**Abstracts **(Incomplete)

**I. Grattan-Guinness** (

**Title: D company: The British community of
operator algebraists**

**Abstract:** The central part of Maria Panteki's thesis is
its most original part, for in

it she revealed a community of British
mathematicians concerned with

developing the D method of solving differential equations and related

topics. At the time of her death she was making a paper out of these

chapters; this lecture must serve as a
substitute.

D = d/dx, and powers of D denote orders of differentiation. This speculative

new algebra has largely French origins in the late 18th century, growing

especially out of work by (the Italian) Lagrange, but the French largely

dropped it after criticisms by Cauchy in the 1820s. However, it had already

become part of the British attachment to algebras of various kinds.

Functional equation were another new algebra of French
background and

British fascination.

After some attention to both algebras from Charles Babbage and John

Herschel in the 1810s and 1820s, the British took up operators again around

1840. Duncan Gregory was an important pioneer, George Boole became the

leading practitioner, and around a score of others took part up to W. H. L.

Russell in the 1870s. Some focus fell upon solving two differential

equations due to

textbooks on the calculus and differential equations of this period.

**T. Crilly** (

**Title:
Nineteenth century British algebra – the careers of Arthur Cayley and Thomas P.
Kirkman.**

**Abstract:** The 1840s saw a
British revival of mathematics after a long period of isolation from the
rest of

significant
steps were made in the algebra of quaternions (William Rowan Hamilton) and
invariant theory (Arthur Cayley ).

To a lesser
extent advances were also made in newly emerging areas such as group theory and combinatorics in which both

Cayley and
Kirkman played a part. In this talk the careers and contributions of Arthur
Cayley (1821-1895) and

Thomas P.
Kirkman (1806-1895) will be compared and contrasted and the nature of their
collaboration examined.

**L. Corry** (

**Title:** **From Algebra (1895**

A Guided
Tour Using the Jahrbuch über die Fortschritte der
Mathematik.

**Abstract:** The rise of modern, structural algebra may be characterized in
terms of the

consolidation of a certain image of the discipline that developed gradually

since the turn of the century, received special impetus with the work of

Emmy Noether beginning in 1920, and eventually became epitomized in van der

Waerden’s famous textbook of 1930, *Moderne Algebra*. The paradigmatic

presentation of the discipline as conceived at the turn of the century is

the one presented in Heinrich Weber’s classical *Lehrbuch der Algebra*,

whose first volume appeared in 1895.

The way from the “classical” to the “modern” conception of the discipline

can be investigated from several perspectives. The most immediate, and

perhaps necessary, one is to look at the milestone articles that

progressively produced the main concepts, theorems and techniques that came

to stand at the center of algebraic research as it was practiced along the

1920s.

Parallel to this, however, one may look for additional hints that clarify

how the practitioners of the discipline interpreted this progressive

evolution and how their image of algebra changed accordingly. One

illuminating way to do so is to look at the leading, German review journal

of the period, the *Jahrbuch über die Fortschritte der Mathematik*.
It turns

out that the changing classificatory schemes adopted by the journal to

account for the current situation at various, important crossroads of this

story add significant insights to our understanding of it.

**W. Hodges** (

**Title:
How Boole broke through the top syntactic level**

**Abstract:** One persistent difference
between algebra and aristotelian logic was that in
algebra

one could
make substitutions at any syntactic depth in a formula, whereas the logicians
were

never
willing to make such substitutions except under very restrictive
conditions. It seems

that
Boole broke through this logical restriction by simply ignoring it. What
did he think he was doing?

**Jean Christianidis** (

**Title: Some reflections on the
historiography of classical algebra**

**Abstract:** Historians of mathematics have proposed since
the middle of the nineteenth century a range

of views regarding
the main phases of the evolution of algebra, the description of each phase, its
starting

point,
and therefore the characterization of works and undertakings of individual
mathematicians

according
to the scheme each historian propounds. Within this context two issues are
easily discerned in

the
recent scholarship for they have given rise to contradictory views among the
historians. The first

concerns
the work and the enterprise of Diophantus, the relevance of which with the
history of classical

algebra
is being denied by some historians. The second concerns the thesis that the
family tree of classical

algebra,
and thereby of the algebra as a whole, starts with the work of
al-Khwārizmī. In the present paper

the above
views are critically discussed, and a scheme of trichotomic periodization of
the history of

algebra
is proposed. According to this scheme the algebraic mode of thought came into
being within

the
ancient traditions of problem solving, and passed from a first phase that might
be called

“from problem to equation” to the phase where algebra was
thought as “the art of solving equations”,

so as to
arrive in its third phase, the phase of modern algebra.

**MJ Durand-Richard (**Universités Paris 8-Paris 13,
CNRS-Paris 7**)**

**Title: Boole’s investigation on
Symbolical methods in his last 1859 and 1860 Treatises**

**Abstract:** George Boole (1804-1864) openly referred to
the theory of Symbolical Algebra in the introduction

of his *Mathematical Analysis of Logic* (1847).
He abundantly investigated it when he worked out for the

first time an algebraical structure of logic. This symbolical mode of algebraic reasoning is often praised

as one of the origins of
modern algebra, even though the effective birth of modern algebra woudl not
occur

before the 1930s, as Leo Corry
rightly underlines in his 1996 book, *Modern Algebra and the Rise of *

*Mathematical Structures..*

In order to understand this apparent delay, I
wish to investigate some of the possible reasons for which these

symbolical methods were not fully explored in
British mathematics of the second half of the 19th century.

As we know, Boole not only investigated the
symbolical methods in logic, he also developped them in

his treatment of differential equations in several
papers of the *Cambridge Mathematical Journal,* as well as

in* *his « .General Method in Analysis » which led the Royal
Society to grant him a Gold Medal in 1844.

But Boole still referred to symbolical methods
in his later *Treatise an Differential Equations *(1859)

and* Calculus of Finite Differences *(1860).For Boole as for the whole
algebraical network of

this symbolical treatment of differential equations
was certainly more essential than for founding

« impossible
quantities », as it used to be praised in history of mathematics. The
difficulties that

it raised in this field, particularly with
regards to the use of the transfer principle asserted by

Duncan F. Gregory (1813-44) and Boole could
help us to understand why the idea of structure was

not more developed in British Mathematics by the
following authors.

**C. Phili** (

**Title:** **On the extension of
the calculus of linear substitutions
" by Kyparissos Stephanos**

**Abstract:** In 1899, the article of K.
Stephanos "Sur une extension du calcul des
substitutions

lineaires" appears at the Proceedings of the

article will be published in length at the " Journal de
Mathématiques Pures et Appliquées" edited by

Continuing the research on bilinear and quadratic forms and
emphasizing the

contributions of Frobenius in the notion of composition of bilinear forms
K. Stephanos

will
follow the footprints of the important German mathematician and will
extend even

more
this symbolic calculus. Thus beyound the common composition of bilinear forms he introduces

two more
operations which he names conjuction (conjonction) and
bialternate composition (composition bialternée) of bilinear
forms.

**A.
Moktefi** (

**Title:** **Who cared about Boole's algebra of logic in
the nineteenth century?**

**Abstract: **In this presentation, we will discuss how
logicians dealt with Boole’s algebra of logic in the second half of the
nineteenth century. The traditional account tells that it has been almost
unnoticed until Jevons revived it, and later Venn popularised it. This account
is incomplete however because it focuses on Boole’s followers (and
semi-followers) without paying attention to his opponents, not to say the
majority of the logicians who either didn’t understand him or didn’t know him
at all. Boole was of course considered as the father of symbolic logic, but
what place did symbolic logic hold within the logical studies of the time? By
answering this question, we determine how Boole’s algebra was considered by logicians,
both mathematicians and philosophers.

**V.
Peckhaus** (

**Title:
What is Algebra of Logic?**

**Abstract:** The German mathematician Ernst
Schröder (1841-1902) is one of the pioneers of the algebra of logic. His
monumental "Vorlesungen über die Algebra der Logik" (1890-1905)
seemed to provide some sort of sum of this field. Comming from Combinatorics
and Combinatorial Analysis he developed his first ideas on logic completely
independent from Boole and the British logicians. He was mainly influenced by
Hermann Günther Grassmann's theory of forms opening Grassmann's
"Ausdehnungslehre" (1844), and by the logic of Robert Grassmann. Schröder's
conception of a formal, and in its last step of development absolute algebra,
can be seen as an early precursor not only of Lattice Theory, but also of
Universal Algebra and Model Theory. In his combination of a general algebraic
theory of structures with an iterated series of interpretation logic played the
role of an intermediate layer between algebra and arithmetic. So "Algebra
of Logic" is indeed no logic, but the algebra of logic.

**A.
Tokmakidis** (

**Title: ****The Establishment of the
Mathematical Profession in 19 ^{th} Century Europe.**

**Abstract:** We will attempt to present briefly the development
of the mathematical profession in Europe – mainly in ^{th}
century

**A. Walsh** (

**Title:
The algebraic logic of Charles S. Peirce (1839-1914)**

**Abstract:** The American Charles Peirce was one
of the most important logicians of the nineteenth century.

This talk
deals with the development of his algebraic logic. An important aspect in
the history of logic is

the part
played by the algebraic logic of English mathematicians of the nineteenth
century, namely George Boole

(1815-1864)
and Augustus De Morgan (1806-1871). They attempted to express the laws of
thought or the

processes
of thinking and logical deduction in the form of algebraic mathematical
equations. The early

influences of George Boole on the algebraic logic of Peirce
are examined, including the areas where George

Boole’s logic
departed from an arithmetic system and where Peirce extended Boole’s calculus
by providing

the
missing operation of division. Whereas Boole used a part/whole theory of
classes and algebraic analogies

involving
symbols, operations and equations to produce a method of deducing consequences
from premises in

logic,
Augustus De Morgan had realised the inadequacies of syllogistic logic and
claimed that some way of

representing relations other than the identity relation was needed. His theory
of relations involved expressing

inferences
in logic in terms of the composition of relations. I will also introduce
how Peirce developed

De Morgan’s
work on the theory of relations and how this was combined with Boole’s
part/whole class

calculus
to form an algebraic logic equivalent to today’s predicate logic.

**P. Wolfson** (

**Title: Resolvents of
Polynomial Equation**

**Abstract:** In 1771 Lagrange published an
analysis of the known methods for solving polynomial equations of

degrees
two, three, and four. He showed how all of the known techniques for
solving these equations could

be
understood in terms of a resolvent of the equation—a certain polynomial in the
equation’s roots—and that

an
analogous solution was not possible for the general fifth degree
equation. This work influenced the general

investigations of Abel and Galois into solvability, but research on the resolvents
themselves lasted through the

nineteenth
century and up to the present. This talk will present some
background on resolvents and will introduce

some of
that research, explaining several reasons why mathematicians continued to study
resolvents after major

questions
about solvability had been answered.