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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
"... Abstract. 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 first ..."
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Abstract. 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
"... Abstract. 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 ..."
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Abstract. 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
A FORMALISATION OF KANT’S TRANSCENDENTAL LOGIC
"... ABSTRACT. Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason [12], logicians and philosophers have generally judged Kant’s logic negatively. What Kant called ‘general ’ or ‘formal ’ logic has been dismissed as a fairly arbitrary subsystem ..."
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ABSTRACT. Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason [12], logicians and philosophers have generally judged Kant’s logic negatively. What Kant called ‘general ’ or ‘formal ’ logic has been dismissed as a fairly arbitrary subsystem of first order logic, and what he called ‘transcendental logic ’ is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant’s ‘transcendental logic ’ is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first order logic. The main technical application of the formalism developed here is a formal proof that Kant’s Table of Judgements in §9 of the Critique of pure reason, is indeed, as Kant claimed, complete for the kind of semantics he had in mind. This result implies that Kant’s ’general ’ logic is after all a distinguished subsystem of first order logic, namely what is known as geometric logic. 1.
Understanding, formal verification, and the philosophy of mathematics
, 2010
"... The philosophy of mathematics has long been concerned with determining the means that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader ra ..."
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The philosophy of mathematics has long been concerned with determining the means that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary mathematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The associated values are often loosely classified as aspects of “mathematical understanding.” Meanwhile, in a branch of computer science known as “formal verification,” the practice of interactive theorem proving has given rise to software tools and systems designed to support the development of complex formal axiomatic proofs. Such efforts require one to develop models of mathematical language and inference that are more robust than the the simple foundational models of the last century. This essay explores some of the insights that emerge from this work, and some of the ways that these insights can inform, and be informed by, philosophical theories of mathematical understanding.