基础数学的本质特征

基础数学的本质特征

    2019年7月12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   8月19日

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〔2018〕4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1～2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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Areas of mathematics

Categories: Fields of mathematicsAbstraction

 

 

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