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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Images des Maths
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.
La revanche des infinitésimaux
Vignette écrite par Michèle Artigue.
Les infinitésimaux ont joué un rôle essentiel dans l’émergence et le développement du calcul différentiel et intégral. La productivité évidente de ce calcul n’empêcha pas cependant des débats récurrents et parfois féroces sur la nature de ces objets et la légitimité de leur usage. A la fin du 19ème siècle, quand la construction des nombres réels à partir des entiers, la définition moderne de la notion de limite, eurent fourni des fondations solides au calcul différentiel et intégral, les infinitésimaux et la métaphysique qui les avait entourés furent rejetés et leur usage perçu synonyme de pratiques peu rigoureuses désormais révolues. Pourtant, le langage des infinitésimaux continua à être utilisé par exemple en physique et, même en mathématiques, il n’a jamais complètement disparu du discours informel, de la pensée heuristique de nombre de chercheurs.
Alors ce langage est-il réellement incompatible avec la rigueur mathématique ? Et qu’offre-t-il d’intéressant, de spécifique qui explique sa permanence ? L’Analyse non standard développée au 20ème siècle a permis de répondre à ces questions, et aux infinitésimaux de prendre leur revanche.
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Cryptographie à clé publique
Vignette écrite par Graeme L. Cohen (University of Technology, Sydney), Steven Galbraith (University of Auckland) et Edoardo Persichetti (University of Auckland).
How can we safely send our credit card details over the internet, or using a mobile phone, when others can intercept our messages? How can we trust software updates, when we know that computer viruses are common? Cryptography (the study of techniques for secure communication in the presence of adversaries) provides answers to these questions, and mathematics provides its foundations.
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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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