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 (...) class='Hi'>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)
On an ordinary view of the relation of philosophy of science to science, science serves only as a topic for philosophical reflection, reflection that proceeds by its own methods and according to its own standards. This ordinary view suggests a way of writing a global history of philosophy of science that finds substantially the same philosophical projects being pursued across widely divergent scientific eras. While not denying that this view is of some use regarding certain themes of and (...) particular time periods, this essay argues that much of the epistemology and philosophy of science in the early twentieth century in a variety of projects (neo-Kantianism, logical empiricism, pragmatism, phenomenology) looked to the then current context of the exact sciences, especially geometry and physics, not merely for its topics but also for its conceptual resources and technical tools. This suggests a more variable project of philosophy of science, a deeper connection between early twentieth-century philosophy of science and its contemporary science, and a more interesting and richer history of philosophy of science than is ordinarily offered. (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.
Peg Rawes examines a "minor tradition" of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject she explores a trajectory of geometric thinking and geometric figurations--reflective subjects, folds, passages, plenums, envelopes and horizons--in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodies subjectivities are constructed. Six chapters, explore the construction of these aesthetic geometric methods and figures in a series of "geometric" texts by Kant, Plato, Proclus, Spinoza, Leibniz, Bergson, Husserl and Deleuze. (...) In each text, geometry is expressed as a uniquely embodies aesthetic activity because each respective geometric method and figure is imbued with aesthetic sensibility and geometric sense (rather than as disembodies scientific methods). An ontology of aesthetic geometric methods and figures is therefore traced from Kant’s Critical writings, back to Plato and Proclus Greek philosophy, Spinoza and Leibniz’s post-Cartesian philosophies, and forwards to Bergson’s "duration" and Husserl’s "horizons" towards Deleuze’s philosophy of sense. (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)
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)
This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and (...) class='Hi'>geometry and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)
Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry (...) as an inexact non-demonstrative science is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)
This work examines the unique way in which Benedict de Spinoza (1632-77) combines two significant philosophical principles: that real existence requires causal power and that geometrical objects display exceptionally clearly how things have properties in virtue of their essences. Valtteri Viljanen argues that underlying Spinoza's psychology and ethics is a compelling metaphysical theory according to which each and every genuine thing is an entity of power endowed with an internal structure akin to that of geometrical objects. This allows Spinoza to (...) offer a theory of existence and of action - human and non-human alike - as dynamic striving that takes place with the same kind of necessity and intelligibility that pertain to geometry. Viljanen's fresh and original study will interest a wide range of readers in Spinoza studies and early modern philosophy more generally. (shrink)
High-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether, covers Einstein’s electrodynamics of moving bodies, Minkowski geometry and other topics. A rich exposition of the elements of the Special and General Theory of Relativity.
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)
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)
In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to (...) the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above. (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)
Hobbes' geometrical disputes are significant since they highlight several important strands in his thought - issues concerning the right to make definitions, his anti-clericalism, the maker's knowledge argument and his objections to algebra. These are examined, and the foundational position, according to Hobbes, of geomentry in relation to philosophy, science and technology, explained and discussed.
Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...) mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)
In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology -after Merleau-Ponty, Heidegger and Rosen- and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further (...) related to the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Mus`es which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classicalquantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succintly described above. (shrink)
Based on the author's Introduction to Philosophy lectures in Hong Kong, this book has been translated into Chinese and Indonesian and has sold over 10,000 copies. Unlike a typical textbook, the author punctuates his objective descriptions of the classical philosophical theories in metaphysics, logic, applied philosophy and ontology, with highly personal examples of how philosophical reflection can stimulate insights. Like a typical textbook, every chapter ends with a list of questions for further thought and a list of recommended (...) further readings. The first of twelve chapters is shown here as an example. (shrink)
In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept (...) crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs. (shrink)
The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (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.
Two seemingly contradictory tendencies have accompanied the development of the natural sciences in the past 150 years. On the one hand, the natural sciences have been instrumental in effecting a thoroughgoing transformation of social structures and have made a permanent impact on the conceptual world of human beings. This histori¬cal period has, on the other hand, also brought to light the merely hypothetical validity of scientific knowledge. As late as the middle of the 19th century the truth-pathos in the natural (...) sciences was still unbroken. Yet in the succeeding years these claims to certain knowledge underwent a fundamental crisis. For scientists today, of course, the fact that their knowledge can possess only relative validity is a matter of self-evidence. The present analysis investigates the early phase of this fundamental change in the concept of science through an examination of Hermann von Helmholtz's conception of science and his mechanistic interpretation of nature. Helmholtz (1821-1894) was one of the most important natural scientists in Germany. The development of this thoughts offers an impressive but, until now, relatively little considered report from the field of the experimental sciences chronicling the erosion of certainty. (shrink)
Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, (...) from the central position it used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)
In the Introduction to the Treatise Hume very enthusiastically announces his project to provide a secure and solid foundation for the sciences by grounding them on his science of man. And Hume indicates in the Abstract that he carries out this project in the Treatise. But most interpreters do not believe that Hume's project comes to fruition. In this paper, I offer a general reading of what I call Hume's ‘foundational project’ in the Treatise, but I focus especially on Book (...) 1. I argue that in Book 1 much of Hume's logic is put in the service of the other sciences, in particular, mathematics and natural philosophy. I concentrate on Hume's negative thesis that many of the ideas central to the sciences are ideas that we cannot form. For Hume, this negative thesis has implications for the sciences, as many of the texts I discuss make evident. I consider and criticize different proposals for understanding these implications: the Criterion of Meaning and the ‘Inconceivability Principle’. I introduce what I call Hume's ‘No Reason to Believe’ Principle, which I argue captures more adequately the link Hume envisions between his logic, in particular his examination of ideas, and the other sciences. (shrink)
Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay oN phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work (...) of Quine, Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincare; and Frege. (shrink)
I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics , I explain how such objects exist in the sensibles in a special way. I conclude (...) by considering the passages that lead Jonathan Lear to his fictionalist reading of Met . M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk. (shrink)