Search

This paper re-constructs Plato's 'philosophy of geometry' by arguing that he uses a geometrical method of hypothesis in his account of the cosmos' generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between (metaxu) Forms and sensible particulars (Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions of mathematical ontology. My key passage of interest is Timaeus' account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth (Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies' origin by hypothesising two right-angled triangles as their starting-point (arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world. [ABSTRACT FROM AUTHOR]

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas's cube duplication was originally a single-motion machine; Diocles's cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. [ABSTRACT FROM AUTHOR]

In Albert the Great's 13th century, a larger circulation of the complete Arabic-Latin translations of Euclid's Elements made by Adelard of Bath, Robert Chester and John of Tinemue begins and joins Gerald of Cremona's translation of Al-Nayrizi's Commentary on Euclid's Elements. The aim of this paper is to present how Albert the Great deals with the combination of these two traditions, i.e., the Arabic-Latin Euclid's translations, and the Latin medieval geometrical practice. [ABSTRACT FROM AUTHOR]

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space-time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: (1) by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and strong and weak nuclear forces; (2) by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space-time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: (1) the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; (2) the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space-time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. [ABSTRACT FROM AUTHOR]

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 facultyof intuition in his dissertation (1873) 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 (1874) 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 (provable from) 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 (alleged) 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. [ABSTRACT FROM AUTHOR]

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (1) it supports the thesis that Euclidean geometry is a priori, (2) it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, (3) it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. [ABSTRACT FROM AUTHOR]

This paper is intended as a contribution to the longstanding debate1 about how to understand Kant's appeal to pure intuition in his philosophical account of geometry. It proceeds by focusing on a notion that has been at the center of this debate, namely, the "construction" or "exhibition" of a concept in intuition. I claim that not only does Kant appeal to this notion in two different ways--in support of both an epistemological and a semantic thesis, each based in his broader philosophical project--but also that the demands placed on this notion by these appeals are in conflict. The tension between the epistemological and semantic appeals to construction-in-intuition can be clearly seen in geometrical proofs that proceed by reductio ad absurdum2. I conclude by considering the prospects for resolving this tension in a manner consistent with Kant's broader philosophical ambitions. [ABSTRACT FROM AUTHOR]

As in standard knowledge bases, hybrid knowledge bases (i.e. sets of information specified by hybrid formulas) may contain inconsistencies arising from different sources, namely from the many mechanisms used to collect relevant information. Being a fact, rather than a queer anomaly, inconsistency also needs to be addressed in the context of hybrid logic applications. This article introduces a paraconsistent version of hybrid logic which is able to accommodate inconsistencies at local points without implying global failure. A main feature of the resulting logic, crucial to our approach, is the fact that every hybrid formula has an equivalent formula in negation normal form. The article also provides a measure to quantify the inconsistency of a hybrid knowledge base, useful as a possible basis for comparing knowledge bases. Finally, the concepts of extrinsic and intrinsic inconsistency of a theory are discussed. [ABSTRACT FROM AUTHOR]

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 tomethodological 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. [ABSTRACT FROM AUTHOR]

The article explores the relationship between the philosopher and historian of mathematics Jacob Klein’s account of the transformation of the concept of number coincident with the invention of algebra , together with Husserl’s early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein’s research is shown to present the historical context for Husserl’s twin failures in thePhilosophy of Arithmetic: to provide a psychological foundation for the proper concept of number (Anzahl), and to show how this concept of number functions as the mathematical foundation of universal (symbolic) arithmetic. This context establishes that Husserl’s failures are ultimately rooted in the historical transformation of number documented in Klein’s research, from its premodern meaning as the unity of a multitude of determinate objects to its modern meaning as a symbolic representation with no immediate relation to a concrete multiplicity. The argument is advanced that one significant result of bringing together Klein’s and Husserl’s thought on these issues is the need to fine-tune Husserl’s project inThe Crisis of European Sciences and Transcendental Phenomenologyof de-sedimenting the mathematization of nature. Specifically, Klein’s research shows that “a ‘sedimented’ understanding of numbers” “is superposed upon the first stratum of ‘sedimented’ geometrical ‘evidences’” uncovered by Husserl’s fragmentary analyses of geometry in theCrisis.In addition then to the task of “the intentional-historical reactivation of the origin of geometry,” recognized by Husserl as intrinsic to the reactivation of the origin of mathematical physics, Klein discloses a second task, that of “the reactivation” of the “complicated network of sedimented significances” that “underlies the ‘arithmetical’ understanding of geometry.” [ABSTRACT FROM AUTHOR]

This article introduces Harvey Brown and Oliver Pooley's 'dynamical approach' to special relativity, and argues that it may be construed as a relationalist form of Einstein's 'practical geometry'. This construal of the dynamical approach is shown to be compatible with related chapters of Brown's text and also with recent descriptions of the dynamical approach by Pooley and others. [ABSTRACT FROM AUTHOR]

The article explores the philosophy of mathematics and a preface by mathematician Federico Commandino to a translation of the "Elements" by philosopher Euclid. The author reflects on the role and influence of a geometrical imagination as developed by the Christian commentator John Philoponus. Emphasis is given to topics such as the Neoplatonic idea of mathematical objects as concepts of the soul, Aristotelian abstractionism, and the hermeneutics of the imagination.

I offer in this paper a contextual analysis of Frege'sGrundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools (including a textbook Frege had used when teaching in a Privatschule in Jena in 1882–1884). In addition, I put Frege's considerations in the context of a widespread debate in Germany on ‘directions’ as a central notion in the theory of parallels. [ABSTRACT FROM PUBLISHER]

This paper examines the second geometrical problem in the Meno. Its purpose is to explore the implication of Cook Wilson's interpretation, which has been most widely accepted by scholars, in relation to the nature of hypothesis. I argue that (a) the geometrical hypothesis in question is a tentative answer to a more basic problem, which could not be solved by available methods at that time, and that (b) despite the temporary nature of a hypothesis, there is a rational process for formulating it. The paper also contains discussion of the method of analysis, problem reduction and a diorism, which have often been ambiguously explained in relation to the geometrical problem in question. [ABSTRACT FROM AUTHOR]

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions. [ABSTRACT FROM PUBLISHER]

The article discusses the German philosopher Gottfried Wilhelm Leibniz's project in developing an encyclopedia and his perspective on the accumulation of knowledge. An overview of Leibniz's philosophy on the natural order of thinking, including in regard to mathematics, geometry and numerical progression, is provided. An overview of Leibniz's perspective on the role that his encyclopedia would play in learning and education is also provided.

The paper reports and analyzes the vicissitudes around Hilbert's inclusion of his famous axiom of completeness, into his axiomatic system for Euclidean geometry. This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894-1905. It is argued that this historical and conceptual analysis not only sheds new light on how Hilbert conceived originally the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings regarding this axiom and the metalogical property of completeness of an axiomatic system, as it was understood by Hilbert in this initial stage. [ABSTRACT FROM AUTHOR]

The article explores the nature of the entities which are the objects in the second division of the divided line in Plato's "Republic." The author argues that the objects of dianoia are in fact the mathematical objects as attributed to Plato by Aristotle in the "Metaphysics." A solution to the problem of interpreting the divided line simile is presented.

Copyright of Dialogue: Canadian Philosophical Review is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper outlines an interpretation of David Hilbert's early axiomatic approach to geometry, i.e., the one developed between 1891 and 1905. It is argued that several aspects of this approach, which could be initially seen as problematic, can be better understood when Hilbert's view is contrasted with one of its most important influences in this period: Heinrich Hertz's “picture theory" [Bildtheorie]. Especially, it will be claimed that an examination of Hilbert's axiomatic conception in the light of the Hertzian view of mechanical theories can be used to relieve the former from certain manifest tensions; more precisely, the tensions that arise when maintaining, at the same time, an abstract or formal axiomatic position and an empiricist view of geometry, which regards it as a kind of natural science. [ABSTRACT FROM AUTHOR]

In §155 of his New Theory of Vision Berkeley explains that a hypothetical ‘unbodied spirit’ ‘cannot comprehend the manner wherein geometers describe a right line or circle’.1The reason for this, Berkeley continues, is that ‘the rule and compass with their use being things of which it is impossible he should have any notion.’ This reference to geometrical tools has led virtually all commentators to conclude that at least one reason why the unbodied spirit cannot have knowledge of plane geometry is because it cannot manipulate a ruler or a compass. In this article I will show that such an interpretation is flawed. I will instead argue that Berkeley's understanding of Euclidian geometry was based on Isaac Barrow's account of the foundations of geometry. On this view geometrical objects are conceived in terms of the idealized motion that generates the objects of geometry. Consequently, that what the unbodied spirit cannot do in this context is to form an idea of motion rather than being unable to handle geometrical tools. 1All references to Berkeley are from, A. A. Luce and T. E. Jessop (eds.), The Works of George Berkeley, Bishop of Cloyne (London: Thomas Nelson and Sons, Ltd., 1948) The following abbreviations are used: An Essay Towards A New Theory of Vision, section x = New Theory x; Philosophical Commentaries, entry x = Commentaries x; Part I of A Treatise concerning the Principles of Knowledge, section x = Principles x. All other references to Berkeley's works are of the form The Works of George Berkeley, Bishop of Cloyne, volume x, page y = Works, x, y. [ABSTRACT FROM PUBLISHER]

Copyright of Matériaux et Techniques is the property of EDP Sciences and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Not simply set out in accompaniment of the Greek geometrical text, the diagram also is coaxed into existence manually (using straightedge and compasses) by commands in the text. The marks that a diligent reader thus sequentially produces typically sum, however, to a figure more complex than the provided one and also not (as it is) artful for being synoptically instructive. To provide a figure artfully is to balance multiple desiderata, interlocking the timelessness of insight with the temporality of construction. Our account of the diagram complements those of Manders and Macbeth by more strongly emphasizing practical synthesis. [ABSTRACT FROM PUBLISHER]

Paolo Parrini discusses here the links between science and philosophy from an epistemological viewpoint, addressing the problem by distinguishing between its historical and theoretical dimensions. He discusses the necessary link between formal epistemology and history, and in this frame identifies different areas of intersection between science and philosophy. [ABSTRACT FROM AUTHOR]

Kant's theory of space includes the idea that straight lines and planes can be defined in Euclidean geometry by a concept which nowadays has been revived in the field of fractal geometry: the concept of self-similarity. Absolute self-similarity of straight lines and planes distinguishes Euclidean space from any other geometrical space. Einstein missed this fact in his attempt to refute Kant's theory of space in his article ‘Geometrie und Erfahrung’. Following Hilbert and Schlick he took it for granted, mistakenly, that purely geometrical definitions of the concepts of straight line and plane are not feasible, except as “implicit definitions”. Therefore Einstein's criticism is not persuasive. [ABSTRACT FROM AUTHOR]

The article offers the author's insights on the geometry of visibles of Thomas Reid. The author explores the separation of the real basis of geometry of visibles from several irrelevant and fallacious considerations. The author presents a list of twelve propositions with regards to tangible figures including of every right-lined triangle, the three angles taken together, are greater than two right angles.

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. [ABSTRACT FROM AUTHOR]

The article discusses several aspects of logic, time, and geometry. Topics discussed include the nature of temporality, and the notion of simultaneously deforming and modeling time (time of the logos). In addition, reference is made to the operational time, experience of life, and the notions of wisdom.