9th November Conference
HISTORY OF MATHEMATICS
Nijmegen, Thursday October 29 -- Sunday November 1
Danny Beckers, Katja Peters, Carsten Vollmers
The ninth ``Novembertagung zur Geschichte der Mathematik'' (November Conference
on the History of Mathematics) took place from 29-10 until 1-11-1998 in
Nijmegen (The Netherlands). Although the weather was not optimally, the
conference took place in a friendly atmosphere. The talks were interesting and
provoked discussions, which often were continued at the dinner table.
The programme of the conference was
followed almost exactly. Only the stroll through Nijmegen ended up in a caffee
quite soon due to the atmospheric conditions. Abstracts, as far as they are
available, may be found at the bottom of this page.
Financial contributions were made by the Dutch Society for the History and
Social Function of Mathematics (GMFW)
and the Nijmegen Department of Mathematics. The
Mathematics in Nijmegen also served as the host
of this years conference by granting us access to their conference facilities.
A final contribution was made by the Nijmegen Mathematics
Student-association (Desda), who
kindly provided us with their coffee-machine, thus taking care of the
coffee-supply on sunday.
On friday the conference took place in the deans room of the Nijmegen science
department: a very luxerious room with very comfortable chairs, nice view from
the fourth floor down upon the garden of the science department, lots of space
and an awfull tapistry at the wall. On saturday, due to security regulations
within the building, this room was not available and we used one of the
ordinary rooms on the ground level, with chairs less comfy etc. Although it
suited our status much, much better, several of us were already longing back to
the ``good old days''. Wenn on sunday (in this same room), due to the holidays
of the canteen personel, the coffee came no longer in real cups, but had to be
consumed from plastic bekers, this provoked one of the participants (who
probably wants to remain anonymous) to the remark ``that in some way a
continuous steadily decreasing function could be spotted in the quality of the
facilities of this years conference''. The way in which this remark was made
(teasing in a friendly way) characterises the atmosphere of the 1998
conference in Nijmegen.
Discovering the discovered integral: William Henry Young und das
Elisabeth Mühlhausen (Berlin)
In 1902 Henri Lebesgue (1875-1941) published his thesis containing a new
theory of integration which was based on Borel's theory of measure.
Independently of this William Henry Young (1863-1942) together with his
wife Grace Chrisholm Young (1868-1944) developed a similar theory of
measure and integration. Only after submitting their papers on this subject
to the London Mathematical Society did they learn about Lebesgue's results.
Consequently the Youngs decided to publish a revised version in which the
concept of Lebesgue was taken into consideration and discussed.
This parallel discovery will be analysed both from a mathematical and a
psychological point of view. The previously unpublished primary sources
from the private correspondence of the Youngs will be used to illuminate
the collaboration between the Youngs and their reaction to Lebesgue.
Theorems with exceptions and the use of proof
Henrik Kragh Sørensen (Århus)
In the early 19th century the concepts of theorem and proof in analysis
differed from our present day standards in two related fundamental ways:
The proofs used other standards of rigour, and theorems admitting
exceptions were not uncommon. The standards of proof depended on the
situation in which they were needed and almost incommensurable standards
were applied in textbooks and state-of-the-art research. In the prehistory
of proof analysis and in the earliest counter-examples in analysis
interesting fields of historical enquiry emerge which can shed light on
both the mathematical practice of the time and the general formation and
evolution of mathematical concepts. I will focus on a famous footnote by
Abel concerning an exception to a theorem of Cauchy and analyse the status
of theorems with exceptions at the time of Abel.
Developmental research on pre-algebra. A proposed learning trajectory from
arithmetic to elementary algebra for 11 to 14 year old students
Barbara van Amerom (Utrecht)
Many secondary school students experience great difficulty in learning how
to construct and solve equations. Arithmetic and elementary algebra appear
to be a world apart: the former deals with straightforward calculations
with known numbers, the latter requires reasoning about unknown or variable
quantities and recognizing the difference between specific and general
situations. In recent years, research has been done on the difficulties
students have with translating arithmetic word problems into algebraic
equations. In the transition from arithmetic to algebra there is claimed to
be a discrepancy called epistemological break point (Chevallard 1991) or
didactical cut (Filloy & Rojano 1989), hampering manipulations of algebraic
expressions. A good starting point for an investigation into this issue
could be a return to the roots. By looking into the past we can gain
insight into the differences and similarities between arithmetic and
algebra and learn from the experiences of others. Recent research on the
advantages and possibilities of using and implementing history of
mathematics in the classroom has led to a growing interest in the role of
history of mathematics in the learning and teaching of mathematics.
Streefland emphasized the value of ``reciprocal shifting'' changing one's
point of view, looking back at the origins in order to anticipate. Such a
change of perspective can propel the learning process of the researcher,
the teacher and the student alike. Inspired by Streefland's work as well as
the HIMED (History in Mathematics Education) movement, a developmental
researchproject called ``Reinvention of Algebra as started at the
Freudenthal Institute in 1995 to investigate which didactical means will
enablestudents to make a smooth transition from arithmetic to
elementaryalgebra. Specifically, the ``invention'' of algebra from a
historical perspective will be compared with possibilities of
``re-invention'' by the students.
``Reinvention of Geometry'',
a study into the use and value of history in teaching geometry.
Klaske Blom (Groningen)
Several arguments are being used in the discussion on the use and value of
history of mathematics in mathematical classrooms in general and in
teaching geometry specifically. These arguments correspond with different
ways in which the history of mathematics can be used in classrooms. I
intend to give first a survey of the recent literature pertaining to this
discussion. (In August 1998 I just started my PhD-project, so I would
appreciate it if you supplement this survey with other sources). In my view
the arguments in favour of the didactical use of the history of mathematics
split into three categories: moral, didactical and mathematical arguments.
A second point I want to address is the position of the teachers who want
to use the history of mathematics in classroom. How do they find useful
material? Do teachers know enough about the history of mathematics and
where to find and how to use primary sources to teach their pupils? Can
historical articles and books of historians of mathematics be usefully
applied by teachers in their lessons? It seems to me that there is a gap
between the two groups, i.e. historians and teachers and that they have a
difficulty in reaching each other. In general, historians think in terms of
mathematical and historical correctness and rigour; teachers look for
practical ideas that they can use. In the talk I shall discuss this
phenomenon by studying examples of material used in teaching geometry.
Afterwards I would like to discuss with you the responsability of
historians in writing historical texts, useful for an educational setting.
Minimum Spanning Tree: A Neglected Paper
Helena Durnová (Brno)
In this talk, I would like to show the relationship between different
papers dealing with the MST problem. The Borùvka's algorithm will be
described at greater length. (The reason is the language of origin.) I
shall also mention other algorithms and try to describe shortly their
method (Kruskal, Prim, Dijkstra). The connection of the MST problem with
cluster analysis will also be considered.
Die Mathematisierung von Brot und Bier im Alten Ägypten
Annette Imhausen (Hochheim)
Innerhalb der altägyptischen mathematischen Aufgabentexte nehmen die
sogenannten Brot- und Bieraufgaben aufgrund ihrer Häufigkeit einen
herausragenden Platz ein. Die dort verwendeten Anweisungen und die
zugehörigen Rechnungen sind für einen heutigen Bearbeiter nicht ohne
weiteres zu verstehen, für eine der vorkommenden Formulierungen konnte
bisher keine befriedigende Deutung gefunden werden. Eine Berücksichtigung
nichtmathematischer Texte ermöglicht eine neue Interpretation.
A History of the Minimax Theorem: a journey through different mathematical
Tinne Hoff Kjeldsen (Roskilde)
In his book Five Golden Rules: Great Theories of 20th-Century
Mathematics --and Why They Matter John L. Casti includes the minimax
theorem in game theory among ``five of the finest achievements of the
mathematician's art in this century''. In the talk I will follow the history
of the minimax theorem from the first proof by John von Neumann in 1928
until 1944 when the proof appeared in a quite different form in the famous
book Theory of Games and Economic Behavior by von Neumann and Oscar
Morgenstern. It will be a story about the minimax theorem in different
mathematical contexts. The main focus will be on von Neumann's contribution
and I will also touch upon the relation to an extension of Brouwers
``Die Mathematisierung der Getriebenkinematik im 19ten Jahrhundert''
Teun Koetsier (Amsterdam)
Die Mathematisierung der Getriebenkinematik im 19ten Jahrhundert ist ein
Prozess der auf der einen Seite eng zusammenhängt mit der Industriellen
Revolution und auf der anderen Seite mit Entwicklungen in der Geometrie.
Ich werde das zeigen an Hand der Problematik der Geradfuehrungen.
Norbert Wiener und die Mathematisierung der Brownschen Bewegung
Carolin Kosiol (Mainz)
Mit dem Artikel `Differential Space' gelang Wiener 1923 der Durchbruch in
seiner Mathematisierung der Brownschen Bewegung. Der Vortrag beschäftigt
sich mit der Frage, inwieweit sich Einflüsse aus der Physik, insbesondere
die Arbeiten Einsteins und Perrins, in den Beweisideen und -strukturen des
Artikels `Differential Space' wiedererkennen lassen.