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18
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...
SET THEORY FROM CANTOR TO COHEN
"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."
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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The
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
Two Moments in the Philosophical Life of 1
, 2007
"... We examine two sets of Gödel’s writings, from the very beginning of his ..."
A matemática de Kurt Gödel ∗
"... I would like to convey to you, most of all, my admiration: You solved this enormous problem with a truly masterful simplicity. (...) Reading your investigation was really a first class aesthetic pleasure. ..."
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I would like to convey to you, most of all, my admiration: You solved this enormous problem with a truly masterful simplicity. (...) Reading your investigation was really a first class aesthetic pleasure.
Completeness Theorems and the Separation of the First and HigherOrder Logic
, 2007
"... With his 1929 thesis Gödel delivers himself to us almost fully formed. He gives in it a definitive, mathematical treatment of the completeness theorem; ..."
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With his 1929 thesis Gödel delivers himself to us almost fully formed. He gives in it a definitive, mathematical treatment of the completeness theorem;