In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition ; the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception. A circle drawn in geometry and the space occupied by an object such as a book are paradigm (...) examples of extensive magnitudes, while the intensity of a light is a paradigm example of an intensive magnitude. These principles justify and explain the possibility of applying mathematics to objects of experience. The Axioms principle also explains the possibility of any mathematical cognition at all. (shrink)

Kant's Critique of Pure Reason makes important claims about space, time and mathematics in both the Transcendental Aesthetic and the Axioms of Intuition, claims that appear to overlap in some ways and contradict in others. Various interpretations have been offered to resolve these tensions; I argue for an interpretation that accords the Axioms of Intuition a more important role in explaining mathematical cognition than it is usually given. Appreciation for this larger role reveals that magnitudes are central to Kant's philosophy (...) of mathematics and to the part that intuition plays in it. (shrink)

The aggregate EIRP of an N-element antenna array is proportional to N 2. This observation illustrates an effective approach for providing deep space networks with very powerful uplinks. The increased aggregate EIRP can be employed in a number of ways, including improved emergency communications, reaching farther into deep space, increased uplink data rates, and the flexibility of simultaneously providing more than one uplink beam with the array. Furthermore, potential for cost savings also exists since the array can be formed using (...) small apertures. (shrink)

In his argument for the possibility of knowledge of spatial objects, in the Transcendental Deduction of the B-version of the Critique of Pure Reason, Kant makes a crucial distinction between space as “form of intuition” and space as “formal intuition.” The traditional interpretation regards the distinction between the two notions as reflecting a distinction between indeterminate space and determinations of space by the understanding, respectively. By contrast, a recent influential reading has argued that the two notions can be fused into (...) one and that space as such is first generated by the understanding through an act of synthesis of the imagination. Against this reading, this article argues that a key characteristic of space as a form of intuition is its nonconceptual unity, which defines the properties of space and is as such necessarily independent of determination by the understanding through the transcendental synthesis of the imagination. The conceptual unity that the understanding prescribes to the manifold in intuition, by means of the categories, defines the formal intuition. Furthermore, this article argues that it is the sui generis, nonconceptual unity of space, when takenas a unity for the understanding by means of conceptual determination, that first enables geometric knowledge and knowledge of spatially located particulars. (shrink)

Following ideas elaborated by Hering in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (...) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated. (shrink)

In a series of papers published in the sixties and seventies, Jaakko Hintikka, drawing upon Kant’s conception, defines an argument to be analytic whenever it does not introduce new individuals into the discussion and argues that there exists a class of arguments in polyadic first-order logic that are to be synthetic according to this sense. His work has been utterly overlooked in the literature. In this paper, I claim that the value of Hintikka’s contribution has been obscured by his formalisation (...) of the original definition. Therefore, I provide (i) a brief reconstruction of the historical framework of the problem and the revolutionary import of Hintikka’s contribution, (ii) a clarification of the most complicated steps of Hintikka’s elaboration of his insight, (iii) a criticism of several features that play a fundamental role in Hintikka’s formalisation and (iv) a selection from Hintikka’s own material of some valuable suggestions towards a clear and workable formalisation. As for the pars construens, I isolate in the approach of depth-bounded first-order logics (D'Agostino et al. 2021) an alternative formalisation of the notion of syntheticity as the introduction of new individuals in the reasoning, and I show that it is not affected by the same difficulties as Hintikka’s proposal. In so doing, I hope to have contributed to the realisation of the project of rehabilitating Kant’s analytic–synthetic distinction in the context of modern first-order logic with the purpose of showing, against the logical empiricist movement, that logic is not analytic. (shrink)

The aim of this paper is to show that attention to Kant's philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he (...) calls the productive imagination. Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude. (shrink)

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; (...) and not only as they were practiced in Kant’s time, but also as architects use them today. (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)

Matthias Neuber’s book represents an important contribution to the relatively young discipline of the History of Philosophy of Science. Starting roughly in the 1980s, increasing attention has been devoted not only to the relationship between philosophy and the history of science, but to an accurate historical reconstruction of earlier projects within philosophy of science. One of the most outstanding results of these investigations has probably been the radical reshaping of the rather caricatural image of logical empiricism—for better or worse the (...) core of the philosophical heritage of many philosophers of science—summarized in the so-called ‘standard view’. By analyzing the historical, sociological, and philosophical questions surrounding logical empiricism new light has been shed on the sense of a cultural, social, and political mission that characterized it before its emigration from Europe to North America in the 1930s and 1940s. What came to be known as .. (shrink)

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant (...) takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a (...) taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

Hermann von Helmholtz (1821-1894) participated in two of the most significant developments in physics and in the philosophy of science in the 19th century: the proof that Euclidean geometry does not describe the only possible visualizable and physical space, and the shift from physics based on actions between particles at a distance to the field theory. Helmholtz achieved a staggering number of scientific results, including the formulation of energy conservation, the vortex equations for fluid dynamics, the notion of free energy (...) in thermodynamics, and the invention of the ophthalmoscope. His constant interest in the epistemology of science guarantees his enduring significance for philosophy. (shrink)

In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical inferences. By developing (...) such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (shrink)