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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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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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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
S.: Passages of proof
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2004
"... Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs w ..."
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Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computerassisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical
A Book of Verses underneath the Bough, A Jug of Wine, a Loaf of Breadand Thou Beside me singing
"... This article is an investigation of humanism in philosophy of mathematics from the point of view of postmodernism. We claim that humanistic mathematics is compatible with postmodernism which is taking over everything we have, do, or wish. Omar Khayyam, an Iranian mathematician and poet: ..."
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This article is an investigation of humanism in philosophy of mathematics from the point of view of postmodernism. We claim that humanistic mathematics is compatible with postmodernism which is taking over everything we have, do, or wish. Omar Khayyam, an Iranian mathematician and poet:
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
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