How to get rid of quantifiers?

Quantifiers_5Originating authors are Reinhard Oldenburg and Michele Artigue.

How do computer packages do abstract algebraic problems such as proving statements “for all x” or finding whether a Real Number x 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 P(x)=x^2+cx+c have two distinct real roots ? », and algorithmically get the answer: c<0 or c>4. Doing so, they have in some sense, found a way of transforming the sentence expressing the problem (\exists x_1 \, \exists x_2 \, (x_1\not= x_2 and P(x_1)=0 and P(x_2)=0) involving the two existential quantifiers ‘there exists x_1’ and ‘there exists x_2’ into the sentence c<0 or c>4 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

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Featured Book of the Month: Mathematical Models

Mathematical Models

September’s featured book of the month is “Mathematical Models” by H.M. Cundy and A.P. Rolett.

This classic was first published in 1952 by Oxford University Press, but was republished in paperback by Tarquin in 1981.

As well as nets polyhedra, it has a wide variety of linkages and dissections, as well as several mechanical models.

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Site of the Month: Bridges

Bridges Org

September’s site of the month: Bridges

Bridges is an organisation that oversees the annual Bridges conference on Mathematics and Art. It contains images and resources of many different kinds of artistic representations, from poetry to models, from dance to origami, from juggling to painting. The -Resources- link on the homepage contains links to other websites of interest to teachers.

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Featured Book of the Month: Indra’s Pearls – The Vision of Felix Klein

Indra's_Pearls_book_coverAugust’s featured book of the month is “Indra’s Pearls: The Vision of Felix Klein” by David Mumford, Caroline Series and David Wright.

Wikipedia says:
“The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity.

The book’s title refers to Indra’s net, a metaphorical object described in the Buddhist text of the Flower Garland Sutra. Indra’s net consists of an infinite array of gossamer strands and pearls. The frontispiece to Indra’s Pearls quotes the following description:

In the glistening surface of each pearl are reflected all the other pearls … In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.

The allusion to Felix Klein’s “vision” is a reference to Klein’s early investigations of Schottky groups and hand-drawn plots of their limit sets. It also refers to Klein’s wider vision of the connections between group theory, symmetry and geometry.”

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Featured Book of the Month: Mathematical Puzzles and Diversions

Gardner

July’s featured book of the month is “Mathematical Puzzles and Diversions” by Martin Gardner.

This book, which was originally published by Simon & Schuster in 1959, and later by University of Chigaco Press in 1988, is the first of several collections of Martin Gardner’s column in the Scientific American. These books represent a very small proportion of the total writings of Martin Garnder — for a full bibliography go to Martin Gardner’s website.

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Site of the Month: Mathematikum

Mathematikum_Gebaude

August’s site of the month: Mathematikum

This is the website of the Mathematics Museum in Gießen — it includes models and activities of interest to people of all ages.

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Site of the Month: Images des Maths

image-1-2

Verhulst dynamics, by Jean-François Colonna

July’s site of the month: Images des Maths

This site is affiliated to the French National Center for Scientific Research. It contains reports about the latest events in mathematical research, and up-to-date articles that contain interesting discussions and applications of mathematics (such as a modelling of the Ebola virus outbreak). The articles are written in concise and comprehensive fashion.

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The Revenge of the Infinitesimals

Originating author is Michèle Artigue.

Infinitesimals played an essential role in the emergence and development of differential and integral calculus. The evident productivity of this calculus did not prevent recurrent and fierce debates about the nature of these objects and the legitimacy of their use. At the end of the 19th century, when the construction of real numbers from integers and the modern definition of the concept of limit provided a solid foundation for differential and integral calculus, infinitesimals and the associated metaphysics was rejected and their use perceived synonymous with bygone and poorly rigorous practices. However, the language of infinitesimals continued to be used, for example in physics and even in mathematics. It never completely disappeared from the informal discourse and heuristic thinking of a number of researchers.

Is this language thus really incompatible with mathematical rigour? What does it offer that is interesting and specific, which explains its permanence? Non-Standard Analysis developed in the 20th century and provided answers to these questions and enabled infinitesimals to take their revenge.

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Featured Book of the Month: Elementary Mathematics from an Advanced Standpoint

From now on we will feature a different book every month that is likely to be of interest to secondary teachers wanting to know more about mathematics. We have made the decision to use this feature to bring older books to the attention of a new generation of teachers (rather than to add to recent book promotions). All books must therefore be older than 10 years.
For our first featured book we turn to Felix Klein’s original works that stimulated the Klein Project, his three volume work Elementary Mathematics from an Advanced Standpoint. Only two volumes have been published in English, although all three are available in German, and have been published in Portuguese. It is exactly ten years since Dover reprinted the English versions.
These books are essentially Klein’s own notes for a series of lectures he gave to graduates of mathematics preparing to become teachers in the gymnasium’s of the time. Of course, Klein’s books discuss mathematics that is more than 100 years old (they were first published in German in 1908), but remain extraordinarily relevant for today’s world.

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Site of the Month: Il Giardino di Archimede

Archimede.

This site of a museum in Florence not only encourages us to visit that beautiful city, but also contains materials for schools. It is easily navigated by those who do not speak Italian.
Il Giardino di Archimede

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