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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
Frege on knowing the foundations
 Mind
, 1998
"... The paper scrutinizes Frege’s Euclideanism—his view of arithmetic and geometry as resting on a small number of selfevident axioms from which nonselfevident theorems can be proved. Frege’s notions of selfevidence and axiom are discussed in some detail. Elements in Frege’s position that are in ap ..."
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The paper scrutinizes Frege’s Euclideanism—his view of arithmetic and geometry as resting on a small number of selfevident axioms from which nonselfevident theorems can be proved. Frege’s notions of selfevidence and axiom are discussed in some detail. Elements in Frege’s position that are in apparent tension with his Euclideanism are considered—his introduction of axioms in The Basic Laws of Arithmetic through argument, his fallibilism about mathematical understanding, and his view that understanding is closely associated with inferential abilities. The resolution of the tensions indicates that Frege maintained a sophisticated and challenging form of rationalism, one relevant to current epistemology and parts of the philosophy of mathematics. From the start of his career Frege motivated his logicism epistemologically. He saw arithmetical judgments as resting on a foundation of logical principles, and he saw the discovery of this foundation as a discovery of the nature and structure of the justification of arithmetical truths and judg
The role of ontology in semantic integration
 In: Second International Workshop on Semantics of Enterprise Integration at OOPSLA
, 2002
"... Abstract. 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 other applications. Currently there are few methodo ..."
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Abstract. 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 other applications. Currently there are few methodologies for undertaking this task – most commercial projects just rely on experience and intuition. Taking semantically heterogeneous databases as the prototypical situation, this paper describes how ontology (in the traditional metaphysical sense) can contribute to delivering a more efficient and effective process of matching by providing a framework for the analysis, and so the basis for a methodology. It delivers not only a better process for matching, but the process also gives a better result. This paper describes a couple of examples of this: how the analysis encourages a kind of generalisation that reduces complexity. Finally, it suggests that the benefits are not just restricted to individual integration projects: that the process produces models which can be used as to construct a universal reference ontology – for general use in a variety of types of projects. 1
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
Lagrange’s theory of analytical functions and his ideal of purity of method. Archive for History of Exact Sciences. Forthcoming
"... ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm ..."
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ABSTRACT. We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and concentrate on powerseries expansions, on the algorithm for derivative functions, and the remainder theorem—especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. hal00614606, version 1 12 Aug 2011 1. PRELIMINARIES AND PROPOSALS Foundation of mathematics was a crucial topic for 18thcentury mathematicians. A pivotal aspect of it was the interpretation of the algoritihms of the calculus. This was often referred to as the question of the “metaphysics of the calculus ” 1 (see Carnot 1797, as an example). Around 1800 Lagrange devoted two large treatises to the matter, both of which went through two editions in Lagrange’s
Amending Frege’s Grundgesetze der Arithmetik
, 2002
"... 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 Grundgeset ..."
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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.
Fragments of Frege’s Grundgesetze and Gödel’s constructible universe
 Unpublished. Dated July
"... Frege’s Grundgesetze ([18], [20]) was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent ([41], [29], [17]). On ..."
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Frege’s Grundgesetze ([18], [20]) was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent ([41], [29], [17]). One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem (Theorem 2.10) shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of ZermeloFraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets, which Gödel famously used to show the relative consistency of the continuum hypothesis ([23, 24, 25]). More specifically, our proofs appeal to Kripke and Platek’s idea of the projectum within the constructible universe ([38], [42]) as well as Jensen’s uniformization theorem ([35]). The axioms of the Grundgesetze are examples of abstraction principles ([10]), and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.
From Lagrange to Frege: Functions and Expressions
, 2011
"... Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is e ..."
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Part 1 I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung] ” of his formal system. It opens with the following claim ([34], § I.1, p. 5; [40], p. 33) 2: When one is concerned with specifying the original reference [Bedeutung] of the word ‘function’ in its mathematical usage, it is easy to fall into calling function of x an expression [Ausdruck] formed from ‘x ’ and particular numbers by means of the notations [Bezeichnungen] for sum, product, power, difference, and so on. This is inappropriate [unzutreffend] because in this way a function is depicted [hingestellt] as an expression—that is, as a concatenation of signs [Verbindung von Zeichen]—not as what is designated [Bezeichnete] thereby. Hence, instead of ‘expression’, one should say ‘reference of an expression’. Frege does not explicitly ascribe this inappropriateness to anyone, though he could have ascribed it to many 3. One is Lagrange, which, a little less than one century later, defined functions as follows, both in the Théorie des fonctions analytiques and in the Leçons sur le calcul des fonctions ([58], § 1, p. 1;
summed up in the title of chapter IX of Raymond Wilder’s Introduction to the Foundations of
"... The “textbook ” account of the movement in the philosophy of mathematics called “logicism ” is ..."
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The “textbook ” account of the movement in the philosophy of mathematics called “logicism ” is