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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 9 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
"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 show how ..."
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Cited by 5 (2 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.
The History and Concept of Mathematical Proof
, 2007
"... A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Intern ..."
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Cited by 1 (0 self)
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A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Internet design, engineering, chemistry, biological science, social science, anthropology, genetics, warfare, cryptography, plastic surgery, security analysis, data manipulation, computer science, and in many other disciplines and endeavors as well. The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof. It is the proof concept that makes the subject cohere, that gives it its timelessness, and that enables it to travel well. The purpose of this discussion is to describe proof, to put it in context, to give its history, and to explain its significance. There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject. This is the reason that we can depend on mathematics that was done by Euclid 2300 years ago as readily as we believe in the mathematics that is done today. No other discipline can make such an assertion.
www.elsevier.com/locate/entcs Automatic Proofs of Termination With Elementary Interpretations
"... Symbolic constraints arising in proofs of termination of programs are often translated into numeric constraints before checking them for satisfiability. In this setting, polynomial interpretations are a simple and popular choice. In the nineties, Lescanne introduced the elementary algebraic interpre ..."
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Symbolic constraints arising in proofs of termination of programs are often translated into numeric constraints before checking them for satisfiability. In this setting, polynomial interpretations are a simple and popular choice. In the nineties, Lescanne introduced the elementary algebraic interpretations as a suitable alternative to polynomial interpretations in proofs of termination of term rewriting. Here, not only addition and product but also exponential expressions are allowed. Lescanne investigated the use of elementary interpretations for witnessing satisfiability of a given set of symbolic constraints. He also motivated the usefulness of elementary interpretations in proofs of termination by means of several examples. Unfortunately, he did not consider the automatic generation of such interpretations for a given termination problem. This is an important drawback for using these interpretations in practice. In this paper we show how to solve this problem by using a combination of rewriting, CLP, and CSP techniques for handling the elementary constraints which are obtained when giving the symbols parametric elementary interpretations. Keywords: Constraint solving, elementary interpretations, program analysis, termination.
Hilbert’s Second Problem: Foundations of Arithmetic
"... of 23 problems that he considered crucial to the development of mathematics. The problems concerned various topics ranging from number theory to analysis to geometry. The second problem Hilbert presented concerned the foundations of arithmetic itself – and, as recent results of that time suggested, ..."
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of 23 problems that he considered crucial to the development of mathematics. The problems concerned various topics ranging from number theory to analysis to geometry. The second problem Hilbert presented concerned the foundations of arithmetic itself – and, as recent results of that time suggested, perhaps the foundations of all of mathematics. Hilbert’s second problem concerns the axioms of arithmetic – in particular, Hilbert was interested in showing that the axioms are independent and more importantly, not contradictory. In his words: “Upon closer consideration the question arises: Whether, in any way, certain statements of individual axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results. ” 12 The first question is fairly clear – Hilbert wishes to decide if any one axiom of a system is
BOOK REVIEWS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thous ..."
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The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year [28]. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. In addition, many works also formulate unsolved problems, often in the form of precise conjectures. How essential is it for the development of mathematical science to draw the readers’ attention unceasingly to open problems? Maybe it would suffice to publish only new results? The firstrank mathematicians of the present time give a definitive answer to this question. In his preface to the first Russian edition [20] of the book under review, 1 V. I. Arnold reminisced: “I. G. Petrovskiĭ, who was one of my teachers in Mathematics, taught me that the most important thing that a student should learn from his supervisor is that some question is still open. Further choice of the problem from
$132.00 [Gol05]. $14.00 [Hug07b].
, 2013
"... Version 1.15 Title word crossreference $1 [Dea02]. $10.95 [Mac00a]. $100.00 [Web08]. $102.50 [Aga04]. ..."
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Version 1.15 Title word crossreference $1 [Dea02]. $10.95 [Mac00a]. $100.00 [Web08]. $102.50 [Aga04].
FIVE THEORIES OF REASONING: INTERCONNECTIONS AND APPLICATIONS TO MATHEMATICS
"... Abstract. The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connection ..."
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Abstract. The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [37], Toulmin’s argumentation layout [50], Lakatos’s theory of reasoning in mathematics [21], Pollock’s notions of counterexample [42] and argumentation schemes constructed by Walton et al [52], and explore some connections between, as well as within, the theories. For instance, we investigate Peirce’s abduction to deal with surprising situations in mathematics, represent Pollock’s examples in terms of Toulmin’s layout, discuss connections between Toulmin’s layout and Walton’s argumentation schemes, and suggest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove’s [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community.