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Topic: [HM] Historia Matematica: our mailing list
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Alfred Ross

Posts: 56
Registered: 12/3/04
Re: [HM] CSHPM talks
Posted: Jun 19, 2000 1:29 PM
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<quote>
History of Mathematics at the Dawn of a New Millennium /
L'histoire des mathematiques a l'aube d'un nouveau millenaire
(Tom Archibald, Organizer)


FRANCINE ABELES, Kean University, Union, New Jersey, USA
Game theory and politics: a note on C. L. Dodgson

In 1884, C. L. Dodgson wrote The Principles of Parliamentary
Representation where he showed, for the first time, that in an
election if each of two political parties utilizes a maximin
strategy in a two person zero sum game model, the result would
be a "best" electoral system in the sense of providing the
greatest degree of voter representation.


AMY ACKERBERG-HASTINGS, Iowa State University, Iowa, USA
The semi-secret history of Charles Davies

Charles Davies (1798-1876) and his textbooks invariably show up
in discussions touching on mathematics education in the United
States in the nineteenth century. Yet, due to the generality, age,
or inaccessibility of most of even the relevant secondary sources,
many mathematicians and historians have been unable to learn the
reasons why Davies exerted such a pervasive influence. Thus, this
paper draws upon the first modern biography of Davies to outline
his life and career. The paper will also note some of the
perennial themes in mathematics education manifested in Davies'
textbook series.


CHRISTOPHER BALTUS, State University of New York, Oswego,
New York, USA
Gauss's algebraic proof of the fundamental theorem of algebra

Gauss offered a second proof of the Fundamental Theorem of Algebra,
some years after his 1799 proof, resting on `purely analytic
principles'-we would now call it algebraic-but avoiding the
fundamental defect of previous proofs, the `supposition that the
function can be resolved into simple factors.' Gauss's argument
has been interpreted in a variety of ways, including the claim by
Bachmacova (1960) that he created a field of decomposition for the
polynomial along lines that Kronecker would later follow. I propose
to simply reexamine Gauss's original argument, in its original
context.


EDWARD COHEN, University of Ottawa, Ottawa, Ontario
The leap-year problem has not gone away

In Julius Caesar's time (45BCE), it was thought that the length
of a year was 365 1/4 days; i.e., Caesar made one leap year
every 4 years or 100 years in 400 years. Pope Gregory XIII
in 1582 changed that to 97 years in 400 years because the
astronomers in his time had a more accurate picture of how
long a period it took for the earth to go around the sun. This
is more precise; however, in the year 2000CE, the Gregorian
calendar overstates the length of the year by approximately
26 seconds. This means that in approximately 1582 + 3330 years,
we would have to drop one leap year. Also, the length of the
solar year is not a constant. This further complicates the
situation. One of the first astronomers to consider the leap-year
problem was Simon Newcomb (1835-1909). We try to state the
problem as he saw it and consider how it might be solved.


TOM DRUCKER, Modern Logic Inc.
Language, truth, and logic in Russell's mathematics

Bertrand Russell's distaste for ordinary language philosophy was
often expressed, and he sought a language with which to express
mathematics in an incorrigible fashion. This talk will look at
Russell's idea of mathematical truth in the first decade of the
1900's and the forms of linguistic expression this truth could
take. His views will be viewed against the background of the
algebraic tradition in logic as well as the subsequent
model-theoretic lines of research.


CRAIG FRASER, University of Toronto
Hilbert's Grundlagen der geometrie and its relation to Euclid's
elements

Hilbert's Grundlagen der Geometrie (1899) is widely regarded as
a canonical work of modern mathematics. Howard Eves and Carroll
V. Newson write, "By developing a postulate set for plane and
solid geometry that does not depart too greatly in spirit from
Euclid's own, and by employing a minimum of symbolism, Hilbert
succeeded in convincing mathematicians, to a far greater extent
than had Pasch and Peano, of the purely hypothetico-deductive
nature of geometry" (Foundations and Fundamental Concepts of
Mathematics (1966, p. 94), my emphasis). The origins and
historical influence of Hilbert's book have been explored in the
writings of Michael Toepell and Leo Corry. The purpose of the
present paper is to provide a comparative study of the Grundlagen
and Elements I-VI in order to elucidate points of similarity and
difference in approach, concept and outlook between the two works.
The paper also explores the meaning of deduction in the modern
mathematical tradition.


ROGER GODARD, Royal Military College
Interpolation theory at the dawn of a new millennium: an
historical approach

This paper comments on the old interpolation problem, and
particularly, the contribution of the twentieth century.
However, we shall start our study with Lagrange (1792-1793).
Here we emphasize the mathematical, numerical, and philosophical
problems, and the evolution of proofs linked to interpolation.
After the discovery of the Runge phenomenon (1901), the problem
of approximation of equidistant data by polynomials was better
understood (de la Vallie-Poussin, Bernstein, Montel). In this
work, we also present trigonometric interpolation, which
represents efforts towards the development of orthogonal functions
and generalized Fourier series, and certain classes of functions,
which can't be developed in Taylor's series. We reexamine the
tools of the Fourier transform in the interpolation theory, and
the sampling theorem (Wiener, 1934; J. M. Whittaker, 1935;
Shannon, 1950) which is also a global interpolation formula.
Schoenberg (1946) tried to divide the problem of interpolation
into two categories. He distinguished the case where the ordinates
belong to known analytical functions and the case where the
ordinates come from empirical observations. In the later case,
he suggested the spline interpolation.

This truly empirical approach was to be extremely fruitful for
the Theory of Approximation, Schoenberg did not realize that he
transformed the problem of interpolation into a problem of
linear algebra, where the results are obtained by the numerical
solution of a sparse system of linear equations, which is very
stable.


HARDY GRANT, York University, York, Ontario
Greek mathematics in cultural context

In classic Greece the claims of various pursuits, including
mathematics, to the status of an "art" or a "science", with
the attendant prestige, were much in the air. I shall try to
set mathematics in this context by considering its possible
influences on, and influences from, its rivals in the
competition-two such rivals in particular.


NICHOLAS GRIFFIN, McMaster University
Russell's logicism is not `If-Thenism'

At the beginning of Principles of Mathematics Russell defines
pure mathematics in terms of a special set of conditional
statements. This remark, together with other things he says
about geometry and rational mechanics, have led many people
to suppose that his logicism is a type of "if-thenism"-that
is, that the theses of his logicism are all of the form "if p
then q" where p is a logical axiom and q a mathematical theorem.
I show that this is not the case and that the conditional form
that Russell insists on at the beginning of the Principles is
imposed for quite different reasons, having to do with the
unrestricted nature of the variables.


AGNES KALEMARIS, SUNY Farmingdale, Farmingdale, New York, USA
Grace Murray Hopper was a mathematician

Grace Murray Hopper (1906-1992) had justly received recognition
for her pioneering work in computer science. However, she began
her career as a mathematician, earning a bachelor's degree in
mathematics and physics at Vassar in 1928, and a masters and
Ph.D. at Yale in 1930 and 1934, respectively. This paper will
discuss some of her accomplishments in mathematics and her
unconventional methods of teaching it.


ISRAEL KLEINER, York University, York, Ontario
Aspects of the evolution of field theory

I will discuss highlights of the evolution of field theory,
including some of its sources and its emergence as a mature,
abstract theory.


ERWIN KREYSZIG, Carleton University, Ottawa
"Modern" starts

This paper concerns the roots and the early period of "modern
mathematics", a short term for the mathematics of the twentieth
century, as opposed to the "classical mathematics" of the
nineteenth century. It explores the principal reasons for the
main differences of modern mathematics from classical, in both
form and content. This includes advances in formalization,
axiomatization, and the emphasis of structures, as well as the
appearance of totally new areas, mainly topology (general as well
as algebraic), functional analysis and algebra (as in van der
Waerden's classic and beyond). We shall concentrate on the first
two of these three areas and show their closely related evolution,
whose systematic beginnings are usually considered to be marked
by Volterra's work on special functionals in 1887, Poincare's
introduction of combinatorial complexes in 1895, and Frechet's
and F. Riesz's (independent) works on abstract spaces in 1906.
It will be shown that the period from 1880 to 1915 (roughly) had
transitional character in the sense that areas and their problems,
mainly in the calculus of variations, spectral theory, and
integral equations, that motivated and paved the way in functional
analysis, were developed by means of classical analysis.

The title of the paper is borrowed from a contemporary exhibition
in the Museum of Modern Art in New York City, and the paper will
be concluded with a few comparative remarks on analogies and
differences.


SHARON KUNOFF, C.W. Post Campus, Long Island University, New York, USA
A commentary on the first Hebrew geometry and its relationship
to the first arabic geometry

In 1932 Solomon Gandz published a version of the Mishnat Ha
Middot containing a fragment which had recently come to light.
This new data enabled him to date the text c. 150. The Geometry
of Al-Khowarizmi C. 820 contains much of the same facts, with
some additions. In this paper we will look at some material from
each and compare the results. We will also consider the practical
nature of the material, seeing how it was written to demonstrate
how to do various geometric computations, rather than as a
theoretical theorem-proof geometry.


GREGORY LAVERS, University of Western Ontario
Goedel, Carnap and Friedman on analyticity

Michael Friedman, in his paper "Analyticity and Logical syntax:
Carnap vs. Goedel", admits that his previous argument concerning
the success of Carnap's project in The LogicalSyntax of Language
was mistaken. He admits, as well, that although he developed
this argument independently it is essentially the same argument
that Goedel put forward in the paper that he wrote for, but was
not included in the Schlipp volume on Carnap. However, he
maintains that the Goedelean argument, although not in itself fatal
to Carnap's program, can be used to point out what is viciously
circular in this program. In this paper I show that Goedel's
argument does point to a flaw in Carnap's project, if one is
willing to accept Goedel somewhat mystical presuppositions. I then
show that Friedman's case against Carnap, which can be seen as
a demystification of Goedel's argument fails to address Carnap.


DAVID LAVERTY, University of Western Ontario
Kit Fine's theory of variable objects

In his Cantorian Abstraction: A Reconstruction and Defense, Kit
Fine argues that if we treat the units resulting from a Cantorian
abstraction as `variable objects' we avoid the traditional
problems associated with Cantorian abstraction and we are left
with an account of number which rivals the competing accounts of
Zermelo/von Neumann on one side, and Frege/Russell on another.
According to Fine, the Zermelo/von Neumann account has the
advantage of being representational, but suffers from being
arbitrary, whereas the Frege/Russell account, while not being
arbitrary, nonetheless is not representational. Fines
reconstruction of Cantor, however, provides us with an account
that combines both advantages. We are given an account of number
that is both representational and nonarbitrary. In this paper,
I argue that the benefits gained from a representational account
are, however, greatly outweighed by the costs involved in
adopting Fines Theory of Variable Objects.


DARYN LEHOUX, University of Toronto
The Zodiacal days in the Geminus and Miletus parapegmata

The Geminus and "Miletus I" parapegmata are astronomical
instruments that give day-by-day predictions for the annual
risings and settings of the fixed stars. In the Geminus
parapegma these stellar `phases' are tied to daily weather
predictions as well. In both parapegmata the stellar phases
are organized according to the sun's motion through the
zodiac. Thus, for example, Geminus has "On the first day of
Leo, Sirius appears, according to Euctemon; the hot weather
begins." The zodiacal days ("first day of Leo, second day
of Leo," etc.) have been interpreted as betraying the
existence of a special Greek (astronomical) zodiacal calendar.
I will argue that this interpretation is untenable. Instead we
should see the zodiacal days as being a kind of extra-calendrical
calibration mechanism for these instruments, which would allow
the Greeks to use the same parapegma from year to year and
in different cities, in spite of the vagaries of the various
observational lunar calendars in use.


ALBERT LEWIS, Indiana University, Indiana, USA
The contrasting views of Charles S. Peirce and Bertrand Russell
on Cantor's transfinite paradise

Russell and Peirce were opposites in many ways, but both saw great
philosophical significance in Georg Cantor's theory of transfinite
numbers. Russell thought Cantor's notions invalid at first but by
1900 came to unreserved acceptance of them. Peirce, the American
pragmatist, in the same year more readily accepted their mathematical
validity but regarded them as products of a curtailed view of the
continuum. Is it possible a century later to evaluate which view
has better stood the test of time?


HUGH MCCAGUE, York University, York, Ontario
The mathematics of building and analysing a medieval cathedral

The designing and building of a medieval cathedral applied
mathematics in a variety of ways. The main application was
practical geometry. A master mason could adeptly and repeatedly
apply a few simple geometric tools and operations to produce a
myriad of sophisticated designs as attested by the cathedrals
themselves, by extant late medieval design manuscripts, and by
full-scale drawings still etched on church floors and walls.
Another applied mathematical element that worked hand in hand
with the geometry was the use of measurement units, such as the
English royal foot and perch. The rediscovery of the mathematical
schema employed at a specific church, such as Durham Cathedral,
is a challenging problem within architectural history. This issue
is beginning to get much needed assistance from recent statistical
methods including circular data analysis and bootstrapping
techniques. The meaning and symbolism of the cathedral also
applied mathematics through such means as the mathematical schema
of the Heavenly Jerusalem which was the key dedicatory
identification for the church. Further, stonemasonry was reverently
known as the Art of Geometry, and was one of the mechanical arts
which in complement with the liberals arts formed the means for
the human's attainment of wisdom. Like Creation, the building of
a medieval cathedral was to follow the law of Wisdom 11:21: "Thou
madest all things in measure, number and weight."


DUNCAN MELVILLE, St. Lawrence University
Third millennium mathematics: A brief survey

While many people are aware of the origins of mathematics in
tokens in the Near East and its flowering into a powerful
technical discipline in the Old Babylonian period, few except
specialists study the history linking these two points. In this
talk, we shall give a brief survey of how mathematics developed
in the crucial third millennium. We will pay particular
attention to the development of cuneiform and the famous
place-value sexagesimal system of scientific computation.


MIKE MILLAR, Northern Iowa


GREGORY MOORE, McMaster University
Editing mathematicians: Bertrand Russell and Kurt Goedel

This talk discusses editing the collected papers of the two
mathematicians mentioned above (both of whom were philosophers
as well), particularly in regard to their mathematical logic.
Questions as to what unpublished materials to include in the
respective volumes are central to this talk. In Russell's case,
the use of his evolving logical symbolism in dating previously
undated manuscripts and in determining which order manuscripts
were composed will be treated in some detail. The speaker was
one of six editors for two volumes of Goedel's works, has
previously published one volume of Russell's and is now
completing a second. For the two Russell volumes, he is the
sole editor.


RAM MURTY, Queen's University
Euclid, Brahmagupta, and ABC

The purpose is to survey the problem of finding integer solutions
(x,y) for ax^n + by^n = 1 with a,b given integers. Of course, for
n = 1, this is Euclid (c. 300 B.C.). For n = 2, it is Brahmagupta
(c. 600 A.D.) and for larger n, one needs the ABC conjecture of
Masser and Oesterle (1980 A.D.) to solve it effectively. (In the
latter case, there are only finitely many solutions.)


ADRIAN RICE, Randolph-Macon College


ARIANE ROBITAILLE, Universiti de Nantes, France
Can we learn something about combinatorics from review journals?

Mathematical review journals, such as Mathematical Reviews and
Zentralblatt f|r Mathematik und ihre Grenzgebiete, are full of
clues helping us to put in a global perspective a particular
mathematical subject seen as a whole within Mathematics. We can
do a quantitative analysis of the number of articles published
about the subject, look where those articles have been published,
scrutinize the evolution of the classification systems used, etc.
Armed with all that material, we can establish a tentative
periodization that will need a deeper look, and we can make a
comparison with the rest of mathematics. In my talk, I will
discuss my work done with review journals in the case of
combinatorics.


DONNA SPRAGGON, McMaster University
Felix Klein's "Erlanger programm" and its influence

Acting on his belief that the study of geometry had become too
fragmented, Klein distributed his bold exposition Vergleichende
Betrachtung ueber neuere geometrische Forschungen on December 17,
1872 at his inaugural address at Friedrich-Alexander-Universitaet
in Erlangen, Germany. This pamphlet suggested that the use of
algebra, or more specifically the group theory of the time, to
classify all of the known geometries. The influence of the
contents, more commonly referred to as the Erlanger Programm (EP),
has been subject to debate by many historians of mathematics.
Although Klein's unifying concept had not been developed to its
full potential, one may gain an appreciation for its significance.
To this end, it is necessary to approach the analysis of the
influence in several different ways. We will examine the years
1872-1889, the influx of republications of the EP, the effects
on the Italian School of geometry, Riemannian geometry, Relativity
theory and the influence on the teaching of geometry. In
considering these factors, this paper exposes a clear, yet complex,
look at the influence of Felix Klein's Erlanger Programm.


JIM TATTERSALL, Providence College
Vignettes from Gerbert's mathematics

Gerbert d'Aurillac (940-1003) distinguished himself as scholar,
mathematician, and cleric. He was an avid proponent of the
Hindu-Arabic system. We investigate several interesting geometric
and number theoretic problems proposed in his treatises and
correspondence.


RUDIGER THIELE, University of Leipzig, Germany
On Hilbert's 24th problem

At the end of the year 1899 David Hilbert was invited to make one
of the major addresses at the second International Congress of
Mathematicians in August 1900 in Paris. Hilbert hesitated and at
last decided to lecture on some open mathematical problems, but
he decided so late (in July) that his talk "Mathematical Problems"
was only included in the History section instead of the opening
session. However, Hilbert's problems have proved to be central
in the 20th century and became famous. All in all he discussed 23
important problems, although he did not present all of them in his
address. Furthermore, he had even cancelled a problem on proof
theory. The talk will give a short prehistory of Hilbert's famous
talk and then a short overview of the problems Hilbert published
in the Proceedings. I will go into some detail on the 23rd problem
and on the (cancelled) 24th problem.


ROBERT THOMAS, University of Manitoba, Winnipeg, Manitoba, R3T 2N2
Mathematics and fiction: A pedagogical comparison

Mathematics is often compared to music and poetry. Another
comparison is presented here, that to simple stories. Most persons
do not write either music or poetry, have no idea how, but do tell
stories. Perhaps this comparison can help those not willing or
able to learn mathematics to appreciate some of how the art of
mathematics is practised.


GLEN VAN BRUMMELEN, Bennington College
sin 10: From Ptolemy to al-Kashi

Trigonometric tables were fundamental to the work of practicing
astronomers from Hipparchus onward; poorly-computed values could
compromise almost all predictions of the positions of the
heavenly bodies. Geometric considerations permit the computation
of 1/3 of the values in a typical sine table; a good estimate of
sin 10 is needed to determine the remaining sines. Ptolemy's
estimate of the (almost equivalent) chord of 10 in the Almagest
is well-known; Jamshid al-Kashi's early 15th-century iterative
scheme is almost as famous. We shall emphasize intervening
accomplishments, including the history behind al-Kashi's
little-known original method and its eventual use in Ulugh Beg's
monumental sine table. Other techniques that might have been used
to generate the thousands of entries in these tables will be
presented. This paper represents joint work with two undergraduate
researchers: Abe Buckingham (The King's University College) and
Micah Leamer (Bennington College).

</quote>

Reference: http://www.cms.math.ca/CMS/Events/math2000/




Date Subject Author
6/15/00
Read [HM] Historia Matematica: our mailing list
Julio Gonzalez Cabillon
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Ralph A. Raimi
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Tony Mann
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Colin Mclarty
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Prof. Dr. Ivo Schneider
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Ken.Pledger@vuw.ac.nz
6/16/00
Read Re: [HM] Historia Matematica: our mailing list
Tony Mann
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Udai Venedem
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Martin Davis
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Arturo Mena
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Bonnie Shulman
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Trudy Thorgeirson
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Daniel Curtin
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Hans Lausch
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Otavio Bueno
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Bob Berghout
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Ivan Van Laningham
6/15/00
Read Re: [HM] Historia Matematica: our mailing list
Elena Marchisotto
6/16/00
Read Re: [HM] Historia Matematica: our mailing list
Nimish Shah
6/16/00
Read Re: [HM] Historia Matematica: our mailing list
Samuel S. Kutler
6/16/00
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Nimish Shah
6/16/00
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Jonathan Seldin
6/17/00
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F. Xavier Noria
6/16/00
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Hardy Grant
6/16/00
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Manoel de Campos Almeida
6/16/00
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Brendan Larvor
6/17/00
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Ivan Van Laningham
6/16/00
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Daina Taimina
6/16/00
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Olivier Souan
6/16/00
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Paul Yiu
6/16/00
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Victor Albis
6/16/00
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M. Robert Showalter
6/16/00
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Ubiratan DAmbrosio iee - 815-7216
6/16/00
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Martin Flashman
6/16/00
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Carlos Sanchez Fernandez
6/16/00
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Dinesh Maheshwari
6/17/00
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Wayne C. Myrvold
6/17/00
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Ed Mares
6/18/00
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Ken.Pledger@vuw.ac.nz
6/18/00
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Nimish Shah
6/19/00
Read Re: [HM] Historia Matematica: our mailing list
Herbert E. Kasube
6/16/00
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Oleg Yu. VOROB'OV
6/16/00
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Emmanuel Attard Cassar
6/17/00
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MANN@vms.huji.ac.il
6/17/00
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Abe Shenitzer
6/17/00
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James T. Smith
6/17/00
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Patricia Linn
6/17/00
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Michael Kinyon
6/18/00
Read [HM] CSHPM talks
Edward Cohen
6/19/00
Read Re: [HM] CSHPM talks
Alfred Ross
6/20/00
Read [HM] CSHPM talks (mishnat hamidot)
MANN@vms.huji.ac.il
6/19/00
Read [HM] Forthcoming meetings
Tony Mann
6/18/00
Read Re: [HM] Historia Matematica: our mailing list
bjorns@saturn.hifm.no
6/18/00
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Julio Gonzalez Cabillon
6/19/00
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Jaime Carvalho e Silva
6/19/00
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Toni Carroll
6/19/00
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Gerard Emch
6/20/00
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Luigi Borzacchini
6/23/00
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Fernando Q. Gouvea
6/24/00
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Nimish Shah
6/26/00
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Luigi Borzacchini
6/26/00
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MANN@vms.huji.ac.il
6/27/00
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Jonathan Seldin
6/30/00
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Nimish Shah
6/30/00
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John Conway
6/27/00
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Christian Marinus Taisbak
6/30/00
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Nimish Shah
6/28/00
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Fernando Q. Gouvea
6/28/00
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Julio Gonzalez Cabillon
6/28/00
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Prof. Peter Schreiber
6/28/00
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Siegmund Probst
7/1/00
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Samuel S. Kutler
7/2/00
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John Conway
7/2/00
Read Re: [HM] Parallel Postulate
Samuel S. Kutler
7/2/00
Read Re: [HM] Parallel Postulate
John Conway
7/4/00
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6/20/00
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2/16/01
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6/25/01
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