Bar Ilan University IMU Session on History and Philosophy of Mathematics Date: monday, 28 may '12, at Bar Ilan building 507 Time: 2:30 pm Leo Corry (Tel Aviv University) Axiomatics Between Hilbert and the New Math: Diverging Views on Mathematical Research and Their Consequences on Education Abstract: David Hilbert is widely acknowledged as the father of the modern axiomatic approach in mathematics. The methodology and point of view put forward in his epoch-making Foundations of Geometry (1899) had lasting influences on research and education throughout the twentieth century. Nevertheless, his own conception of the role of axiomatic thinking in mathematics and in science in general was significantly different from the way in which it came to be understood and practiced by mathematicians of the following generations, including some who believed they were developing Hilbert's original line of thought. The topologist Robert L. Moore was prominent among those who put at the center of their research an approach derived from Hilbert's recently introduced axiomatic methodology. Moreover, he actively put forward a view according to which the axiomatic method would serve as a most useful teaching device in both graduate and undergraduate teaching mathematics and as a tool for identifying and developing creative mathematical talent. Some of the basic tenets of the Moore Method for teaching mathematics to prospective research mathematicians were adopted by the promoters of the New Math movement. A different thread of ideas leading indirectly from Hilbert to the New Math is the one going through the Bourbaki group. Like Moore, Bourbaki interpreted Hilbert's view on the role of axiomatics in a way that deviated from Hilbert's own, and their views were strongly influential on the rise of the New Math movement. The talk traces the genealogy of ideas leading from Hilbert to the New Math via Moore and via Bourbaki. Alexandre Borovik (University of Manchester) Protomathematics, metamathematics, and some challenges in philosophy of mathematics Abstract: For a number of years I was collecting stories (told by adults) about challenges and difficulties they have encountered in their your early learning of mathematics; some of these stories are analysed in my draft book "Shadows of the Truth", <http://www.maths.manchester.ac.uk/~avb/ST.pdf> . They provide a fascinating insight into the psychology of mathematical thinking and frequently lead to deep mathematics -- but they also show how difficult it could be to describe and analyse obstacles to understanding of mathematics. Unfortunately, it appears that neither the research mathematics community, nor mathematical educationalist possess an adequate language for description of the real mathematical content of basic mathematical activities. In particular, I argue that classical approaches to the matter (such as, for example, the one developed in Hans Freudenthal's classical treatise "Didactical Phenomenology of Mathematical Structures") would benefit from injection of some concepts of (very elementary) category-theoretic nature and some (pretty basic) ideas from computer science. The talk will focus on principal challenges in development of such a language for description of actual mathematical thinking, many of which have deep philosophical roots. Carl Posy (Hebrew University) Category Theory and Mathematical Intuition Abstract: The notion of mathematical intuition was made prominent by Kant as part of his overall account of how mathematics supports empirical science. The notion was adopted by some of the 20th century foundational movements and used to ground their versions of mathematical knowledge and their theories about mathematical objects. However this philosophical notion is not the same as the common sense meaning of "intuition", and it does not conform to mathematical practice. We will propose a more natural notion of mathematical intuition, one adapted to mathematical practice. We will suggest that category theory codifies this notion, and that the relation between category theory and mathematical practice in fact captures some of the same aspects as the 'foundational' notion, but in a more natural setting. Joint work with David Kazhdan. Michael Fried (Ben Gurion University) Opposite and Conjugate Sections in Apollonius' Conica Abstract: Apollonius of Perga was born sometime in the middle of the 3rd century BCE. The exact dates are uncertain, but what can be said without a doubt is that Apollonius was much admired as a mathematician in his own time and afterwards, and this was mostly on the basis of his great work, the Conica. As his commentator from late antiquity, Eutocius of Ascalon (c.480-540 CE), tells us, it was "because of the remarkable theorems [Apollonius] proved about conics he was called the Great Geometer." The Conica, as Apollonius himself describes it, includes fundamental theorems and constructions related to diameters, tangents and asymptotes of conic sections, an account of how conic sections "touch and intersect one another," "minima and maxima of lines in conic sections," the similarity and equality of conic sections, and relations between conjugate diameters. So much of what Apollonius proved about conic sections can be shown to correspond to what one proves in a thorough course in analytic geometry that it is tempting to view Apollonius' work as a kind of proto-analytic geometry text, algebraic in spirit though geometric in style. But there are aspects of the work make this view of the Conica as a modern text in ancient form problematic. Among those things refractory to modernization is that most peculiar entity in the Conica, the opposite sections. In this talk, I will discuss the status of the opposite sections in the Conica as a mathematical object and show, among other things, why their status, why their place in the Conica, is different from that of the conjugate sections. I hope to persuade you that not only is the style of the Conica geometric, but so is its spirit as well. For additional details see <http://imu.org.il/Meetings/IMUmeeting2012/program.html#20> --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Colloquium no-reply <colloquium@macs.biu.ac.il>