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>
 
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