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24
Granular Computing: An Emerging Paradigm
, 2001
"... We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their developme ..."
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We provide an overview of Granular Computing a rapidly growing area of information processing aimed at the construction of intelligent systems. We highlight the main features of Granular Computing, elaborate on the underlying formalisms of information granulation and discuss ways of their development. We also discuss the concept of granular modeling and present the issues of communication between formal frameworks of Granular Computing. © 2007 World Academic Press, UK. All rights reserved.
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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Cited by 9 (3 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
Mind change efficient learning
 Info. & Comp
, 2005
"... Abstract. This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evi ..."
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Abstract. This paper studies efficient learning with respect to mind changes. Our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem, but also locally in subproblems after receiving some evidence. Formalizing this idea leads to the notion of uniform mind change optimality. We characterize the structure of language classes that can be identified with at most α mind changes by some learner (not necessarily effective): A language class L is identifiable with α mind changes iff the accumulation order of L is at most α. Accumulation order is a classic concept from pointset topology. To aid the construction of learning algorithms, we show that the characteristic property of uniformly mind change optimal learners is that they output conjectures (languages) with maximal accumulation order. We illustrate the theory by describing mind change optimal learners for various problems such as identifying linear subspaces and onevariable patterns. 1
Methodology and metaphysics in the development of Dedekind’s theory of ideals
 In The architecture of modern mathematics, 159–186
, 2006
"... Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a cer ..."
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Cited by 6 (2 self)
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Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Cited by 2 (1 self)
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
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
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"... We live in the world of digital technology that surrounds us and without which we can barely function. There are myriads of examples (which we take for granted) in which computers bring a wealth of services. Computers constitute an omnipresent fabric of the society (Vasilakos and Pedrycz, 2006). As ..."
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We live in the world of digital technology that surrounds us and without which we can barely function. There are myriads of examples (which we take for granted) in which computers bring a wealth of services. Computers constitute an omnipresent fabric of the society (Vasilakos and Pedrycz, 2006). As once
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"... Methodology and metaphysics in the development of Dedekind’s theory of ideals ..."
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Methodology and metaphysics in the development of Dedekind’s theory of ideals