Kritik der Reinen Vernunft
(Critique of Pure Reason)
first edition 1781 (denoted by “A”), second edition 1787 (denoted by “B”)1781/1787
The science of all the principles of sensibility a priori, I call transcendental aesthetic.* There must, then, be such a science forming the first part of the transcendental doctrine of elements, in contradistinction to that part which contains the principles of pure thought, and which is called transcendental logic.
For Kant space is not a concept but an intuition, more precisely an intuition, “which serves for the foundation of all external intuitions”. Kant develops this view in the following steps (A24-25/B38-40)
Space is not a conception which has been derived from outward experiences. For, in order that certain sensations may relate to something without me (that is, to something which occupies a different part of space from that in which I am); in like manner, in order that I may represent them not merely as without, of, and near to each other, but also in separate places, the representation of space must already exist as a foundation. …
Space then is a necessary representation a priori, which serves for the foundation of all external intuitions. We never can imagine or make a representation to ourselves of the non-existence of space, though we may easily enough think that no objects are found in it. …
Space is no discursive, or as we say, general conception of the relations of things, but a pure intuition. For, in the first place, we can only represent to ourselves one space, and, when we talk of divers spaces, we mean only parts of one and the same space. Moreover, these parts cannot antecede this one all-embracing space, as the component parts from which the aggregate can be made up, but can be cogitated only as existing in it. Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations. …
The intuition (in the ordinary sense) Kant has concerning space is similar to a number theorist who considers “the” natural numbers as the object of her studies, though certainly she certainly has limitations in specifying what she means. Indeed, common way to do so have limitation: for instance, one can not uniquely determine the natural numbers by a list of axioms, as Gödel's fist incompleteness theorem demonstrates, nor is it satisfying to describe the natural numbers by a concrete model, because there are several ways to do so.
Due to his intuition that there is “one” space, Kant discards the first two out of the three possibilities he discusses. Since Kant this intuition has been dropped mostly by geometers in favor of the first introducing pioneering concepts like Riemannian manifolds in 1854. In his very early work Bertrand Russell tried to adapt Kant’s work to this new view (see An Essay on the Foundations of Geometry).
Actually, all three possibilities discussed by Kant appear in modern mathematics:
the “discursive, or .., general conception” is just the most common way to describe a space by a list of axioms, e.g. a topological space or a metric space.
a space that “antecede this one all-embracing space, as the component parts from which the aggregate can be made up” (German: “vor dem einigen allbefassenden Raume gleichsam als dessen Bestandtheile (daraus seine Zusammensetzung möglich sei) vorhergehen”). A modern concept that matches this description quite well is the concept of a classifying space, that does not contain each member of a certain class of spaces as a subspace, but from which “antecede” each such space by pullback.
“Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations.” (German: “Er [Der Raum] ist wesentlich einig, das Mannigfaltige in ihm, mithin auch der allgemeine Begriff von Räumen überhaupt beruht lediglich auf Einschränkungen.”) A situation like this is found if any space in a geometrical theory can be considered as a subspace of one fix space. Good examples of such situations are given by descriptive set theory that can be understood as the study of subsets of the real line.
Achourioti and van Lambalgen (June and November 2011) offered a formalization of Kant’s logic in terms of geometric logic.
Regarding Aristotelian logic:
Concepts [Begriffe] … serve as predicates of possible judgements. (Critique of Pure Reason A69/B94)
Compare to the identification of concepts with the types in the sense of type theory, see the references here.
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Wikipedia, Critique of Pure Reason
Wikisource, First edition (1881) and several English translations
Achourioti and van Lambalgen June 2011, A formalization of Kant’s transcendental logic.
Achourioti and van Lambalgen November 2011 (Video), The Completeness of Kant’s Table of Judgments & Consequences for Philosophy of Mathematics.
Last revised on January 27, 2019 at 05:03:59. See the history of this page for a list of all contributions to it.