When it comes to the relation of modern mathematics and philosophy, most people tend to think of the three major schools of thought—i.e. logicism, formalism, and intuitionism—that emerged as profound researches on the foundations and nature of mathematics in the beginning of the 20th century and have shaped the dominant discourse of an autonomous discipline of analytic philosophy, generally known under the rubric of “philosophy of mathematics” since then. What has been completely disregarded by these philosophical attitudes, these foundational researches (...) which seek to provide pure mathematics with a philosophically plausible justification by founding it on firm logico-philosophical bases, is that the genuine self-foundation of pure mathematics had been done before, namely during the 19th century, when it was developing into an entirely new and independent discipline as a concomitant of the continuous dissociation of mathematics from the physical world. This self-foundation of the 19th-century pure mathematics, however, was more akin to the German-idealist interpretations of Kant’s transcendental philosophy, than the post-factum, retrospective 20th-century researches on the foundations of mathematics. This article aims to demonstrate this neglected historical fact via delving into the philosophical inclinations of the three major founders of the 19th-century pure mathematics, Riemann, Dedekind and Cantor. Consequently, pure mathematics, with respect to its idealist origins, proves to be a formalization and idealization of certain activities specific to a self-conscious transcendental subjectivity. (shrink)

I advance a novel interpretation of Kant's argument that our original representation of space must be intuitive, according to which the intuitive status of spatial representation is secured by its infinitary structure. I defend a conception of intuitive representation as what must be given to the mind in order to be thought at all. Discursive representation, as modelled on the specific division of a highest genus into species, cannot account for infinite complexity. Because we represent space as infinitely complex, the (...) spatial manifold cannot be generated discursively and must therefore be given to the mind, i.e. represented in intuition. (shrink)

I argue for a novel, non-subjectivist interpretation of Kant’s transcendental idealism. Kant’s idealism is often interpreted as specifying how we must experience objects or how objects must appear to us. I argue to the contrary by appealing to Kant’s Transcendental Deduction. Kant’s Deduction is the proof that the categories are not merely subjectively necessary conditions we need for our cognition, but objectively valid conditions necessary for objects to be appearances. My interpretation centres on two claims. First, Kant’s method of self-knowledge (...) consists in his determining what makes our cognitive faculty finite in contrast to God’s infinite cognitive faculty. Second, Kant’s limitation of our knowledge to appearances consists in his developing an account according to which appearances and our finite cognitive faculty are conceived of in terms of each other and in contrast to noumena in the positive sense and God’s infinite cognitive faculty. (shrink)

This paper analyses the features of Kant’s view of memory, which Kant himself described explicitly in his lectures on anthropology and implicitly in the A edition of the Critique of Pure Reason. I shall offer a review of literature on Kant’s view of memory up to this day. I suggest that memory is a cognitive faculty that has the power to store and reproduce representations. Kant distinguishes among three different kinds of memorization which are relevant for human cognition. I offer (...) reasons to hold that imagination and memory must be differentiated by their functioning, although the first one grounds the second one. Finally, I hold that certain functions of memory need to be presupposed at a transcendental level, in which memory would play a fundamental function with regard to the possibility of experience. Resumen Este artículo analiza los rasgos de la concepción de la memoria de Kant, los cuales él describió explícitamente en sus lecciones sobre antropología e implícitamente en la edición A de la Crítica de la Razón Pura. Proporcionaré una revisión de la literatura sobre la concepción de Kant acerca de la memoria hasta el presente. Sugiero que la memoria es una facultad cognitiva que tiene el poder de almacenar y reproducir representaciones. Kant distingue tres tipos diferentes de memorización, los cuales son relevantes para el conocimiento humano. Asimismo, proporciono razones para sostener que la imaginación y la memoria deben ser diferenciadas por su funcionamiento, aunque la primera es el fundamento de la segunda. Finalmente, sostengo que ciertas funciones de la memoria necesitan ser presupuestas a un nivel trascendental, en el cual la memoria cumpliría una función fundamental con respecto a la posibilidad de la experiencia. (shrink)

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...) that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense. (shrink)

This paper attempts to shed light on three sets of issues that bear directly on our understanding of Locke and Kant. The first is whether Kant believes Locke merely anticipates his distinction between analytic and synthetic judgments or also believes Locke anticipates his notion of synthetic a priori cognition. The second is what should we as readers of Kant and Locke should think about Kant’s view whatever it turns out to be, and the third is the nature of Kant’s justification (...) for the comparison he draws between his philosophy and Locke’s. I argue (1) that Kant believes Locke anticipates both the analytic-synthetic distinction and Kant’s notion of synthetic a priori cognition, (2) that the best justification for Kant’s claim draws on Locke’s distinction between trifling and instructive knowledge, (3) that the arguments against this claim developed by Carson, Allison, and Newman fail to undermine it, and (4) that Kant’s own justification for his claim is quite different from what many commentators have thought it was (or should have been). (shrink)

In the preface to the second edition of his Critique of Pure Reason, Kant laments thatit still remains a scandal to philosophy and to human reason in general that the existence of things outside us … must be accepted on faith, and if anyone thinks good to doubt their existence, we are unable to counter his doubts by any satisfactory proof. The two editions of the Critique each contain a celebrated refutation of epistemological scepticisms like those of Descartes and Hume. (...) The first edition refutation has been widely decried as relying on an objectionable sort of idealism. The refutation of the second edition, though rather more difficult to interpret than the first, has usually been read as a mere reworking of the first. (shrink)

Of the questions concerning the many great minds of the European Enlightenment, the question of what constitutes right and proper government perhaps had the most enduring influence on the world stage. Both Thomas Jefferson and Immanuel Kant attempted to answer the question of what constitutes right government, in particular by basing the system upon the idea of human freedom as an inalienable right. This project is an attempt to compare the systems proposed by these two authors, as well as to (...) critique each on its ability to protect and foster individual freedom. It is my opinion that neither manages to do what it is constructed to do, as each fails to fully protect individual freedom, and each has as part of it a component which conflicts with individual freedom. (shrink)