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Reason and intuition
 Synthese
, 2000
"... In this paper I will approach the subject of intuition from a different angle from what has been usual in the philosophy of mathematics, by beginning with some descriptive remarks about Reason and observing that something that has been called intuition arises naturally in that context. These conside ..."
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In this paper I will approach the subject of intuition from a different angle from what has been usual in the philosophy of mathematics, by beginning with some descriptive remarks about Reason and observing that something that has been called intuition arises naturally in that context. These considerations are quite general, not specific to mathematics. The conception of intuition might be called that of rational intuition; indeed the conception is a much more modest version of conceptions of intuition held by rationalist philosophers. Moreover, it answers to a quite widespread use of the word “intuition ” in philosophy and elsewhere. But it does not obviously satisfy conditions associated with other conceptions of intuition that have been applied to mathematics. Intuition in a sense like this has, in writing about mathematics, repeatedly been run together with intuition in other senses. In the last part of the paper a little will be said about the connections that give rise to this phenomenon. * An abridgement of an earlier version of this paper was presented to a session on Mathematical Intuition at the 20th World Congress of Philosophy in
Slim models of Zermelo set theory
, 1996
"... Working in Z+KP , we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula \Phi(; a) such that for any sequence hA j a limit ordinal i where for each , A ..."
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Working in Z+KP , we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula \Phi(; a) such that for any sequence hA j a limit ordinal i where for each , A ` 2, there is a supertransitive inner model of Zermelo containing all ordinals in which for every A = fa j \Phi(; a)g. Preliminaries This paper explores the weakness of Zermelo set theory, Z, as a vehicle for recursive definitions. We work in the system Z+KP , which adds to the axioms of Zermelo those of KripkePlatek set theory KP . Z + KP is of course a subsystem of the familiar system ZF of ZermeloFraenkel. Mention is made of the axiom of choice, but our constructions do not rely on that Axiom. It is known that Z+KP +AC is consistent relative to Z: see [M2], to appear as [M3], which describes inter alia a method of extending models of Z +AC to models of Z +AC + KP . We begin by r...
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
COHEN AND SET THEORY
"... Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ..."
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Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. That is, he established that Con(ZF) implies Con(ZF+¬AC) and Con(ZFC) implies Con(ZFC+¬CH). Already prominent as an analyst, Cohen had ventured into set theory with fresh eyes and an openmindedness about possibilities. These results delimited ZF and ZFC in terms of the two fundamental issues at the beginnings of set theory. But beyond that, Cohen’s proofs were the inaugural examples of a new technique, forcing, which was to become a remarkably general and flexible method for extending models of set theory. Forcing has strong intuitive underpinnings and reinforces the notion of set as given by the firstorder ZF axioms with conspicuous uses of Replacement and Foundation. If Gödel’s construction of L had launched set theory as a distinctive field of mathematics, then Cohen’s forcing began its transformation into a modern, sophisticated one. The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. Soon, ZFC became quite unlike Euclidean geometry and much like group theory, with a wide range of models of set theory being investigated for their own sake. Set theory had undergone a seachange, and with the subject so enriched, it is difficult to convey the strangeness of it. Received April 24, 2008. This is the full text of an invited address given at the annual meeting of the Association
Logical Consequence and Natural Language ∗
"... One of the great successes of the past fifty or so years of the study of language has been the application of formal methods. This has yielded a flood of results in many areas, both of linguistics and philosophy, and has spawned fruitful research programs with names like ‘formal semantics ’ or ‘form ..."
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One of the great successes of the past fifty or so years of the study of language has been the application of formal methods. This has yielded a flood of results in many areas, both of linguistics and philosophy, and has spawned fruitful research programs with names like ‘formal semantics ’ or ‘formal syntax ’ or ‘formal pragmatics’. 1 ‘Formal ’ here often means the tools and methods of formal logic are used (though other areas of mathematics have played important roles as well). The success of applying logical methods to natural language has led some to see the connection between the two as extremely close. To put the idea somewhat roughly, logic studies various languages, and the only special feature of the study of natural language is its focus on the languages humans happen to speak. This idea, I shall argue, is too much of a good thing. To make my point, I shall focus on consequence relations. Though they hardly constitute the full range of issues, tools, or techniques studied in logic, a consequence relation is the core feature of a logic. Thus, seeing how consequence relations relate to natural language is a good way to measure how closely related logic and natural language are. I shall argue here that what we find in natural language is not really logical consequence. In particular, I shall argue that studying the semantics of a natural language is not to study a genuinely logical consequence relation. There is indeed a lot we can glean Thanks to Jc Beall for valuable discussions of the topics explored in this paper. Versions of this material were presented at the Conference on the Foundations of Logical