RESUMEN Se defiende la tesis de que la teoría kantiana del espacio, en el contexto del idealismo trascendental, constituye un marco adecuado para solucionar la paradoja de las contrapartidas incongruentes. Se incluyen dos hipótesis: a) la solución kantiana implica una referencia a la conciencia de la propia corporalidad, para comprender cómo poseemos la capacidad para orientarnos en el espacio; b) no existe contradicción en el uso kantiano de las contrapartidas incongruentes en diferentes estadios de su pensamiento. La solución kantiana de (...) 1768 constituye una versión germinal del desarrollo que culminará en la teoría idealista trascendental acerca del espacio como una forma de nuestra sensibilidad. ABSTRACT The article argues that the Kantian theory of space, in the context of transcendental idealism, constitutes an adequate framework to resolve the paradox of incongruent counterparts. Two hypotheses are considered: a) the Kantian solution implies a reference to the awareness of one's corporeity in order to understand how we have the capacity to orient ourselves in space; b) there is no contradiction in the Kantian use of incongruent counterparts throughout the different stages of his thought. The Kantian solution of 1768 is a germinal version of the development leading up to the transcendental idealist theory of space as a form of our sensibility. (shrink)

Kant’s treatments of incongruent counterparts have been criticized in the recent literature. His 1768 essay has been charged with an ambiguous use of the notion of ‘inner ground’, and his 1770 claim that those differences cannot be apprehended conceptually is thought to be false. The author argues that those two charges rest on an uncharitable reading. ‘Inner ground’ is equivocal only if misread as mapping onto Leibniz notion of quality. Concepts suffice to distinguish counterparts, but are insufficient to specify their (...) spatial forms. Kant’s claims are reasonable and plausible, and have been reaffirmed repeatedly in contemporary discussions of demonstrative identification. (shrink)

Kant uses incongruent counterparts in his work before and after 1781, but not in the first Critique. Given the relevance that incongruent counterparts had for his thought on space, and their persistence in his work during the 1780s, it is plausible to think that he had a reason for leaving them out of both editions of the Critique. Two implausible conjectures for their absence are here considered and rejected. A more plausible alternative is put forth, which explains that textual absence (...) as a result of the synthetic method of presentation intended for the Critique. (shrink)

I propose that we interpret Kant’s argument from incongruent counterparts in the 1768 article ‘Concerning the Ultimate Ground of the Differentiation of Directions in Space’ in light of a theory of dynamic absolute space that he accepted throughout the 1750s and 1760s. This force-based or material conception of space was not an unusual interpretation of the Newtonian notion of absolute space. Nevertheless, commentators have continually argued that Kant’s argument is an utter failure that shifts from the metaphysics of space to (...) its epistemology, because he has no way to connect ‘directionality’ and ‘handedness’ to absolute space. This supposed failure is based on an understanding of absolute space in purely mathematical terms and as an absolute void that lacks any qualitative or dynamic features. If we recognize that Kant held that space had an intrinsic directional asymmetry then his argument successfully connects incongruent counterparts to absolute space. The presence of this notion in Kant’s pre-Critical thought is rarely noted, and its necessity in understanding his incongruence argument is novel. (shrink)

The concept of athleticism seems, at first glance, to be a minor component of Deleuze's analysis in Francis Bacon, appearing by name in the text only six times. In this article, I draw out the close link between athleticism and Deleuze's fundamental concept of intensity, arguing that this ostensibly minor term is in fact central to his account of the ‘clear and durable sensation’ produced by Bacon's painting. In tracing links between athleticism, Deleuze's aesthetic concept of ‘the fall’, and his (...) reliance on and critique of Kant's theory of intensive magnitudes, I show that athleticism in Francis Bacon names the form of intensity proper to painting. (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

This review of contemporary discussions of Kantian philosophy of mathematics is timed for the publication of the essay Kant’s Philosophy of Mathematics. Volume 1: The Critical Philosophy and Its Roots (2020) edited by Carl Posy and Ofra Rechter. The main discussions and comments are based on the texts contained in this collection. I first examine the more general questions which have to do not only with the philosophy of mathematics, but also with related areas of Kant’s philosophy, e. g. the (...) question: What is intuition and singular term? Then I look at more specific questions, e. g.: What is the subject of arithmetic and what is the significance of diagrams in mathematical reasoning? As a result, the reader is presented with a fairly complete overview of modern discussions which can be used as an introduction to the problem field of Kant’s philosophy of mathematics. (shrink)

I argue that, by the time of his essay "Concerning the Ultimate Ground of the Differentiation of Directions in Space" (1768), Kant had come to question the status of the Principle of Sufficient Reason as a result, at least in part, of his recognition of the existence of incongruent counterparts. Though Kant's argument against absolute space based on the existence of incongruent counterparts has been much discussed in recent years, its importance as a useful benchmark by which to judge the (...) direction of Kant’s thinking during this period has gone unnoticed. This paper is a modest attempt to remedy this deficiency. (shrink)

This paper develops a novel interpretation of Kant's argument from incongruent counterparts to the effect that the representations of space and time are intuitions rather than concepts. When properly understood, the argument anticipates the contemporary position whereby the meaning of indexicals cannot be captured by descriptive contents.