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      13小时前 北大袁萌最新国内数学守旧派为何惧怕超实微积分?

国内数学守旧派为何惧怕超实微积分?

   坦率地说,进入二十一世纪,数学形态发生巨变,形式化方法广泛流行。

国内数学守旧派,故步自封,沉迷于十九世纪数学,“亲”不够,惧怕

“对接”现时代超实微积分。 

   对超实微积分有兴趣读者可见本文附件文章。

袁萌  陈启清   824

附件:

Hyperreal calculus

Abstract This project deals with doing calculus not by using epsilons and deltas, but by using a number system called the hyperreal numbers. The hyperreal numbers is an extension of the normal real numbers with both innitely small and innitely large numbers added. We will rst show how this systemcanbecreated,andthenshowsomebasicpropertiesofthehyperreal numbers. Then we will show how one can treat the topics of convergence, continuity, limits and dierentiation in this system and we will show that the two approaches give rise to the same denitions and results.

Contents 1 1 Construction of the hyperreal numbers 3 1.1 Intuitive construction .. . . . .. . 3

1.2 Ultralters . 3

1.3 Formal construction . . . . . . .. . . 4

1.4 Innitely small and large numbers . . . . . . . . . . . . . . . . . 5

1.5 Enlarging sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Extending functions . . . . . . . . . . . . . . . . . . 6

 

2 The transfer principle 6

2.1 Stating the transfer principle . . . . . . . . 6

2.2 Using the transfer principle . . . . . . . . . . . . . . . 7

3 Properties of the hyperreals 8

3.1 Terminology and notation . . . . . . . . . . . . . . 8

3.2 Arithmetic of hyperreals .  . . . . . . . . 9

3.3 Halos . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Shadows . . . . . . . . . . . . . . . . . . . 10

4 Convergence 11 4.1 Convergence in hyperreal calculus. . . . . . .. 11 4.2 Monotone convergence . . . .  . . . . . 12

5 Continuity 13



 

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1天前 北大袁萌数学的逻辑基础?

数学的逻辑基础?

   20197月,国家4部委联合发文,要求全国加强数学研究的《通知》。

  该《通知》开宗明义,说:“数学是自然科学的基础。”

   然而,英国逻辑学家罗素认为:逻辑才是数学的基础。为此,罗素花费了十年时间,于1910-13年,发表经典专著“数学原理”,严格论证数学的逻辑基础。由此,我们应该关注数理逻辑学科。

  请见本文附件文章。

袁萌   陈启清  823

附件:

Principia Mathematica

  For Isaac Newton's book containing basic laws of physics, see Philosophiæ Naturalis Principia Mathematica.

Not to be confused with The Principles of Mathematics—another book of Russell published in 1903.

The title page of the shortened Principia Mathematica to 56

 

 

54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful." Τhey go on to say "It is used at least three times, in 113.66 and 120.123.472.")

I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated....

Hardy, G. H. (2004) [1940]. A Mathematician's Apology. Cambridge: University Press. p. 83. ISBN 978-0-521-42706-7.

He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one

Littlewood, J. E. (1985). A Mathematician's Miscellany. Cambridge: University Press. p. 130.

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced 9 and all-new Appendix B and Appendix C. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM was originally conceived as a sequel volume to Russe

 

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3天前 北大袁萌数理逻辑奠基人--弗雷格

数理逻辑奠基人--弗雷格

  坦率地说,站在无穷小背后的人有一大串数学巨人:弗雷格、希尔伯特、罗素、塔哥德尔、塔尔斯基与鲁宾逊,以及J.Keisler 

    老实说,离开数理逻辑模型的严格证明,超实数无穷小理论就无从谈起。

弗雷格是什么人,请见本文明附件。 

袁萌   陈启清  821附附件:

戈特洛布•弗雷格

出生

1848118

德意志邦联维斯马

逝世

1925726

魏玛共和国巴特克莱嫩

 

时代

十九世纪哲学

地区

西方哲学

学派

分析哲学

主要领域

数学哲学、数理逻辑、 语言哲学

著名思想

谓词演算, 逻辑主义, Sense and reference, Mediated reference theory

受影响于

显示

影响于

显示

弗里德里希•路德维希•戈特洛布•弗雷格(德语:Friedrich Ludwig Gottlob Frege,宽式IPA/tlop fe/1848118日-1925726日),著名德国数学家、逻辑学家和哲学家。是数理逻辑和分析哲学的奠基人。

 

目录

1生平

2 逻辑学家

3 思想

4 参考

4.1 主要的

4.2 次要的

5 外部链接

生平

弗雷格的父亲是擅长数学的学校教师。1869年弗雷格进入耶拿大学学习,两年后转至哥廷根大学,1873年在那里得到了他在数学领域的哲学博士学位。 根据Sluga的资料(1980) 弗雷格在大学所受的逻辑和哲学教育仍是未知。1875年,他回到耶拿担任讲师,并于1879年成为助理教授, 1896年成为教授。弗雷格只有一名注册学生,鲁道夫•卡尔纳普。 弗雷格的孩子都在成年前死去,而他于1905年领养了一名男孩。

弗雷格的工作没有在有生之年得到广泛的赞誉,但是受到伯特兰•罗素和路德维希•维特根斯坦和卡尔纳普的称赞,认为他注定会产生重大的影响。二战后他的工作才在英语世界广为人知,部分原因是一些哲学家和逻辑学家移居到了美国——例如卡尔纳普,塔尔斯基,和哥德尔——那些了解尊敬弗雷格工作并将他的主要著作翻译成英文的人。弗雷格的工作对分析哲学产生了巨大的影响。

逻辑学家

主条目:概念文字

弗雷格被公认为伟大的逻辑学家,如同亚里士多德,哥德尔,塔尔斯基。他于1879年出版的《概念文字》标志着逻辑学史的转折。《概念文字》开辟了新的领域。

思想

弗雷格是政治立场保守的德国数学家,他重新激起人们对逻辑学的哲学兴趣。他试图找出算术的“基础”,以演绎的方式证明“二加二等于四”这类基本恒等式必然为真。从亚里斯多德以降,逻辑学一直是研究命题与命题彼此关系的学问,弗雷格则扩大逻辑学的内容,创造了“量化”逻辑 ( 与“全部”、“有些”、“无”等范畴有关),使其成为今日哲学家熟知与沿用的知识。正如笛卡儿与洛克沿着知识论大道发展现代哲学,弗雷格也沿着逻辑学与语言分析之路发展当代哲学。“语言学转向”是个令人兴奋的突破,它试图以“分析”哲学为基础,解释所有的理论。

参考[编辑]

主要的[编辑]

Online bibliography of Frege's works and their English translations.

1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press.

1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell.

1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980).

1892a. "Über Sinn und Bedeutung" in Zeitschrift für Philosophie und philosophische Kritik 100: 25-50. Translation: "On Sense and Reference" in Geach and Black (1980).

1892b. "Über Begriff und Gegenstand" in Vierteljahresschrift für wissenschaftliche Philosophie 16: 192-205. Translation: "Concept and Object" in Geach and Black (1980).

1893. Grundgesetze der Arithmetik, Band I. Jena: Verlag Hermann Pohle. Band II, 1903. Partial translation: Furth, M, 1964. The Basic Laws of Arithmetic. Uni. of California Press.

1904. "Was ist eine Funktion?" in Meyer, S., ed., 1904. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904. Leipzig: Barth: 656-666. Translation: "What is a Function?" in Geach and Black (1980).

Peter Geach and Max Black, eds., and trans., 1980. Translations from the Philosophical Writings of Gottlob Frege, 3rd ed. Blackwell.

Frege intended that the following three papers be published together in a book titled Logical Investigations. The English translati/ns thereof were so published in 1975.

1918-19. "Der Gedanke: Eine logische Untersuchung (Thought: A Logical Investigation)" in Beiträge zur Philosophie des Deutschen Idealismus I: 58-77.

1918-19. "Die Verneinung" (Negation)" in Beiträge zur Philosophie des deutschen Idealismus I: 143-157.

1923. "Gedankengefüge (Compound Thought)" in Beiträge zur Philosophie des Deutschen Idealismus III: 36-51.

次要的

George Boolos, 1998. Logic, Logic, and Logic. MIT Press. Contains several influential papers on Frege's philosophy of arithmetic and logic.

Michael Dummett, 1973. Frege: Philosophy of Language. Harvard University Press.

Michael Dummett, 1991. Frege: Philosophy of Mathematics. Harvard University Press.

Demopoulos, William, 1995. "Frege's Philosophy of Mathematics". Harvard University Press. A nice collection that explores the significance of Frege's theorem, and his mathematical and intellectural background.

Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.

Ferreira, F. and Wehmeier, K., 2002, "On the consistency of the Delta-1-1-CA fragment of Frege's Grundgesetze," Journal of Philosophic Logic 31: 301–11.

Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press. Fair to the mathematician, less so to the philosopher.

Hatcher, William, 1982. The Logical Foundations of Mathematics. Pergamon. Uses natural deduction to rederive Peano's axioms from the Grundgesetze system, recast in modern notation.

Hill, C. O. Word and Object in Husserl, Frege and Russell: The Roots of Twentieth-Century Philosophy. Athens: Ohio University Press, 1991.

Hill, C. O., and Rosado Haddock, G. E., 2000. Husserl or Frege: Meaning, Objectivity, and Mathematics. Open Court. The Frege-Husserl-Cantor triangle.

Hans Sluga, 1980. Gottlob Frege. Routledge.

 



 

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5天前 北大袁萌基础数学的代表:数论与数理逻辑

基础数学的代表:数论与数理逻辑

   当前已经进入二十一世纪,谈论基础数学而不涉及数学公理系统及其应用,荒谬至极。

   国内数学守旧派

整天守着自己的一一亩三分地(老古董极限微积分),从来不提基础数学的公理化,不知他们如何应对国家4部委联合下发《关于加强数学科学研究工作方案》的通知?

   为什么说“基础数学的代表:数论与数理逻辑”?有何根据?请见本文附件。

注:《方案》的第一条款是“ 一、持续稳定支持基础数学科学”,清楚地表明了国家支持基础数学科学研究的意志。

袁萌  陈启清  820

附件:

纯粹数学

维基百科,自由的百科全书

此条目没有列出任何参考或来源。 (201183)

维基百科所有的内容都应该可供查证。  

 

请协助添加来自可靠来源的引用以改善这篇条目。无法查证的内容可能被提出异议而移除。

一般而言,纯粹数学是一门专门研究数学本身,不以应用为目的的学问(至少可见范围内无法应用),相对于应用数学而言。纯粹数学以其严格、抽象和美丽著称。自18世纪以来,纯粹数学成为数学研究的一个特定种类,并随着探险、天文学、物理学、工程学等的发展而发展。

纯粹数学以数论,数理逻辑为其代表。

 

目录

1 历史

1.1 19世纪

1.2 20世纪

2 一般化与抽象

3 纯粹主义

4 参考

历史

19世纪

“纯粹数学”这个词是从Sadleirian Chair这个19世纪中期建立的教授职位的全名而来的。“纯粹”数学作为一门独立的学科的想法可能就是从那个时候发展起来的。高斯一代的数学家没有彻底地区分过“纯粹”和“应用”。之后,专门化和专业化,特别是魏尔施特拉斯研究数学分析的方法,使得两者的区别越来越大。

20世纪

进入20世纪,数学家们受到希尔伯特的影响,开始使用公理系统。罗素提出了“纯粹数学”的逻辑公式化方法,以量化的命题为形式。随着数学的公理化,这些公式变得越来越抽象,“严格证明”成为了简单的标准。

实际上在公理系统中,“严格”在“证明”中没有任何新意。以布尔巴基小组的观点,纯粹数学就是已经被证明了的公理。纯粹数学家成为普遍接受的职业,可以通过训练而取得。

一般化与抽象

纯粹数学的一个核心思想就是一般化,它常常有一种更加一般化的趋势。

将定理或数学结构一般化能使对其理解更深

一般化能够简化表达,使证明更短

利用一般化可避免重复证明

一般化可为不同数学分支的联系带来便利。范畴论即是探索这种关联和共性的一个数学领域。

纯粹主义

关于纯粹数学和应用数学,数学家们总有不同的见解。有人认为,最有名的现代例子莫过于戈弗雷·哈罗德·哈代的一个数学家的辩白。

通常认为,哈代认为应用数学非常丑陋和枯燥。哈代偏爱纯粹数学,常把纯粹数学跟画和诗相提并论。他认为应用数学只不过是在数学框架内寻求世界的物理原理,而纯粹数学则表达了独立于物理世界的另一种真实。在他眼中,“真实”数学“具有永恒的美学价值”,而“数学的基本和枯燥的部分”拥有实用价值。

 

 



 

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5天前 北大袁萌基础数学的本质特征

基础数学的本质特征

    2019712日,国家4部委联合下发《关于加强数学科学研究工作方案

》的通知(附件1),要求全国高校及科研院所建立基础数学中心,大规模开展当代基础数学研究。

   什么是基础数学?尤其是,什么是当代的基础数学?请见本文附件2

    二十世纪初,基础数学的本质特点是:形式化与公;理化,受到希尔伯特计划的深远影响(请见附件2)。。

   4部委《方案》与国内数学守旧派格格不入,… …(详见附件2)。

袁萌  陈启清   819

附件1

关于加强数学科学研究工作方案

数学是自然科学的基础,也是重大技术创新发展的基础。数学实力往往影响着国家实力,几乎所有的重大发现都与数学的发展与进步相关,数学已成为航空航天、国防安全、生物医药、信息、能源、海洋、人工智能、先进制造等领域不可或缺的重要支撑。

 

  2018年,国务院发布《关于全面加强基础科学研究的若干意见》(国发〔20184号),提出“潜心加强基础科学研究,对数学、物理等重点基础学科给予更多倾斜”。为切实加强我国数学科学研究,科技部、教育部、中科院、自然科学基金委共同制定本工作方案。

 

  一、持续稳定支持基础数学科学

 

  (一)稳定支持基础数学研究。鼓励科研人员瞄准数学科学重大国际前沿问题和学科发展方向开展创新性研究,鼓励探索新思想、新理论和新方法,强化优秀人才培养,争取取得重大突破。

 

  国家自然科学基金继续加强对基础数学研究的支持,稳定自由探索类项目经费占比,保障基础数学各分支学科均衡协调可持续发展。加大面向科学前沿和国家需求的项目部署力度,提升数学支撑经济社会发展的能力。教育部、中科院支持学科建设和相关基础数学发展。

 

  (二)支持高校和科研院所建设基础数学中心。基础数学中心围绕数学学科重大前沿问题开展基础研究,稳定支持一批高水平科研人员潜心探索,争取重大原创性突破;进行数学科普和数学文化建设,与12所数学教学有特色的中学建立对口交流联系机制,采取数学家科普授课、优秀中学生参与实习、导师制培养等方式进行挂钩指导和支持,培育优秀数学后备人才。

 

  高校和科研院所负责基础数学中心的建设、组织管理和考核评价,为中心提供人才、经费、场地和环境等基础条件,支持中心围绕建设任务开展相关工作。教育部、中科院对所属建设基础数学中心的单位予以相应经费支持。科技部支持基础数学中心开展重要前沿方向项目研究。

 

  二、加强应用数学和数学的应用研究

 

  (三)加大支持应用数学研究。支持科研人员面向国家重大需求和国际前沿研究,面向制约核心产业发展的瓶颈问题,针对重点领域、重大工程、国防安全等国家重大战略需求中的关键数学问题开展研究。

 

  在国家重点研发计划中设立“数学与交叉科学”重点专项,统筹支持数学及交叉科学研究,围绕科学与工程计算、大数据与人工智能的数学理论与方法、复杂系统优化与控制、计算机数学等重点方向,以及信息技术、能源与环境、海洋、生物医药、经济与金融安全等国家重大战略需求中的关键数学问题进行项目部署。

 

  (四)支持地方政府依托高校、科研院所和企业建设应用数学中心。应用数学中心要搭建数学科学与数学应用领域的交流平台,加强数学家与其它领域科学家及企业家的合作与交流,聚焦、提出、凝练和解决一批国家重大科技任务、重大工程、区域及企业发展重大需求中的数学问题。打破单位界限和学科壁垒,鼓励和引导地方、企业及社会资金加大对数学研究的经费投入,推进数学与工程应用、产业化的对接融通,提升数学支撑创新发展的能力和水平。

 

  地方政府负责中心的建设、组织管理和考核评价,为中心提供人才、经费、场地和环境等基础条件,支持中心围绕建设任务开展相关工作,支持关系地方区域发展重大需求的应用数学问题研究。科技部对应用数学中心提出的面向国家战略需求,具有重大社会、经济意义的重要数学问题给予支持。

 

  三、持续推进和深化高层次的国内外交流与合作

 

  (五)加强交流研讨与科学问题凝练。针对若干数学及其交叉领域,通过“香山科学会议”“双清论坛”等平台开展学术交流研讨,聚焦问题、深化合作,解决重大关键科学问题,激发并形成新的学科方向和研究群体。加强天元数学交流中心建设,加大对数学天元基金项目的持续稳定支持。

 

  (六)加强国际合作。积极推动高层次的国际学术交流与合作,提升我国数学水平和国际影响力。充分发挥国家自然科学基金数学天元基金的作用,促进国际交流合作。“走出去”与“请进来”相结合,鼓励数学领域科研人员赴国外深造交流,吸引更多高水平外国学者和学生来华开展合作研究和交流。

附件2:(基础数学又叫纯粹数学)

Pure mathematics

Pure mathematics studies the properties and structure of abstract objects, such as the E8 group, in group theory. This may be done without focusing on concrete applications of the concepts in the physical world

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900,[1] after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need of renewing the concept of mathematical rigor and rewriting all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.[2]

It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.

 

Contents

1

History

1.1

Ancient Greece

1.2

19th century

1.3

20th century

2

Generality and abstraction

3

Purism

4

See also

5

References

6

External links

History[edit]

Ancient Greece[edit]

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, and "logistic", now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being."[3] Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns."[4] The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted,[5]

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that the subject is one of those that "...seem worthy of study for their own sake."[5]

19th century[edit]

The term itself is enshrined in the full title of the Sadleirian Chair, Sadleirian Professor of Pure Mathematics, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

20th century[edit]

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.

Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training.

The case was made that pure mathematics is useful in engineering education:[6]

There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.

Generality and abstraction[edit]

 

 

An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures

Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.

One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.

Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. The point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central.

Purism[edit]

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy's A Mathematician's Apology.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.

Hardy considered some physicists, such as Einstein, and Dirac, to be among the "real" mathematicians, but at the time that he was writing the Apology he also considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well.

Another insightful view is offered by Magid:

I've always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and non-commutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative ring is a not-necessarily-commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we mean not-necessarily-applied mathematics... [emphasis added][7]

See also[edit]

Applied mathematics

Logic

Metalogic

Metamathematics

References[edit]

^ Piaggio, H. T. H., "Sadleirian Professors", in O'Connor, John J.; Robertson, Edmund F. (eds.), MacTutor History of Mathematics archive, University of St Andrews.

^ Robinson, Sara (June 2003). "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders" (PDF). SIAM News. 36 (5).

^ Boyer, Carl B. (1991). "The age of Plato and Aristotle". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 86. ISBN 0-471-54397-7. Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who "must learn the art of numbers or he will not know how to array his troops." The philosopher, on the other hand, must be an arithmetician "because he has to arise out of the sea of change and lay hold of true being."

^ Boyer, Carl B. (1991). "Euclid of Alexandria". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 101. ISBN 0-471-54397-7. Evidently Euclid did not stress the practical aspects of his subject, for there is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to give the student threepence, "since he must make gain of what he learns."

^

Jump up to:

a b Boyer, Carl B. (1991). "Apollonius of Perga". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 152. ISBN 0-471-54397-7. It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, as in ours, there were narrow-minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: "They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." (Heath 1961, p.lxxiv).

The preface to Book V, relating to maximum and minimum straight lines drawn to a conic, again argues that the subject is one of those that seem "worthy of study for their own sake." While one must admire the author for his lofty intellectual attitude, it may be pertinently pointed out that s day was beautiful theory, with no prospect of applicability to the science or engineering of his time, has since become fundamental in such fields as terrestrial dynamics and celestial mechanics.

^ A. S. Hathaway (1901) "Pure mathematics for engineering students", Bulletin of the American Mathematical Society 7(6):266–71.

^ Andy Magid (November 2005) Letter from the Editor, Notices of the American Mathematical Society, page 1173

External links[edit]

 

Wikiquote has quotations related to: Pure mathematics

What is Pure Mathematics? – Department of Pure Mathematics, University of Waterloo

What is Pure Mathematics? by Professor P. J. Giblin The University of Liverpool

The Principles of Mathematics by Bertrand Russell

How to Become a Pure Mathematician (or Statistician), a list of undergraduate and basic graduate textbooks and lecture notes, with several comments and links to solutions, companion sites, data sets, errata pages, etc.

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7天前 北大袁萌值得纪念的这六年

值得纪念的这六年
   当面,在中国的互联网上,阿基米德微积分“老古董”,在中国互联网上,四处泛滥,
毒害读者。
六年前,2013年8月15日,我们将知名数学家J.Keisler教授精心撰写的非阿基米德微积分教科书“知识共享”的第1.5节全文袖珍电子版上传互联网,内容是“无穷小,有限超实数与无穷大”。
至此,国内广大读者可以原汁原味上网学习非阿基米德含有无穷小的微积分了。因此,这是值得纪念的一天;
袁萌  陈启清   8月17日
注:请读者搜索关键字“无穷小,有限超实数与无穷大。”查阅原文。
第1.5节 无穷小,有限超实数与无穷大 (2013-08-15)


 

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8天前 北大袁萌为了数学的明天,,穿越时空,重返南大(III)

为了数学的明天,,穿越时空,重返南大(III

    进入二十一世纪,非阿基米德数学(比如:含有无穷小的连续统)逐渐兴起,我们用该如何面对?

    这是一个基本问题,必须彻底搞清楚,事实求是.

  请见本文附件。

 

袁萌  陈启清   818

附件:

Continuity and Infinitesimals

First published Wed Jul 27, 2005; substantive revision Fri Sep 6, 2013

The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity—a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th century this definition came to be replaced by one employing the more precise concept of limit.

Traditionally, an infinitesimal quantity is one which, while not necessarily coinciding with zero, is in some sense smaller than any finite quantity. For engineers, an infinitesimal is a quantity so small that its square and all higher powers can be neglected. In the theory of limits the term “infinitesimal” is sometimes applied to any sequence whose limit is zero. An infinitesimal magnitude may be regarded as what remains after a continuum has been subjected to an exhaustive analysis, in other words, as a continuum “viewed in the small.” It is in this sense that continuous curves have sometimes been held to be “composed” of infinitesimal straight lines.

Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,” they reappear in the mathematics of the late middle ages and later played an important role in the development of the calculus. Their doubtful logical status led in the nineteenth century to their abandonment and replacement by the limit concept. In recent years, however, the concept of infinitesimal has been refounded on a rigorous basis.

1. Introduction: The Continuous, the Discrete, and the Infinitesimal

2. The Continuum and the Infinitesimal in the Ancient Period

3. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods

4. The Continuum and the Infinitesimal in the 17th and 18th Centuries

5. The Continuum and the Infinitesimal in the 19th Century

6. Critical Reactions to Arithmetization

7. Nonstandard Analysis

8. The Constructive Real Line and the Intuitionistic Continuum

9. Smooth Infinitesimal Analysis

Bibliography

Academic Tools

Other Internet Resources

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9天前 北大袁萌0为了数学的明天,,穿越时空,重返南大(II)

为了数学的明天,,穿越时空,重返南大(II

今年7月,国家4部委联合下发《关于加强数学科学研究工作方案》的重要通知,说明国家决定专项财政拨款资助基础数学研究,希尔伯特计划并不除外。此举,在世界范围内,开创了国家(级)专项资助基础数学研究之先河。

    为了数学的明天,,穿越时空,重返南大。记得,在数学分析的课堂上,何旭初先生教导我们:极限定义需要阿基米德公理的支持,因为实数系统是阿基米德有序域。

   过了两年,1960年,鲁宾逊证明事实并非如此,非阿基米德无穷小微积分成立,因为,超实数系统是非阿基米德有序域。

回到现在,希尔伯特与哥德尔关于数学形式化(公理化)的研究成果,将世界数学研究推进到一个本质上全新的高度。我们要顺应历史发展潮流(比如超实数方向),不做数守旧派。

袁萌   陈启清  815

附件:

阿基米德公理

在抽象代数和分析学中,以古希腊数学家阿基米德命名的阿基米德公理(又称阿基米德性质),是一些赋范的群、域和代数结构具有的一个性质。粗略地讲,它是指没有无穷大或无穷小的元素的性质。由于它出现在阿基米德的《论球体和圆柱体》的公理五,1883年,奥地利数学家Otto Stolz赋予它这个名字[1]

这个概念源于古希腊对量的理论;如大卫·希尔伯特的几何公理,有序群、有序域和局部域的理论在现代数学中仍然起着重要的作用。

阿基米德公理可表述为如下的现代记法: 对于任何实数

x,存在自然数n  

n > x

在现代实分析中,这不是一个公理。它退却为实数具完备性的结果。基于这理由,常以阿基米德性质的叫法取而代之。

0

目录

1 形式叙述以及证明

1.1 解释

1.2 与实数的完备性的关系

2 参看

形式叙述以及证明

解释

简单地说,阿基米德性质可以认为以下二句叙述的任一句:

给出任何数,你总能够挑选出一个整数大过原来的数。

给出任何正数,你总能够挑选出一个整数其倒数小过原来的数。

这等价于说,对于任何正实数a b ,如果

a < b

,则存在自然数

n ,有

a + + a n  terms > b {\displaystyle \underbrace {a+\cdots +a} _{n{\text{ terms}}}>b}

 

与实数的完备性的关系

实数的完备性蕴含了阿基米德性,证明利用了反证法:

假设对所有

n  

n a < b

(注意

n a  

表示

n  

a  

相加),令

S = { n a | n = 1 , 2 , 3 , . . . }  

,则

b  

S  

的上界(

S  

上方有界,依实数完备性,必存在最小上界,令其为

α  

),于是

n = 1 , 2 , 3 , . . . {\displaystyle \forall n=1,2,3,...}

n a < α ( n + 1 ) a < α n a < α − a {\displaystyle na<\alpha \Rightarrow (\Rightarrow na<

 

得出

α − a

也是

S {\displaystyle S}

的一个上界,这与

α {\displaystyle \alpha }

是最小上界矛盾。这样就由实数的完备性推出了阿基米德性质,但阿基米德性推不出实数的完备性,因为有理数满足阿基米德性,但并不是完备的。

 

 

 

 



 

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10天前 北大袁萌为了明天,,穿越时空,重返南大(I)

为了明天,,穿越时空,重返南大(I

   1957年,袁萌考入南京大学数学天文系学习数学。

   第一学期,学习“三高”(数学分析、线性代数与解析几何),由何旭初(已故)、莫绍揆(已故)与韩继昌(已故)三位数学启萌先生分别任教。

    现在可以肯定的是:那时世界数学界没有超实数的概念;学微积,用手机,更是闻所未闻。

 

   今年,国家4部委下发《关于加强数学科学研究工作方案》的通知,那是62年以后的事情了。在当时,想都不敢想。

回到现在,希尔伯特与哥德尔关于数学形式化(公理化)的研究成果,将世界数学研究推进到一个本质上全新的高度。我们要顺应历史发展潮流(比如超实数方向),不做数守旧派。

袁萌   陈启清  814



 

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11天前 北大袁萌超实数系统存在性的严格数学证明

超实数系统存在性的严格数学证明

   1929年,23岁“小毛头”哥德尔证明了“完全性定理”(if… then..)如下:

The completeness theorem says that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula.(后来,到了1953年,该定理的证明过程简化)。

基本思路是,由该定理出发,导出紧致性定理,而紧致行定理的一个简单推论就是超实数系统存在性的数学证明(由鲁宾逊完成)。

   什么是紧致性定理?请见本文附件。

   面对如此情景。莱布尼兹望而生叹。

袁萌  陈启清  812

附件?

Compactness theorem

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces[1], hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.

The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.

 

Contents

1 History

2 Applications

3 Proofs

4 See also

5 Notes

6 References

History

Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.[2][3]

Applications

The compactness theorem has many applications in model theory; a few typical results are sketched here.

The compactness theorem implies Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. This can be seen as follows: suppose φ is a sentence that holds in every field of characteristic zero. Then its negation ¬φ, together with the field axioms and the infinite sequence of sentences 1+1 0, 1+1+1 0, , is not satisfiable (because there is no field of characteristic 0 in which ¬φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset A of these sentences that is not satisfiable. We can assume that A contains ¬φ, the field axioms, and, for some k, the first k sentences of the form 1+1+...+1 0 (because adding more sentences doesn't change unsatisfiability). Let B contain all the sentences of A except ¬φ. Then any field with a characteristic greater than k is a model of B, and ¬φ together with B is not satisfiable. This means that φ must hold in every model of B, which means precisely that φ holds in every field of characteristic greater than k.

A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So, for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'. To achieve this, let T be the initial theory and let κ be any cardinal number. Add to the language of T one constant symbol for every element of κ. Then add to T a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of κ2 sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of T, or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least κ

A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let Σ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1/n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε. By the compactness theorem, there is a model *R that satisfies Σ and also contains an infinitesimal element ε. A similar argument, adjoining axioms ω > 0, ω > 1, etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization Σ of the reals.[4]

Proofs

One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.[5]

Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found, i.e., proofs that refer to truth but not to provability. One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i Σ of it has a model

M i {\displaystyle {\mathcal {M}}_{i}}

. Also let

i Σ M i {\displaystyle \prod _{i\subseteq \Sigma }{\mathcal {M}}_{i}}

 be the direct product of the structures and I be the collection of finite subsets of Σ. For each i in I let Ai := { j I : j i}. The family of all of these sets Ai generates a proper filter, so there is an ultrafilter U containing all sets of the form Ai.

Now for any formula φ in Σ we have:

the set A{φ} is in U

whenever j A{φ}, then φ j, hence φ holds in

M j {\displaystyle {\mathcal {M}}_{j}}

 

the set of all j with the property that φ holds in

M j {\displaystyle {\mathcal {M}}_{j}}

 is a superset of A{φ}, hence also in U

Using o's theorem we see that φ holds in the ultraproduct

i Σ M i / U {\displaystyle \prod _{i\subseteq \Sigma }{\mathcal {M}}_{i}/U}

. So this ultraproduct satisfies all formulas in Σ.

See also

…..   



 

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13天前 北大袁萌非阿基米德几何,从何而来??

非阿基米德几何,从何而来??  

在几何学中,不存在无穷小量,被人们称为阿基米德原理,但是,1899年,希尔伯特在《几何基础》中指出:几何学含有无穷小量不会导致逻辑矛盾,预言了非阿基米德几何学的合理性。

   然而,半个世纪过去了。非阿基米德几何学没有铺出现。1960年,鲁宾逊引入了超实数系统(直线),至此,非阿基米德几何学应运而生。

    本文附件2是我们在今年321日发表的“倒数与非阿基米德几何”;附件1是进入21世纪非阿基米德几何学迅速发展的趋势。

  让我们国内数百万大学生知晓这段历史,不做“小书呆”(宝贝)。

袁萌  陈启清  811

附件1

Several approaches to non-archimedean geometry

Brian Conrad1

Introduction Let k be a non-archimedean eld: a eld that is complete with respect to a specied nontrivial non-archimedean absolute value |·|. There is a classical theory of k-analytic manifolds (often used in the theory of algebraic groups with k a local eld), and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C. Serre’s Harvard lectures [S] on Lie groups and Lie algebras develop this point of view, for example. However, these kinds of spaces have limited geometric interest because they are totally disconnected. For global geometric applications (such as uniformization questions, as rst arose in Tate’s study of elliptic curves with split multiplicative reduction over a non-archimedean eld), it is desirable to have a much richer theory, one in which there is a meaningful way to say that the closed unit ball is “connected”. More generally, we want a satisfactory theory of coherent sheaves (and hence a theory of “analytic continuation”). Such a theory was rst introduced by Tate in the early 1960’s, and then systematically developed (building on Tate’s remarkable results) by a number of mathematicians. Though it was initially a subject of specialized interest, in recent years the importance and power of Tate’s theory of rigid-analytic spaces (and its variants, due especially to the work of Raynaud, Berkovich, and Huber) has become ever more apparent. To name but a few striking applications, the proof of the local Langlands conjecture for GLn by Harris–Taylor uses ´etale cohomology on non-archimedean analytic spaces (in the sense of Be

ural maps Tn Tn+n0 andT n0 Tn+n0 onto the rst n and last n0 variables, it makes sense to let J,J0 Tn+n0 be the ideals generated by I and I0 respectively. Consider the k-anoid algebra Tn+n0/(J + J0). There are evident k-algebra maps ι : A Tn+n0/(J +J0) and ι0 : A0 Tn+n0/(J +J0). Prove that this pair ofmapsisuniversalinthefollowingsense: forany k-Banachalgebra B and any k-Banach algebra maps φ : A B and φ0 : A0 B, there is a unique k-Banach algebra map h : Tn+n0/J B so that hι = φ and hι0 = φ0. In view of this universal property, the triple (Tn+n0/(J+J0),ι,ι0) is unique up to unique isomorphism, so we may denote Tn+n0/(J + J0) as Ab kA0.The product ι(a)ι0(a0) is usually denoted ab a0.(2) Let j : A00 A and j0 : A00 A0 be a pair of maps of k-anoid algebras. Dene the k-anoid algebra Ab A00A0 := (Ab kA0)/(j(a00)b 1−1b j0(a00)|a00

(全文请见CSDN网站文章)

附件2

百度一下“无穷小微积分”,访问该网站,下载“Elementary Calculus,查找第二章46Figure 2.1.2,此时,在你的眼前就是非阿基米德几何的导数示意图了。

 

在无穷小微积分教材中到处都是非阿基米德几何示意图,因为,无穷小就是一种非阿基米德量。

 

坦率地,学习微积分,,离不开希尔伯特5组几何公理。

 

其实,超实平面几何就是一种非阿基米德几何。

 

告别传统(阿基米德几何),迎向未来(非阿基米德几何)。

袁萌  陈启清  2019321

 

 

 



 

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16天前 北大袁萌无穷小,现在终于走出来了

无穷小,现在终于走出来了

   经过近六十年的努力,现在,无穷小终于走出来了得到世人的认可。

当今,国际无穷小的现代定义(wiki)反映在附件1之中。 

   相对而言,我们国内无穷小的官方定义反映在附件2之中。

   请读者仔细对比研究。

袁萌   陈启清   88

附件1:(开门见山,第一句话)

Infinitesimals (ε) and infinites (ω) on the hyperreal number line (ε = 1/ω

    

Functions tending to zero

In a related  \but somewhat different sense, which evolved from the

original definition(注意!) of "infinitesimal" as an infinitely small quantity, the term has also(注意! been used to refer to a function tending to zero.

附件2

无穷小量

同义词 无穷小一般指无穷小量

本词条由“科普中国”科学百科词条编写与应用工作项目 审核

无穷小量是数学分析中的一个概念,在经典的微积分或数学分析中,无穷小量通常以函数、序列等形式出现。 [1]  无穷小量即以数0为极限的变量,无限接近于0。确切地说,当自变量x无限接近x0(或x的绝对值无限增大)时,函数值f(x)0无限接近,即f(x)0(f(x)=0),则称f(x)为当xx0(x→∞)时的无穷小量。特别要指出的是,切不可把很小的数与无穷小量混为一谈。  

 

 

 



 

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17天前 北大袁萌无穷小会导致逻辑矛盾吗?

无穷小会导致逻辑矛盾吗?

   新学年就要开始了。国内数百万大学新生都必须面对“无穷小会导致逻辑矛盾吗?”这是一

个回避不了的问题。

   1960年,41岁的鲁宾逊站在美国普林斯顿研讨班上第一次宣读了自己的研究成果:无穷小不会导致逻辑矛盾

  鲁宾逊的根据是什么?

   810日,新一届向全国普通高校投放无穷小微积分电子教科书大行动即将开始。

袁萌  陈启清  87

 



 

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18天前 北大袁萌关于非标准数学的感想

关于非标准数学的感想

   一般认为,希尔伯特计划中的数学只有一种,包括非欧几何在内。

   如今,出现了“非标准”数学,真是有点儿匪夷所思也。什么是非标准数学?它是什么东西?从何而来?

   希尔伯特想把全部数学形式化、公理化,哥德尔不完全性定理表明希尔伯特的想法是行不通的。 

   实际情况是,在数学的形式化世界里面潜伏着数学非非标准模型。这一事实等待着人们去发现。谁是幸运儿呢?鲁宾逊也!

    请见本文附件,其中有50多篇“非标准数学”的珍贵历史文献。

袁萌  陈启清  86

附件:

On the Foundations of Nonstandard Mathematics

Mauro Di Nasso Dipartimento di Matematica Applicata, Universit`a di Pisa, Italy E-mail: dinasso@dma.unipi.it

Abstract In this paper we survey various set-theoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics.

Introduction. Since the early developments of calculus, innitely small and innitely large numbers have been the object of constant interest and great controversy in the history of mathematics. In fact, while on the one hand fundamental results in the dierential and integral calculus were rst obtained by reasoning informally withinnitesimalquantities, itwaseasilyseenthattheirusewithoutrestrictions led to contradictions. For instance, Leibnitz constantly used innitesimals in his studies (the dierential notation dx is due to him), and also formulated the so-called transfer principle, stating that those laws that hold about the real numbers also hold about the extended number system including innitesimals. Unfortunately, neitherhenorhisfollowerswereabletogiveaformaljustication of the transfer principle. Eventually, in order to provide a rigorous logical frameworkforthetreatmentoftherealline, innitesimal numbers were banished from calculus and replaced by the εδ-method during the second half of the nineteenth century. 1 A correct treatment of the innitesimals had to wait for developments of a new eld of mathematics, namely mathematical logic and, in particular, of its branch called model theory. A basic fact in model theory is that every innite mathematical structure has nonstandard models, i.e. non-isomorphic structures which satisfy the same elementary properties. In other words, there are dierent but equivalent structures, in the sense that they cannot be distinguished by means of the elementary properties they satisfy. In a slogan, one could say that in mathematics “words are not enough to describe reality”.

1An interesting review of the history of calculus can be found in Robinson’s book [R2], chapter X.



 

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19天前 北大袁萌二十世纪数学主流的发展-

二十世纪数学主流的发展

   希尔伯特计划引导了二十世纪数学的发展方向。

   但是,1931年,哥德尔不完全性定理的证明,深化了希尔伯特计划的含义。

   随后,数学模型理论、证明理论、递归理论数学发展起来。

相比而言,无穷小微积分(模型理论分支)只是一朵浪花而已。

:请见本文数学简介。

袁萌   陈启清  85

附件:

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.

The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).

 

Contents

1 General principles

1.1 Use of second-order arithmetic

1.2 Use of higher-order arithmetic

2 The big five subsystems of second-order arithmetic

2.1 The base system RCA0

2.2 Weak Knig's lemma WKL0

2.3 Arithmetical comprehension ACA0

2.4 Arithmetical transfinite recursion ATR0

2.5 Π11 comprehension Π11-CA0

3 Additional systems

4 ω-models and β-models

5 See also

6 References

7 External links

General principles

In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.

For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a system S is required to prove a theorem T, two proofs are required. The first proof shows T is provable from S; this is an ordinary mathematical proof along with a justification that it can be carried out in the system S. The second proof, known as a reversal, shows that T itself implies S; this proof is carried out in the base system. The reversal establishes that no axiom system S that extends the base system can be weaker than S while still proving T.

Use of second-order arithmetic[edit]

Most reverse mathematics research focuses on subsystems of second-order arithmetic. The body of research in reverse mathematics has established that weak subsystems of second-order arithmetic suffice to formalize almost all undergraduate-level mathematics. In second-order arithmetic, all objects can be represented as either natural numbers or sets of natural numbers. For example, in order to prove theorems about real numbers, the real numbers can be represented as Cauchy sequences of rational numbers, each of which can be represented as a set of natural numbers.

The axiom systems most often considered in reverse mathematics are defined using axiom schemes called comprehension schemes. Such a scheme states that any set of natural numbers definable by a formula of a given complexity exists. In this context, the complexity of formulas is measured using the arithmetical hierarchy and analytical hierarchy.

The reason that reverse mathematics is not carried out using set theory as a base system is that the language of set theory is too expressive. Extremely complex sets of natural numbers can be defined by simple formulas in the language of set theory (which can quantify over arbitrary sets). In the context of second-order arithmetic, results such as Post's theorem establish a close link between the complexity of a formula and the (non)computability of the set it defines.

Another effect of using second-order arithmetic is the need to restrict general mathematical theorems to forms that can be expressed within arithmetic. For example, second-order arithmetic can express the principle "Every countable vector space has a basis" but it cannot express the principle "Every vector space has a basis". In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces. Many principles that imply the axiom of choice in their general form (such as "Every vector space has a basis") become provable in weak subsystems of second-order arithmetic when they are restricted. For example, "every field has an algebraic closure" is not provable in ZF set theory, but the restricted form "every countable field has an algebraic closure" is provable in RCA0, the weakest system typically employed in reverse mathematics.

Use of higher-order arithmetic[edit]

A recent strand of higher-order reverse mathematics research, initiated by Ulrich Kohlenbach, focuses on subsystems of higher-order arithmetic (Kohlenbach (2005)). Due to the richer language of higher-order arithmetic, the use of representations (aka 'codes') common in second-order arithmetic, is greatly reduced. For example, a continuous function on the Cantor space is just a function that maps binary sequences to binary sequences, and that also satisfies the usual 'epsilon-delta'-definition of continuity.

Higher-order reverse mathematics includes higher-order versions of (second-order) comprehension schemes. Such an higher-order axiom states the existence of a functional that decides the truth or falsity of formulas of a given complexity. In this context, the complexity of formulas is also measured using the arithmetical hierarchy and analytical hierarchy. The higher-order counterparts of the major subsystems of second-order arithmetic generally prove the same second-order sentences (or a large subset) as the original second-order systems (see Kohlenbach (2005) and Hunter (2008)). For instance, the base theory of higher-order reverse mathematics, called RCA

ω {\displaystyle \omega }



 

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21天前 北大袁萌逆数学:数学实现公理化的有效工具

逆数学:数学实现公理化的有效工具

    众所周知,任何数学分支都应该具有自己的“公理组”。

    但是,我们怎么找出有效的“公理组”

呢?这个问题(二阶算术的子系统)相当复杂,是国内数学研究的一个“空白”。

答案是,逆数学:数学实现公理化的有效工具。

  请读者查阅“subsystems of second order arithmetic ”(共计462页)

另外,请见本文附件。

袁萌  陈启清  83

附件:

逆数学

逆数学(Reverse mathematics)是数学的一个分支,大致可以看成是“从定理导向公理”而不是通常的方向(从公理到定理)。更精确一点,它试图通过找出证明所需的充分和必要的公理来评价一批常用数学结果(即数学分支)的逻辑有效性。

该领域由Harvey Friedman1954- )在其文章“二阶算术系统及其应用(Some systems of second order arithmetic and their use)”中创立。它被Stephen G. Simpson和他的学生以及其他一些人所追随。Simpson写了关于该主题的参考教科书二阶算数的子系统(Subsystems of Second Order Arithmetic);本条目大部分内容取自该书的简介性质的第一章。其他参考读物的细节参看参考。

 

目录

1 原则

1.1 一般性

1.2 语言和基系统的选择

2 二阶算术

2.1 语言

2.2 子系统

3 参考

原则

一般性

逆数学的原则如下:从一个框架语言和一个基理论—一个核心(公理)体系—开始,它可能弱到无法证明大部分我们感兴趣的定理,但是它要强到足以证明一些特定的其区别和所研究的课题不相关的命题之间的等效性或者足以建立一些足够明显的事实(例如加法的可交换性)。在该弱的基系统之上有一个全理论,强到足以证明我们感兴趣的定理,而正常的数学直觉在该理论中又不受侵害。

在基系统和全系统之间,逆数学家需求给一些公理集标上中间的强度,它们(在基系统上)互相不等价:每个系统不仅要证明这个或那个经典定理而且需要在核心体系上等价于该定理。这保证定理的逻辑强度可以被精确的衡量(至少对所选的框架语言和核心系统来说):更弱的公理系统无法证明该定理,而更强的公理系统不被该定理所蕴涵。

语言和基系统的选择[编辑]

若基系统选得太强(作为极端情况,选它为完整的策墨罗-富兰科集合论),则逆数学没有什么信息:很多(全系统的,也就是说通常的数学定理)定理会成为核心系统的定理,所以他们全都等价,我们对于他们的强度一无所知。若基系统选得太弱(作为极端情况,选它为谓词演算),则定理间的等价关系过于细化:没有任何东西等价除了很明显的,同样我们一无所知。如何选取框架语言也是一个问题:它需要不用太多翻译便足以表达通常的数学思想,而它不应预设太强的公理否则我们会碰到和核心系统太强一样的麻烦。

例如,虽然通常(正向的)数学使用集合论的语言,并在策墨罗-富兰科集合论的系统中实现(这个系统,如果不加显式的否认,被数学工作者假设为缺省基础系统),事实上这个系统比真正所需要的强很多—这也是逆数学给我们的教训之一。虽然逆数学特定的结果可以用集合论的框架表达,通常这不是很合适,因为这预设了太强的假定(例如任何阶的集合的存在性和构造它们的一致性)

在逆数学根据Friedman,Simpson和其他人的现在的实现中,框架语言(通常)选为二阶算术,而核心理论选为递归理解,而全理论则为经典分析。

二阶算术

本节有点技术性,主要试图精确描述逆数学的通常框架(也就是,二阶算数子系统)

语言

二阶算数的语言是一种分为两类的(一阶谓词演算的)语言。一些术语和变量,通常用小写字母表示,用于指代个体/数字,它们可以视为自然数。其他变量,称为类变量或者谓词,并经常用大写表示,指代个体的类/谓词/属性,它们可以视为自然数的集合。个体和谓词都可以量化,所有或者存在。一个公式如果有未限定的类变量,(虽然它可能有自由类变量和确定个体变量,)称为算式(arithmetical)

个体术语可以用常数0,单元函数S (后续函数)和二元操作+和· (加和乘)组成。后续函数产生一个比输入大一的自然数。关系 = (相等) < (自然数的比较) 可以关联两个个体,而关系 (属于) 关联一个个体和一个类。

例如

n ( n X S n X ) {\displaystyle \forall n(n\in X\rightarrow Sn\in X)}

是二阶算数定义严谨的公式,它是一个算式,有一个自由类变量X和一个确定个体变量n (但是没有确定类变量,这是算术共识的要求),

X n ( n X ↔ n < S S S S S S 0 S S S S S S S 0 ) {\displaystyle \exists X\forall n(n\in X\leftrightarrow n

是一个定义严谨的公式却不是算式,它有一个确定类变量X和一个确定个体变量n

 

子系统[编辑]

常用的5个子系统按照强度(stength)分别是 RCA0 (Recursive comprehension axiom); WKL0 (Weak König's lemma); ACA0 (Arithmetical comprehension axiom); ATR0 (Arithmetical transfinite recursion); Π11-CA0 (Π11 comprehension axiom);

参考[编辑]

Harvey Friedman's home page

Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 235–242. Canad. Math. Congress, Montreal, Que., 1975.

Stephen G. Simpson's home page

Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999. ISBN 3-540-64882-8.

分类:数理逻辑

 



 

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  • 匿名人士 江苏省南通市 (112.85.165.*): Mathematical Logic, ……   (2019-08-04 22:16:22)

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22天前 北大袁萌逆数学是什么?

逆数学是什么?

   在我们国内,以往对数理逻辑学科注意不够,导致许多知识“盲区”,比如

:逆数学(Reverse mathematics)。

  现在,讲到希尔伯特计划及其近代发展就回避不”Reverse mathematics”的基本概念。它是希尔伯特计划的现代

延伸之一。

  请见本文附件。

袁萌  陈启清 82”

附件:

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.

Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.

The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).

 

Contents

1 General principles

1.1

Use of second-order arithmetic

1.2

Use of higher-order arithmetic

2

The big five subsystems of second-order arithmetic

2.1

The base system RCA0

2.2

Weak Knig's lemma WKL0

2.3

Arithmetical comprehension ACA0

2.4

Arithmetical transfinite recursion ATR0

2.5

Π11 comprehension Π11-CA0

3

Additional systems

4ω-models and β-models

5 See also

6 References

7 External links

General principles

In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a base system which can speak of re

 

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23天前 北大袁萌现代数学的发端

现代数学的发端

  附件1代表现代数学发展观;附件2代表了传统数学发展观。

   希尔伯特计划就是现代数学的发端。希尔伯特是数学形式化的开山鼻祖。

袁萌 陈启 81

附件1

Hilbert's program

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.

Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore anything else.

 

Contents

1 Statement of Hilbert's program

2 ödel's incompleteness theorems

3 ilbert's program after Gödel

4 ee also

5 eferences

6  ternal link

Statement of Hilbert's program

The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular this should include:

A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.

Completeness: a proof that all true mathematical statements can be proved in the formalism.

Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.

Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.

Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

Gödel's incompleteness theorems[edit]

Main article: Gödel's incompleteness theorems

Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way. Gödel's second incompleteness theorem shows that any consistent theory powerful enough to encode addition and multiplication of integers cannot prove its own consistency. This presents a challenge to Hilbert's program:

It is not possible to formalize all mathematical true statements within a formal system, as any attempt at such a formalism will omit some true mathematical statements. There is no complete, consistent extension of even Peano arithmetic based on a recursively enumerable set of axioms.

A theory such as Peano arithmetic cannot even prove its own consistency, so a restricted "finitistic" subset of it certainly cannot prove the consistency of more powerful theories such as set theory.

There is no algorithm to decide the truth (or provability) of statements in any consistent extension of Peano arithmetic. Strictly speaking, this negative solution to the Entscheidungsproblem appeared a few years after Gödel's theorem, because at the time the notion of an algorithm had not been precisely defined.

Hilbert's program after Gödel[edit]

Many current lines of research in mathematical logic, such as proof theory and reverse mathematics, can be viewed as natural continuations of Hilbert's original program. Much of it can be salvaged by changing its goals slightly (Zach 2005), and with the following modifications some of it was successfully completed:

Although it is not possible to formalize all mathematics, it is possible to formalize essentially all the mathematics that anyone uses. In particular Zermelo–Fraenkel set theory, combined with first-order logic, gives a satisfactory and generally accepted formalism for almost all current mathematics.

Although it is not possible to prove completeness for systems at least as powerful as Peano arithmetic (at least if they have a computable set of axioms), it is possible to prove forms of completeness for many other interesting systems. The first big success was by Gödel himself (before he proved the incompleteness theorems) who proved the completeness theorem for first-order logic, showing that any logical consequence of a series of axioms is provable. An example of a non-trivial theory for which completeness has been proved is the theory of algebraically closed fields of given characteristic.

The question of whether there are finitary consistency proofs of strong theories is difficult to answer, mainly because there is no generally accepted definition of a "finitary proof". Most mathematicians in proof theory seem to regard finitary mathematics as being contained in Peano arithmetic, and in this case it is not possible to give finitary proofs of reasonably strong theories. On the other hand, Gödel himself suggested the possibility of giving finitary consistency proofs using finitary methods that cannot be formalized in Peano arithmetic, so he seems to have had a more liberal view of what finitary methods might be allowed. A few years later, Gentzen gave a consistency proof for Peano arithmetic. The only part of this proof that was not clearly finitary was a certain transfinite induction up to the ordinal ε0. If this transfinite induction is accepted as a finitary method, then one can assert that there is a finitary proof of the consistency of Peano arithmetic. More powerful subsets of second order arithmetic have been given consistency proofs by Gaisi Takeuti and others, and one can again debate about exactly how finitary or constructive these proofs are. (The theories that have been proved consistent by these methods are quite strong, and include most "ordinary" mathematics.)

Although there is no algorithm for deciding the truth of statements in Peano arithmetic, there are many interesting and non-trivial theories for which such algorithms have been found. For example, Tarski found an algorithm that can decide the truth of any statement in analytic geometry (more precisely, he proved that the theory of real closed fields is decidable). Given the Cantor–Dedekind axiom, this algorithm can be regarded as an algorithm to decide the truth of any statement in Euclidean geometry. This is substantial as few people would consider Euclidean geometry a trivial theory.

See also[edit]

Grundlagen der Mathematik

Foundational crisis of mathematics

Atomism

References[edit]

G. Gentzen, 1936/1969. Die Widerspruchfreiheit der reinen Zahlentheorie. Mathematische Annalen 112:493–565. Translated as 'The consistency of arithmetic', in The collected papers of Gerhard Gentzen, M. E. Szabo (ed.), 1969.

D. Hilbert. 'Die Grundlagen Der Elementaren Zahlentheorie'. Mathematische Annalen 104:485–94. Translated by W. Ewald as 'The Grounding of Elementary Number Theory', pp. 266–273 in Mancosu (ed., 1998) From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s, Oxford University Press. New York.

S.G. Simpson, 1988. Partial realizations of Hilbert's program. Journal of Symbolic Logic 53:349–363.

R. Zach, 2006. Hilbert's Program Then and Now. Philosophy of Logic 5:411–447, arXiv:math/0508572 [math.LO].

External links[edit]

Entry on Hilbert's program by Richard Zach at the Stanford Encyclopedia of Philosophy.

 

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ZairjaArs Magna (1300)An Essay towards a Real Character, and a Philosophical Language (1688)Calculus ratiocinator and characteristica universalis (1700)Dewey Decimal Classification (1876)Begriffsschrift (1879)Mundaneum (1910)Logical atomism (1918)Tractatus Logico-Philosophicus (1921)Hilbert's program (1920s)Incompleteness theorem (1931)World Brain (1938)Memex (1945)General Problem Solver (1959)Prolog (1972)Cyc (1984)Semantic Web (2001)Evi (2007)Wolfram Alpha (2009)Watson (2011)Siri (2011)Knowledge Graph (2012)Wikidata (2012)Cortana (2014)Viv (2016)

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See also: Logic machines in fiction and List of fictional computers

Categories: Mathematical logicProof theoryHilbert's problems

 

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 附件2:“作业帮”的看法

《普通高中数学课程标准》指出:“形式化是数学的基本特征之一.在数学教学中,学习形式化的表达是一项基本要求,但是不能只限于形式化的表述,要强调对数学本质的认识,否则会将生动活泼的数学思维活动淹没在形式化的海洋里.数学的现代发展也表明,全盘形式化是不可能的.因此,高中数学课程应该返璞归真,努力揭示数学概念、法则、结论的发展过程的本质.

所谓“数学形式”,就是用特定的数学语言,包括数学的符号语言、图象语言和文字语言,表达自然现象和社会现象的空间结构和数量关系,即具有相对固定样式的数学概念、法则、结论,它具有如下特征:

1)稳定性.数学概念、法则、结论等内容一旦成为“形式”,就有相对稳定的特征,决不会因环境、条件的变更而发生变化.

2)概括性.数学形式是无数具体事物经抽象概括的结果,应该是研究数量关系或图形本质属性的反应.

3)简洁性.最简单的往往是最深刻的,越简洁的东西就越具有生命力,越具使用价值.数学形式就以其表述方式的简洁而称道.

4)广泛性.数学形式的概括性决定了它具有广泛性,可真正达到华罗庚教授所说的“数学是一个原则,无数内容,一个方法,到处有用.

5)可操作性.按照相关数学形式进行的程式化操作可称为行为模式.人的行为模式有两种,一种是需要智力投入、思维参与的行为模式;一种是较少需要智力投入、思维参与的行为模式.在数学学习和解决数学问题的所有活动中,创造性思维的含量只占少部分,运用更多的是程式化的操作.这种操作讲究的是熟练、准确、快速、高效.学生大多数解题是按既定法则进行模式化操作.即使是难度较大的需要一定的创造思维,但创造的“根”仍然扎在坚实的基本数学形式的土壤中.基本数学形式是创造的源泉与原型.当然,即便进行的是简单化、机械化、程序化的操作,也要在其中努力加大智力与思维的含量.

形式化有着不可否认的弊端:

1)形式化可能掩盖事物的本质,学生只会机械操作.

2)形式化会轻视过程,只知结论,不知来龙去脉.

3)形式化不利于学生对基础知识和基本能力的记忆及养成,教学中容易出现“开门见山,直达结论”的现象.

4)形式化会使学生产生思维惰性.

对概念、定理、法则和解题技法等若都能达到本质的理解固然很好,但毕竟有些内容要求学生在形式化的基础上形成机械记忆,并能投入操作应用即可.问题的关键是,哪些内容应保留形式,哪些内容需要否定形式,哪些内容需要形式和本质的和谐共处,这些不能靠主观臆断,而要靠我们老师在吃透新课程标准和新教材的基础上科学合理地来确定.一般来讲,数学教学之初,应该充分展示数学知识发生发展的过程,引导学生弄清本质,在熟练的基础上适度形式化,形成自己的技能,这样的知识学得牢固一些,对于大面积提高数学成绩也有帮助.再说行为模式,包括某些解题方法,必须引领学生在解题实践的过程中总结有典型意义的重要形式,且注意思维的参与,使这些行为模式的操作更有效.

 

作业帮用户 2016-12-07

 

 



 

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24天前 北大袁萌希尔伯特计划是什么?

希尔伯特计划是什么?

    近日,国家4部委制定、颁布关于加强数学科学工作方案的通知,要求全国高校与科研院所组建“基础数学中心”,国家提供人、财、物的支持。

由此可见,千余所“基础数学中心”即将“挂牌(公布IP地址)。

既然国家出资组建大批“基础数学中心”,其首要历史使命必然是:保卫、发展国家数学事业的安全与健康发展。

进入二十世纪,数学基础的安全离不开“希尔伯特计划”的实行。                          

因此,希尔伯特计划是什么?必须搞明白。请见本文附件。

袁萌   陈启清  731

附件:

希尔伯特计划是由德国数学家大卫希尔伯特在1920年代提出的一个数学计划(Program)。它是一个关于公理系统相容性(无矛盾性)的严谨证明的一项计划。

这个计划不应该和希尔伯特的二十三个问题混淆,不过这个计划对数学的发展也有着重要的影响。

希尔伯特计划的陈述

这个计划的主要目标,是为全部的数学提供一个安全的理论基础。具体地,这个基础应该包括:

所有数学的形式化。意思是,所有数学应该用一种统一的严格形式化的语言,并且按照一套严格的规则来使用。

完备性。我们必须证明以下命题:在形式化之后,数学里所有的真命题都可以被证明(根据上述规则)。

相容性。我们必须证明:运用这一套形式化和它的规则,不可能推导出矛盾。

保守性。我们需要证明:如果某个关于“实际物”的结论用到了“假想物”(如不可数集合)来证明,那么不用“假想物”的话我们依然可以   证明同样的结论。

确定性。应该有一个算法,来确定每一个形式化的命题是真命题还是假命题。

 

 



 

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25天前 北大袁萌基础数学中心招聘人才,函数序偶定义必考

基础数学中心招聘人才,函数序偶定义必考

    根据国家4部委的部署,全国高校与科研院所即将设立大批“基础数学中心”,必定招聘数学专业人才。

   基础数学中心招聘人才,为什么函数序偶定义必考?

    因为国内微积分教科书没有函数的序偶定义,所以,,应聘者不上互联网,死读书,读死书,是个大书呆子。

我们希望:基础数学中心建立在互联网之上,敞开大门,百花齐放,… …

学习微积分,大书呆。小糊涂,都很可爱。

袁萌  陈启清  729



 

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