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Visual Thinking in Mathematics: An Epistemological Study
, 2007
"... aimed to “prepare the scientific foundations for a future construction of that discipline. ” His goals should seem reasonable to contemporary philosophers of mathematics:... through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, ..."
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aimed to “prepare the scientific foundations for a future construction of that discipline. ” His goals should seem reasonable to contemporary philosophers of mathematics:... through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. [7, p. 5]2 But the ensuing strategy for grounding mathematical knowledge sounds strange to the modern ear. For Husserl cast his work as a sequence of “psychological and logical investigations, ” providing a psychological analysis... of the concepts multiplicity, unity, and number, insofar as they are given to use authentically and not through indirect symbolizations. (ibid., pp. 6–7) This emphasis on psychology is a reflection of Husserl’s training. As a teenager studying in Leipzig, he attended the lectures of Wilhelm Wundt, a seminal figure in the field of experimental psychology. Wundt held that, via introspection, we can study and classify our inner experiences, in much the same way that scientists study the natural world.3 People working in his laboratory were therefore trained in procedures for observing and reporting on their own thought processes, as a means of gathering scientific data regarding our cognitive faculties. Bridging the gap between psychology and epistemology, Wundt felt that the results of such inquiry could have normative consequences, since the principles of reasoning employed in the
The twofold role of diagrams in Euclid’s plane geometry
"... Proposition I.1 of Euclid’s Elements requires to “construct ” an equilateral triangle on a “given finite straight line”, or on a given segment, in modern parlance1. To achieve this, Euclid takes this segment to be AB (fig. 1), then describes two circles with its two extremities A and B as centres, a ..."
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Proposition I.1 of Euclid’s Elements requires to “construct ” an equilateral triangle on a “given finite straight line”, or on a given segment, in modern parlance1. To achieve this, Euclid takes this segment to be AB (fig. 1), then describes two circles with its two extremities A and B as centres, and takes for granted that these circles intersect each other in a point C distinct from A and B. This last step is not warranted by his explicit stipulations (definitions, postulates, common notions). Hence, either his argument is flawed, or it is warranted on other grounds. According to a classical view, “the Principle of Continuity ” provides such another ground, insofar as it ensures “the actual existence of points of intersection ” of lines ([7], I, ∗Some views expounded in the present paper have been previously presented in [30], whose first version was written in 1996, during a visiting professorship at the Universidad Nacional Autónoma de México. I thank all the people who supported me during my stay there. Several preliminary versions of the present paper have circulated in different forms and one of them is available online at
FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY
"... Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a firstorder the ..."
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Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a firstorder theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. Using Gentzen’s cutelimination theorem, we show that when ECG proves an existential theorem, then the things proved to exist can be constructed by Euclidean rulerandcompass constructions. In the second part of the paper we take up the formal relationships between three versions of Euclid’s parallel postulate: Euclid’s own formulation in his Postulate 5, Playfair’s 1795 version, which is the one usually used in modern axiomatizations, and the version used in ECG. We completely settle the questions about which versions imply which others using only constructive logic: ECG’s version implies Euclid 5, which implies Playfair, and none of the reverse implications are provable. The proofs use Kripke models based on carefully constructed rings of realvalued functions. “Points ” in these models are realvalued functions. We also characterize these theories in
Proof and Computation in Geometry
"... We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs directl ..."
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We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations. But this does not produce computercheckable firstorder proofs in geometry. We might try to produce such proofs directly, or we might try to develop a “backtranslation” from algebra to geometry, following Descartes but with computer in hand. This paper discusses the relations between the two approaches, the attempts that have been made, and the obstacles remaining. On the theoretical side we give a new firstorder theory of “vector geometry”, suitable for formalizing geometry and algebra and the relations between them. On the practical side we report on some experiments in automated deduction in these areas.
MSc in Logic
, 2012
"... Friedman [1, 2] claims that Kant’s constructive approach to geometry was developed as a means to circumvent the limitations of his logic, which has been widely regarded by various commentators as nothing more than a glossa to Aristotelian subjectpredicate logic. Contra Friedman, and building on the ..."
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Friedman [1, 2] claims that Kant’s constructive approach to geometry was developed as a means to circumvent the limitations of his logic, which has been widely regarded by various commentators as nothing more than a glossa to Aristotelian subjectpredicate logic. Contra Friedman, and building on the work of Achourioti and van Lambalgen [3], we purport to show that Kant’s constructivism draws its independent motivation from his general theory of cognition. We thus propose an exegesis of the Transcendental Deduction according to which the consciousness of space as a formal intuition of outer sense (with its properties of, e.g., infinity and continuity) is produced by means of the activity of the transcendental synthesis of the imagination in the construction of geometrical concepts, which synthesis must be in thoroughgoing agreement with the categories. In order to substantiate these claims, we provide an analysis of Kant’s characterization of geometrical inferences and of geometrical continuity, along with a formal argument illustrating how the representation of space as a continuum can be constructed from Kantian principles. Contents
SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Preface
, 2012
"... geometry [180] when I was preparing my dissertation on Euclid’s Elements and was focused on studying Greek mathematics and classical Greek philosophy. Then I convinced myself that the mathematical category theory is philosophically relevant not only because of its name but also because of its conten ..."
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geometry [180] when I was preparing my dissertation on Euclid’s Elements and was focused on studying Greek mathematics and classical Greek philosophy. Then I convinced myself that the mathematical category theory is philosophically relevant not only because of its name but also because of its content and because of its special role in the contemporary mathematics, which I privately compared to the role of the notion of figure in Euclid’s geometry. Today I have more to say about these matters. The broad historical and philosophical context, in which I studied category theory, is made explicit throughout the present book. My interest to the Axiomatic Method stems from my work on Euclid and extends through Hilbert and axiomatic set theories to Lawvere’s axiomatic topos theory to the Univalent Foundations of mathematics recently proposed by Vladimir Voevodsky. This explains what the two subjects appearing in the title of this book share in common. The next crucial biographical episode took place in 1999 when I was a young scholar visiting Columbia University on the Fulbright grant working on ontology of events under the supervision of Achille Varzi. As a part of my Fulbright program I had to make a presentation in a different American university, and I decided to use this opportunity for talking about the philosophical