{"title":"Godel: A Life of Logic","authors":"J. Rauff","doi":"10.5860/choice.38-4504","DOIUrl":"https://doi.org/10.5860/choice.38-4504","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122955024","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}
{"title":"Triumph of the Nerds","authors":"C. Ashbacher","doi":"10.5860/choice.37-3938","DOIUrl":"https://doi.org/10.5860/choice.37-3938","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124026613","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}
{"title":"The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity","authors":"J. Rauff","doi":"10.5860/choice.38-4502","DOIUrl":"https://doi.org/10.5860/choice.38-4502","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122155645","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}
THE LIFE AND LEGACY OF G. I. TAYLOR by George Batchelor, Cambridge University Press, 1996 When students encounter the work of Geoffrey Ingram (G. I.) Taylor in their fluid mechanics courses (Taylor-Couette flows and Rayleigh-Taylor instabilities), they are generally unaware of the extraordinary scope and depth of Taylor's contributions to modern classical physics. Indeed, G. I. Taylor is one of the great applied scientists of the 20th century. He ranks with von Karman, Prandtl, and Burgers as one of the foremost leaders in mechanics. Taylor's numerous contributions include fundamental research in fluid dynamics, turbulence theory, and plasticity. He made discoveries related to shock formations in gases and to the mechanics of explosions, as well as developing basic principles in oceanography, meteorology, and aerodynamics. Contrary to the popular notion that mathematicians and scientists do their most consequential work during their early years, Taylor was in his 70's when he produced results that helped launch the field of electro-hydrodynamics. Many of his results continue to influence the course of research in modern classical physics today. Taylor was active during that extraordinary period in physics when the fields of quantum mechanics and relativity were emerging. Taylor was the first to demonstrate one of the basic results of quantum mechanics: namely, that the diffraction patterns from light shining on a needle do not change with the intensity of the light. However, it became his habit to eschew fashionable research topics such as quantum mechanics and to devote himself to the exploration of more classical mechanics and less popular subjects. Taylor was often instrumental in establishing an area of research, but would drop it and begin something different when the subject became popular. Taylor's approach to research was simple yet elegant, and usually involved a complimentary blend of theory and experiment. He brought originality and insight to problems, as well as a fabulous intuition, which enabled him to construct models that elucidated the important features of a problem. This biography focuses primarily on Taylor's scientific contributions and less so on his personal life. The technical descriptions of Taylor's work are sometimes at the advanced undergraduate or beginning graduate level. The author does an excellent job of communicating Taylor's work in descriptive, qualitative terms. Mathematical formulas appear rarely and derivations not at all; therefore, most of the text is readable by a general reader. …
{"title":"The Life and Legacy of G. I. Taylor","authors":"C. M. Kirk","doi":"10.5860/choice.34-3274","DOIUrl":"https://doi.org/10.5860/choice.34-3274","url":null,"abstract":"THE LIFE AND LEGACY OF G. I. TAYLOR by George Batchelor, Cambridge University Press, 1996 When students encounter the work of Geoffrey Ingram (G. I.) Taylor in their fluid mechanics courses (Taylor-Couette flows and Rayleigh-Taylor instabilities), they are generally unaware of the extraordinary scope and depth of Taylor's contributions to modern classical physics. Indeed, G. I. Taylor is one of the great applied scientists of the 20th century. He ranks with von Karman, Prandtl, and Burgers as one of the foremost leaders in mechanics. Taylor's numerous contributions include fundamental research in fluid dynamics, turbulence theory, and plasticity. He made discoveries related to shock formations in gases and to the mechanics of explosions, as well as developing basic principles in oceanography, meteorology, and aerodynamics. Contrary to the popular notion that mathematicians and scientists do their most consequential work during their early years, Taylor was in his 70's when he produced results that helped launch the field of electro-hydrodynamics. Many of his results continue to influence the course of research in modern classical physics today. Taylor was active during that extraordinary period in physics when the fields of quantum mechanics and relativity were emerging. Taylor was the first to demonstrate one of the basic results of quantum mechanics: namely, that the diffraction patterns from light shining on a needle do not change with the intensity of the light. However, it became his habit to eschew fashionable research topics such as quantum mechanics and to devote himself to the exploration of more classical mechanics and less popular subjects. Taylor was often instrumental in establishing an area of research, but would drop it and begin something different when the subject became popular. Taylor's approach to research was simple yet elegant, and usually involved a complimentary blend of theory and experiment. He brought originality and insight to problems, as well as a fabulous intuition, which enabled him to construct models that elucidated the important features of a problem. This biography focuses primarily on Taylor's scientific contributions and less so on his personal life. The technical descriptions of Taylor's work are sometimes at the advanced undergraduate or beginning graduate level. The author does an excellent job of communicating Taylor's work in descriptive, qualitative terms. Mathematical formulas appear rarely and derivations not at all; therefore, most of the text is readable by a general reader. …","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131086247","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}
{"title":"Introduction to Ordinary Differential Equations","authors":"T. H. Fay","doi":"10.1016/c2013-0-08204-7","DOIUrl":"https://doi.org/10.1016/c2013-0-08204-7","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"334 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115877007","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 : 1996-02-01DOI: 10.1080/0025570X.1996.11996388
J. O. Chilaka
{"title":"Proof without words","authors":"J. O. Chilaka","doi":"10.1080/0025570X.1996.11996388","DOIUrl":"https://doi.org/10.1080/0025570X.1996.11996388","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131981098","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}
ROADS TO INFINITY: THE MATHEMATICS OF TRUTH AND PROOF by John Stillwell A. K. Peters, 2010, 203 pp. ISBN: 978-1-56881-466-7 Roads to Infinity: The Mathematics of Truth and Proof 'is an account of the discovery of the uncountably infinite; the interaction between set theory and logic in the realm of the irifinite; and the mathematical consequences of accepting the infinite levels of infinity. The book follows essentially two roads to infinity: Cantor's diagonal argument and Cantor's construction of the ordinals. Stillwell shows how these two themes intertwine and influence a wide range of mathematical questions including consistency, provability, computability, and existence. Roads to Infinity comprises seven chapters, each based upon a mathematical question. The historical responses to the question are explored and the concepts and theorems resulting from these responses are explained in essentially non-technical language. However, the abiiity to read mathematical symbolism and understand mathematical argumentation is required. Still well begins with Cantor's diagonal argument. His focus is the uncountability of the real continuum and he includes in the discussion the ever-amazing uncountability of transcendental numbers, an application of the diagonal argument to the rate of growth of functions, the paradoxes of set theory, and the axioms of Zermelo-Fraenkel set theory. Next, the book examines the transfinite ordinals, the continuum hypothesis, the axiom of choice and well-ordering, measurability of sets, and Cohen's technique of forcing. Also included here is a discussion of how Cantor's set theory arose from his investigation Fourier series. In the third chapter, Stillwell turns his attention to questions of computability and provability. Here we encounter Godei' s first and second incompleteness theorems, Turing machines, the Halting Problem (which Stillwell finds lurking in Cervantes' Don Quixote^!), and Hubert's Entscheidungsproblem for predicate logic. The chapter leads nicely into Chapter 4 on the consistency and completeness of propositional and predicate logic. A major theme in this chapter on logic is "cut-elimination", a way of inference in logic that replaces modus ponens by falsification trees. Chapter 5 focuses on arithmetic. Here we find a detailed discussion of Peano Aritiimetic, and an infinite extension of Peano Arithmetic and how the extension may be used to prove the consistency of Peano Arithmetic (which cannot prove its own consistency). The diagonal argument theme is reinforced in this chapter as Stillwell shows how the unprovability of consistency of Peano Arithmetic within Peano Arithmetic is related to the argument that 2N" is uncountable. …
John Stillwell A. K. Peters, 2010, 203页。ISBN: 978-1-56881-466-7《通往无限的道路:真理与证明的数学》是关于发现不可数无限的叙述;集合论与逻辑在无限域中的相互作用以及接受无限层次的数学结果。这本书基本上遵循了通向无穷的两条道路:康托的对角线论证和康托的序数构造。Stillwell展示了这两个主题是如何交织在一起，并影响了广泛的数学问题，包括一致性、可证明性、可计算性和存在性。《无限之路》由七章组成，每一章都是基于一个数学问题。对这个问题的历史反应进行了探索，从这些反应中产生的概念和定理基本上用非技术语言进行了解释。然而，阅读数学符号和理解数学论证的能力是必需的。还是从康托尔的对角线论证开始。他的重点是实连续统的不可数性，他在讨论中包括了惊人的超越数的不可数性，对角线论证在函数增长率中的应用，集合论的悖论，以及Zermelo-Fraenkel集合论的公理。接下来，本书考察了超限序数、连续统假设、选择和有序公理、集合的可测量性以及科恩的强迫技术。这里还讨论了康托尔的集合论是如何从他对傅里叶级数的研究中产生的。在第三章中，史迪威将注意力转向可计算性和可证明性问题。在这里，我们遇到了Godei的第一和第二不完备定理，图灵机，停止问题(Stillwell发现潜伏在塞万提斯的堂吉诃德中)，以及Hubert的谓词逻辑的Entscheidungsproblem。这一章很好地引出了关于命题逻辑和谓词逻辑的一致性和完备性的第四章。本章关于逻辑的一个主要主题是“切-消”，这是一种逻辑推理方法，用证伪树代替了模态。第五章着重于算术。这里我们详细讨论了Peano算法，以及Peano算法的无限扩展，以及如何使用扩展来证明Peano算法的一致性(它不能证明自己的一致性)。对角线论证的主题在本章中得到加强，因为Stillwell展示了Peano算术中Peano算术的一致性的不可证明性是如何与2N不可数的论证相关联的。．．．
{"title":"Roads to Infinity: The Mathematics of Truth and Proof","authors":"J. Rauff","doi":"10.5860/choice.48-3928","DOIUrl":"https://doi.org/10.5860/choice.48-3928","url":null,"abstract":"ROADS TO INFINITY: THE MATHEMATICS OF TRUTH AND PROOF by John Stillwell A. K. Peters, 2010, 203 pp. ISBN: 978-1-56881-466-7 Roads to Infinity: The Mathematics of Truth and Proof 'is an account of the discovery of the uncountably infinite; the interaction between set theory and logic in the realm of the irifinite; and the mathematical consequences of accepting the infinite levels of infinity. The book follows essentially two roads to infinity: Cantor's diagonal argument and Cantor's construction of the ordinals. Stillwell shows how these two themes intertwine and influence a wide range of mathematical questions including consistency, provability, computability, and existence. Roads to Infinity comprises seven chapters, each based upon a mathematical question. The historical responses to the question are explored and the concepts and theorems resulting from these responses are explained in essentially non-technical language. However, the abiiity to read mathematical symbolism and understand mathematical argumentation is required. Still well begins with Cantor's diagonal argument. His focus is the uncountability of the real continuum and he includes in the discussion the ever-amazing uncountability of transcendental numbers, an application of the diagonal argument to the rate of growth of functions, the paradoxes of set theory, and the axioms of Zermelo-Fraenkel set theory. Next, the book examines the transfinite ordinals, the continuum hypothesis, the axiom of choice and well-ordering, measurability of sets, and Cohen's technique of forcing. Also included here is a discussion of how Cantor's set theory arose from his investigation Fourier series. In the third chapter, Stillwell turns his attention to questions of computability and provability. Here we encounter Godei' s first and second incompleteness theorems, Turing machines, the Halting Problem (which Stillwell finds lurking in Cervantes' Don Quixote^!), and Hubert's Entscheidungsproblem for predicate logic. The chapter leads nicely into Chapter 4 on the consistency and completeness of propositional and predicate logic. A major theme in this chapter on logic is \"cut-elimination\", a way of inference in logic that replaces modus ponens by falsification trees. Chapter 5 focuses on arithmetic. Here we find a detailed discussion of Peano Aritiimetic, and an infinite extension of Peano Arithmetic and how the extension may be used to prove the consistency of Peano Arithmetic (which cannot prove its own consistency). The diagonal argument theme is reinforced in this chapter as Stillwell shows how the unprovability of consistency of Peano Arithmetic within Peano Arithmetic is related to the argument that 2N\" is uncountable. …","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115583115","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}
If you are searched for a book by John Derbyshire Unknown Quantity: A Real and Imaginary History of Algebra in pdf form, in that case you come on to the faithful site. We furnish full edition of this ebook in ePub, txt, doc, DjVu, PDF formats. You can read Unknown Quantity: A Real and Imaginary History of Algebra online either download. In addition to this ebook, on our site you may reading the guides and different artistic eBooks online, or download their. We will draw on attention that our site not store the book itself, but we grant url to the website wherever you can load either read online. So if want to load Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire pdf, then you have come on to loyal website. We own Unknown Quantity: A Real and Imaginary History of Algebra txt, DjVu, PDF, ePub, doc formats. We will be happy if you return over.
{"title":"Unknown Quantity: A Real and Imaginary History of Algebra","authors":"J. Rauff","doi":"10.5860/choice.44-3328","DOIUrl":"https://doi.org/10.5860/choice.44-3328","url":null,"abstract":"If you are searched for a book by John Derbyshire Unknown Quantity: A Real and Imaginary History of Algebra in pdf form, in that case you come on to the faithful site. We furnish full edition of this ebook in ePub, txt, doc, DjVu, PDF formats. You can read Unknown Quantity: A Real and Imaginary History of Algebra online either download. In addition to this ebook, on our site you may reading the guides and different artistic eBooks online, or download their. We will draw on attention that our site not store the book itself, but we grant url to the website wherever you can load either read online. So if want to load Unknown Quantity: A Real and Imaginary History of Algebra by John Derbyshire pdf, then you have come on to loyal website. We own Unknown Quantity: A Real and Imaginary History of Algebra txt, DjVu, PDF, ePub, doc formats. We will be happy if you return over.","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126890697","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}
{"title":"Algebraic and Discrete Mathematical Methods for Modern Biology","authors":"Chris Arney","doi":"10.1016/c2013-0-18496-6","DOIUrl":"https://doi.org/10.1016/c2013-0-18496-6","url":null,"abstract":"","PeriodicalId":365977,"journal":{"name":"Mathematics and Computer Education","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134524070","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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