Kant: Science, Logic, and Mathematics, Misc

Kant's obscure essay entitled An Attempt to Introduce the Concept of Negative Quantities into Philosophy has received virtually no attention in the Kant literature. The essay has been in English translation for over twenty years, though not widely available. In his original 1983 translation, Gordon Treash argues that the Negative Quantities essay should be understood as part of an ongoing response to the philosophy of Christian Wolff. Like Hoffmann and Crusius before him, the Kant of 1763 is at odds with (...) the Leibnizian-Wolffian tradition of deductive metaphysics. He joins his predecessors in rejecting the assumption that the law of contradiction alone can provide proof of the principle of sufficient reason: -/- In his rejection of the possibility of deducing all philosophic truth from the law of contradiction, however, and in the clear recognition that this impossibility has immediate consequences for defense of the law of sufficient reason, Kant's work most definitely and positively constitutes a line of succession from Hoffmann and Crusius (Treash, 1983, p. 25). -/- The recognition that Kant's Negative Quantities essay is part of a response to the tradition of deductive metaphysics is, without a doubt, an important contribution to the Kant literature. However, there is still more to be said about this neglected essay. The full significance of the paper becomes known through its ties to a second, empiricist line of succession. Clues to this second line of succession can be found in Kant's prefatory remarks concerning Euler's 1748 Reflections on Space and Time and Crusius' 1749 Guidance in the Orderly and Careful Consideration of Natural Events. As I will show, these prefatory remarks suggest a reading of Kant's Negative Quantities paper that reaches beyond German deductive metaphysics to engage a debate regarding the application of mathematics in philosophy initiated by George Berkeley. (shrink)

Since his Metaphysische Anfangsgründe der Naturwissenschaft was first published in 1786, controversy has surrounded Immanuel Kant’s conception of matter. In particular, the justification for both his dynamical theory of matter and the related dismissal of mechanical philosophy are obscure. In this paper, I address these longstanding issues and establish that Kant’s dynamism rests upon Leibnizian, metaphysical commitments held by Kant from his early pre-Critical texts on natural philosophy to his major critical works. I demonstrate that, throughout his corpus and inspired (...) by Leibniz, Kant endorses the a priori law of continuity of alteration as a truth of metaphysics, according to which all alterations in experience must occur gradually through all intervening degrees. The principle thus legislates against mechanical philosophy’s absolutely impenetrable atoms, as they would would involve instantaneous changes of velocity in impact. This reveals the metaphysical incoherencies in mechanical philosophy and leaves Kant’s own dynamical theory of matter, grounded on material forces, as the only viable approach to physical explanation. Subsequently, I demonstrate that Kant nevertheless made conceptual space in his system for the theoretical consideration of mechanical explanations, which makes manifest one of the positive roles that the faculty of reason can play with respect to natural science. (shrink)

Kant’s geographical theory, which was informed by contemporary travel reports, diaries, and journals, developed before his so-called “critical turn.” There are several reasons to study Kant’s lectures and material on geography. The geography provided Kant with terms, concepts, and metaphors which he employed in order to present or elucidate the critical philosophy. Some of the germs of what would become Kant’s critical philosophy can already be detected in the geography course. Finally, Kant’s geography is also one source of some of (...) the empirical claims in his philosophical works, including the Critique of the Power of Judgment. To give an example of this, I examine his account of the sublime. (shrink)

I argue that the key dynamical concepts and laws of Newton's Principia never gained a solid foothold in Germany before Kant in the 1750s. I explain this absence as due to Leibniz. Thus I make a case for a robust Leibnizian legacy for Enlightenment science, and I solve what Jonathan Israel called “a meaningful historical problem on its own,” viz. the slow and hesitant reception of Newton in pre-Kantian Germany.

Maimon’s theory of the differential has proved to be a rather enigmatic aspect of his philosophy. By drawing upon mathematical developments that had occurred earlier in the century and that, by virtue of the arguments presented in the Essay and comments elsewhere in his writing, I suggest Maimon would have been aware of, what I propose to offer in this paper is a study of the differential and the role that it plays in the Essay on Transcendental Philosophy (1790). In (...) order to do so, this paper focuses upon Maimon’s criticism of the role played by mathematics in Kant’s philosophy, to which Maimon offers a Leibnizian solution based on the infinitesimal calculus. The main difficulties that Maimon has with Kant’s system, the second of which will be the focus of this paper, include the presumption of the existence of synthetic a priori judgments, i.e. the question quid facti, and the question of whether the fact of our use of a priori concepts in experience is justified, i.e. the question quid juris. Maimon deploys mathematics, specifically arithmetic, against Kant to show how it is possible to understand objects as having been constituted by the very relations between them, and he proposes an alternative solution to the question quid juris, which relies on the concept of the differential. However, despite these arguments, Maimon remains sceptical with respect to the question quid facti. (shrink)

One of Bolzano’s objections to Kant’s way of drawing the analytic-synthetic distinction is that it only applies to judgments within a narrow range of syntactic forms, namely, universal affirmative judgments. According to Bolzano, Kant cannot account for judgments of other syntactic forms that, intuitively, are analytic. A recent paper by Ian Proops also attributes to Kant the view that analytic judgments beyond a limited range of syntactic forms are impossible. I argue that, correctly understood, Kant’s conception of analyticity allows for (...) analytic judgments of a wider range of syntactic forms. (shrink)

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, namely what (...) is known as geometric logic. (shrink)

In the Critique of Pure Reason, Kant defends the mathematically deterministic world of physics by arguing that its essential features arise necessarily from innate forms of intuition and rules of understanding through combinatory acts of imagination. Knowing is active: it constructs the unity of nature by combining appearances in certain mandatory ways. What is mandated is that sensible awareness provide objects that conform to the structure of ostensive judgment: “This (S) is P.” -/- Sensibility alone provides no such objects, so (...) the imagination compensates by combining passing point-data into “pure” referents for the subject-position, predicate-position, and copula. The result is a cognitive encounter with a generic physical object whose characteristics—magnitude, substance, property, quality, and causality—are abstracted as the Kantian categories. Each characteristic is a product of “sensible synthesis” that has been “determined” by a “function of unity” in judgment. -/- Understanding the possibility of such determination by judgment is the chief difficulty for any rehabilitative reconstruction of Kant’s theory. I will show that Kant conceives of figurative synthesis as an act of line-drawing, and of the functions of unity as rules for attending to this act. The subject-position refers to substance, identified as the objective time-continuum; the predicate-position, to quality, identified as the continuum of property values (constituting the second-order type named by the predicate concept). The upshot is that both positions refer to continuous magnitudes, related so that one (time-value) is the condition of the other (property-value). -/- Kant’s theory of physically constructive grammar is thus equivalent to the analytic-geometric formalism at work in the practice of mathematical physics, which schematizes time and state as lines related by an algebraic formula. Kant theorizes the subject–predicate relation in ostensive judgment as an algebraic time–state function. When aimed towards sensibility, “S is P” functions as the algebraic relation “t → ƒ(t).”. (shrink)

There are at least five ‘core’ notions of community found in Kant's works: 1. The scientific notion of interaction. This concept is introduced in the Third Analogy and developed in the Metaphysical Foundations of Natural Science. 2. A metaphysical idea. The idea of a world of individuals (monads) in interaction. This idea was developed in Kant’s precritical period and can be found in his metaphysics lectures. 3. A moral ideal. The idea of a realm of ends. 4. A political ideal. (...) The idea of a juridical community (or community of communities) governed by juridical laws. 5. A theological ideal. What Kant calls ‘the kingdom of heaven’, and which can be thought of as a community of holy beings, or angels. In this paper I focus on the relationship between the first, second and fourth of these notions. My argument is that Kant’s notion of a juridical community governed by juridical laws is modelled on the metaphysical idea of the world. This metaphysical idea of a world is, in turn, modelled on the category of community introduced in the first Critique and developed in his logic lectures. (shrink)

Recent perspectival interpretations of Kant suggest a way of relating his epistemology to empirical science that makes it plausible to regard Einstein’stheory of relativity as having a Kantian grounding. This first of two articles exploring this topic focuses on how the foregoing hypothesis accounts for variousresonances between Kant’s philosophy and Einstein’s science. The great attention young Einstein paid to Kant in his early intellectual development demonstrates the plausibility of this hypothesis, while certain features of Einstein’s cultural-political context account for his (...) reluctance to acknowledge Kant’s influence, even though contemporary philosophers who regarded themselves as Kantians urged him to do so. The sequel argues that this Kantian grounding probably had a formative influence not only on Einstein’s discovery of the theory of relativity and his view of the nature of science, but also on his quasi-mystical, religious disposition. (shrink)

As most of its readers are aware, the Critique of Pure Reason is primarily concerned not with empirical, but with a priori knowledge. For the most part, the Kant of the first Critique tends to assume that experience, and the knowledge that is based on it, is unproblematic. The problem with which he is concerned is that of how we can be capable of substantive knowledge independently of experience. At the same time, however, the notion of experience plays a crucial (...) role in the central arguments of the Critique. For, again as most readers of the Critique know, Kant aims to show how we can have synthetic a priori knowledge by showing that the categories, or pure concepts of the understanding, are conditions of the possibility of experience. This means that, whether or not Kant is concerned with the notion of experience for its own sake, his account of a priori knowledge carries with it at least some commitments regarding the character of experience. If the account of a priori knowledge is to be successful, then experience has to be the kind of thing for which the categories can, in principle, serve as conditions of possibility. More specifically, experience must involve not only the senses, but also thought or understanding, for otherwise the claim that it presupposes a certain specific set of concepts is simply unintelligible. And indeed at least some parts of the Critique, in particular the so-called subjective deduction in the first edition, and the briefer passages which correspond to it in the second edition, seem to be intended to show how this requirement is met. That is, they are concerned not so much with showing that experience is governed by the categories, as with elaborating a view of experience as involving conceptual activity überhaupt. (shrink)