Topology has been characterized as an unsuitable mathematical framework for the investigation of geometric-perceptual phenomena. This has been attributed to the highly abstract nature of topology leading to failures in tasks such as making distinctions between geometrical figures (e.g., a cube versus a sphere) in which the human perceptual system succeeds easily. An alternative thesis is proposed on both philosophical and empirical grounds. The present analysis applies the Müller-Lyer (ML) illusion as a method of investigation to examine the suitability of (...) topology in raising questions about geometric-perceptual phenomena and formulating potential answers to these questions. Not only will it be concluded that topology is a suitable mathematical framework in which one can formulate relevant questions and potential answers with respect to perceptual phenomena such as the ML illusion, it will also be discussed that topology will not be replaced by other means of spatial representation consistent with the findings that indicate its contributions to our daily spatial behavior along with other means such as Euclidean geometry. In addition, the analysis will be discussed with respect to two relevant themes: 1) the implications of the proposed view in relation to Husserl’s phenomenology and 2) the effect of developmental stages and education that may change the involvement of topology in such perceptual responses over time. (shrink)

One of the central debates in contemporary Kant scholarship concerns whether Kant endorses a “conceptualist” account of the nature of sensory experience. Understanding the debate is crucial for getting a full grasp of Kant's theory of mind, cognition, perception, and epistemology. This paper situates the debate in the context of Kant's broader theory of cognition and surveys some of the major arguments for conceptualist and non-conceptualist interpretations of his critical philosophy.

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)

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation. (shrink)

Kant's theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant's theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant's theory of arithmetic can (...) be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or "formulas"—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant's theory of addition as a form of synthetic operation. (shrink)

The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the framework (...) of Maimon’s philosophy, arithmetical judgments cannot be analytical, nor is arithmetic an objectively necessary science, but rather only subjectively necessary. We should also cast doubt on his claim that we can create real objects from pure concepts of the understanding. (shrink)

This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.