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Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Cited by 7 (7 self)
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Information is a Physical Entity
"... This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Infor mation is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possib ..."
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Cited by 3 (0 self)
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This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Infor mation is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that, on the ultimate nature of the laws of physics are included. 1
Springer, New York 2009. Indiscrete Variations on GianCarlo Rota’s Themes
"... I never met GianCarlo Rota but I have often made references to his writings on the philosophy of mathematics, sometimes agreeing, sometimes disagreeing. In this paper I will discuss his views concerning four questions: the existence of mathematical objects, definition in mathematics, the notion of ..."
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I never met GianCarlo Rota but I have often made references to his writings on the philosophy of mathematics, sometimes agreeing, sometimes disagreeing. In this paper I will discuss his views concerning four questions: the existence of mathematical objects, definition in mathematics, the notion of
Informal Prologue
, 2000
"... This paper is born out of many years of thinking about the issues relating science and theology. I started out as an undergraduate, pondering such questions as those regarding the nature of physical law and the puzzle of why mathematics works in describing nature. Over the years of reflection, I hav ..."
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This paper is born out of many years of thinking about the issues relating science and theology. I started out as an undergraduate, pondering such questions as those regarding the nature of physical law and the puzzle of why mathematics works in describing nature. Over the years of reflection, I have come to believe that if Christians are to take their faith seriously into the realm of science, then a thorough reexamining of the relation between theology and science is warranted. In particular, as we learn from the philosophy of science and from the Dutch Reformed tradition that we cannot avoid our presuppositions when theorizing about science, for Christians it becomes all the more obvious that whatever lies at the foundation of faith commitments for any scientist cannot be avoided. Thus we must ask the proactive question: just how does our faith give a foundation for our own way of understanding science? This paper is an attempt to address that question from the perspective of the Reformed tradition. My task is of course a highly integrative effort, combining ideas from science with those from philosophy, theology and history. As a physicist without formal training in these other disciplines, I fully expect that my story is incomplete; I expect that there are important sources I have missed while writing this paper which would provide an even fuller picture. On the other hand, integration by its very nature should be viewed as a community effort, so I welcome comments and suggestions which might serve to add to the story and
Mathematical Explanation: Problems and Prospects
"... Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution. ..."
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Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution.
Last revised 01.22.08 Why Proof? What is a Proof?
"... This paper is concerned with real proofs as opposed to formal proofs, and specifically with the ultimate reason of real proofs (‘Why Proof?’) and with the notion of real proof (‘What is a Proof?’). Several people believed and still believe that real proofs can be represented by formal proofs. A rece ..."
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This paper is concerned with real proofs as opposed to formal proofs, and specifically with the ultimate reason of real proofs (‘Why Proof?’) and with the notion of real proof (‘What is a Proof?’). Several people believed and still believe that real proofs can be represented by formal proofs. A recent example is provided by Macintyre who claims that “one could go on to translate ” all “classical informal proofs into formal proofs of some accepted formal system”, where such translations “do map informal proofs to formal proofs ” (Macintyre 2005, p. 2420). This view is to a certain extent implicit in Frege – to a certain extent only, because according to Frege in a sense “every inference is nonformal in that the premises as well as the conclusions have their thoughtcontents which occur in this particular manner of connection only in that inference ” (Frege 1984, p. 318). Anyway, the view that real proofs can be represented by formal proofs is explicitly stated by Hilbert and Gentzen.
Centre for Discrete Mathematics and Theoretical Computer ScienceSearching for Spanning kCaterpillars and kTrees
, 2008
"... We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also ..."
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We consider the problems of finding spanning kcaterpillars and ktrees in graphs. We first show that the problem of whether a graph has a spanning kcaterpillar is NPcomplete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1caterpillar in a graph with treewidth k. Also, as a generalized versions of the depthfirst search and the breadthfirst search algorithms, we introduce the ktree search (KTS) algorithm and we use it in a heuristic algorithm for finding a large kcaterpillar in a graph. 1
Beyond the axioms: The question of objectivity in mathematics
"... I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert... ..."
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I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert...