredo Ferrarin; [Ferrarin, Alfredo]
Language: eng
Format: epub
ISBN: 9780226243290
Publisher: University of Chicago Press
Published: 2015-03-12T05:00:00+00:00

6. The A Priori

The mathematician obviously need not look to appearances to construct intuitive objects. There is nothing like mathematical abstraction from experience. When, contrasting mathematical and philosophical knowledge, Kant writes that the mathematician constructs both the empirical image of a triangle and its pure intuition, he stresses that he/she does so “in both cases completely a priori, without having had to borrow the pattern [das Muster] for it from any experience” (KrV A 713/B 741). It is natural to infer that the mathematician constructs a priori the triangle, whereas in my experience of triangles, i.e., when the geometric form is applied to an object (the side of the pyramid), I neither see anything a priori nor construct a triangle. I am as passive with regard to the perception of the given appearance as mathematics is wholly active in its activities and constructions. The philosopher does not enjoy the mathematician’s privilege and can hope to explain only the concepts involved in experience.

This apparently natural conclusion is mistaken.

1. Were it so, mathematics would give rise to a world of forms, if not separate from then surely independent of the world of experience. Reconciling the two worlds would be a subsequent but distinct problem, one that would consist in accounting for not simply the (“top-down,” as it were) possibility of applying mathematics to appearances but also the inverse problem, the “bottom-up” possibility of the mathematical apprehension of objects. For in everyday and scientific experience it so happens that I apprehend appearances according to geometric forms (as in “this table is rectangular”). How that can happen if mathematics is an a priori construction while experience is experience of given objects is bound to remain unintelligible. As with Aristotle’s criticism of Plato’s mathematical forms, forms (mathematical or otherwise) are not independent of matter. Forms are always forms-of-matter, principles of spatiotemporal or conceptual organization of appearances. To be sure, for Kant mathematical objects are not Aristotle’s hulē noētē (intelligible matter), nor does the mathematician abstract forms from matter. But certainly mathematical objects do not exist in the mind (or in a Platonic plane of truth). Construction speaks against this givenness of mathematical objects. If we drove this wedge between a priori mathematical knowledge and a posteriori empirical knowledge, the purity of rational mathematical cognitions would be distinct from but in truth irreconcilable with our access to empirical reality. Between a priori and a posteriori there would be an alternative rather than a duality, for they would denote two mutually exclusive worlds.

2. Were it so, mathematical knowledge would be opposed to empirical knowledge and to transcendental philosophy alike. We would get the amazing result that the first Critique would not even come close to—indeed has probably even forgotten along the way—a major part of the problem it set out to solve. That is, it could not reply to its leading question—how are synthetic a priori judgments possible?—through an explanation of the lawfulness of experience. Instead of the fundamental investigation of the principles making possible all forms

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