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13
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
Mathematical method and proof
"... Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that ..."
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Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not wellequipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.
Strong Statements of Analysis
 Bulletin of the London Mathematical Society
"... Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong settheoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong settheoretical concepts. 1. ..."
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Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong settheoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong settheoretical concepts. 1.
How Modern Algebra was used in Economic Science in the 1950s: Breaking the Glass Wall to the Scientific Acceptance (General Equilibrium Theory (2): the Existence Question)
, 2009
"... This paper investigates how Japanese mathematical economists studied the questions relating to the existence of a general equilibrium and fixed point theorems (FPTs), which were keys to the proof, from the 1940s till the early 1960s. We focus on Hukukane Nikaido (19232001) and Hirofumi Uzawa (b.192 ..."
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This paper investigates how Japanese mathematical economists studied the questions relating to the existence of a general equilibrium and fixed point theorems (FPTs), which were keys to the proof, from the 1940s till the early 1960s. We focus on Hukukane Nikaido (19232001) and Hirofumi Uzawa (b.1928) and trace their direct connection with John von Neumann (19031957) and Kenneth J. Arrow (b.1921). Then we first reconstruct the process in which the cannon of modern neoclassical economics, namely Walrasian general equilibrium theory, was established through the use of modern algebra in the 1950s. Second, we show how the Japanese overcame the glass wall to the international community which had been established by the swift circulation of refereed economics journals like Econometrica. They owed much to von Neumann and active members of the Econometric Society including Arrow. 2 1. Japanese Economists and General Equilibrium Approach in the 194060s The proof of existence, stability and uniqueness are important topics for the study of general equilibrium theory. In the 1950s, the proof of the existence of a general equilibrium utilized topology and fixed point theorems (FPTs) or set theory and the convex set method,
Category Theory and Structuralism
, 2009
"... The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond ..."
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The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond Queneau) and in Mathematics (Nicolas Bourbaki). To the layman the structuralist movement in mathematics was perhaps most visible the form of New Math, which was strongly influenced by the Bourbaki school. It has been argued in (Aubin 1997) that there were cultural connections between these movements. (See also A. Aczel 2007.) Some of these interactions may be regarded as rather superficial. The epistemologist Piaget however was very much influenced by Bourbaki, and seems to have suggested that those patterns of thought used to explain cognitive development were closely related to the mathematical “mother structures ” found by Bourbaki. On a very general level, structuralism refers to a mode of thinking involving abstraction from specifics and systematic identification and naming of common patterns. It is the relation of objects under study to each other that is of importance rather than their specific appearance, or “nature”. In mathematics, Richard Dedekind may be said to be the first structuralist. He described the positive integers (1, 2, 3,...) as positions in an infinite progression of elements (a socalled simply infinite system) 1 � � 2
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... We begin with a short (obviously incomplete) historical summary related to the book being reviewed. A complex number α is said to be “algebraic ” if there is a polynomial 0 ̸ = p(x) with integer coefficients such that p(α) = 0; otherwise α is said to be “transcendental”. The first recorded instance ..."
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We begin with a short (obviously incomplete) historical summary related to the book being reviewed. A complex number α is said to be “algebraic ” if there is a polynomial 0 ̸ = p(x) with integer coefficients such that p(α) = 0; otherwise α is said to be “transcendental”. The first recorded instance of mathematicians encountering an “irrational ” algebraic number appears to be the ancient Greeks. Indeed, if one considers a square with unit sides, then, of course, by the Pythagorean Theorem a diagonal must have length √ 2. To Pythagoras (and his followers) is also due the result that √ 2 cannot be expressed as the quotient of integers. One story has it that Hippasus, the person who revealed this irrationality, died at sea in a shipwreck, “struck by the wrath of the gods.” As history evolved, so did mathematicians ’ views about numbers. During the sixteenth century cubic equations were discussed by the Italian algebraists G. Cardano and R. Bombelli ([v1], Part C §2). In particular a very curious phenomenon appeared with equations like x 3 =15x +4. Here one readily finds that x =4isa root and, after division by x−4, that the other two roots are also real. However, the
On Rereading van Heijenoort’s Selected Essays
"... Abstract. This is a critical reexamination of several pieces in van Heijenoort’s Selected Essays that are directly or indirectly concerned with the philosophy of logic or the relation of logic to natural language. Among the topics discussed are absolutism and relativism in logic, mass terms, the ide ..."
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Abstract. This is a critical reexamination of several pieces in van Heijenoort’s Selected Essays that are directly or indirectly concerned with the philosophy of logic or the relation of logic to natural language. Among the topics discussed are absolutism and relativism in logic, mass terms, the idea of a rational dictionary, and sense and identity of sense in Frege. Mathematics Subject Classification. 0303, 03A05.
Forthcoming in: Perspectives on Mathematical Practices, Vol. II, J.P. van Bendegem and B. van Kerkhove (eds.). Bridging Theories with Axioms:
, 2008
"... In discussions of mathematical practice the role axiomatics has often been confined to providing the starting points for formal proofs, with little or no effect on the discovery or creation of new mathematics. For example, quite recently Patras wrote that the axiomatic method “never allows for authe ..."
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In discussions of mathematical practice the role axiomatics has often been confined to providing the starting points for formal proofs, with little or no effect on the discovery or creation of new mathematics. For example, quite recently Patras wrote that the axiomatic method “never allows for authentic creation ” (Patras 2001, 159), and similar
Dual Aspects of Abduction and Induction
"... Abstract. A new characterization of abduction and induction is proposed, which is based on the idea that the various aspects of the two kinds of inference rest on the essential features of increment of (so to speak, extensionalized) comprehension and, respectively, of extension of the terms involved ..."
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Abstract. A new characterization of abduction and induction is proposed, which is based on the idea that the various aspects of the two kinds of inference rest on the essential features of increment of (so to speak, extensionalized) comprehension and, respectively, of extension of the terms involved. These two essential features are in a reciprocal relation of duality, whence the highlighting of the dual aspects of abduction and induction. Remarkably, the increment of comprehension and of extension are dual ways to realize, in the limit, a `deductivization ' of abduction and induction in a similar way as the Closed World Assumption does in the case of the latter.