Nature Of Mathematical Proof Research Articles

C.K. RAJU. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE.

C.K. RAJU. Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE.

Zermelo: a Well Founded Antiskolemism

Zermelo had in mind here his works listed below. They concern a foundational program formulated by him with special emphasis put on the infinitary (though always well founded) nature of mathematical proof. This idea is of course in sharp opposition to the (quite well established at that time) common uderstanding of the notion of finitary formal proof. Zermelo rejects what he himself calls Skolemism and the finitary prejudices: the views that set theory should be axiomatized in a first order language (which implies that quantification over propositional functions in the comprehension axiom would be out of question) and that mathematical theories in general should be codified solely in terms of finitary logic.

Abstracts of additional presentations made at the Royal Society Discussion Meeting ‘The nature of mathematical proof’

Social processes and mathematical proof in mathematics & computing: a quarter-century perspective By Richard Lipton Georgia Institute of Technology, Atlanta, GA, USA Twenty-five years ago we (DeMillo, Lipton, Perlis) wrote a paper on how mathematics is a ‘social process’. In particular,

Computing and the cultures of proving

This article discusses the relationship between mathematical proof and the digital computer from the viewpoint of the 'sociology of proof': that is, an understanding of what kinds of procedures and arguments count for whom, under what circumstances, as proofs. After describing briefly the first instance of litigation focusing on the nature of mathematical proof, the article describes a variety of 'cultures of proving' that are distinguished by whether the proofs they conduct and prefer are (i) mechanized or non-mechanized and (ii) formal proofs or 'rigorous arguments'. Although these 'cultures' mostly coexist peacefully, the occasional attacks from within one on another are of interest in respect to what they reveal about presuppositions and preferences. A variety of factors underpinning the diverse cultures of proving are discussed.

Answering junior ant's ‘why’ for Pythagoras' Theorem

A seemingly simple question in a cartoon about Pythagoras' Theorem is shown to lead to questions about the nature of mathematical proof and the profound relationship between mathematics and science. It is suggested that an analysis of the issues involved could provide a good vehicle for classroom discussions or projects for senior students.

Proofs: Teaching and testing-a tragedy in three acts

Student in examination: 'This theorem is proved in three tragic steps.' (Lecture notes: 'This theorem is proved in three strategic steps.') This article discusses the nature of mathematical proof, the problems of communicating and teaching proofs and the important part played by the methods of assessment in students' choice of learning strategies. The traditional method of asking students to reproduce theorems and proofs in examinations simply encourages rote learning. Examples are given of alternative methods of assessment within the confines of the traditional examination system.

Negotiating Arithmetic, Constructing Proof: The Sociology of Mathematics and Information Technology

The first part of this paper discusses floating-point arithmetic, as performed by computers and advanced digital calculators. It shows that different computer arithmetics have been proposed, and that the nearest approximation to a consensual computer arithmetic — the Institute of Electrical and Electronics Engineers' standard for floating-point arithmetic — had to be negotiated, rather than deduced from existing human arithmetic. The second part of the paper discusses mathematical proof of the correctness of programs and hardware designs, which is increasingly being demanded for systems crucial to safety or security. It argues that this extension of the domain of application of mathematical proof involves the negotiation of what a proof consists in. In 1987, this argument led the author and colleagues to predict a legal case concerning the nature of mathematical proof. Such litigation took place in 1991; the key point at issue is described. More general disagreement about proof as applied to computer systems is also discussed. A further prediction is made: that there will be litigation concerning not just what kind of argument counts as a proof, but over the internal structure of formal proofs.

Making room for mathematicians in the philosophy of mathematics

We began this essay with the concept of mathematician and the realization that a concept of mathematician could be idealized in various ways. We drew an elementary distinction between private and public conceptions. Private theories of mathematicians applied, in principle, to the case of a single mathematician practicing in isolation. Public theories made essential use of the fact that mathematicians occur in communities. We argued that tradition encourages us to prefer private theories and that competition among traditional schools in the philosophy of mathematics is competition among private theories of mathematicians. The rest of the essay tried to establish public theories as an alternative in serious philosophy of mathematics. We considered the practice of teaching and argued that it was an important feature of mathematical experience, intimately related to other central aspects of mathematics and finally, that teaching could only be accounted for by public models. Teaching is not traditionally regarded as a central topic in the philosophy of mathematics, but the nature of mathematical proof is. So we turned to the concept of proof and tried to suggest how it might appear from the perspective of public epistemology. We found that the proofs of a mathematical community could be characterized very differently from the formal proofs of private, isolated mathematicians. These arguments are very suggestive. They raise for the philosophy of mathematics some issues that are basic to contemporary epistemology and the philosophy of science. More important, they offer to philosophy a new way of looking at and characterizing mathematics, one that better coheres with the experience of mathematicians. In making room for the concept of mathematicians in the philosophy of mathematics, perhaps we are also making more room for mathematicians themselves.