Prolegomena for a Kantian Formal Theory of Space

(Metaphysik & Ontologie, Englisch)

ant's formal logic has traditionally been considered as "terrifyingly narrow-minded and mathematically trivial" (Hazen, A., 1999, ''Logic and Analyticity,'' in A.C. Varzi (ed.), The Nature of Logic, Stanford, CA: CSLI, pp. 79-110). This negative assessment can be traced back to Frege's ''Begriffschrift'', where, in light of the then newly developed symbolic logic, it is argued that Kant's logic is plagued by irremediable defects, such as, e.g., the reliance on the obsolete Aristotelian subject-predicate framework, the distinction between the negative and infinite judgment, between the categorical and hypotetical judgment, et cetera.

Recently, however, some philosophers have challenged this received view. Achourioti and van Lambalgen ( T. Achourioti and M. van Lambalgen (2011). A formalization of Kant's Transcendental Logic. The Review of Symbolic Logic, 4, pp 254-289), (T. Achourioti and M. van Lambalgen (forthcoming). Kant and Logical Theory, Oxford University Press), building on Longuenesse's hermeneutical work (B. Longuenesse (1998), Kant and the Capacity to Judge, Princeton University Press), have developed a formalization of Kant's transcendental logic in terms of contemporary mathematics. They argue that the logical form of Kant's judgments and their intended semantics are inherently more complex than what is generally believed, and that it is thus misguided to interpret transcendental logic in light of classical logic. According to their interpretation, transcendental logic constitutes a non-classical formal logic in its own right, where the logical forms of judgments can be identified with the geometric fragment of first-order logic, while the intended semantics of objects of synthesis is captured by means of inverse systems of first-order models. The outcome of the formalization is that Kant's table of judgments is sound and complete with respect to its intended semantics.

According to Kant, transcendental logic differs from general logic in that the former does not abstract from the cognition of the object, i.e., it is a logic by means of which the spatio-temporal objects of experience are actually constructed from the material of appearances. Achourioti and van Lambalgen's formalization in (T. Achourioti and M. van Lambalgen (2011). A formalization of Kant's Transcendental Logic. The Review of Symbolic Logic, 4, pp 254-289) does not explicitly represent spatial nor temporal information. In order to improve the formalization of Kant's transcendental logic, it becomes thus necessary to obtain a better formal understanding of Kant's theory of space-time. Achourioti and van Lambalgen (T. Achourioti and M. van Lambalgen (forthcoming). Kant and Logical Theory, Oxford University Press.) have attempted to develop a formalization of Kant's notion of time as pure intuition. According to Longuenesse, the Transcendental Analytics, in its chapter on the synthesis speciosa, is supposed to explain how the intuitions of space and time, which in the Transcendental Aesthetics had been presented as given, must instead be seen as produced, or constructed by the subject by means of imagination. Relying on this interpretation, Achourioti and van Lambalgen propose a formalization in which the Kantian temporal continuum is constructed from given appearances by means of the action of the categories, which are interpreted as functions for the ordering of these appearances, while formally they are nothing else than functors between categories (in the categoy theory sense).

The purpose of this talk is to present a paper extending the formal analysis to the intuition of space. In particular, we shall argue that Kant sharply distinguished between "metaphysical" space and geometrical space, where the ground of the distinction lies in the fact that metaphysical space is characterized by means of purely topological properties, while geometrical space requires the construction of geometrical concepts. The topological properties that characterize metaphysical space are (i) unity, (ii) infinity, (iii) continuity, (iv) dimnesionality.

We shall analyse in detail Kant's rendition of these topological notions, and we shall examine, in light of contemporary mathematics, other importantant aspects of Kant's theory of space such as the notions of boundary, part and whole, contact and location, as well as the role of the categories in the construction of the spatial continuum. The outcome of the paper is to provide precise guidelines as to how a Kantian formal theory of space must look like, and to investigate possible formal correlates of Kant's spatial notions. This work thus provides the philosophical groundwork for a subsequent paper, currently under development, in which the mathematical theory is to be fully developed.

Chair: Laurenz Hudetz

Zeit: 11:15-11:45, 14. September 2013 (Samstag)

Ort: HS 101

Anmerkung: ÄNDERUNG. Der Vortrag entfällt!

Riccardo Pinosio

(Institute for Logic, Language and Computation, University of Amsterdam, Niederlande)

Riccardo Pinosio (MSc Logic). Institute for Logic, Language and Computation, University of Amsterdam. 2010 BA in philosophy (thesis on the Church-Fitch knowability paradox). 2012 MSc in Logic (thesis on Kant's theory of space and constructive Euclidean geometry). I work mostly on the following two topics: (I) formal theories of space-time from a phenomenological and computational perspective (spatial logics, mereotopologies, etc) (II) non-classical logics as models of reasoning (counterfactuals, defeasible reasoning).

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