Revolution In Mathematics Research Articles

Setting the Stage for Disciplinary Transformations: Riker, McKenzie, and the Case of the University of Rochester

This article offers a historical reconstruction of the birth and shaping of two research communities at the University of Rochester, that of economists and that of political scientists. Starting from the late 1950s, the arrival at Rochester of the renowned mathematical economist Lionel W. McKenzie and, later, of the innovative political scientist William H. Riker transformed this university into a leading research institution in economics and political science. The simultaneous presence of Riker and McKenzie at Rochester introduces a “local dimension” in this story, especially since Riker's aim was to convert political science into a theory-driven, highly formal discipline, which he labeled “positive political theory,” apparently along similar lines as the postwar mathematical revolution in economics, of which McKenzie was one of the main characters. This article tries to assess the existence, direction, and effects of this “local” element in the case of Rochester.

‘to find out the pathe’: Mapping the Universal Machine in William Cuningham’s Cosmographical Glasse (1559)

The article places The Cosmographical Glasse (1559), William Cuningham’s magnum opus, in its English and European context. The Cosmographical Glasse appeared during the early modern revolution in mathematics that turned mathematics to practical use by applying it to geography for a better conceptualisation of the globe and universe. Despite the work’s encyclopaedic scope and its author’s pioneering attempt to help readers retrieve the growing body of data that was being amassed by scholars and explorers, the article argues that this first book in English to deal with navigation in relation to astronomy and cosmography stood at a crossroads between the old and new epistemologies as geography both resorted to and confirmed traditional sources of knowledge while challenging them. Limited and flawed as it may be, The Cosmographical Glasse nevertheless added another building block in the construction of a modern sense of cosmography.

Deep Disagreement in Mathematics

Disagreements that resist rational resolution, often termed ``deep disagreements'', have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki's work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice.

의사 인지 이론 관점에서 본 2015 개정 고등학교 수학에서 대학 수학으로의 정적분 담론의 이행

According to the 2015 national curriculum revision, the high school mathematics 2 textbook introduces the definite integral as the difference of the anti-derivatives at the endpoints. However, because most calculus and elementary analysis textbooks at the university level define definite integral as the limit of Riemann sums, students who learned mathematics according to 2015 curriculum revision in high school are forced to use the word “definite integral” differently in university. Sfard views mathematics as a discourse, and she explains the historical development of mathematics and mathematics learning as the change of discourse in the participationist’s perspective. This article looks at the transition problem of the definite integral from school and university in the commognitive framework of Sfard. The comparative analysis of high school and university mathematics discourse about definite integral demonstrates that the two discourses are incommensurable not only in the different word use of definite integral but also the different meta-rule of using definitions and proving. Furthermore, the historical analysis shows that this meta-level change in the transition can be traced to the mathematical revolution from 18th result-oriented calculus to 19th rigorous calculus based on precise definitions and proof. This analysis helps instructors to understand the nature and meaning of the challenges of students in the transition of definite integral between school and university mathematics, to be aware of the commognitive conflict in the communication with students about definite integral, and to design a lesson that helps student change their discourse. The commognitive perspective can contribute to instructors seeing the didactical possibilities in the transition and improve their teaching approach.

The Mathematical Revolution in Economics: Factor Flows in the Production of Academics

The main theme of this essay is that the mathematical revolution in economics precipitated factor flows in the production of academics, wherein able, if not creative, mathematicians moved into economics to earn large rents. This “technical migration” set in motion a counterproductive path dependency from which the discipline may find it difficult to extricate itself. A secondary theme explores the disparate views of Keynes and Schumpeter on economics as an “exact science” and the role that mathematics should play. The seminal question is not whether mathematics is an important tool for economic analysis, because this is not a debatable proposition, but whether the deification of mathematics has come at too high a cost. This cost includes elevating elegance over substance, discounting the importance of other disciplines, and dissuading gifted undergraduates from pursuing advanced study in economics because they do not aspire to be “mathematicians.” JEL Classifications: A22, A23, B31, B41

Revolusi Pembelajaran dalam Matematika sebagai Konsekuensi dari Implementasi Physical Distancing

The proliferation of the corona virus affects all aspects of life, including education. This led to a revolution in the learning from conventional to digital. The purpose of this study is to 1) describe the general review of learning in mathematics and; 2) analyzing the alternatives to the learning revolution in mathematics with the physical policy of physical distancing. This research uses type of reference research. The validity test uses source data triangulation. Data validity obtained with various sources and extensions. Data analysis techniques use constant methods of comparison. The result of this article, which is 1) in general, math learning has been more conventional (face to face) and; 2) there is an alternative to math related to physical distancing, which is online/long-distance learning by making good use of accessible applications through Internet service. Online math learning by giving both a learning video, the settings are as interesting as possible, online discussions, and assessments through online quizzes.

Assessing the influence of allometry on sexual and non-sexual traits: An example in Cicindelidia trifasciata (Coleoptera: Cicindelinae) using geometric morphometrics

Based on a quantitative mathematical revolution, the study of morphology has had an important emphasis by developing statistical shape analysis. This made possible the combination of multivariate statistical methods and new ways to visualize a morphological structure. Patterns of allometry (shape influence on size) in sexual and non-sexual traits were examined in adults of Cicindelidia trifasciata Fabricius collected in north Chilean wetland, using geometric morphometric techniques. Abdomen, mandible and wing shapes were analyzed using 16, 12 and 19 morphological landmarks. Sexual and non-sexual traits differences were found after a principal component analysis, however, most of them were ascribable to sexual size dimorphism. A multivariate regression analysis of shape on size of the abdomen, mandible and wing shapes showed high levels of allometry in C. trifasciata. Static allometry in the sexual traits of C. trifasciata is a common pattern, being the size an important trait that plays an important role during the sexual selection in this species. These differences raise the question of whether sexual dimorphism of sexual traits may be modulated by natural selection.

Революции в математике: возвращаясь к старому спору. Часть 2

Настоящая статья представляет собой вторую (завершающую) часть исследования, посвященного анализу спора о революциях в математике, который возник в 1970-е гг. и был вызван к жизни популярностью концепции научных революций Т. Куна. В первой части исследования было рассмотрено инициировавшее полемику противостояние двух известных историков математики, М. Кроу и Дж. Даубена. Кроу сформулировал десять «законов» развития математики, последний из которых утверждал отсутствие в математике революций, Даубен же предложил исторические свидетельства в пользу противоположной точки зрения. На основании анализа дальнейшего развития этой полемики – определенный итог которого постарался подвести Д. Джиллис, выступивший в 1992 г. редактором книги «Революции в математике» – был сделан предварительный вывод, что победа в споре Кроу и Даубена осталась, скорее, за Даубеном, поскольку подавляющее большинство участников, включая и самого Кроу, в итоге признало существование революций в математике. Затем было начато продолжающееся во второй части исследования обсуждение позиции Б. Порсиоу, поставившего под вопрос такой вывод. Согласно Порсиоу, следует говорить не столько о «споре», сколько о «согласии» между Кроу и Даубеном, поскольку ни один из них не признает нарушения кумулятивности в накоплении математических результатов и не сомневается в существовании сквозного прогресса математического знания. Рассмотрение многочисленных источников, не вошедших в книгу 1992 г., заставляет признать правоту Порсиоу и сделать вывод, что победу Даубена следует считать лишь номинальной, тогда как подлинную победу нужно признать за исходной позицией Кроу, поскольку кумулятивистский тезис так и остался незыблемым, а следовательно, существование «куновских» революций в математике не было признано. В ряде работ спор о революциях в математике объявляется безрезультатным. В данной статье делается иной вывод: этот спор выявил ряд интеллектуальных тенденций, обнажающих компромиссный характер доминирующих представлений о революциях в математике, а тем самым и о математике в целом.

To Outdo Kuhn: on Some Prerequisites for Treating the Computer Revolution as a Revolution in Mathematics

The paper deals with some conceptual trends in the philosophy of science of the 1980‒90s, which being evolved simultaneously with the computer revolution, make room for treating it as a revolution in mathematics. The immense and widespread popularity of Thomas Kuhn’s theory of scientific revolutions had made a demand for overcoming this theory, at least in some aspects, just inevitable. Two of such aspects are brought into focus in this paper. Firstly, it is the shift from theoretical to instrumental revolutions which are sometimes called “Galisonian revolutions” after Peter Galison. Secondly, it is the shift from local (“little”) to global (“big”) scientific revolutions now connected with the name of Ian Hacking; such global, transdisciplinary revolutions are at times called “Hacking-type revolutions”. The computer revolution provides a typical example of both global and instrumental revolutions. That change of accents in the post-Kuhnian perspective on scientific revolutions was closely correlated with the general tendency to treat science as far more pluralistic and transdisciplinary. That tendency is primarily associated with the so-called Stanford School; Peter Galison and Ian Hacking are often seen as its representatives. In particular, that new image of science gave no support to a clear-cut separation of mathematics from other sciences. Moreover, it has formed prerequisites for the recognition of material and technical revolutions in the history of mathematics. Especially, the computer revolution can be considered in the new framework as a revolution in mathematics par excellence.

Революции в математике: возвращаясь к старому спору Часть 1

1970‒1990-е гг. – особый период в истории философии математики. Именно тогда сфор­мировалось влиятельное направление исследований, известное сегодня как «философия математической практики». Один из наиболее заметных эпизодов его становления был связан с так называемым спором о революциях в математике, спровоцированным вопро­сом о том, можно ли применить к математике концепцию развития науки Т. Куна. В ста­тье представлен ретроспективный анализ этого спора, призванный ответить на вопрос о его результатах и последствиях для современной философии математики. Отправной точкой развернувшейся дискуссии стало противостояние М. Кроу, утверждавшего, что в математике никогда не происходит революций, и Дж. Даубена, стоящего на противопо­ложной позиции. На основе разбора аргументов каждой из сторон в статье обосновыва­ется вывод, согласно которому победа в споре номинально осталась за Даубеном, по­скольку большинство его участников, в том числе и сам Кроу, в итоге признало, что революции в математике все-таки происходят. Однако в действительности это большин­ство солидаризировалось с исходной позицией Кроу, поскольку готово было признать наличие в истории математики только таких «революций», которые не нарушают куму­лятивного характера накопления математических результатов, что плохо согласуется с исходным куновским пониманием научной революции. В противоположность тем ра­ботам, в которых утверждается, что спор о революциях в ма­тематике оказался безре­зультатным, в данной статье его польза и поучительность связывается с обнаружением нескольких характерных интеллектуальных тенденций, требующих либо дальнейшего развития, либо осознанного противодействия, но в любом случае сохраняющих свою ак­туальность в контексте современной философии математической практики.

On the Significance of the Copernican Revolution: Transcendental Philosophy and the Object of Metaphysics

This paper argues that the famous passage that compares Kant’s efforts to reform metaphysics with his transcendental idealism to the earlier Copernican revolution in astronomy has a more systematic significance than many recognize. By examining the totality of Kant’s references to Copernicus, one can see that Kant’s analogy points to more than just a similar reversal of perspective. By situating Kant’s comments about Copernicus in relation to his understanding of the logic implicit in the great revolutions in mathematics and natural science, this paper argues that Kant’s appeal to the Copernican revolution in astronomy as a forebear to his own transcendental project indicates that his attempt to revolutionize metaphysics by setting it on the secure path of the sciences demands a shift in how we think of the proper object of metaphysics.

C. S. Peirce and the Square Root of Minus One: Quaternions and a Complex Approach to Classes of Signs and Categorical Degeneration

The beginning for C. S. Peirce was the reduction of the traditional categories in a list composed of a fundamental triad: quality, respect and representation. Thus, these three would be named as Firstness, Secondness and Thirdness, as well given the ability to degeneration. Here we show how this degeneration categorical is related to mathematical revolution which Peirce family, especially his father Benjamin Peirce, took part: the advent of quaternions by William Rowan Hamilton, a number system that extends the complex numbers, i.e. those numbers which consists of an imaginary unit built by the square root of minus one. This is a debate that can, and should, have contributions that take into account the role that mathematical analysis and linear algebra had in C. S. Peirce’s past.

Santilli Isomathematics for Generalizing Modern Mathematics

The establishment of isomathematics, as proposed by R. M. Santilli thirty years ago in the USA, and contributed to by Jiang Chun-Xuan in China during the past 12 years, is significant and has changed modern mathematics. At present, the primary teaching of mathematics is based on the simple operations of addition, subtraction, multiplication and division; a middle level teaching ofmathematics takes these four operations to a higher level, while the university teaching of mathematics extends them to an even higher level. These four arithmetic operations form the foundation of modern mathematics. Santilli isomathematics is a generalisation of these four fundamental operations and heralds a great revolution in mathematics. HIn this paper, we study the four generalized arithmetic operations of isoaddition, isosubtraction, isomultiplication and isodivision at the primary level of isomathematics. The material introduced here should be readily understandable by middle school pupils and university students.Santilli’s isomathematics [1] ssis based on a generalisation of modern mathematics. Isomultiplication is defined by a× a=abT , isodivision by a÷ b=a/b I , where I ≠1 is called an isounit; T I =1, where T is the inverse of the isounit. If addition and subtraction remain unchanged, (+ ,- ,× ,÷ )are the four arithmetic operations in Santilli’s isomathematics[1-5]. Isoaddition a+ b=a+b+0 and isosubtraction a+ b=a+b+0 , where 0 ≠0 is called the isozero, together with the operations of isomultiplication and isodivision introduced above, form the four arithmetic operations(+ ,- ,× ,÷ ) in Santilli-Jiang isomathematics[6]. Santilli [1] suggests isomathematics based on a generalisation of multiplication ×, division ÷, and the multiplicative unit 1 of modern mathematics. It is an epoch making suggestion. From modern mathematics, the foundations of Santilli’s isomathematics will be established

A Silent revolution in mathematics

A Silent revolution in mathematics

El Surgimiento de las Geometrías no Euclidianas y su Influencia en la Cosmología y en la Filosofía de la Matemática


 En este artículo se describe el nacimiento en el siglo XIX de las geometrías no euclidianas y el impacto que al doblar el siglo esta revolución matemática tuvo en la reflexión metacientífica sistemática y, particularmente, en la filosofía de la matemática, generando, desde una concepción formalista, un nuevo territorio teórico denominado Metamatemática. Asimismo se muestra la influencia en lacosmología y la lógica-matemática. Protagonistas de esta historia: Eudoxo, Euclides, Saccheri, Kant, Lobachevsky, Bolyai, Gauss, Riemann, Beltrami, Klein, Poincaré, Frege, Peano, Hilbert, Russell, Whitehead, Minkowski, Einstein, Wittgenstein, Carnap, Gödel.
 

The tangled origins of the Leibnizian calculus: a case study of a mathematical revolution

Evolution or Revolution in Mathematics: The Case of Leibniz Some Issues Raised by Seventeenth Century Mathematics Isaac Barrow, a Foil to Leibniz A Young Central European Polymath between the Scholastics and the Moderns First Steps in Mathematics (1673 - 1675) The Creation of Calculus (1676 - 1686) Logic The Universal Characteristic The Baroque Cultural Context Epilogue Some Concluding Remarks on Mathematical Change Appendices: A: A Transmutation Theorem of Leibniz B: Leibniz's Series Quadrature of a Conic C: Syllogistic Logic D: The Vis Viva Controversy E: Some Applications of Conic Sections, Mechanical Curves and Neusis to Greek Geometry.

Pavel Florenskij mellom geometriske utopier og ikonkosmologi

Pavel Florensky Between Geometrical Utopies and Icon Cosmology On the occasion of the 130th anniversary of the birth of Pavel Florensky, the aim of this article is to analyse the connection between Florensky's mathematical approach to the physical space in the book Imaginations in Geometry and his influence on new the- ories of the space of the icon. The 19th and early 20th centuries witnessed a revolution in mathematics through the development of non-Euclidean geometry and the fourth dimension, which Florensky interpreted as rehabilitation of the philosophy of the Middle Ages. In Imaginations in Geometry he conceptualized a «Ptolemeic-Dantic view of the world» where by means of the Special Theory of Relativity he made an exact approach to the physical characteristics of objects in the border between Heaven and Earth. Although Florensky did not introduce any «divine geometry» when it came to the icon, the concurrent art theorist Nikolay Tarabukin was influenced by Imagina- tions in Geometry. Tarabukin interpreted the so-called «inverse perspective» as a non- Euclidean geometrical visualization of the metaphysical worldview of the Middle Ages that got lost in Europe in the Renaissance, and in Russia under Peter the Great, but is reborn in the 20th century, inspired by the experiments in mathematics.

A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today

A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today

A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today

T he physical sciences all went through “revolutions”: wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a “revolution”, and the second describes turmoil and resistance to the changes at the time. The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, while old mathematics needed precision upgrades but was mostly correct. The sciences were completely transformed while mathematics split, with the core changing profoundly but many applied areas, and mathematical science outside the core, relatively unchanged. The strangest difference is that the scientific revolutions were highly visible, while the significance of the mathematical event is essentially unrecognized. The section “Obscurity” explores factors contributing to this situation and suggests historical turning points that might have changed it. The main point of this article is not that a revolution occurred, but that there are penalties for not being aware of it. First, precollege mathematics education is still based on nineteenth-century methodology, and it seems to me that we will not get satisfactory outcomes until this changes [9]. Second, the mathematical community is adapted to the social and intellectual environment of the midand late twentieth century, and this environment is changing in ways likely to marginalize core mathematics. But core mathematics provides the skeleton that supports the muscles and sinews of science and technology; marginalization will lead to a scientific analogue of osteoporosis. Deliberate management [2] might avoid this, but only if the disease is recognized.

Beyond patient safety Flatland

When Edwin A Abbott published in 1884 his famous fantasy in geometry, Flatland: A romance of many dimensions, his imagination sparked not so much a revolution in mathematics or physics as of perspective. Flatland is a universe in two dimensions – length and width – and the inhabitants (lines, triangles, squares and multisided polygons) in the plane cannot apprehend the complexities of the third dimension: height. It remains beyond their cognitive reach. This analogously describes the problem of patient safety, where we know a lot about two dimensions – the magnitude of the predicament (the length of the problem) and the categories of harm (its width) – but the nature of the third dimension, the lofty heights of what to do, and how to address problems, has proven elusive. We are on the brink of understanding this dimension more thoroughly, and thus better positioned to tackle the challenges it presents.