It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

How does the modern scientific conception of time constrain the project of assigning the mind its proper place in nature? On the scientific conception, it makes no sense to speak of the duration of a pain, or the simultaneity of sensations occurring in different parts of the brain. Such considerations led Henri Poincaré, one of the founders of the modern conception, to conclude that consciousness does not exist in spacetime, but serves as the basic material out of which we must (...) create the physical world. The central claim of Sensorama is that Poincaré was substantially correct. The best way to reconcile the scientific conception of time with the evidence of introspection is through a phenomenalist metaphysic according to which consciousness exists in neither time nor space, but serves as a basis for the logical construction of spacetime and its contents. (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 examine how Kant argues for the transcendental ideality of space. I defend a reading on which Kant accepts the ideality of space because it explains our (actual) knowledge that mathematical judgments are necessarily true. I argue that this reading is preferable over the alternative suggestion that Kant can infer the ideality of space directly from the fact that we have an a priori intuition of space. Moreover, I argue that the reading I propose does not commit Kant to incoherent (...) modal views. If we carefully distinguish between different senses of modality, the fact that our spatial form of intuition is (in some sense) contingent does not undermine the claim that this form can explain how our mathematical judgments are (in some sense) necessary. (shrink)

By way of a reply to Charles Parsons's paper in the Nagel Festschrift, Kant's notion of intuition (Anschauung) is examined. It is argued that for Kant the immediate relation which an intuition has to its object is a mere corollary to its singularity. It does not presuppose (as Parsons suggests) any presence of the object to the mind. This is shown, e.g., by the Prolegomena § 8, where the objects of intuitions a priori are denied by Kant to be so (...) present. They yield knowledge, not in virtue of their immediacy but in virtue of their ideality. (shrink)

In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in order to clarify Ernst Zermelo’s reading of Kant’s antinomy, according to which the latter is deeply rooted in the (...) tendency of the mind of producing “creative progress” and “inclusive closure”. The aim of this paper is to explain in which sense and why Kant’s treatment of the antinomies attracts the attention of Zermelo in the early 1900s and which aspects of his second axiomatic system have been inspired by Kant’s philosophy. Thus, by reading Kant’s antinomy ‘through Zermelo’s eyes’—with emphasis on the concept of regressive series in indefinitum and on that of regressive series ad infinitum – this paper identifies the echoes of Kant’s work in the making of the ZFC set theory. (shrink)

SummaryThe paper discusses major issues concerning the A104‐10 transcendental‐object theory. For that theory, our de re knowledge becomes related to its object just because our understanding thinks a certain object to stand related to the intuition via which we know. Employing an apparatus of intensional logic, I argue that this thought of an object is to be understood as a certain sort of intuition‐related, de dicto thought. Then I explore how, via such a de dicto thought, we can nevertheless achieve (...) de re knowledge. This question involves an important Kantian reduction of de re to de dicto outer‐object thinking, which I consider. Finally, I investigate some further topics about the transcendental object. I endeavor to show, throughout, that Kant's theory of that object is crucially related to matters of intensionality. (shrink)

Famously, Kant describes space and time as infinite “given” magnitudes. An influential interpretative tradition reads this as a claim about phenomenological presence to the mind: in claiming that space and time are given, this reading holds, Kant means to claim that we have phenomenological access to space and time in our original intuitions of them. In this paper, I argue that we should instead understand givenness as a metaphysical notion. For Kant, space and time are ‘given’ in virtue of three (...) related facts: (i) they are necessary grounds of the existence of all other spatial and temporal possibilities; (ii) in virtue of being such grounds, they are metaphysically linked to all other represented spatiotemporal things; and (iii) as representations, space and time issue from the nature of our faculty of sensibility rather than from an arbitrary act of choice. Understanding givenness in this non-epistemological way helps us recover what is plausible in both ‘intellectualist’ and ‘anti-intellectualist’ readings of the Transcendental Aesthetic. Anti-intellectualists are correct that space and time are given in mere sensibility, but intellectualists are correct that we depend on the understanding for consciousness of space and time. (shrink)

I examine central points in the 1787 deduction, Including the question of how kant can demonstrate his crucial claim that if I know via intuition "i", Then any element of "i"'s manifold is such that I am or can become conscious that that element is mine. I also consider the deduction's overall strategy, Kant's theory of synthesis and of our use of 'i', And some recent interpretations. See, Further, My 1981 "dialectica" transcendental-Object paper.

It is argued that mathematical statements are "a posteriori synthetic" statements of a very special sort, To be called "structure-Analytic" statements. They follow logically from the axioms defining the mathematical structure they are describing--Provided that these axioms are "consistent". Yet, Consistency of these axioms is an empirical claim: it may be "empirically verifiable" by existence of a finite model, Or may have the nature of an "empirically falsifiable hypothesis" that no contradiction can be derived from the axioms.

The ArgumentThe main subject examined in this paper is Immanuel Kant's controversy withPhilosophisches Magazinregarding Kant's new theory of judgments. J. A. Eberhard, editor ofPhilosophisches Magazin, and his colleagues wanted to vindicate the Wollfian traditional concept of judgments by undermining Kant's claims. As will be demonstrated, their arguments were effective mainly in exposing the ambiguity that was inherent in Kant's concept of the synthetic a priori; an ambiguity that resulted from Kant's desire—central to his critique of metaphysics—to present judgments pertaining to (...) mathematics, (dogmatic) metaphysics, and pure natural science as judgments which shared a common form. Exposing this ambiguity was not the intended result, and it was insufficient for the purpose of vindicating the Wollfian tradition. The contributors toPhilosophisches Magazinignored the important properties shared by the class of judgments falling under Kant's concept of synthetic a priori judgments. They also ignored the fact that their position was unable to account for the logical phenomena that motivated Kant to present a new theory of judgments. On the other hand, Kant's theory of judgments was insensitive to the important differences that exist among the distinct types of judgments falling under his concept of a synthetic a priori judgment. This latter point is clearly shown in the controversy regarding the novelty of Kant's concept of a synthetic a priori judgment, and in the controversy regarding the function of intuitions within synthetic judgments.A result of the controversy was that Kant's concept of the synthetic a priori, which he believed to be an exact concept, was revealed to be a metaphor: no more than an invitation to view certain intellectual fields in the light of others. On the other hand, Eberhard and his colleagues failed to come up with satisfactory answers to Kant's questions within their traditional concept of judgment. Both parties refused to acknowledge this result. Consequently, the search for a new logic, a new architectonic order, and a new unity within reason became a general problem for the new generation of philosophers. (shrink)

The quest to understand the natural and the mathematical as well as philosophical principles of dynamics of life forms are ancient in the human history of science. In ancient times, Pythagoras and Plato, and later, Copernicus and Galileo, correctly observed that the grand book of nature is written in the language of mathematics. Platonism, Aristotelian logism, neo-realism, monadism of Leibniz, Hegelian idealism and others have made efforts to understand reasons of existence of life forms in nature and the underlying principles (...) through the lenses of philosophy and mathematics. In this paper, an approach is made to treat the similar question about nature and existential life forms in view of mathematical philosophy. The approach follows constructivism to formulate an abstract model to understand existential life forms in nature and its dynamics by selectively combining the elements of various schools of thoughts. The formalisms of predicate logic, probabilistic inference and homotopy theory of algebraic topology are employed to construct a structure in local time-scale horizon and in cosmological time-scale horizon. It aims to resolve the relative and apparent conflicts present in various thoughts in the process, and it has made an effort to establish a logically coherent interpretation. (shrink)