One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but (...) instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant. (shrink)

In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls faculty of intuition in his dissertation is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift it is through spatial intuition (...) that we come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry. (shrink)

A review of Roberto Torretti's book, "Philosophy of Geometry from Riemann to Poincare.".

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)

In my dissertation, I argue that contemporary interpretive work on Kant's philosophy of geometry has failed to understand properly the diagrammatic aspects of Euclidean reasoning. Attention to these aspects is amply repaid, not only because it provides substantial insight into the role of intuition in Kant's philosophy of mathematics, but also because it brings out both the force and the limitations of Kant's philosophical account of geometry. ;Kant characterizes the predecessors with which he was engaged as (...) agreeing that mathematical judgments are analytic a priori. This is an inadequate account of geometrical judgments for Kant, because it cannot be reconciled with the evident success of the Euclidean practice in producing a priori knowledge of the nature of space. The ubiquitous presence of diagrams in Euclidean proofs suggests that, at least in Kant's time, any plausible account of the establishment of geometrical judgments must provide for the role of quasi-perceptual representations of individuals. Kant's claim that mathematical judgments are synthetic a priori reflects, in its positive aspect, his recognition of the importance of these 'intuitive' representations to Euclidean reasoning. Furthermore, Kant's doctrine that geometrical knowledge is established by the construction of concepts in intuition can be usefully understood as an attempt to accommodate the role of diagrams in the establishment of synthetic a priori geometrical judgments. ;Both Kant's rejection of the analytic a priori account of mathematics and his positive account of geometry depend on the plausibility of his appeals to Euclidean practice. Attention to the role of diagrams in Euclidean proof supports Kant's contention that intuitive representations are an essential component of Euclidean reasoning; however, the role of diagrams in both reductio proofs and in proofs by cases brings out tensions in Kant's positive account. The use of diagrams to represent impossible geometrical situations in reductio proofs indicates that Kant's notion of 'construction in intuition' can have a broader application than the construction of instances of geometrical concepts . Similarly, proofs by cases undermine Kant's attempts to explain the generality of judgments established through the consideration of individuals by appeal to the rules for the construction of those individuals. ;Contemporary interpretive debates about Kant's philosophy of geometry have centered on the philosophical role played by his appeal to intuition. This appeal cannot be primarily understood as an attempt to establish that the fundamental assumptions of Euclidean geometry are necessary truths about space. Nor can the role for intuition be regarded as merely licensing moves in geometrical proofs like an inference rule in modern logic. Instead, by understanding Kant's appeal to intuition as, at least in part, an attempt to accommodate the diagrammatic features of Euclidean reasoning, a much richer role for intuition is revealed. Intuition is required to play a logical role by expressing spatial relationships and establishing the existence of geometrical entities. In addition, it plays a quasi-perceptual role by supporting the truth of certain claims appealed to during the course of Euclidean proof. (shrink)

This deeply researched, carefully constructed and very thoughtful book is fascinating in its own right as well as being indispensable background material for anyone interested in current philosophical thought about space, time, and geometry.

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential (...) compass-and-straightedge drawings qua quantitative and continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it (...) is through imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

This paper is an exposition and defense of Kant’s philosophy of geometry. The main thesis is that Euclidean geometry investigates the properties of geometrical objects in an inner space that is given to us a priori (pure space) and hence is a priori and synthetic. This thesis is supported by arguing that Euclidean geometry requires certain intuitive objects (Sect. 1), that these objects are a priori constructions in pure space (Sect. 2), and finally that the role (...) of geometrical construction is to provide geometrical objects, not concepts, as some have claimed (Sect. 3). (shrink)

Graduate Papers from the 2022 Joint Session. It is often said that Aristotle takes geometrical objects to be absolutely unmovable and unchangeable. However, Greek geometrical practice does appeal to motion and change, and geometers seem to consider their objects apt to be manipulated. In this paper, I examine if and how Aristotle’s philosophy of geometry can account for the geometers’ practices and way of talking. First, I illustrate three different ways in which Greek geometry appeals to change. (...) Second, I examine Aristotle’s ontology of geometrical objects and argue that although the truth-makers of geometrical statements are in fact unmovable because they are properties of sensible objects, geometers ‘separate them in thought’ and treat them as substances apt to be modified. Finally, I examine whether allowing for the possibility of manipulating these semi-fictional geometrical individuals creates problems for the applicability of geometry. I find that it does not, insofar as one accepts that geometry is not meant to track physical change but merely to study the instantaneous geometrical configuration of sensible bodies, and is thus only applicable at the instant. (shrink)

Kant's philosophy of geometry rests upon his doctrine of the "schematism" which I argue is formally identical to the ability to grass the middle term of an Aristotelian syllogism. The doctrine fails to avoid obscurities which were already present in Plato, Aristotle, and Hume.

In the preface to the Principia (1687) Newton famously states that “geometry is founded on mechanical practice.” Several commentators have taken this and similar remarks as an indication that Newton was firmly situated in the constructivist tradition of geometry that was prevalent in the seventeenth century. By drawing on a selection of Newton's unpublished texts, I hope to show the faults of such an interpretation. In these texts, Newton not only rejects the constructivism that took its birth in (...) Descartes's Géométrie (1637); he also presents the science of geometry as being more powerful than his Principia remarks may lead us to believe. (shrink)

The notion of the "construction" or "exhibition" of a concept in intuition is central to Kant's philosophical account of geometry. Kant invokes this notion in all of his major Critical Era discussions of mathematics. The most extended discussion of mathematics, and geometry more specifically, occurs in "The Discipline of Pure Reason in its Dogmatic Employment." In this later section of the Critique, Kant makes it clear that construction-in-intuition is central to his philosophy of mathematics by presenting it (...) as the defining characteristic of mathematical knowledge. He claims: [M]athematical knowledge is the knowledge gained by reason from the construction of concepts. To construct a concept means to exhibit a... (shrink)

Hans Reichenbach's so-called geometrical conventionalism is often taken as an example of a positivistic philosophy of science, based on a verificationist theory of meaning. By contrast, we shall argue that this view rests on a misinterpretation of Reichenbach's major work in this area, the Philosophy of Space and Time (1928). The conception of equivalent descriptions, which lies at the heart of Reichenbach's conventionalism, should be seen as an attempt to refute Poincaré's geometrical relativism. Based upon an examination of (...) the reasons Reichenbach gives for the cognitive equivalence of geometrical descriptions, the paper argues that his conventionalism is a specific form of scientific realism. At the same time we shall argue against those interpretations which lead to a trivialization of Reichenbach's conventionalism or deny it entirely. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and Poincaré), and discussing conventionalist and neopositivist perspectives and the problem of the origins of (...) geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field. (shrink)

The total irrelevance of absolute space to scientific observation and experiment led him early to a most radical conclusion: experience cannot teach us anything about the true structure of space; consequently, the choice of a geometry for the ...

The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of (...) class='Hi'>geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. (shrink)

The critique of my protophysical approaches to operational foundation of geometry by Lucas Amiras (Journal for General Philosophy of Science Vol. 34 (2003)) concerns my first publication from 1976 but not the further 30 years of work. It does not offer any argument leading from the (erroneous) judgement “lacking success” to the conclusion “impossible”. And it is, in general, based on a philosophical defect: it ignores the principle of methodical order as leading for constructivist protophysics.

The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These (...) figures are combined in the emblematic statement of Salomon Maimon, "In mathematical construction we are, as it were, gods." Construction is the mark of modernity. The disciplines of classical philology, literary interpretation and the history of philosophy and of mathematics are woven together in this volume. (shrink)