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

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

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

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

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## Recursive Synthesis and the Foundations of Mathematics

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

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## Idempotent mathematics and interval analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Addressing the underrepresentation of women in mathematics conferences

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/02/2015
Português

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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)

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## Illustrating Mathematics using 3D Printers

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

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## Unity and Disunity in Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/12/2013
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/10/2013
Português

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#Mathematics - History and Overview#Mathematics - Dynamical Systems#Nonlinear Sciences - Chaotic Dynamics#Physics - History and Philosophy of Physics

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.

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## Experimental library of univalent formalization of mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/10/2003
Português

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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

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## Reverse mathematics and properties of finite character

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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...

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## Current Mathematics Appears to Be Inconsistent

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/12/2006
Português

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#Mathematical Physics#Mathematics - General Mathematics#35D05,35D10,35L67,35Q72,35R05,46F30,65M99,73J30,82D20,83C57

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

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## Tropical mathematics, classical mechanics and geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Rings and Algebras#Mathematics - Metric Geometry#81Q20, 14M25, 16S80, 70H20, 14T05, 46S10, 51P05, 52A20

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

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## Dequantization of Mathematics, idempotent semirings and fuzzy sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/09/2012
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2008
Português

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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

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## Algebraic structures of tropical mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/05/2013
Português

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#Mathematics - Rings and Algebras#Mathematics - Commutative Algebra#06F20, 11C08, 12K10, 14T05, 14T99, 16Y60

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...

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## Mathematics and group theory in music

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/2014
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/05/2014
Português

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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?

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