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
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
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) to Moderne Algebra (1930): Changing Conceptions of a Discipline.
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
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
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
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.
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.
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.
Title: The Establishment of the
Mathematical Profession in 19th Century
Abstract: We will attempt to present briefly the development
of the mathematical profession in Europe – mainly in
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.