Iconic Proofs more geometrico

2007

2010

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the "axiomatic conception" of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books. Along with Cellucci, Rav, Grattan-Guinness and Grosholz, we deplore this view of mathematical proof, and favour instead the "analytic conception" of mathematical proof, where the axiomatic proof, when it exists at all, is only the core of a proof. An analytic proof solves a problem, by making hypotheses and using a mixture of deductive moves and induction (loosely construed to include diagrams, etc.) to present a solution to the problem. This implies that proofs are not always finite, that it might involve much more than axioms and straight logical inferences from these deductions and a proof can always be questioned. Moreover, this is where a lot of the interesting conceptual work of mathematics takes place. We view proofs as communicative acts made within the mathematical community which ensures correctness through application, context and standards of rigor.

Springer eBooks, 2009

Proceedings of the Aristotelian Society, 2018

With the discovery of consistent non-Euclidean geometries, the a priori status of Euclidean proof was radically undermined. In response, philosophers proposed two revisionary interpretations of the practice: some argued that Euclidean proof is a purely formal system of deductive logic; others suggested that Euclidean reasoning is empirical, employing concepts derived from experience. I argue that both interpretations fail to capture the true nature of our geometrical thought. Euclidean proof is not a system of pure logic, but one in which our grasp of the content of geometrical concepts plays a central role; moreover, our grasp of this content is a priori.

Synthese, 2003

2016

Abstract: Through the Standards documents, NCTM has called for changes related to Reasoning and Proof and Geometry. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to realize the goals of the Standards, we must pose problems to our students that allow them to play a greater role in proving. We offer nine such problems and discuss how using multiple proof representations moves us toward more authentic proof practices in geometry.

International research journal of MMC, 2022

The objective of the study is to find the learning effectiveness of given the theorem of geometry of class10 in the book published by government of Nepal MOEST which has been the subject of debate among teachers and learners. The study primarily focus Makawanpur district and the informants were selected convinently for the study. There are 95 community schools and 41 private schools in the district. The teachers were selected from both Nepali and English mediums. I conducted the questionnaire survey of the selected teachers and a case study of students. So, there were 4o mathematic teachers for the data collection and five students. Primary data had been collected visiting personally to fill up a set questionnaire with six objectives yes/no and true/false questions. There was a pilot test before the collection of the data. It was found that the present diagram and used strategy of class 10 book has to be reviewed immediately before the new session starts for the coming academic session. The conclusion of the study is that the debatable proof of theorem can be replaced with the one that matches the principle of theorem "Figures which are satisfied with a statement of a theorem the proofs satisfy all of them".

Journal of Humanistic Mathematics, 2016

Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to go together. To make headway we need to have some grip on what it is for a proof to be beautiful, and for that we need some account of judgements of beauty in general. That is the concern of the first section. The second section faces the problem that it is far from obvious how abstract entities, such as mathematical proofs, can be beautiful, strictly and literally speaking. Reasons are given for the view that they can be. The third section introduces the distinction between proofs which explain their conclusions and proofs which do not. Finally, the question whether, for mathematical proofs, the beautiful and the explanatory tend to coincide is addressed. It is argued that we have reason to doubt that explanatory proofs tend to be beautiful, and insufficient reason to believe or disbelieve that beautiful proofs tend to be explanatory.

2008

International Journal of Scientific Research in Mathematical and Statistical Sciences, 2020

This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid’s Postulate 1), and Axiom 8 (an extended version of Euclid’s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid's First Postulate, Euclid's Second Postulate, Hilbert's Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid's Postulate 4, Posidonius-Geminus' Axiom, Proclus' Axiom, Cataldi's Axiom, Tacquet's Axiom 11, Khayyam's Axiom, Playfair's Axiom, and an extended version of Euclid's Fifth Postulate.