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## The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics

Sanders, Sam
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/11/2015 Português
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The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis pertaining to Reverse Mathematics (RM). In particular, we shall establish RM-equivalences between theorems from Nonstandard Analysis in a fragment of Nelson's internal set theory. We then extract primitive recursive terms from Goedel's system T (not involving Nonstandard Analysis) from the proofs of the aforementioned nonstandard equivalences. The resulting terms turn out to be witnesses for effective1 equivalences in Kohlenbach's higher-order RM. In other words, from an RM-equivalence in Nonstandard Analysis, we can extract the associated effective higher-order RM-equivalence which does not involve Nonstandard Analysis anymore. Finally, we show that certain effective equivalences in turn give rise to the original nonstandard theorems from which they were derived.; Comment: arXiv admin note: substantial text overlap with arXiv:1508.07434

## Webbing and orchestration. Two interrelated views on digital tools in mathematics education

Trouche, Luc; Drijvers, Paul
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/08/2014 Português
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The integration of digital tools in mathematics education is considered both promising and problematic. To deal with this issue, notions of webbing and instrumental orchestration are developed. However, the two seemed to be disconnected, and having different cultural and theoretical roots. In this article, we investigate the distinct and joint journeys of these two theoretical perspectives. Taking some key moments in recent history as points of de- parture, we conclude that the two perspectives share an importance attributed to digital tools, and that initial differences, such as different views on the role of digital tools and the role of the teacher, have become more nuances. The two approaches share future chal- lenges to the organization of teachers'collaborative work and their use of digital resources.; Comment: Teaching Mathematics and its Applications (2014) to be completed

## Recursive Synthesis and the Foundations of Mathematics

Hohti, Aarno
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/08/2002 Português
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This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in its precise formulation. The requirement on precise, formal content discloses the relational structure of (mathematical) experience, and gives a new meaning to the `ideal' objects beyond concrete forms. The paper also provides a systematic reason why set theory represents an ultimate stage in mathematical technology.; Comment: 20 pages

## Idempotent mathematics and interval analysis

Litvinov, Grigori; Maslov, Viktor; Sobolevskii, Andrei
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.; Comment: 22 pages, no figures; to be published in an abridged form in Reliable Computing (Kluwer); the bibliography is updated in this version

## Addressing the underrepresentation of women in mathematics conferences

Martin, Greg
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/02/2015 Português
Relevância na Pesquisa
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Despite significant improvements over the last few generations, the discipline of mathematics still counts a disproportionately small number of women among its practitioners. These women are underrepresented as conference speakers, even more so than the underrepresentation of women among PhD-earners as a whole. This underrepresentation is the result of implicit biases present within all of us, which cause us (on average) to perceive and treat women and men differently and unfairly. These mutually reinforcing biases begin in primary school, remain active through university study, and continue to oppose women's careers through their effects on hiring, evaluation, awarding of prizes, and inclusion in journal editorial boards and conference organization committees. Underrepresentation of women as conference speakers is a symptom of these biases, but it also serves to perpetuate them; therefore, addressing the inequity at conferences is valuable and necessary for countering this underrepresentation. We describe in detail the biases against women in mathematics, knowing that greater awareness of them leads to a better ability to mitigate them. Finally, we make explicit suggestions for organizing conferences in ways that are equitable for female mathematicians.; Comment: 26 pages. See also "An annotated bibliography of work related to gender in science" (arXiv:1412.4104)

## Illustrating Mathematics using 3D Printers

Knill, Oliver; Slavkovsky, Elizabeth
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/06/2013 Português
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3D printing technology can help to visualize proofs in mathematics. In this document we aim to illustrate how 3D printing can help to visualize concepts and mathematical proofs. As already known to educators in ancient Greece, models allow to bring mathematics closer to the public. The new 3D printing technology makes the realization of such tools more accessible than ever. This is an updated version of a paper included in book Low-Cost 3D Printing for science, education and Sustainable Development, ICTP, 2013 edited by Carlo Fonda Enrique Canessa and Marco Zennaro.; Comment: 95 figures, 6 tables with example source code. An earlier version of a paper was included in book Low-Cost 3D Printing for science, education and Sustainable Development, ICTP, 2013 edited by Carlo Fonda Enrique Canessa and Marco Zennaro

## Unity and Disunity in Mathematics

Booss-Bavnbek, Bernhelm; Davis, Philip J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/12/2013 Português
Relevância na Pesquisa
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The unity of mathematics has its power to compactify experiences in a form capable of being transferred and modified or adapted to new mathematical situations. Yet, we believe that the phrase "Unity of Mathematics" expresses a dream, an ideal that doesn't exist. We shall point to diachronic and cross cultural disunities, to semantic, semiotic and philosophic ambiguities and to the non-acceptance of certain mathematical texts by some practitioners of the subject.; Comment: 4 pages. This is a condensed, sharpened and polemic outcome of our previous discussions on that topic, disseminated in arXiv:1204.5143

## Justifying Definitions in Mathematics---Going Beyond Lakatos

Werndl, Charlotte
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/10/2013 Português
Relevância na Pesquisa
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This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world-justification, condition-justification and redundancy-justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show that various kinds of justification can be found and can be reasonable, and they fail to acknowledge the interplay between the different kinds of justification.

## Experimental library of univalent formalization of mathematics

Voevodsky, Vladimir
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.

## Why I don't like "pure mathematics"

Runde, Volker
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/10/2003 Português
Relevância na Pesquisa
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An opiniated essay on what pure mathematics is and why the adjective "pure" in "pure mathematics" is not a good choice.; Comment: 4 pages; written for high school students and teachers

## Reverse mathematics and properties of finite character

Dzhafarov, Damir D.; Mummert, Carl
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.; Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse mathematics and equivalents of the axiom of choice", which has been abbreviated and divided into two pieces for publication

## Mathematics of Knowledge Refinement: Probabilistic Arithmetic, with no unknowns and no infinity. Part I. Generalized Probabilistic Arithmetic. Basic definitions and properties

Luboschinsky, Mikhail
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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An approach to build Probabilistic Arithmetic in which initial values of all correlated random variables are known, but with varying degrees of accuracy. As a result of the proposed Probabilistic Arithmetic operations, variable values, degrees of their accuracy and correlations are refined. Probabilistic Generalized Addition (PGA) and Probabilistic Generalized Multiplication (PGM) operations on correlated random variables are defined and their basic properties identified and described: \bullet Proposed PGA and PGM operations possess inverse operations - subtraction and division. \bullet There is no difference between direct and inverse operations: addition and subtraction, multiplication and division (this is why these operations are called "Generalized"). \bullet Division by approximately zero is possible and the result never equals to \infty, making this approach promising in computational and computer mathematics. \bullet Unlike the usual hyperbola, which, when the argument is changing from + \infty to 0, has a gap at 0, the Generalized Probabilistic Hyperbola, under certain combination of initial accuracies, is continuous at 0. First, as the argument changes from +\infty to 0, it behaves like a typical hyperbola, monotonically increasing. However...

## Current Mathematics Appears to Be Inconsistent

Li, Guang-Liang; Li, Victor O. K.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
We show that some mathematical results and their negations are both deducible. The derived contradictions indicate the inconsistency of current mathematics. This paper is an updated version of arXiv:math/0606635v3 with additional results and proofs.; Comment: 5 pages and 1 figure, updated version of math.GM/0606635v3 with additional results and proofs

## Generalized functions as a tool for nonsmooth nonlinear problems in mathematics and physics

Colombeau, J. F.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/12/2006 Português
Relevância na Pesquisa
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A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final result is often a classical function or distribution which is not solution in the classical or distributional sense. The aim of this text is to invite the readers in applying this tool in their own research without significant prerequisites by presenting its use on a sample of elementary applications from mathematics and physics. This text is the written version of a talk at the 2007 annual meeting of the AMS, NewOrleans, january 2007.; Comment: 11 pages;research-expository paper

## Tropical mathematics, classical mechanics and geometry

Litvinov, G. L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.; Comment: 21 pages, 4 figures

## Dequantization of Mathematics, idempotent semirings and fuzzy sets

Litvinov, G. L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/09/2012 Português
Relevância na Pesquisa
35.41%
In this talk a brief survey of basic ideas of Idempotent Mathematics is presented. Relations between this theory and the theory of fuzzy sets as well as the possibility theory and some applications (including computer applications) are discussed.; Comment: 6 pages, a talk for the 25th Linz Seminar on Fuzzy Set Theory Linz, Austria, 2004

## In whose mind is Mathematics an "a priori cognition"?

Berti, Massimiliano; Suarez, Antoine; Tarchini, Rocco
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/09/2008 Português
Relevância na Pesquisa
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According to the philosopher Kant, Mathematics is an "a priori cognition". Kant's assumption, together with the unsolvability of Hilbert's 10th problem, implies an astonishing result.; Comment: Philosophy of Mathematics, 11 pages

## Algebraic structures of tropical mathematics

Izhakian, Zur; Knebusch, Manfred; Rowen, Louis
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/05/2013 Português
Relevância na Pesquisa
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Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid $\tM$ is replaced by $R: = L\times \tM$, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the "standard" supertropical structure with its "ghost" layer. When $L = \NN$ we can describe multiple roots of polynomials via a "layering function" $s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination $\sum \a_i v_i$ is a ghost...

## Mathematics and group theory in music

Papadopoulos, Athanase
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/07/2014 Português
Relevância na Pesquisa
35.41%
The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.; Comment: To appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press

## What international studies say about the importance and limitations of using computers to teach mathematics in secondary schools

Silva, Jaime Carvalho e
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/05/2014 Português
Relevância na Pesquisa
35.41%
The use of technology in schools has been one of the most debated topics around mathematics education. In some countries there is a huge investment, in others there is a down-scaling. Malaysia decided in 2013 to put its 10 million students to use Google laptops and Google apps, while Australia in the same year decided it would not continue funding their own high school laptop program. Who is right from the educational point of view? The last major curriculum document written in the world to date, the Common Core State Standards (CCSS) in the United States, whose mathematics part is coordinated by the well known mathematician William McCallum, sets as one of its standards for mathematical practice: "Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software." Strong moves need substantiation from research, including the analysis of the existing situation in different countries. What does research say about the use of computers in schools in present time and the use of different pieces of software from spreadsheets to computer algebra systems?