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Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Scientific Representation and the Semantic View of Theories”, Theoria 55: 49–65
, 2006
"... ABSTRACT: It is now part and parcel of the official philosophical wisdom that models are essential to the acquisition and organisation of scientific knowledge. It is also generally accepted that most models represent their target systems in one way or another. But what does it mean for a model to re ..."
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ABSTRACT: It is now part and parcel of the official philosophical wisdom that models are essential to the acquisition and organisation of scientific knowledge. It is also generally accepted that most models represent their target systems in one way or another. But what does it mean for a model to represent its target system? I begin by introducing three conundrums that a theory of scientific representation has to come to terms with and then address the question of whether the semantic view of theories, which is the currently most widely accepted account of theories and models, provides us with adequate answers to these questions. After having argued in some detail that it does not, I conclude by pointing out in what direction a tenable account of scientific representation might be sought.
2003: ‘Everything
 in J. Howthorne and D
"... I am about to take a flight. Everything is packed into my carryon baggage. On reading the last sentence, did you interpret me as saying falsely that everything — everything in the entire universe — was packed into my carryon baggage? Probably not. In ordinary language, ‘everything ’ and other quan ..."
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I am about to take a flight. Everything is packed into my carryon baggage. On reading the last sentence, did you interpret me as saying falsely that everything — everything in the entire universe — was packed into my carryon baggage? Probably not. In ordinary language, ‘everything ’ and other quantifiers (‘something’, ‘nothing’, ‘every dog’,...) often carry a tacit restriction to a domain of contextually relevant objects, such as the things that I need to take with me on my journey. Thus a sentence of the form ‘Everything Fs ’ is true as uttered in a context C if and only if everything that is relevant in C satisfies the predicate ‘F’; ‘everything ’ ranges just over the contextually relevant things. Such generality is restricted in a contextrelative way. Is there also absolute generality, without contextual restrictions? In that sense, absolutely everything Fs only if everything that is relevant in any context Fs. To use ‘everything ’ to express absolute generality, we need a context in which absolutely nothing is excluded as irrelevant. Are there such contexts? Bradley describes metaphysics as ‘the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole ’ ([AR]: 1). How could we comprehend
The Role of Ontology in Integrating Semantically Heterogeneous Databases
, 2002
"... More and more enterprises are currently undertaking projects to integrate their applications. They are finding that one of the more difficult tasks facing them is determining how the data from one application matches semantically with the data from the other applications. Currently there are few ..."
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More and more enterprises are currently undertaking projects to integrate their applications. They are finding that one of the more difficult tasks facing them is determining how the data from one application matches semantically with the data from the other applications. Currently there are few methodologies for undertaking this task  most commercial projects just rely on experience and intuition. Taking
Natural Logicism via the Logic of Orderly Pairing by
, 2008
"... Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arith ..."
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Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neoFregean done so. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the GentzenPrawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free firstorder logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any settheoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on senseconstituting rules of natural deduction, the completed account can be described as ‘natural logicism’. 2 1 Introduction: historical
A Solution Based H Norm Triangular Mesh Quality Indicator. Grid Generation and Adaptive
 Mathematics: Proceedings of the IMA Workshop on Parallel and Adaptive Methods
, 1999
"... Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, s ..."
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Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, since they too bring in plural terms—terms such as ‘�4 ’ or the descriptive ‘the inhabitants of London ’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously if only because mathematicians take manyvalued functions seriously. We assess the objection (by Russell, Frege and others) that manyvalued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection illfounded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. 1. Russell’s theory of plural descriptions Everybody knows that Russell had a theory of definite descriptions. Not everybody realizes that he had two: one for singular descriptions, another for plural descriptions. The contents of ‘On Denoting ’ have blinkered the popular conception of his agenda. 1.1 The Principles of Mathematics In the Principles class talk is plural talk: ‘soandso’s children, or the children of Londoners, afford illustrations ’ of classes; ‘the children of Israel are a class ’ (1903c, pp. 24, 83). Readers brought up on modern set theory must beware. Russell’s plural descriptions each stand for many things, and accordingly his classes are ‘classes as many’: they are many things—the children of Israel are a class—not one. (Unless, of course, they only have a single member. Throughout this paper we use the plural idiom inclusively, to cover the singular as a limiting case. Purists should read ‘the Fs ’ as ‘the F or Fs ’ and adjust the context to suit.) Russell first investigates plural idioms in the chapter on ‘Denoting’. His exposition is complicated, however, by his insistence that distribu
Amending Frege’s Grundgesetze der Arithmetik, Synthese Vol.147
, 2005
"... Abstract. Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the ..."
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Abstract. Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
What is Frege’s Theory of Descriptions?
"... When prompted to consider Frege’s views about definite descriptions, many philosophers think about the meaning of proper names, and some of them can cite the following quotation taken from a footnote Frege’s 1892 article “ Über Sinn und Bedeutung.”2 ..."
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When prompted to consider Frege’s views about definite descriptions, many philosophers think about the meaning of proper names, and some of them can cite the following quotation taken from a footnote Frege’s 1892 article “ Über Sinn und Bedeutung.”2