In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. The difficulty to understand it lies in the close intertwining of different fields we found in it: philosophy, history and mathematics. But conversely, correctly understood, it gives some evidences both about the question of the origins of the irrationals in Greek mathematics and some points concerning Plato's thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text, which far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, is ending on an aporia, in accordance with the whole dialogue.
{"title":"Comprendre les math'ematiques pour comprendre Platon - th'e'et`ete (147d-148b)","authors":"S. Ofman","doi":"10.20416/LSRSPS.V1I1.53","DOIUrl":"https://doi.org/10.20416/LSRSPS.V1I1.53","url":null,"abstract":"In this paper, we study the so-called 'Mathematical part' of Plato's Theaetetus. Its subject concerns the incommensurability of certain magnitudes, in modern terms the question of the rationality or irrationality of the square roots of integers. As the most ancient text on the subject, and on Greek mathematics and mathematicians as well, its historical importance is enormous. The difficulty to understand it lies in the close intertwining of different fields we found in it: philosophy, history and mathematics. But conversely, correctly understood, it gives some evidences both about the question of the origins of the irrationals in Greek mathematics and some points concerning Plato's thought. Taking into account the historical context and the philosophical background generally forgotten in mathematical analyses, we get a new interpretation of this text, which far from being a tribute to some mathematicians, is a radical criticism of their ways of thinking. And the mathematical lesson, far from being a tribute to some future mathematical achievements, is ending on an aporia, in accordance with the whole dialogue.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132485765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-06-21DOI: 10.17654/FJMSJUL2015_583_615
F. Hazama
A directed graph, called an M-graph, is attached to every melody. Our chief concern in this paper is to investigate (1) how the positivity of the slope of the M-graph is related to singability of the melody, (2) when the M-graph has a symmetry, and (3) how we can detect a similarity between two melodies. For the third theme, we introduce the notion of transposed discrete Frechet distance, and show its relevance in the study of similarity detection among an arbitrary set of melodies.
{"title":"Mathematical Analysis of Melodies: Slope and Discrete Frechet Distance","authors":"F. Hazama","doi":"10.17654/FJMSJUL2015_583_615","DOIUrl":"https://doi.org/10.17654/FJMSJUL2015_583_615","url":null,"abstract":"A directed graph, called an M-graph, is attached to every melody. Our chief concern in this paper is to investigate (1) how the positivity of the slope of the M-graph is related to singability of the melody, (2) when the M-graph has a symmetry, and (3) how we can detect a similarity between two melodies. For the third theme, we introduce the notion of transposed discrete Frechet distance, and show its relevance in the study of similarity detection among an arbitrary set of melodies.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116808876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-05-29DOI: 10.1007/978-3-319-08440-4_6
Steven J. Miller, Taylor C. Corcoran, Jennifer Gossels, Victor Luo, Jaclyn D. Porfilio
{"title":"Pythagoras at the Bat","authors":"Steven J. Miller, Taylor C. Corcoran, Jennifer Gossels, Victor Luo, Jaclyn D. Porfilio","doi":"10.1007/978-3-319-08440-4_6","DOIUrl":"https://doi.org/10.1007/978-3-319-08440-4_6","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129464918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-05-16DOI: 10.1515/9783110565959-013
S. Negrepontis
In the present work it is shown, by an examination of the Platonic dialogues Theaetetus, Sophistes, Politicus, and Philebus, that (a) a Platonic Idea is the philosophic analogue of a pair of lines incommensurable in length only, (b) the Division and Collection, the method by which humans obtain knowledge of a Platonic Idea, is the philosophic analogue of the palindromically periodic anthyphairesis of this pair, and (c) a Platonic Idea is One in the sense of the self-similarity induced by periodic anthyphairesis. A byproduct of the above analysis is that (d) Theaetetus had obtained a proof of the Proposition: The anthyphairesis of a dyad of lines incommensurable in length only is palindromically periodic. It is further verified that the concepts and tools contained in the Theaetetean Book X of the Elements suffice for the proof of the Proposition.
{"title":"The Anthyphairetic Revolutions of the Platonic Ideas","authors":"S. Negrepontis","doi":"10.1515/9783110565959-013","DOIUrl":"https://doi.org/10.1515/9783110565959-013","url":null,"abstract":"In the present work it is shown, by an examination of the Platonic dialogues Theaetetus, Sophistes, Politicus, and Philebus, that (a) a Platonic Idea is the philosophic analogue of a pair of lines incommensurable in length only, (b) the Division and Collection, the method by which humans obtain knowledge of a Platonic Idea, is the philosophic analogue of the palindromically periodic anthyphairesis of this pair, and (c) a Platonic Idea is One in the sense of the self-similarity induced by periodic anthyphairesis. \u0000A byproduct of the above analysis is that (d) Theaetetus had obtained a proof of the Proposition: The anthyphairesis of a dyad of lines incommensurable in length only is palindromically periodic. It is further verified that the concepts and tools contained in the Theaetetean Book X of the Elements suffice for the proof of the Proposition.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124446964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful group with eight elements) has not appeared in this fashion. We describe the quaternion group, both formally and intuitively, and give our strategy for representing the quaternion group as the symmetry group of a physical sculpture.
{"title":"The Quaternion Group as a Symmetry Group","authors":"Vi Hart, Henry Segerman","doi":"10.2307/j.ctvc778jw.17","DOIUrl":"https://doi.org/10.2307/j.ctvc778jw.17","url":null,"abstract":"We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful group with eight elements) has not appeared in this fashion. We describe the quaternion group, both formally and intuitively, and give our strategy for representing the quaternion group as the symmetry group of a physical sculpture.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124547587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-03-27DOI: 10.1007/978-1-4419-0576-5_9
N. Balacheff
{"title":"Bridging knowing and proving in mathematics An essay from a didactical perspective","authors":"N. Balacheff","doi":"10.1007/978-1-4419-0576-5_9","DOIUrl":"https://doi.org/10.1007/978-1-4419-0576-5_9","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"58 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113941076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reminiscences about I. M. Gel'fand on the 100th anniversary of his birth, and about mathematical life in Moscow in the former Soviet Union.
关于格尔方诞辰100周年的回忆，以及前苏联莫斯科的数学生活。
{"title":"An American mathematician in Moscow, or how I destroyed the Soviet Union","authors":"M. Nathanson","doi":"10.1090/NOTI1076","DOIUrl":"https://doi.org/10.1090/NOTI1076","url":null,"abstract":"Reminiscences about I. M. Gel'fand on the 100th anniversary of his birth, and about mathematical life in Moscow in the former Soviet Union.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"685 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116955933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.
{"title":"Some Thoughts on the Teaching of Mathematics -- ten years later","authors":"Igor Rivin","doi":"10.1090/NOTI1131","DOIUrl":"https://doi.org/10.1090/NOTI1131","url":null,"abstract":"I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131227805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In celebration of the distinguished achievements of Professor Tsuyoshi Ando in matrix analysis and operator theory, we conducted an interview with him via email. This paper presents Professor Ando's responses to several questions we gave him regarding his education and life as a mathematician.
{"title":"An interview with Tsuyoshi Ando","authors":"R. Brualdi, M. Moslehian","doi":"10.15352/AFA/1396833513","DOIUrl":"https://doi.org/10.15352/AFA/1396833513","url":null,"abstract":"In celebration of the distinguished achievements of Professor Tsuyoshi Ando in matrix analysis and operator theory, we conducted an interview with him via email. This paper presents Professor Ando's responses to several questions we gave him regarding his education and life as a mathematician.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120858301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-27DOI: 10.4310/ICCM.2014.V2.N1.A9
A. Collino, A. Conte, A. Verra
A collection of selected papers of Gino Fano, containing most of his research papers in Italian, is now available on line at the site of Biblioteca Digitale Italiana di Matematica. The address is: this http URL The realization of this collection on line is part of a joint project of Unione Matematica Italiana (UMI) and Societa' Italiana per la Matematica Applicata e Industriale (SIMAI). The present submission to arXiv is the introduction to the collection and an announcement of it as well.
吉诺·法诺的论文选集，其中包括他的大部分意大利语研究论文，现在可以在意大利数字数学图书馆的网站上找到。这个在线集合的实现是Unione Matematica Italiana (UMI)和Societa' Italiana per la Matematica application and Industriale (SIMAI)联合项目的一部分。目前提交给arXiv的是对该系列的介绍和公告。
{"title":"On the life and scientific work of Gino Fano","authors":"A. Collino, A. Conte, A. Verra","doi":"10.4310/ICCM.2014.V2.N1.A9","DOIUrl":"https://doi.org/10.4310/ICCM.2014.V2.N1.A9","url":null,"abstract":"A collection of selected papers of Gino Fano, containing most of his research papers in Italian, is now available on line at the site of Biblioteca Digitale Italiana di Matematica. The address is: this http URL The realization of this collection on line is part of a joint project of Unione Matematica Italiana (UMI) and Societa' Italiana per la Matematica Applicata e Industriale (SIMAI). The present submission to arXiv is the introduction to the collection and an announcement of it as well.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115106160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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