Receive notice of every new vignette.
Search
Meta
متتاليات غودشتاين : Goodstein قوة الالتفاف عبر اللانهاية
Publié dans غير مصنف
Laisser un commentaire
La revancha de los infinitésimos
Autor original: Michèle Artigue.
Traducción: Tomás Recio, Universidad de Cantabria.
Los infinitésimos han jugado un papel clave en la emergencia y desarrollo del cálculo diferencial e integral. La evidente productividad de este cálculo no ha impedido que se produzcan debates recurrentes y, a veces, feroces sobre la naturaleza de estos objetos (los infinitésimos) y sobre la legitimidad de su uso. A finales del siglo XIX, cuando la construcción de los números reales a partir de los números enteros y la definición moderna del limite proporcionaron fundamentos sólidos al cálculo diferencial e integral, los infinitésimos, y la metafísica que los rodeaba, fueron rechazados y su uso se volvió sinónimo de prácticas poco rigorosas, ya superadas. Sin embargo el lenguaje de los infinitésimos siguió siendo utilizado, por ejemplo, en la física; e incluso en las matemáticas nunca desapareció por completo del discurso informal, del pensamiento heurístico de muchos investigadores.
¿Es, pues, este lenguaje realmente incompatible con el rigor matemático? ¿Qué es lo que ofrece, interesante o específico, que explique su permanencia? El Análisis no estándar, que se desarrolló en el siglo XX, ha permitido responder a estas preguntas y, a los infinitésimos, tomarse su revancha.
Publié dans Vignettes @es
Laisser un commentaire
Libro del Mes: Cuentos con Cuentas
El libro del mes de octubre es Cuentos con Cuentas, por Miguel de Guzmán (Labor, Barcelona, 1984).
También está publicado en inglés bajo el título The Countingbury Tales, traducido por Jody Doran y publicado por World Scientific (2000).
Miguel de Guzmán escribió libros similares, el mas conocido es Aventuras Matemáticas, que también está traducido al chino, finlandés, francés y portugués.
Publié dans Español
Laisser un commentaire
A revanche dos infinitesimais
A autora original é Michèle Artigue. Traduzido por Pedro Malagutti (UFSCar, Brasil) e Cydara Cavedon Ripoll (UFRGS, Brasil).
Os infinitesimais desempenharam um papel essencial no surgimento e desenvolvimento do cálculo diferencial e integral. O ganho evidente com esse cálculo não impediu, contudo, recorrentes debates sobre a natureza de tais objetos e a legitimidade de seu uso. Ao final do século XIX, quando a construção dos números reais a partir dos inteiros e a definição moderna da noção de limite forneceram as fundações sólidas ao cálculo diferencial e integral, os infinitesimais e a metafísica que os cerca foram rechaçados, e o seu uso passou a ser percebido como sinônimo de práticas pouco rigorosas e doravante então encerradas. No entanto, a linguagem dos infinitesimais continuou sendo utilizada, por exemplo, na física e, mesmo na matemática, ela nunca desapareceu completamente do discurso informal e do pensamento heurístico de muitos pesquisadores.
Afinal, é esta linguagem realmente incompatível com o rigor matemático? E o que ela oferece de interessante, de específico que explica sua permanência?
A análise não standard desenvolvida no século XX permitiu responder estas questões, e, aos infinitesimais, permitiu terem a sua revanche.
Publié dans Vignettes @pt-br
Laisser un commentaire
Mathematics Today
October’s site of the month: Mathematics Today
Mathematics Today is a magazine published by the UK Institute of Mathematics and its Applications.
It is a general interest magazine about mathematics intended for those interested in the subject (by christopher). Much material is UK-specific, however it contains many general articles that are downloadable from the website.
—————————————————————————————————————————————-
Publié dans Site of the Month
Laisser un commentaire
Cuentos Con Cuentas
October’s Book of the Month is Cuentos Con Cuentas by Miguel de Guzman (Labor, Barcelona, 1984).
This is also published in English under the title The Countingbury Tales, translated by Jody Doran and published by World Scientific (2000).
Miguel de Guzman wrote other similar books, the most well known being Aventuras Matematicas which is also published in Chinese, Finnish, French and Portuguese.
Publié dans Book of the Month
Laisser un commentaire
Trying to predict a floating leaf: chaos and predictions
Originating author is César R. de Oliveira, Universidade Federal de São Carlos.
What path will a leaf follow floating down a turbulent stream? Is it even possible to make a mathematical model that will predict such motion? Is this the same sort of problem as predicting the path of planets as they move round the sun? Even when we know all the rules governing the motion of an object, and can determine precisely the initial conditions, it turns out that some motions can be predicted and some cannot. And it is not just a matter of complexity: we can model unpredictable systems with very simple equations. In this Vignette we illustrate mathematically the existence of chaotic dynamical systems using the decimal form of real numbers. You will see how the unpredictability can be simply generated. One of the main goals of theoretical models is to make (good) predictions. However, there are deterministic dynamical systems that in practice are unpredictable; they are the so-called chaotic systems. The aim of this text is to discuss how this unpredictability is generated, and the main tool here will be the decimal representation of real numbers.
How to get rid of quantifiers?
Originating authors are Reinhard Oldenburg and Michele Artigue.
How do computer packages do abstract algebraic problems such as proving statements “for all 
” or finding whether a Real Number with certain conditions exists?
Recent advances draw on theorems in mathematical logic, as well as improvements in computing.
High school students learn how to solve problems such as the following: « For what values of the real number c, does the polynomial 
have two distinct real roots ? », and algorithmically get the answer: 
or 
. Doing so, they have in some sense, found a way of transforming the sentence expressing the problem 
and 
and 
involving the two existential quantifiers ‘there exists 
’ and ‘there exists 
’ into the sentence or which no longer includes quantifiers. For what kind of problems is this possible, theoretically but also practically with an effective computer program? In 1938, thanks to a theorem of elimination of quantifiers proved by the logician Alfred Tarski, a decisive step was achieved regarding these questions, but this was not at all the end of the story… READ MORE
Bridges
Septembers Seite des Monats ist: Bridges
Bridges ist eine Organisation, die die jährliche Bridges Konferenz über Mathematik und Kunst leitet. Die Seite beinhaltet viele Bilder und Quellen verschiedenster künstlerischer Darstellungen: von Poesie bis zu Modellen, von Tanz bis zu Origami, von Jonglieren bis zur Malerei. Der Reiter -Resources- auf der Homepage enthält Links zu anderen Seiten, die für Lehrer von Interesse sind.
Publié dans Site of the Month @de
Laisser un commentaire