Research programme 2008-2012
Sébastien Gandon
Subject : Philosophy
Area of research : History of analytical philosophy and philosophy of mathematics
Title : Philosophy and mathematics in the days of ‘the foundational crisis of
mathematics’
My name is Sébastien Gandon, and I am 37 years old. Since 2001, I have been an
assistant professor in the department of philosophy at Blaise Pascal University (UBP), and
also a member of the research centre in Clermont-Ferrand called « Philosophies and
Rationalities » (PHIER ; EA 3297). I have been mainly (along with Elisabeth Schwartz)
involved in developing the « epistemological » side of the laboratory and, within this context,
I have collaborated with the CNRS teams of Nancy (Archives Poincaré ; UMR 7117) and of
Aix-en-Provence (CEPERC ; UMR 6059). Since last year, I have been working regularly with
the CNRS team of Paris (REHSEIS ; UMR 7596) as well.
Two years ago, a new « Maison des Sciences de l’Homme » (MSH) was created in
Clermont. Since then, Laurent Jaffro (a PHIER member, professor at the department of
philosophy) has been at the head of it, and I have also been involved in it. I am in charge of an
interdisciplinary seminar on the history and epistemology of mathematics, which is organised
by the MSH in partnership with the « Institut de Recherche sur l’Enseignement des
Mathématiques » (IREM) and the department of mathematics, thus gathering, on the site of
Clermont, philosophers, hellenists as well as mathematicians and historians of mathematics. I
am also one of the elected members of the MSH committee.
My PhD was about Wittgenstein’s Tractatus, which I regarded as revealing the deep
differences between Frege’s and Russell’s thoughts (who were the two main influences on the
book). But since 2002, I have been working exclusively on Russell and the mathematical
context surrounding his The Principles of Mathematics (1903). My research thus focuses both
on the history of the early analytical philosophy and on the history of mathematics (during the
same period : 1870-1930) – to be more precise, my main interest is the interaction between
philosophy and mathematics in the days of «the foundational crisis ».
The question that drives my thought is that of the distinction between philosophy and
mathematics (this is, in fact, a platonistic, classical question). What distinguishes these two
kinds of purely rational thought ? An extraordinary mutation of the traditional distinction
occurred during the period I am investigating. Some issues which had been traditionally
regarded as belonging to metaphysics are, from this period on, regarded as simple
mathematical questions (for example, see the Russellian reading of Zeno’s paradoxes and of
Kant’s antinomies); on the other side, some mathematical developments are condemned on
the pretence that they would pertain to metaphysics (see Kronecker’s or Weyl’s point of view
concerning Cantor’s mathematics) ; finally, new disciplines which were neither specifically
mathematical nor specifically philosophical arose (the new logic being the most striking
example).
These mutations must be taken into account when recounting the history of this period.
Some whole parts of mathematics were at that time motivated only by « foundational » (that
is : epistemological or philosophical) issues. On the other side, it appears as if Frege’s and
Russell’s thoughts (for example) were aided by the symbolisms they manipulated. The early
analytical philosophy as a whole stems directly from these subversions of the traditional
boundaries between mathematics and philosophy.
But I think that these mutations must be taken into account first and foremost when
one wishes to carry through a philosophical reflection on mathematics today. Indeed, the
foundational debate was not a mere dispute between different views on mathematics. What
was then at stake was the very possibility of articulating a non mathematical and truly
philosophical conception of mathematics. During the « foundational crisis », the difficulty
was not simply knowing on what mathematics must be founded, but rather knowing what the
phrase « to found mathematics » could mean. And this question is still an ongoing issue
around which the present debate revolves.
My research is thus at the same time philosophical and historical. It is historical since
its aim is to understand and to explain some philosophical and mathematical works within a
given period of time. But it is philosophical as well, since recounting the history of
philosophy and mathematics at this period requires to ask once again the old platonistic
question : what is the difference between mathematics and philosophy ? Are they
distinguished by their objects, and/or their methods, and/or their languages ? How does one
explain that certain mathematical reasonings can be used as philosophical arguments ? How
does one redraw the outlines of philosophy of mathematics after the foundational crisis ?
The general topic of my past and future research work is, as it appears from above,
very wide. But the method I use is « local ». Confronting the issues directly, abstracted from
their historical setting, can conceal their importance. Rather than tackling the problems in
their abstract form, I prefer to anchor them in examples, either by studying the works of
mathematicians (Pasch, Klein, …) or of philosophers (Russell, Wittgenstein, Royce, …) of
the beginning of the XXth Century, or by analysing some particular epistemological concepts
(explanatory proof, quantity, …).
More precisely, four main lines, strongly connected to each others, will organise my
research works during the next few years :
1) Russell’s studies : the reconstruction of the scientific and philosophical context of
The Principles of Mathematics (1903)
2) History of early analytical philosophy
3) Epistemology and Philosophy of mathematics
4) The theories of quantity at the crossroad of mathematics, physics and psychology
Each of these projects will give rise to some new research, but will also contain an
institutional side, devoted to collaborations with various partners.
1) Russell’s studies : the reconstruction of the scientific and philosophical context of
The Principles of Mathematics (1903)
Russell is mainly known as the discoverer of paradoxes and as the creator of the
theory of types. From his main work, The Principles, the first part only (as well as the
appendices) is being currently translated into French. The six other parts (which are less
logical and more mathematical, dealing with analysis, geometry and mechanic) are usually
neglected by both the historians of philosophy and the historians of sciences. Russell’s work
is thus often represented as that of a philosophical logician, whose reflection was, already in
his time, outdated by the mathematical practice.
I intend to question this image which, I think, is the fruit of a truncated reading of the
Principles. I would like to show that one cannot grasp Russell’s philosophical project without
inserting it into the very intricate mathematical context of its time – to show also that the
study of the Principles makes it possible to refine our vision of the mathematics and of the
mechanics of that same period.
To me, Russell scholarship is a fundamental and priority work, for it constitutes the
very core of my whole project. The British thinker is constantly reshaping his metaphysics to
take into account the new mathematical theories. On the contrary, he does not hesitate to
modify some classical mathematical presentations in order to fit them to what he regards as a
more rigorous ontology. This uninterrupted movement between metaphysical traditions and
mathematical practices, which is the mark of the Russellian philosophy, represents both a
research field especially adapted to my general project, and a unique point of view on the
scientific and philosophical scene of this time.
Various projects, connected to this research axis, have already been finished or are still
in progress :
a) Studies on the mathematical parts of the Principles
Over the last three years, I have been working on the Russellian theory of geometry
and I have published two articles, [B1] and [B2], on this topic. I am working on [B4] again
for it to be published in English. I still have to study more thoroughly the Russellian theory of
the metrical geometry, which implies the relationship with both Poincaré and the neokantians
(see [D1]).
For one year, I have also focused my research on the Russellian theory of quantity. I
have written two artices on that subject ([B1] and [B5]) and delivered several speeches on this
topic ([D3], [D5], [D6] and [D9]). I intend to start a detailed study of the very technical last
part of the Principia, which is devoted to this notion (on quantity, see also section 4 of this
programme).
In the next five years, I have the intention of studying the part on analysis –
particularly the relation to Cantor even if a lot has already been written on it – and of working
on the conception Russell derives from mechanics.
b) The Russell-Whitehead relationship
While studying the Russellian theory of geometry I realized that Whitehead was very
important to Russell (see [B4] and [B3]). The discussion of Whitehead’s first book, Universal
Algebra (1898), takes a leading part in the development of the Russellian thought from 1899
to 1903. But Whitehead also wrote the last parts of the Principia, particularly the one on
quantity (see [B5]). His influence goes far beyond the years before the publication of the
Principles.
I am planning to study the still unpublished Correspondence between Whitehead and
Russell that are kept in the Russell Archives in McMaster University. Only Whitehead’s
letters have survived but even so, this Correspondance constitutes a unique source of
information on the way the Principia was written and on the evolution of the Russellian
thought between the Principles and the Principia.
I would like to edit (both in French and English) this still unpublished Correspondence
between Russell and Whitehead. Such a publication is likely to renew the field of the
Russellian studies, and even more widely, that of the history of the first analytical philosophy.
c) The translation of The principles of Mathematics
One year ago, together with three collegues, B. Halimi (Lecturer, ParisVII), J. Sackur
(Assistant Professor, Paris X) and I. Smadja (Assistant Professor, Caen), I started to translate
the whole of the Principles. The book is to be published at Hermann Publishing Company in
2009. We have already translated two parts of the book.
I am very keen on achieving this project for it would allow the French-speaking
readers to have access to the mathematical parts of Russell’s masterpiece, and would at the
same time draw attention its philosophical ambition.
In a longer term (2010-2012), I intend to gather the results of my current work on The
Principles in a book that I will write in English. A work that builds bridges between history of
mathematics studies and philosophical scholarship on Russell’s book is still missing. If one
bypasses Russell’s patient analysis of mathematical practices, it is the whole philosophical
meaning of the Russellian project that would be distorted.
As regards institutional aspects, I wish to to organize a research agreement convention
with the Russell Archives (McMaster U., Hamilton), which would allow French scholars to
go to Hamilton. I am already behind an agreement for a student exchange programme
between McMaster University and the Université Blaise Pascal. But we need to widen the
basis of the collaboration in order to instigate a similar convention at a research level. The
Poincaré Archives (Nancy) and the UQAM (Montréal) are interested in this project, which, I
hope, will soon begin to take shape.
In the same spirit, I have applied for the 2008 Bertrand Russell Visiting Professorship,
launched by McMaster University (a one semester invitation; see Appendix 2 for more
details). I have just learnt that I have been elected by the department of Philosophy to this
post. I am now waiting for the offer from the Dean of McMaster’s Faculty of Humanity.
2) History of early analytical philosophy
The time when it was possible to present analytical philosophy as an ahistorical
philosophy, and on this matter, to contrast it with the continental tradition, is past. More and
more work is entirely devoted to the history of analytical philosophy. Moreover, even in the
non-historical studies, the interpretative discussions are often crucial. In the philosophy of
mathematics, to take for one example, the different versions of structuralism are closely
connected with the various interpretations of Dedekind’s Die Zahlen (1888). The history of
philosophy, especially the history of early analytical philosophy, is thus acknowledged by
many as a way of renewing the philosophical issues.
Since my book on the philosophy of the early Wittgenstein and its fregean-russellian
context ([A1]) was published, I have been completely taken up with my work on the
Principles. In the next few years, I intend to extend my research field, and to go back over the
comparison between the three main figures of the first analytical philosophy, namely Frege,
Russell and Wittgenstein. This new look at some topics I have already worked on will, I hope,
benefit from my thorough studies on the scientific context surrounding the Principles (cf.
Section 1).
Two kinds of research work are related to this project :
a) Frege and Russell’s relationship
The relationship between the two logicists is a classic subject in the history of early
analytical philosophy. Yet, some entire fields remain unexplored.
In the first place, some new perspectives have recently been opened by the reappraisal
of the ‘substitutional’ approaches in Russell’s thought (cf. Landini’s and de Rouilhan’s
works). These previously neglected ‘substitutional’ theories are themselves linked with
Russell’s reading of Frege’s works. The recent discovery of their importance calls thus for a
complete reconsideration of the relation between Russellian and Fregean thoughts.
In the second place, some of Frege’s writing did not, until recently, attract the
attention of scholars – I am speaking about Frege’s first mathematical papers on projective
geometry and on complex numbers, and on quantities and the construction of reals. Now, on
these two points, a comparison between Frege and Russell would be enlightening, since the
British philosopher, without knowing anything about Frege’s writings, was interested in
exactly the same issues. Why were both fascinated by the very same geometrical theory
(projective geometry) ? What conclusion is to be drawn from the great similarity between
Russell’s and Frege’s heterodox conception of quantities ?
I regard this work on the two logicists as a priority topic for my future researches.
I am currently writing an article on the impact of Russell’s substitutional theory on
Wittgenstein’s early thought, and I am working, with M. Panza, on the issue concerning
quantity in Russell’s and Frege’s works (on this, some remarks have been made in [B5]).
I intend to organize a regular collaboration around these questions, which would
gather various French and foreign researchers. Spurred on by G. Landini (Iowa U.), some
young scholars of Frege and Russell have started tackling these questions anew – I am
especially thinking of G. Stevens from Manchester and of K. Klement from Amherst. I am in
contact with all of them, and I would like to introduce their work in France.
b) Idealism and early analytical philosophy
N. Griffin showed how British idealism put Russell away from the English empiricist
tradition. Russell’s notorious criticism of psychology would take root in Bradley’s neohegelianism. I intend to extend this idea in three different directions :
- The neo-hegelian impact moves away Russell not only from the empiricist tradition,
but from the whole set of the neo-kantian movements. Thus, Hegel’s theory of quantity (as it
is reworded by Bosanquet) is behind Russell’s criticism of the kantian distinction between
extensive and intensive quantity (See [B1]). I plan to elaborate this topic in a series of studies
that could shed a new light not only on some aspects of Russell’s thought, but also on some
discussions within the neo-kantian movements.
- One of the most important source of Bosanquet’s and Bradley’s thoughts is the work
of the German philosopher H. Lotze. This multitalented man (he made important
contributions in mathematics, physics, psychology), who is quite neglected in France, began,
after the empiricist waves in the 1840’s, to go back to Fichte’s and Hegel’s idealism. His
influence on Husserl and Frege is today widely acknowledged, but his impact on British
idealism and on Russell is less known. I intend to study the content of his Logik, and the
interpretations that Frege, Husserl, Bradley and Bosanquet made of this book (I speak about
Lotze in [B8]).
- I plan to carry on with the work I began in [B8] on the American idealist philosopher
J. Royce. His unique attempt, which is diametrically opposed to Russell’s one, consists in
combining an explicitly hegelian idealism with a well-informed interpretation of the new
mathematics (Dedekind, Cantor, Veblen especially). In my work on Royce, I would like to
show that exactly the same mathematics can give rise to widely distinct if equally convincing
philosophical elaborations.
Generally speaking, the importance of idealism in Russell’s and Frege’s thoughts
have, for various reasons, been minimized. In June 2006, I organized a two-day workshop on
British idealism ([R2]), whose proceedings will appear in a special issue of Philosophiques
(Canada) ([A3]). I intend to continue to explore this path.
These works will be a part of the programme developed by my Research center in
Clermont. For a few years, E. Cattin (Clermont, Professor) has been coordinating various
works on German idealism, elaborated in collaboration with the University of Toulouse, Paris
1 and the Husserl Archives (Paris). L. Jaffro (Clermont, Professor) has been organizing a
research programme on British philosophy that includes some scholars from Nantes, Paris X,
Glasgow and St-Andrews. The relations of early analytical philosophy to idealism are at the
crossroad of these two projects.
3) Epistemology and philosophy of mathematics
I plan to extend my work on logicism by a more general reflection about the
philosophy of mathematics. Compared to the other topics, this project could seem less
developed. However I wrote some papers ([B2], [B7]) and gave several conference papers
([D2], [D8] and [D11]), in which I tried to describe how some mathematicians (Pasch, Klein,
Peano) justified their reasoning. Moreover, for two years, I have coordinated for the MSH a
programme on epistemology and history of mathematics ([R2], [R3]), elaborated in
collaboration with the IREM and the department of mathematics.
In the future, I intend to carry on with these various activities and to extend them in
two directions : first and foremost, towards a reflection on the contemporary philosophy of
mathematics ; secondly, towards a commitment in various teacher training actions.
Three projects are related to this research topic :
a) Interaction between mathematics and philosophy (1800-1930)
A ‘délégation IUF’ would allow me to anchor my interdisciplinary researches on
epistemology and history of mathematics in Clermont.
With regard to the IREM-MSH seminar, a conference about ‘Euclid’s posterities’ has
already been planned in 2007. Moreover, the seminar has been found sufficiently successful
to be renewed by the two regulators, the MSH and the IREM.
With regard to the history of mathematics, I intend to continue the works on the Italian
mathematics (Pieri, Peano, Enriques especially) at the beginning of the XXth Century. This
quite neglected field is very rich, since the differences between the various philosophical
approaches are extremly sharp, but they do not exactly overlap the more usual ones that shape
the French or German philosophical and scientific scene at the same period.
With Y. Perrin (Professor of mathematics, Clermont), I am currently writing a paper
devoted to Peano’s and Lebesgue’s definitions of area (see [D2]). We plan to submit it to
Historia Matematica. We both intend to carry on with this very rewarding interdisciplinary
collaboration.
A translation into French of Peano’s main articles is still missing. With G. Crocco and
E. Audureau (Aix), we plan by 2010-2011 to publish a commented edition of Peano’s greatest
texts.
I believe that working on the history of mathematics is a necessary condition for
elaborating a philosophical reflection on mathematical knowledge – it constitutes a kind of
safeguard, a ground allowing to think out and to test some more general hypothesis.
b) Philosophy of mathematics
With I. Smadja (Assistant Professor, Caen), we are currently working on a volume
([A2]) devoted to the contemporary philosophy of mathematics that will be published in the
collection ‘Textes clés’ by Vrin Publishing Company (our detailed project has been accepted).
This book will collect articles from the 60’s to the 90’s onwards, not translated in French until
now (papers by Benacerraf, Field, Shapiro, Maddy, Kitcher, Awodey, Wilson,…). The aim is
to present to the French public the contemporary issues – it is as well to show that some
traditional oppositions (as for instance, the one between logical foundationalist and more
historical approaches) tend to become less marked.
I intend to extend this collective editorial work by a more individual research work,
which will focus on the notion of explanation.
As M. Steiner has shown, mathematical proofs are not all equally explanatory – the
question is thus to know what singles out the explanatory proof from the other ones. However,
it seems to me that another problem comes with this first question : is the distinction between
explanatory and non explanatory proof purely internal to mathematics, or does it fall within
the province of non mathematical, for example, physical or philosophical, consideration?
In this regard, I plan to elicit H. Field’s great nominalist manifesto Science without
numbers (1980). In this book, the philosopher articulates an idea he does not use again in his
subsequent works: a plea for nominalism should not be grounded on the taste for ‘desert
landscapes’, but on the fact that nominalism explains many things that the others positions do
not explain. There would be a rational benefit to be a nominalist.
I suscribe to this idea, but I would like to know more precisely what the nature of the
nominalist explanations is – what is the difference between these kinds of explanation and the
mathematical explanatory proofs? Which rational interest does the nominalism fulfil?
This research project is a priority one, but I still need some time to articulate it, and I
do not plan any publication before 2009.
c) Applied research toward secondary teaching
The work I am involved in is at the crossroad of philosophy and mathematics, and it is
likely to have some effects on the teaching of both philosophy and mathematics. Even if I do
not regard this applied research as a prioritary goal, it would be a pity not to take the
opportunity provided by the already working collaboration with the colleagues from the
IREM and from the department of mathematics. I intend then, as much as possible, to take
part in elaborating various documents for the secondary school teachers in philosophy and
mathematics. Clermont MSH has announced the launching of a programme devoted to the
‘teacher training’ in 2007. With my colleague, I plan to answer to this call.
More precisely, I intend to work out various documents aimed at making easier the
implementation of a dialogue between teachers of mathematics and teachers of philosophy.
These documents would be immediately understandable by teachers of mathematics without
any philosophical background, and by teachers of philosophy without any mathematical
background. I would like to consider different issues, all of them connected with parts of the
secondary mathematical cursus (‘zero’, ‘negative numbers’, ‘limit’, ‘the infinite’,…), and put
them in relation to a limited number of well chosen philosophical texts (to the philosophical
problem raised by the concept of ‘nothing’, to the issue concerning the ‘quantities smaller
than nothing’, to the questions raised by the notion of ‘limit’, to the paradoxes of the
infinite,…). These various documents would be put online.
Let us point out the fact that the MSH context favours collaborations with other MSH.
I am particularly thinking of the new MSH of Nancy, in which the Poincaré Archives are
involved. I would like to contribute with them to the setting up of a summer school on the
history and epistemology of mathematics. This school would be the ideal place to develop the
project just mentionned.
4) the theories of quantity at the crossroad of mathematics, physics and
psychology1
Several of my research works led me through various ways to come across the concept
of quantity : the analysis of Russell’s works of course, but also the study of the neo-kantian’s
and of Meinong’s writings ; the reading of many early XXth Century mathematicians (such as
Helmholtz, Burali-Forti, Huntington, Hilbert,…) ; and also the more recent discovery for me
of the « Theory of Measurement » by Suppes and his collaborators. I am currently writing up
a lecture on quantity meant for Clermont PhD students and for IFMA (‘Institut Français de
Mécanique Avancée’) engineering students. In the future, I want to devote at least a third of
my research work to the analysis of the problems raised by this concept.
Yet I still have to read up the literature before venturing to publish anything on that
subject. This topic will be mainly developed from 2010 on.
Two main lines will organize this work :
a) The persistence of the theory of magnitude (1880-1950)
The fact that the concept of quantity was, up to the XVIIth Century, used to define the
field of the mathematical sciences is well known. It is also well known that this definition was
completely abandoned at the start of the XXth Century.
My hypothesis is that concession does not amount to the the end of the theorie(s) of
quantity. If mathematical sciences are not any more described as the science of the quantity,
the idea that there is a definite concept of quantity, whose theory can be developed, remains
very lively throughout the first half of the XXth Century. This hypothesis suggests that the
problems linked to measurement did not at the time come first from the intricacy of the new
research fields (psychology, economy, …), but from the theoretical constraints generated by
the belief that only a special kind of object, the quantities, can be measured.
I intend to examine the various ways in which the mathematicians, and also the
physicists, the economists and the psychologists, dealt with the notion of quantity at the turn
of the last Century. One thing is especially striking : the idea that there is something such as a
definite concept of quantity combines with a very great variability in the definitions of
quantity one gives.
1
This sentence is taken from a recent article by O. Darrigol.
In the shorter term, I plan to publish various works on the Russellian theory : in
addition to the already mentioned (see section 1 and 2) ones, I plan a work on N. Wiener’s
‘psychophysic’ use of Russell’s approach [D3], and another critical one, on the ‘ordinal’
interpretation the physicist N. Campbell gives of Russell’s theory.
b) Is the project of constructing a theory of quantity really obsolete?
The break-up with the precedent realist paradigm, in which to know what a magnitude
is constituted a preliminary condition for developing a theory of measurement, occurred only
at the beginning of the fifties. In P. Suppes’s ‘theory of measurement’, the possibility of a
measure is justified by a representation theorem, that is by the demonstration that there is a
isomorphism between an empirical ‘relational structure’ and a ‘numerical model’ (a subset of
the reals provided with the relevant structure). In this approach, no general theory of quantity
occurs any more; on the contrary, Suppes emphasizes that there are many distinct
measurement systems, and accordingly, many different definitions of what quantities are as
well.
Measurement theory unquestionably had a liberating effect, especially on the human
sciences. It remains that this release had a cost. In Suppes’s approach, it seems that one has to
give up the idea, quite natural when all is said and done, that quantity characterizes a
particular phenomenal sphere – that is, one has to give up the idea that quantity is a category
(to talk like Aristotle), whose formal structure can, and must, be described. In the
measurement theory, it is the multiplicity of the measurement systems that is put forward, and
the unity of all these devices that becomes difficult to explain.
If the disconnection between measure and magnitude is undoubtely epistemologically
very efficient, it leaves unanswered some legitimate questions: What is common to the
various phenomena that can be measured? What is the ‘raison d’être’ of measurement? Why
do we assign numbers to objects?
I intend to examine these questions, from the standpoint of the theories developed at
the beginning of the last Century (Russell’s being one of them, but Poincaré’s, Klein’s and
Duhem’s, … being others) – but also directly from the works of contemporary philosophers,
like Kyburg and Mundy, who have raised some relevant criticism of Suppes’s approach.
Moreover, to my knowledge, there is still no presentation in French of the
measurement theory and of the philosophical issues it gives rise to. I would like to fill this
gap, and write a synthetic philosophical introduction to this topic.
On the institutional level, these research works would be done in the PHIER
laboratory in collaboration with the REHSEIS CNRS-team. Last year, the REHSEIS sets up a
research programme gathering historians of mathematics and physics on (among others) the
topic of measurement and quantity.
The various projects I propose can be summarized as follows : My PhD provided me
with a background in early analytical philosophy and its scholarship. My more recent research
on projective geometry revealed to me the works of historians of science from that same
period. This double involvement led me to realize that there are numerous obvious relations
between these two research fields.
However, as surprising as it might appear, not many bridges exist between the two
domains of research. Not many philosophical scholars refer to the works (even classic ones)
of historians of mathematics – likewise, not many historians of mathematics seem interested
in the works of the first analytical philosophers. The main issues that liven up the
philosophical scholarship are still kept separated from the ones that shape the historiography
of mathematics – this is unfortunate, since the questions philosophers and historians of
mathematics ask themselves are often very close.
My aim is to contribute to the reduction of this gap. I think that the history of
philosophy (and philosophy itself) has everything to gain in drawing closer to the studies of
actual scientific practices, if only because the philosophical works it considers were closely
linked to some of these practices. I believe as well that this approach is not destitute of
interest for the history of mathematics itself. Thanks to its unconventional standpoint, this
approach could shed some new light on the mathematical scene at the time of the
‘foundational crisis’.
My institutional affiliations reflect the content of my scientific project. I am a member
of a philosophical research centre, the PHIER, in which a fertile collaboration between
historians of philosophy and philosophers of science is going on, and in which also strong
links have been tied with Aix and Nancy CNRS research centres.
My work is first and foremost a part of PHIER’s activities : it falls within the
framework a philosophical-minded research centre opened to epistemology.
I am also strongly involved in some programmes of Clemont’s new MSH. This
structure constitutes a perfect surrounding in which to develop my interdisciplinary works.
Moreover, the MSH environment could give rise to a new kind of collaboration with the
researchers of the Poincaré archives (Nancy).
Finally, I am involved in the activities of the REHSEIS, a CNRS laboratory working
on the history of science, especially on the history of mathematics. An anchoring in the
history of science, at the highest possible standard, is provided by this last collaboration.
APPENDIX 1
1- List of the projects classified by order of priority :
In bold type, the projects regarded as having priority ; in italic type, the project regarded as
having secondary importance
[1-1] works on the mathematical parts of The Principles (and book on the
mathematical context of The Principles)
[1-2] works on the relation Whitehead/Russell (and edition of the
Correspondence)
[1-3] French translation of The Principles
[2-1] comparison between Frege’s and Russell’s logicism
[2-2] works on idealism and on early analytical philosophy.
[3-1] works on the history of mathematics (edition and translation of Peano’s main
articles)
[3-2] works on the notion of explanation in mathematics (and edition of the
volume « Philosophie des Mathématiques »)
[3-3] applied research toward secondary teaching of philosophy and of mathematics
[4-1] works on magnitudes theories in the first half of the XXth Century
[4-2] works on ‘measurement theory’
2- List of the projects classified according to their collective or individual scope :
With two stars ‘**’, the projects handled in collaboration ; with one star ‘*’, the project
handled in the framework of a collaboration ; without star, the projects handled alone.
[1-3]** French translation of The Principles
[3-1]** works on the history of mathematics (edition and translation of Peano’s main
articles)
[3-3]** applied research in relation to secondary teaching of philosophy and of
mathematics
[2-1]* comparison between Frege’s and Russell’s logicism
[2-2]* works on idealism and on early analytical philosophy.
[4-1]* works on magnitudes theories in the first half of the XXth Century
[1-1] works on the mathematical parts of The Principles (and book on the
mathematical context of The Principles)
[1-2] works on the relation Whitehead/Russell (and edition of the Correspondence*)
[3-2] works on the notion of explanation in mathematics (and edition of the volume
« Philosophie des Mathématiques »**)
[4-2] works on ‘measurement theory’
3- List of the projects classified according to the expected chronological order of their
setting up
2008 : [1-1] ; [1-2] ; [1-3] ; [2-1] ; [3-1] ; [3-2] ;
[3-3]
2009 : [1-1] ; [1-2] ; [1-3] ; [2-1] ; [2- 2] ;
[3-3]
2010 : [1-1] ; [1-2] ; [2-1] ; [2-2] ; [3-1] ; [3-2] ;
[3-3]
2011 : [1-1] ; [2-2] ; [3-1] ; [3-2] ; [4-1] ; [4-2] ;
[3-3]
2012 : [1-1] ; [3-2] ; [4-1] ; [4-2] ;
[3-3]
APPENDIX 2
- Bertrand Russell Visiting Professorship :
The Department of Philosophy invites applications for a Visiting Professorship in Russell and
the History of Early Analytic Philosophy. McMaster University, which houses the Bertrand
Russell Archives and the Bertrand Russell Research Centre, is one of the leading centres for
research on Russell’s philosophy. The Visiting Professorships, one of which will be available
each year, are intended for established scholars whose research would be benefited by access
to the Bertrand Russell Archives for an extended period. They are tenable for either one or
two semesters, and involve the obligation to present at least one paper in the Philosophy
Department’s Speakers Series and teach one graduate course, preferably on the history of
analytic philosophy (although a different topic may be agreed upon with the Chair of the
Department of Philosophy), while undertaking research in the Russell Archives. The stipend
for teaching the course is up to $12,000, dependent on the candidate's qualifications. It is
expected that successful applicants will be on research leave from their home university
during the tenure of their Visiting Professorship and thus can rely on their regular leave salary
for their main financial support. Applicants should send a copy of their CV together with a
description of the research they propose to conduct at the Russell Archives to the Chair,
Department of Philosophy, University Hall 312, McMaster University, Hamilton, Ontario
L8S 4K1, Canada.
- My research project for the B. Russell’s visiting professorship
I am already involved in two distinct projects which would both greatly benefit from an
access to the Russell Archives during a semester or a year :
- I am currently working, with three colleagues (Brice Halimi, Jérôme Sackur, Ivahn Smadja),
on the first complete French translation of The Principles of Mathematics (1903). The
translation will be published by Hermann (http://www.editions-hermann.fr/) publishing
company and it is planned to be completed in 2009.
For this, an introduction will need to be written describing the different stages in the genesis
of Russell's book.. A lot has already been done about the writing process of The Principles
(cf. for instance, I. Grattan-Guinness « How did Russell write The Principles of Mathematics
(1903) ? » (Russell, 16, 1996) and the articles of M. Byrd on the Russell’s manuscript copy of
The Principles (published in the same journal)). But Byrd's articles do not cover all parts of
the book and there is yet to be written a comprehensive account of the book's creation. I
would like to verify all the information and to read all the letters related to the topic.
I will have also some more specific questions about the meaning and the translation of some
idiosyncrasic Russellian terms (as for example, the notion of stretch, ratio, likeness…).
Reading the manuscripts and the letters could give me some hint.
- For four years, I have been investigating the relation between Russell’s logicism and the
development of mathematical sciences (geometry, analysis, algebra…) at the turn of the
century. I have already published some articles on the topic2, and I would like to write a book
on that, which could contribute to fill the gap between the historians of mathematics (who
usually tend to ignore Russell’s work) and the historians of early analytical philosophy (who
often tend to neglect the mathematical part of The Principles).
A lot of manuscripts have been published in the volume 2 and 3 of the Collected Works. But
there are still three sorts of unpublished text I need to consult : 1) the letters where Russell
talked about mathematics (geometry, analysis, …) ; 2) the annotations he put on the
mathematical texts he read ; 3) the notes he took when he read a book (for example, there is
an unpublished manuscript of notes on F. Klein).
I would like to stress that the correspondence and the annotations are essential to my project.
Russell is sometimes clearer about his aim in this kind of text than in his published articles,
which he fills with technical developments. For example, I have understood what Russell was
doing in the axiomatisation he presented in his answer to Poincaré, while reading a letter from
Russell to Couturat.
These two projects are long-term work, that will exceed the time allowed by the visiting
professorship. But I have a less ambitious proposition to submit.
Russell and Whitehead had an intense correspondence during the writing of The Principia,
which is still unpublished. For the most part, only Whitehead’s letters survived, but in spite of
the loss of most of Russell’s, this correspondence is highly interesting, at least for three
reasons :
1
2
3
It sheds some light on the evolution of Russell and Whitehead’s reflection about
geometry, and about what could have been the volume IV of the Principia, never
written. Some work has been done about this issue (cf. for instance, GrattanGuiness « Algebras, Projective Geometry, Mathematical Logic and Constructing
the World: Intersections in the Philosophy of Mathematics of A. N. Whitehead » ),
but I believe that there is still much to say (especially on the importance of
Veblen’s work).
Many letters are devoted to the analysis of the contradiction and to the means of
avoiding it. Thanks to Landini’s and de Rouilhan’s works, there has been recently
a deep renewal of the scholarship on this issue. The neglected substitutional theory
has been placed at the center of the landscape, and the evolution of Russell’s
thought has been adjusted accordingly. Neither Landini nor de Rouilhan mention
the Whitehead-Russell correspondence, and it could be interesting to relate these
letters to the evolution of the theory from 1906 to 1910. Do Whitehead’s letters
confirm the picture made by the recent scholars of the evolution of Russell’s
thought during the period ?
The correspondence constitutes a unique testimony of what could be seen as a
unique philosophical project. Russell and Whitehead do not start from a fixed
philosophical standpoint on which they would base their symbolic presentation (as
did Frege for example) – neither do they adopt whatever technical device could
lead to the expected result (as Peano for instance). Instead, they seem to use
symbolism as an experimental means to discriminate between different competing
theories. There is then a continuous two-sided movement in the correspondence :
« Russell et l’Universal Algebra de Whitehead: la géométrie projective entre ordre et incidence (1898-1903) », Revue
d’Histoire des Mathématiques, 2004 : 10, p. 187-256 ; « Grandeurs, vecteurs et relations chez Russell (1897-1903) »,
Philosophiques, 2006 ; « Which Arithmetisation for which Logicism ? Russell on Quantities and Relations », History and
Philosophy of Logic (accepted for publication).
2
from the philosophy, to the formalism, and vice-versa. This fascinating method of
research is of course not apparent in the finished work, where only the final result
of the investigation is delivered – and, if only for making apparent what was this
astonishing way to do philosophy, it would be worth publishing the
correspondence.
I would like to constitute a kind of synopsis of the correspondence, that is :
1) to review the principal topics being discussed ;
2) to distinguish the various periods in the discussion ;
3) to correlate the topics and the periods to Russell and Whitehead’s published works.
Doing the last two tasks would be a very hard and demanding work. I am not sure that it could
be done in one single term, but I hope to be able to clear the way. This research could be the
first step toward a publication of the correspondence.