D. Rabouin, « On Mathematical Style » (accepté) in K. Chemla & E. Fox-Keller, Culture without
Culturalism, à paraître chez Oxford University Press, 2013.
VERSION DE TRAVAIL. NE PAS CITER SANS ACCORD DE L’AUTEUR
English will be polished on the last version
On Mathematical Style
David Rabouin
Introduction
Although the use of the category of “style” in history and philosophy of science became
in recent years so fashionable that it resembled a kind of “epidemy”, it did not reach the
description of mathematical activity with the same strength than other fields of
research 1. One aim of this paper is to advocate for the fecundity of such an approach.
The main motivation underlying this claim is the following one: history of science in
general, and history of mathematics in particular, provide us with many “objects”,
“theories” and “concepts” which stabilize over time notwithstanding an important
variability of their “meanings” and “interpretations” 2. In other words, historians often
present us with epistemic entities3 which circulate across different contexts (be they
“national cultures”, “schools”, “traditions”, “Denkkollektiv”, “epistemological cultures”,
etc.) 4. The problem, from an epistemological point of view, is that most of the existing
models consider precisely “meaning” as the best way to identify epistemic entities. This
is particularly true in the description of mathematical activity where one lacks at first
sight alternative ways of identification, in particular all the material surroundings
existing in other sciences such as: instruments, experimental apparatus, scientific
artifacts, etc. “Styles” may provide, I shall argue, a solution to these difficulties.
Considered as “ways of writing”, they direct our attention to some material aspects of
1 On the “epidemic” character of “style” see the survey by (Gayon 1996). For a complementary survey of
the (quite rare) literature on “Mathematical Style” (Mancosu 2010).
2 To have an example in mind, see in this volume C. Ehrhardt’s study on the mathematical “concept” of
“group” and its related “theories”. Pages?
3 I use this vague terminology to indicate that the approach advocated in this paper does not commit us to
the existence of “concepts”, “objects”, “theories” and other entities of the same sort which can therefore
remain unspecified.
4 Following a “Geertzian” orientation, I will often designate these contexts as “cultural”, “culture” being
characterized as “an historically transmitted pattern of meanings embodied in symbols, a system of
inherited conceptions expressed in symbolic forms by means of which men communicate, perpetuate, and
develop their knowledge about and their attitudes toward life” (Geertz 1973, 89). What I will be interested
in is precisely the part of the circulation which is independent of these “patterns of meanings” (or, in other
words, what is invariant under a variation of the meanings attached to a given scientific practice).
1
mathematical activity. More than that, these material anchors can circulate across
various contexts of interpretation much in the same way than material artifacts do in
experimental sciences 5.
Amongst the very few works devoted to this issue is an ancient paper by the
mathematician Claude Chevalley, published in 1935 in the Revue de Métaphysique et de
Morale 6. In this paper, entitled “Variations du style mathématique”, Chevalley claimed
that one could detect in mathematics (as in literature) general tendencies in the ways of
writing. These “styles”, he pointed out, were widely circulating across different places at
given historical periods, so that they may serve to characterize these periods. Although
very descriptive and almost naïve in its approach, Chevalley’s article presents (at least)
one originality which deserves particular attention and which will guide me in this
paper: by focusing on the ways of writing in their almost concrete aspect, it put
particular emphasis on external displays as opposed to other traditional “internal” ways
of characterizing “styles” (in terms of “concepts”, “theories”, “reasoning” or “objects”).
But, as I will try to show, it also detached itself from another very common way of
characterizing styles in terms of specific cultural contexts. Under this particular
characterization (style as “ways of writing”), this category reveals itself, I shall argue, not
only as a fruitful tool, but also as a specific one. In particular, it ceases to be a clumsy
reformulation of other classical designations of scientific activity such as “schools”,
“traditions”, “research programs”, “methodologies”, “practices” or “techniques” 7.
Because of this specificity, Chevalley’s notion of style falls so to say in the middle
between the two dominant uses of this category in history and philosophy of science,
namely a “cultural” one (sometimes also called “local”) and “style of reasoning” à la
Crombie-Hacking 8. Paolo Mancosu, in his survey on “mathematical style”, already
credited Chevalley for having delineated the accurate level of inquiry, at least on a
programmatic level: “In the case of mathematics there is good evidence that the most
natural locus for ‘styles’ falls, so to speak, in between these two categories. Indeed, by
and large, mathematical styles go beyond any local community defined in simpler
sociological terms (nationality, direct membership in a school etc.) and are such that the
support group can only be characterized by the specific method of enquiry pursued. On
the other hand, the method is not so universal as to be identifiable as one of the six
methods described by Crombie or in the extended list given by Hacking” 9.
My main motivation for expanding upon Chevalley’s notion of “style” is not, however,
that it seems more accurate in the description of mathematical activity. It is not a simple
On the notion of “material anchor” see (Hutchins 2005) and the conclusion of this paper.
(Chevalley 1935).
7 On the fact that “styles of reasoning” à la Crombie could be equally rendered as “methods of reasoning”,
see (Gayon 1996). (Mancosu 2010) makes a similar comment as regards “cultural style” in reference to
(Rowe 2003): “The scare quotes highlight the problem of trying to grasp the difference between ‘schools’,
‘styles’, ‘methodologies’ etc.”
8 On this typology, see (Gayon 1996).
9 (Mancosu 2010 § 5).
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matter of finding a better scale of description between “local” and “global”, since it could
be easily argued that at a descriptive level both of these approaches are equally
important and complement each other very well. It is rather to address the
epistemological challenge formulated by Mancosu at the end of his paper and to which
no satisfactory answer seems to have been given yet: “The problem of an epistemology
of style can perhaps be roughly put as follows. Are the stylistic elements present in
mathematical discourse devoid of cognitive value and so only part of the coloring of
mathematical discourse or can they be seen as more intimately related to its cognitive
content?” In more basic terms: do we reach, when focusing on “style”, “essential” or
“accidental” characteristics of mathematical knowledge? Chevalley’s reflections are
interesting regarding this question not only because they testify about one
mathematician’s view (and what mathematician!) on the importance of “style” in his
activity. They also allow pointing to an aspect which is sometimes considered as merely
accidental or, in the best case scenario, as purely “expressive”: the role of ways of
writing in mathematical knowledge. More than that, by considering these concrete
aspects of mathematical activity, Chevalley exhibits circulations which are not tied to
what is often considered as the core of the “cognitive content” of mathematics and which
I will designate, following Gert Schubring, as “shared epistemologies” 10, i.e. basic
common understanding about the rough delineation of the theories in play and the
characterization of their objects.
Chevalley’s proposal reverses, in fact, the traditional way of looking at mathematical
knowledge by indicating cases in which what is stable is the way of writing and what is
potentially variable is its understanding. This approach goes in opposite direction to
usual characterizations of this category. Most of the (rare) epistemological descriptions
of “style” (as opposed to merely descriptive uses of it) rely indeed on a classical
approach of mathematical knowledge in terms of “theories” and “concepts”. They insist
therefore on its expressive aspect, i.e. the way in which a given style participates in the
expression of some “content”. One of the most elaborate philosophical analysis, that of
Granger, states for example that “style appears to us (…) on the one hand as a way of
introducing the concepts of a theory, of connecting them, of unifying them; and on the
other hand, as a way of delimiting the what intuition contributes to the determination of
these concepts” 11. Hacking, for his part, insists heavily on the fact that styles coincide
with the introduction of new domains of objects or better, in a clear reference to Kant,
new conditions of possibility for objectivity (which hence function as historicized a
priori): “My styles of reasoning, he claims, eminently public, are part of what we need to
understand what we mean by objectivity. This is not because styles are objective (that is,
that we have found the best impartial ways to get at the truth), but because they have
(Schubring 1996, 363). My use of this notion is however opposite to Schubring since according to him
“shared epistemologies” could be used as a good characterization of what “style” are.
11 (Granger 1968: 20), as translated in (Mancosu 2010). On the characterization in terms of “form” and
“content” see also (Granger 1968, 6-7) and (Otte 1991, 238): “Style does not mean the form or the relation
between form and content but the meaning of a unity of the two. This unity however is not absolute and
static but is a unity of process”.
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settled what it is to be objective (truths of certain sorts are what we obtain by
conducting certain sorts of investigations, answering to certain standards)” 12. In both
these characterizations, “styles” are inseparable of conceptual domains, in which, to
mimic Hacking’s Foucaldian terminology, a certain “regime of truth” is rendered
possible.
What is particularly interesting in Chevalley’s styles is that they are free of such
presuppositions. They do not ask for an identification of “concepts”, “objects”,
“structures”, “regimes of truth” or “theories” expressed through them. As Chevalley
recalls on the example of the modern “axiomatic style”, a style in his sense can be
common to very different “theories” (General Topology, Abstract Algebra, Measure
Theory). It even helps, in this case, to circulate between them and uncover new
conceptual architectures 13; on the other hand, as he indicates on the example of the
Weierstrassian “ε-style” (“style des ε”), the conceptual weight of a style can be seen as
fully captured by the way of writing in and of itself (rather than expressed through it) –
as if, to put it in Leibnizian terms, writings were “reasoning for us”.
One of my aims in this paper is to expand upon these two features in order to elaborate a
richer (and hopefully more useful) notion of style than the one actually existing for the
description of mathematical activity: 1. The relative independence of “styles of writing”
as regards their domains of “interpretation”, and in particular as regards the delineation
of “theories”, “objects” and “concepts” involved; 2. The cognitive value of the ways of
writing in and of themselves (as external displays carrying specific kind of inferences,
which I will also designate, following an indication from Ed Hutchins, as “material
anchors”). The first part of the paper will be dedicated to an exploration of this program
on a purely methodological level: I will first describe “Chevalley’s styles” in more details,
paying particular attention to the context in which they were put forward (section 1). I
will then confront this notion critically with the two other descriptions of “style”
mentioned above (section 2). In the second part of the paper, I will try to give more flesh
to this description by dwelling upon a research program launched with my colleague
Sébastien Maronne: the study of the stabilization of a “Cartesian” style of Geometry
which developed in the second half of the XVIIth Century and is often anachronistically
referred to as the birth of “Analytic Geometry” (section 3). Finally (section 4), I will
indicate other examples on which this category of “style” seems to apply and try to
sketch an epistemological framework adapted to it.
(Hacking 2002, chap. 12, 181)
This first aspect would be accepted by Crombie and Hacking, but on a much larger – way too large in
fact – scale. According to them “axiomatic style” is indeed a “style of reasoning” stable since… the Greeks!
This, as we will see later, prevent them to see the dynamical aspect of modern “axiomatic style”, by which
it distinguishes itself very much from the ancient one and on which Chevalley puts a lot of emphasis
(Chevalley 1935, 384: see note 18??? below).
12
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1. Chevalley’s Styles
Chevalley’s characterization of “style”
As was often pointed out by commentators, the notion of “style”, although quite popular
in recent literature in history and philosophy of science, is rarely defined and its
epistemological grounds are more rarely rendered explicit 14. Even when authors try to
provide a definition, it is very striking that this characterization is most of the time
intelligible only trough a list of examples 15. At first glance, Chevalley is no exception to
this fact: he just states that one can identify “general tendencies” in the ways of writing
mathematics 16 and limits himself to dwell upon two important examples: 1. the “ε-style”
developed by Weierstrass and his School to express basic properties of functions in Real
and Complex Analysis (by contrast to the style of “Algebraic Analysis” which was
dominant in the previous period) 17; 2. The “axiomatic style” initiated by Hilbert for
Plane Geometry in his famous Grundlagen der Geometrie (1899), and then developed in
various fields of research at the beginning of the XXth Century: Measure Theory
(Lebesgue), General topology and Functional Analysis (Fréchet), Abstract Algebra
(Dedekind, Steinitz, Noether) and finally Probability Theory (for which Chevalley gives
no proper names, but clearly alludes to Kolmogorov).
This rather general description is already interesting in and of itself as soon as one
realizes that it is uttered by one of the founders of the (then just created) Bourbaki
group. To acknowledge that there are “revolutions that inflect writing, and thus thought”
seems indeed to go in opposite direction to a certain kind of naïve mathematical
Platonism sometimes ascribed to the “structuralist” ideology (as if “structure” were
some fixed abstract entity living in a platonic realm). To this effect, one could jump
directly to the conclusion of the paper in which Chevalley insists on the fact that the
“axiomatic style”, which puts forth the notion of “structure”, is a style like any other. As a
(Gayon 1996, Mancosu 2010)
See (Granger 1968, 20-21). This strategy is also followed by Hacking, who recalls that “Crombie does
not expressly define ‘style of scientific thinking in the European tradition.’ He explains it ostensively by
pointing to six styles that he then describes in painstaking detail.” Hacking describes then his own
contribution as a way to isolate “at least a necessary condition for there being such a ‘style’.” (Hacking
2002, 181-182).
16 “Mathematical style, just like literary style, is subject to important fluctuations in passing from one
historical age to another. Without doubt, every author possesses an individual style; but one can also
notice in each historical age a general tendency that is quite well recognizable. This style, under the
influence of powerful mathematical personalities, is subject every once in a while to revolutions that
inflect writing, and thus thought, for the following periods” (Chevalley 1935, 375), as translated in
(Mancosu 2010).
17 The canonical example of this style, still familiar to any undergraduate student, is the “modern”
definition of continuity. For a real function (i.e. a function mapping some element x of the set of the real
numbers, IR, to other elements f(x) of IR), it would take the following form:
“Let f be a real function, x0 a point in its domain of definition, and ε, δ, some positive quantities (∈IR+),
The function f would be said to be continuous in x0 if:
For any ε, there exist a δ so that : |𝑥 − x0 | ≤ δ → |𝑓(𝑥) − 𝑓(x0 )| ≤ ε”
I intentionally do not write symbols for quantification, since there were invented after the development of
the Weierstrass style. What is important in this “style” is not only the fact that it provides “rigorous”
definitions for some intuitive notions, such as continuity, but that it commands also a repertoire of
classical techniques of demonstration, which were developed by Weierstrass and his School.
14
15
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consequence, “structure” is not presented as a way to unravel the “real” ontology suited
to mathematics in general, but as a temporary practice loaded with a fruitful heuristic
value: “As a result, contemporary mathematics tends to define mathematical objects in
comprehension, i.e. by their characteristic properties, rather than in extension, i.e.
through a construction. This feature has presumably nothing definite. But it is difficult to
foresee right now in which direction it would develop. In any case, the actual trend
seems far from having exhausted its internal dynamic” 18.
But there is another reason why Chevalley’s proposal is not as naïve as it could seem at
first. To publish a paper on the notion of “style” sounds indeed nothing like innocent for
a European mathematician writing in 1935. The year just before appeared, in fact, an
essay written on the same topic, Stilarten mathematischen Schaffens, by a famous
mathematician enrolled in the Nazi movement: Ludwig Bieberbach 19. In this paper,
which was the continuation of other articles and lectures on the role of “personality
structures” in mathematics, Bieberbach advocated, following Jaensch’s Racial
psychology and the “Blut und Boden” doctrine, for the existence of very different styles
of mathematical creation, namely the “J-Style” (characteristic of the German mind, closer
to intuition) and the “S-Style” (characteristic of the Jewish mind, more abstract and
analytic). This prompted in European mathematical Journals harsh polemics involving
important mathematicians of the time such as Harald Bohr and Geoffrey Hardy 20.
Considering this context, the choice of the category of “style”, and more than that of the
examples given in the paper seems significant. One might first notice that the three great
German mathematicians that Chevalley mentions: Weierstrass, Hilbert and Dedekind,
are not only the ones who were the most difficult (to say the least) for Bieberbach to
integrate in his classification of “styles” (in which abstraction is correlated with the SStyle), but also that they are closely associated by Chevalley to Jewish mathematicians
such as Ernst Steinitz and Emmy Noether. One should also notice that the very purpose
of the paper is to designate two “styles of writing”, that of the “ε-style” and that of the
“axiomatic style”, which circulated between Germany and France at the time. Chevalley
recalls to this effect two important French names: Lebesgue and Frechet.
This context is important for our purpose: first, it reminds us, if needed, that our
analytical categories have a history and that because of this history, their use can be far
from neutral. At the time of Chevalley, there were good strategic reasons to recuperate
the notion of “style”. But these reasons are still valid nowadays: not only do we still have
18 (Chevalley 1935, 384). He then continues: “The various theories, which were until now kept separated,
did not presumably reach yet their final form. Many of them could certainly be analyzed as
superimpositions of more general theories; others are felt to be equivalent or to derive from a same
source. The structural analysis of already known facts is hence far from being achieved, not to speak of
new facts which manifest themselves from time to time”. This opinion does not seem to have been
idiosyncratic to Chevalley in the Bourbaki group and one should notice that the same kind of claims is
made at the end of “L’Architecture des Mathématiques”, the famous manifesto published by the group in
1948 (Bourbaki 1950, for an English translation).
19 (Bieberbach 1934). I thank Moritz Epple for having pointed to me this historical context when I first
presented my ideas on style.
20 On the general context, see (Segal 2003).
6
to fight against religious or nationalistic recuperations of the history of mathematics 21,
but, less dramatically, the danger of “cultural essentialism” is often lurking behind the
use of the category of “style” in history of science. It is therefore important not only to
emphasize that styles can circulate across different cultural determinations, but also that
one can give a positive content to this category by anchoring it in the ways of writing, i.e.
the more material aspect of this circulation (as opposed to what is attached to
“interpretations” and “meanings” shared by a community of actors or, in the opposite
direction, to some platonic “ideas” grounding the supposed “universality” of the
supposed “conceptual” background).
The story that Chevalley is telling about the emergence of the “ε-style” presented as a
consequence of a need for “rigorous” foundations in mathematics and the subsequent
development of the “axiomatic style” may appear nowadays as commonplace, if not as
an oversimplifying myth. However, one should keep in mind that it was a way to recall to
people like Bieberbach the absurdity of an opposition between “French” and “German”
mathematics (which was very common at the time). Whatever the historical reasons for
the emergence of the “ε-style” be, it would remain an interesting historical fact that this
way of writing spread rapidly out of the Berlin School in Germany and in France (to the
point that it is still nowadays one of the standard ways to write mathematics in Real and
Complex Analysis). To take just one example of a French use on which I shall come back
later: Henri Poincaré, although quite opposite in many ways to Weierstrass’s program of
“arithmetization”, had no trouble to adopt the “ε-style” when dealing with complex
functions 22. It should also be noticed at that point that Chevalley was not denying in his
paper the existence of individual styles, which might involve cultural and biographical
specificities (think of Poincaré and Weierstrass again) 23. His point was more likely that
these styles coexist with some “general tendencies” marked by circulation between
individuals and cultures, and particularly between national traditions and schools.
Epistemological issues
These elements of context being recalled, it would certainly be to go too far to claim that
Chevalley’s paper is not naïve as regards its epistemological grounds, which are not – to
say the least – rendered explicit. At first glance, the core of the paper seems to be a quite
commonsensical reaction: there are general trends in the ways of writing mathematics
and these “styles” are not fixed cultural determinations, but vary through history and
circulate between different cultures. To give immediately an example of the distance
which I think one should take with Chevalley’s position: he is quite clear at the beginning
of the paper about the fact that styles are due, according to him, to the influence of
See the articles presented at the workshop "When Nations Shape History of science", organized by
Karine Chemla and Agathe Keller in 2006 (Chemla & Keller 2008) and subsequent work by Keller on
“Vedic Mathematics” (http://www.reseau-asie.com/colloque/thematiques-du-1er-congres-2003/atelier2-histoire-des-sciences-en-asie-history-of-sciences-in-asia/and
http://halshs.archivesouvertes.fr/docs/00/06/92/76/PDF/VM_enjeux_multiples.pdf); See also (Dhruv Raina 1999).
22 See, for example, (Poincaré 1883).
23 “Without doubt, every author possesses an individual style; but one can also notice in each historical
age a general tendency that is quite well recognizable” (Chevalley 1935, 375).
21
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strong personalities in mathematics (“This style, under the influence of powerful
mathematical personalities, is subject every once in a while to revolutions”). This is the
way in which he describes his two examples (attached to the names of Weierstrass and
Hilbert). But he gives no convincing reason (in fact, no reason at all) why this should be
so. Indeed, it is not difficult to find examples of ways of writing which had parallel
developments in several authors coming from different and independent traditions. To
give just one example contemporary to Chevalley’s corpus: Frederic Brechenmacher has
given a nice case study on the development of matricial calculus from 1850 to 1938 in
which it appears that this “style of writing” was developed by several authors
independently and in quite different contexts before merging in a unified theory 24. There
seems to be no clear difference between this example and that of the “ε-style”: it was a
new way of writing, which inflected mathematical thought and it coincided with the
development of a new way not only of doing mathematics (later to be called “Linear
Algebra”), but of conceiving some of its fundamental objects. As I would argue in the
following part of the paper, it is paradoxically also the case of the “Cartesian” style of
geometry. The new “algebraic” style of doing geometry, although attached to the name of
Descartes, is indeed rendered completely unintelligible if one does not see first that it
stabilized itself rapidly as a mixture of (at least) two independent origins: Descartes of
course, but also Fermat. More than that, what we designate under the name “Descartes”
is not referring to a “strong personality” here, but primarily to a book, which has a
history of its own and particularly an editor, Frans Van Schooten, who played a pivotal
role in the shaping of its content.
Another aspect on which Chevalley is unclear is whether or not something like a “shared
epistemology” has to be considered as attached to a given style. As I pointed out before,
he certainly claims that changes in the ways of writing “inflect thought”. This aspect is
rendered explicit in the course of the development through an analysis of the inflexion
produced by the different styles. In the case of the “ε-style”, the main feature is the
emphasis put on inequalities (whereas the previous “style”, that of “algebraic analysis”,
is presented as centered on equalities) 25. In the case of the “axiomatic style”, the
emphasis is put on the comparison between theories and the way in which a finegrained analysis of the role played by each axiom forces us to reshape them. But what is
not clear is whether or not these features are part of more general “shared
epistemologies”, such as, for example, the trend towards “rigorous” foundations which
are also invoked by Chevalley. If it were the case, it would not be difficult to find
counter-examples. Poincaré, as we already saw, adheres occasionally to the “ε-style”
without encompassing – to say the least - Weierstrass’ general epistemology. The same
could be said about the Moderne Algebra mentioned by Chevalley as a prototypical
example of the new “axiomatic style”. As was pointed out by Leo Corry in his seminal
study on this tradition, the fact that Dedekind, Steinitz and Emmy Noether shared a
same “epistemology” (often time described as a kind of “structuralism”) is far from
24
25
(Brechenmacher 2010).
(Chevalley 1935, 378-379).
8
obvious and resembles more likely to a myth created retrospectively by Noether herself
to give a strong identity to the “Göttingen School” 26.
Since Chevalley is not explicit about this fact, it is not easy to know exactly what he has
in mind when claiming that there are “revolutions that inflect writing, and thus thought”.
Considering the previous counter-examples, I would limit myself to a narrow-minded
interpretation: the epistemological content of a “style of writing” do not have to involve
elements of a “shared epistemology” (even if it may of course happen that they do).
When Poincaré uses the “ε-style”, this is not because he shares with Weierstrass a
certain conception of what the objects (“real numbers”, “functions”, etc.) involved in this
manipulation are and what the “good” (or, in this case, “rigorous”) delineation of the
theories are, but because this way of writing allows some powerful inferences, which are
not made possible in the previous “style”. As we will see in section 3, this distinction is
crucial when one wants to give an account of the development of the “Cartesian” style of
Geometry which developed in XVIIth-XVIIIth Centuries.
2. Comparison with other notions of mathematical “style”
Hacking’s program
The two main uses identified in survey essays like that of Gayon, Hacking or Mancosu
are mainly a “local” (also called “cultural”) and a “general” (also called “methodological”)
category of “style”. Due to their philosophical background, these presentations have a
strong tendency to conceive the first use as a merely descriptive category intended to
characterize the epistemic configuration of a given community, if not of a single person,
and used mainly by historians 27. Even if they do not deny that this use has an interest of
its own, they tend to present it as unable to help us in the resolution of epistemological
problems, precisely because they leave us with local descriptions and no way (or even
no desire) to reconnect them on larger point of view.
More generally, their view on the subject, as strongly expressed by Hacking, seems to be
that the “local” studies in history of science, which flourished in recent years, offer a
material, on the ground of which a new kind of epistemology awaits to be built. This
would support an harmonious cooperation between “styles for historians” and “styles
for philosophers” and reestablish the dialog between three kind of inquiries which “have
almost ceased to speak to each other, namely”:
(Corry 2004). One could notice that Fréchet (another name mentioned by Chevalley) developed a kind
of “empiricist” view of Geometry quite different from Hilbert’s “formalism” and the Göttingen protostructuralism, see (Arboleda And Recalde 2003).
27 Another distinction, not equivalent to the first one, is also often mentioned in this context: that between
an “individualizing” and a “generalizing” notion of style (the Style of Shakespeare and the Shakespearian
style).
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A. Newly gained analyses of, and case-studies of, the fleeting ‘micro-social’ interactions
of knowers and discoverers, their ‘macrosocial’ relationship to larger communities,
and the material conditions and objects in which the discoveries are made and which
they are about. At this level, the relevant events last a week or at most a few decades.
B. Current philosophical conceptions of truth, being, logic, meaning, and knowledge.
C.
Models of relatively permanent, growing, self-modulating, revisable features of
science (…). The result of their persistence is a body of what is counted as objective
ways of determining truth, of setting belief, of understanding meaning, a body of
nothing less than logic itself (Hacking 1991, 131).
One could certainly agree with the fact that these fields of research developed largely
independently in recent years. This development is problematic if one considers that
they all aim at giving a description of what scientific activity is. One could also agree that
the traditional use of the category of style by historians were mainly taken into the first
tradition (especially through the notion of “national style” 28 or styles attached to given
“schools”), with the exception of Crombie who was one of the rare tenant of the third –
so that Hacking can strategically present his own program as a way to match A and C by
a detour through B. 29
However, this presentation takes for granted what should be first put into question: do
we really need a philosophical framework (B) in order to match again A and C? Many
historians would not think so. The difficulty appears clearly when Hacking mentions the
“constructionalist” option: “they study the first shift at the factory of facts. Quitting work
early in the day, they leave us in the lurch with a feeling of absolute contingency”. But
what is exactly wrong with this feeling of “absolute contingency”? What if there was no
philosophical framework to “ground” historical contingencies – or, better, what if
“absolute contingency” was, in fact, the only true metaphysical framework? What if, to
take up Clifford Geertz’s beautiful expression, there was nothing more than “local
knowledges” – the history of scientific knowledge being nothing else, to mimic one of
Jacques Lacan’s famous dictum, than the integral of the equivoques left by local language
games? 30
All of these questions should be taken into account in order to develop a unifying notion
of “style” compatible with “analyses of, and case-studies of, the fleeting ‘micro-social’
interactions of knowers and discoverers”. Hacking has something to say to this regard
On which See (Daston and Otte 1991).
In the introduction of the paper, he describes emphatically this important “philosophical task in our
times” as a way to reconnect: “(1) Social studies of knowledge, of the sort pioneered by David Bloor and
Harry Barnes in Edinburgh (…); (2) Metaphysics, particularly the debates that resulted from Hilary
Putnam’s series of revised positions (…); (3) The Braudelian aspect of science, that is, the long-term slowmoving, persistent, and accumulating aspects of the growth of knowledge” (Hacking 1991, 130).
30 See (Geertz 1973). This approach is not yet well developed in the history of mathematics, but some very
interesting indications can be found in the notion of “readings” introduced in (Goldstein 1995). According
to this study (on one of Fermat’s theorem), the history of a mathematical result can be described through
the series of its interpretation, be they from historians (“lectures historiennes”) or from mathematicians
(“lectures mathématiciennes”). No particular epistemological background seems then to be needed to
reconcile “long term” History and local determinations.
28
29
10
when he briefly mentions that the constructionalists “give little sense of what hold the
constructions together beyond the networks of the moment, abetted by human
complacency” (Hacking 1991, 131). As a consequence, he rightly emphasizes the fact
that we need to give an account of a phenomenon which seems as important as the
variability of representations in history of science: that of stabilization and of Braudelian
long-term (longue durée) phenomena 31. Hence the following program: “A proposed
account of self-stabilizing techniques begins by observing that a style becomes
autonomous of the local microsocial incidents that brought it into being. Then there is
the detailed account of how each style does stabilize itself.” 32
Hacking’s own proposal to this effect is nonetheless very disappointing when it comes to
mathematics. The “longue durée” which he proposes to consider as a process of
“stabilization” is indeed nothing less than… the whole history of “deductive
mathematics”! The first of Crombie’s “style of scientific thinking” was indeed “the simple
method of postulation exemplified by the Greek mathematical sciences”. Even if Hacking
claims that “styles do not determine a content, a specific science”, he makes here a
startling exception: “We do tend to restrict ‘mathematics’ to what we establish by
mathematical reasoning, but aside from that, there is only a very modest correlation
between items (a) through (f) and a possible list of fields of knowledge” (Hacking 2002,
182. My emphasis. “(a) to (f)” refer to the six different “styles of reasoning”). In other
words, “Mathematics” is the only case in which a style of reasoning is correlated by
Crombie and Hacking with a science and in which the longue durée is that of a whole
“field of knowledge”!
Exploring in more depth the list of Crombie’s examples, Hacking finally acknowledged
another mathematical “style” which he named “the algorismic style of reasoning” and
which he attributed to the “Indo-Arabic style of applied mathematics” 33. But that did not
really inflect his first characterization. “Styles”, according to him, are for example
strongly connected with the introduction of new objects34, but he did not make explicit
what kind of “objects” were associated – if there were – with the “algorismic” style. We
are left with the preceding characterization in which mathematical objects in general (as
“non-empirical” entities) were supposed to be introduced with the “postulational style”.
This may be the reason why Hacking designates the “algebrizing of geometry” (he does
not say much more about it than naming it an “Arabicizing of the Greek”) not as a third
independent style, but as a combination of the two others (as if it was an evolution of the
31 Hacking talks about “quasi-stability of Science (Hacking 2002, 192). For a very interesting case study on
the persistence of Newtonian physics into contemporary one, see (Smith 2010).
32 (Hacking 2002, 196. My emphasis).
33 (Hacking 2002, 185). Note the important – and very debatable – qualification of “applied” mathematics.
34 “Every style of reasoning introduces a great many novelties including new types of: objects / evidence /
sentences, new ways of being a candidate for truth or falsehood laws, or at any rate modalities /
possibilities”. And later on: “Each style, I say, introduces a number of novel types of entities, as just listed.
Take objects. Every style of reasoning is associated with an ontological debate about a new type of object.
Do the abstract objects of mathematics exist? That is the problem of Platonism in mathematics” (Hacking
2002, 189).
11
type of “objects” supposedly introduced in the first style and characteristic of “pure”
mathematics) 35.
The only fact that Hacking finally acknowledges the existence of another style of
reasoning in mathematics and that he claims that it can “combine” with the postulational
style in order to form a new style (“algebrized” Geometry) opens the door to a
multiplication of styles with no clear limitation to their possible proliferation. If one
accepts the “algebrizing of geometry” as a new style, why not “the style of indivisibles”,
which developed so vividly at the same time in Europe? or the “synthetic” style
advocated at the end of the XVIIIth Century as a reaction to the “algebrized” geometry?
or the modern “axiomatic style” (as differentiated from the ancient one)? 36 What would
be the reasons for rejecting these styles and accept the “algebraico-geometrical” one?
And if we accept these ones, should we say, in a more fine grained description, that
Stevin’s style of symbolical algebra (centered on polynomials) is the same as Vieta’s one
(centered on equations)? That Carnot’s style of synthetic Geometry is the same as that of
Monge? That Poncelet’s sketch of Projective Geometry is in the same “style” than Von
Staudt’s etc., etc.? What would be the limitation in the precision and the delineation of
mathematical “styles of reasoning”?
One can certainly follow Mancosu here in the suggestion that we need a middle-scale
description – or rather that we should at least be able to determine clearly at which scale
we want our category of style to be efficient (a point on which Hacking is finally less
clear than he claims to be). The emphasis on the processes of stabilization is certainly a
good start, but, as we saw, Hacking is unclear about the kind of stabilization he has in
mind: is it that of the postulational way of thinking, coinciding with the whole history of
deductive mathematics? Is it that of the “algorismic style of reasoning” (correlated with
which kind of “longue durée”? from Babylon to the present days? Or more limited to the
so-called “Indo-Arabic style of applied mathematics”?)? Is it finally that of the
“algebrizing of geometry” which began with the Arabs and developed so vividly after
Descartes – and when should we stop the life of this style?
This brings us back to our first question: the supposed deception arose by a “local” study
of “styles”. What is exactly the problem with these “local styles”, such as the one rapidly
reached in any fine grained description of mathematical activity? Why should they be
comprised in a larger (“philosophical”?) point of view? At a descriptive level, we have a
possible answer through the focus on the process of stabilization, especially when it
occurs across different social or cultural contexts. It is true that, in this sense, we should
On the combination of the different styles, see (Hacking 2002, 194): “The historian will want to
distinguish several types of events. There is the extinction of a style, perhaps exemplified by reasoning in
similitudes. There is the insertion of a new style that may then be integrated with another, as has
happened with algorismic reasoning, combined with geometrical and postulational thought”.
36 These “styles” are the one proposed by (De Lorenzo 1971), summarized in (Mancosu 2010). (Granger
1968) also proposed to contrast Descartes’ style of geometry with that of Desargues (See also Otte 1991).
He took as another example the “vectorial style” flourishing at the beginning of the XIXth Century (in a
completely different way of “algebrizing” geometry than what happens with “analytical” geometry).
35
12
at least try to fit the gap between “local” determinations and “longue durée”
descriptions; but, as was pointed out by (Goldstein 1995), this articulation can certainly
be done with no recourse to philosophy. One just needs to follow the long term
trajectories (what Goldstein calls the “readings”) of mathematical documents 37. So, once
again, why should we need a “metaphysical” framework to fill this gap? Why would
“style” precisely occur at this strategic locus?
The role of philosophy
If by “metaphysical”, we understand something as general as what Hacking designates
when he talks about “philosophical conceptions of truth, being, logic, meaning, and
knowledge”, with explicit reference to the type of works done by Hilary Putnam, my
opinion is that it would indeed be very difficult to escape them. But this has nothing to
do with the ridiculous idea that philosophy would here be the welcoming home were
historians could cease to fight against each other (as if philosophers were not!) and
where “local” descriptions would find the way to reconnect one with each other. It is
linked more likely to the specific object of inquiry considered here. At some point, we
have indeed to recognize in our historical documents where, so to speak, “knowledge”
occurs and these “facts” of knowledge are (re)constructed from the documents like any
other facts. As a consequence, they suppose some analytical category to be so. By not
explicating what she considers as the proper characteristics of these facts, the historian
just takes the risk of projecting on the documents her own prejudice. To take up the case
made by George Smith (see note 30???), the persistence of Newtonian results in modern
physics and the cumulative aspect of science were simply rendered invisible to the
“Kuhnian” generation of historians of science raised in a conception of scientific
knowledge as a non-cumulative process. The seeing of a (historical) fact, like any other
fact, is dependent of an epistemological background which renders this fact visible 38.
Even radical “constructionists” have in mind a certain conception of how “meaning” and
“knowledge” are supposed to function (the repeated references to the Wittgensteinian
“language games” in this literature make it quite clear) 39. The necessity of an
“epistemological” dimension has nothing to do with a normative position of philosophy.
It should resemble more a kind of hygiene: a way to render his own conception of
“where” knowledge occurs and how it functions as explicit as possible. Symmetrically, as
rightly emphasized by Hacking, this epistemological move is a good way to contribute to
a better description of what “knowledge” is.
I insist on this methodological issue because this is precisely where an important
problem with the notion of “style” occurs. In the literature on “style”, especially the
See also the paper by Caroline Ehrhardt in this volume.
Ironically enough, this phenomenon of visibility/invisibility is one of the characteristic of a “style of
thought” as characterized by Fleck. Hence a very interesting problem: when historians describe “local
styles”, there is no reason to believe that they are not themselves taken in some “styles of thought”, exactly
in the same way that the facts isolated by their actors are made visible in a certain Denkkolektiv.
39 The evolution of Bruno Latour’s conception toward a full fledge metaphysical framework is also quite
significant to this regard. See (Latour 2012).
37
38
13
“local styles”, it is indeed often assumed that the identification of “knowledge” and its
correlation with a certain “locus” (be it a nation, a school, a laboratory or an individual)
can be expressed in terms of a system of beliefs and understanding and more generally in
terms of the “meaning” of the entities in play. Here is, for example, the way in which one
of the rare explicitation of this notion by an historian goes: “While the notion of style is
intuitively appealing, there have been few attempts to clarify and apply it convincingly
to comparative cases. My preference is to analyse communication processes, to identify
basic units of communication with shared epistemologies of science and to study
interaction between such units; which may either lead to a negociation for a common
understanding, or to misunderstandings” (Schubring 1996, 363). The idea that “styles”
have to do with “shared epistemologies of science”, common understanding or
misunderstanding, is indeed very widespread – be it explicit or not 40. It is also a very
good example, in the case mentioned above, of the danger of hidden epistemological
assumptions. In fact, it is clear from Schubring’s analysis that the system of beliefs which
he would favor is based on agreement/disagreement on the nature of some fundamental
concepts and/or values (as he himself studied on the case of negative numbers and
infinity in the debates about generality and rigor from the end of XVIIth to XIXth Century
in France and in Germany) 41.
It is certainly not my intention to deny that this type of agreement/disagreement plays a
very important role in the history of science, but it so happens that the contrary is also
true. More often than not, actors do indeed share some aspects of their scientific activity
even if they strongly disagree on the nature of the fundamental objects/concepts/theories
and values involved. More interestingly, as was pointed out by Putnam (one of Hacking’s
“metaphysical” hero), it has to be the case if we want any scientific theory to have a
history. If we want representations of a scientific object (say, the “atom”) to change in
such a way that one (say, Bohr’s one) is incompatible with the other (say, Rutherford’s
one), but without abandoning them to totally “incommensurable” theories, we have to
give up the idea that scientific activity is based on transparent concepts or coherent
systems of beliefs about the description of “objects” and “theories” (what Putnam coined
the “principle of the benefit of doubt”)42. We need, in order to give an account of the
historical evolution of concepts and/or theories, to delineate what is invariant under
this evolution: for obvious reasons, “concepts” and “theories” themselves, at least as
classically understood, seem bad candidates. In any case, and this is my only point here,
this is a place where epistemological issues seem inescapable.
I don’t want to claim that Putnam’s “historical” theory of reference and the emphasis put
on inferences more that on objects and concepts is the only proper framework for any
epistemology compatible with the history of science (and vice versa). But, as was
already emphasized by Hacking, it is at least clear that the traditional (“correspondist”)
It is very pregnant, for example, in Ludwig Fleck’s characterization of Denkstils associated with a
Denkkolectiv (Fleck 1935).
41 (Schubring 2005).
42 (Putnam 1975, 275-276).
40
14
view on truth is not and that other tools are needed to account for the material provided
by history of science. This is important because the traditional view still plays a
prominent role in the characterization of “shared epistemology”, in the sense of system
of beliefs concerning fundamental objects, concepts or theories. This is also why
Chevalley’s proposal is of great interest: by characterizing “style” in a completely
different manner (through the ways of writing and their circulation), it allows describing
phenomena of circulation which are not based on “shared epistemologies” or “common
understanding”. It also directs our attention to something which stays invisible in more
traditional account of knowledge, especially when it comes to mathematical knowledge:
the role of material anchors 43.
3. A case Study: the Cartesian Style of Geometry
Position of the problem
In this section, I would like to advocate for the fecundity of Chevalley’s notion of “style”
on a case study: that of the development of “Cartesian” Geometry in XVIIth and XVIIIth
Century 44. By “Cartesian style of Geometry”, I would first designate, in a yet naïve
approach, the mix of algebra and geometry which Descartes and other mathematicians
developed in the first half of the XVIIth Century and which stabilized itself at the
beginning of the next century to finally become, after a long period of time, what would
later be called “analytic geometry” or, better, “algebraic geometry” – I would tackle the
issue of these anachronistic designations later on. The “Cartesian” style of Geometry was
already one of the examples chosen by (Granger 1968) and we saw that Hacking also
alludes to it by mentioning the “algebrizing of geometry”. It will therefore provide a nice
way to indicate the divergence between their approaches and that advocated in this
paper.
According to Granger 45, Descartes’ style has first to be characterized by the
philosophical idea that spatial extension can be reduced to what is clearly and distinctly
knowable in it, i.e. what can be, in Descartes’ parlance, accessed through a process of
“exact” measurement. The control over these geometrical entities is hence insured by
the fact that they obey at the same time to a kind of regular construction 46 and to a form
of computation 47. On this ground, Geometry can be rigorously based on algebraic
43 On the fact that the “representationalist” view prevent to grasp what is at stake in the material
anchoring of inferences, see (Hutchins 2005, 1559-1562).
44 This section summarizes the orientations of a joint program with my colleague Sébastien Maronne on
the development of the Cartesian style of Geometry from 1650 to 1750.
45 (Granger 1968, 47-55).
46
Called by Descartes “continuous movement” and related to the way in which the constructions involved
can be described as combinations of geometrical proportions.
47
This geometrical calculus is technically related to two important mathematical improvements: the
abandonment of the homogeneity constraint when dealing with geometrical magnitudes and, what goes
with it, a proper definition of a multiplication between them, able to bypass the traditional problems of
dimensionality.
15
calculations and the acceptable “geometrical” objects can be redefined as the one
accessible through algebraic manipulations. This allows at the same time an extension of
the realm of Ancient Geometry (what would nowadays be called “algebraic curves of any
degree”), a complete classification of these “geometrical” objects and a clear exclusion of
what is not accessible through these “exact” determinations (what Descartes designates
as “mechanical” curves). One sees clearly why, considered as a “way of reasoning”, it
could be described, as proposed by Hacking, as a combination between the ancient
“postulational” style and the new “algorismic” one. As regards the introduction of new
objects, the Cartesian style appears more like an extension than like a creation stricto
sensu.
This overall description could sound like a neutral characterization of what Descartes’
“new style” of doing Geometry consisted of. From the point of view of the historian,
however, it raises several difficulties:
1. There is a vivid debate amongst historians of mathematics to determine whether
or not the core of Descartes’ Géométrie really consists in treating curves through
algebraic equations. According to commentators like Henk Bos, this feature,
although very important, has to be interpreted in a larger setting: the program of
the geometric “construction” of equations. This context, now quite forgotten, is
the only way to understand, according to Bos, the very puzzling structure of the
treatise 48. There are other important arguments to support this view: for
example, Descartes never treats straight lines and circle by their equations
(whereas circles play a very important role in his construction of normals to
curves) 49. According to other commentators, such as Enrico Giusti, the core of
Descartes’ project is the method for finding normals exposed in the Second Book
of La Géométrie. In this case, the identification of curves with their equation is
central 50.
2. If we take “Cartesian Geometry” as related to the belief that any acceptable curve
in Geometry has to be definable in terms of an algebraic equation, it appears that
this is precisely the point which was not accepted by Descartes’ heirs. Although it
makes no doubt that Descartes did define the only “geometrical” curves in this
way and although he clearly grounded his overall project on a kind of “calculus of
segments” (presented at the beginning of Book I) which limits the realm of
acceptable (constructible) objects, it is a very striking historical fact that he was
not followed by his first readers, even the most faithful one, on these two aspects.
This supposed “shared epistemology” was therefore not the core of the real
change occurring in mathematics in the middle of the XVIIth Century and of the
acceptance of a new style of doing mathematics.
(Bos 1984); (Bos 2001).
(Galuzzi & Rovelli, forthcoming).
50 (Giusti 2000).
48
49
16
3. Finally, any attempt to ground a change of style solely in an internal conceptual
analysis, as is proposed by Granger, is in danger of ignoring some documents
attesting for the same kind of conceptual change occurring before, but with a lack
of transmission. This is precisely the case with the “algebrizing of geometry”
which, as expressed by Hacking, is first an “Arabicizing of the Greeks”. A better
knowledge of the Arabic sources has shown that there were important
predecessors of Descartes as early as the IXth Century: the abandonment of the
homogeneity constraint and the proper definition of an internal product between
geometrical magnitudes were already elaborated by mathematicians such as AlKhayyam, as was the geometrical construction of equations for degree three and
four. We hence need a better way to assess the proper originality of Descartes’
achievement (if there is one) and the real beginning of a new “style” (as opposed
to the extension of an already existing algebraic style in geometry) 51.
What is interesting in the characterization of “style” as ways of writing is that it allows
precisely to shortcut these difficulties. As regards the first one, it is clear that in our view
both interpretations can be true at the same time, since La Géométrie is precisely at the
same time at the crossroad of different lines of development. More than that, these lines,
originating in one and the same book, might go in opposite directions (especially as
regards the role of geometric intuition in mathematics and the identification of curves
with algebraic objects). History, contrary to systematic philosophy, has no problem with
contradiction, since a same fact can be interpreted over time in quite opposite manners.
Even if Bos is certainly right that the treatise obeys a strange composition, which has to
be explained historically by reference to a particular context (that of the “construction of
equations”), and even if it is true that Descartes was not systematically treating curves
through equations (and more generally – as he still explains to Princess Elisabeth in
1643 – still relied heavily on some diagrammatic reasoning 52), Giusti is also right that
many readers did not pay too much attention to that general structure and focused on
the “method of normals” exposed in the second Book. In this method, the identification
of curves with equations and the use of proper algebraic tools are pivotal. Although it
was still linked in Descartes’ practice with some diagrammatic elements, it took with
authors such as Hudde or Newton a more and more purely “algorithmic” nature 53.
This is a good example of the fact that mathematical results cannot be considered in
isolation of the way in which they were received and that the historian, instead of
silently pursuing one of these possible readings, should first try to make them as explicit
as possible 54. But this is also a way to start making our notion of “ways of writing” a little
bit more precise. One important debate on the nature of “Descartes’ Geometry” is to
determine whether or not it marks the beginning of an era in which symbolic
On this issue see (Rashed 2005).
Letter to Elisabeth, November 1643 (AT IV, 38).
53 See (Panza 2005).
54 This is one of the leading intuitions of (Goldstein 1995).
51
52
17
manipulations took the place of diagrammatic thinking in Geometry. On the one hand, it
is clear that Descartes’ method of normals relies heavily on symbolic manipulations and
supposes to treat geometrical objects through their symbolic counterpart (the equation
of the curve); on the other hand, the fact that Descartes spent a lot of time detailing the
graphical construction of algebraic equations in the third part of the treatise (which is
supposed to deal with the “nature” of equations) indicates that the geometrical
construction still plays a crucial role to his eyes. This tension, giving birth to a long
lasting debate, can easily be bypassed by simply realizing that Descartes’ ways of writing
is precisely a combination of diagrammatic reasoning and symbolic ones. Sharp
conceptual divergences occur on the basis of this common material setting. What
modern commentators do is exactly what the first readers did in their various
interpretations: they put emphasis on different aspects of one and the same “style” 55.
Finally, and related to the second and third debates, the attention paid to practices of
writing appears as a good way to escape difficulties involved in the conflation of what
could called conceptual and stylistic changes. Descartes is indeed at the same time in the
continuity of important conceptual changes, which already occurred with the
“algebrizisation” of geometry in Arabic mathematics and at the origin of a new way of
writing mathematics (I will specify later more precisely what this change consisted of).
The best way to indicate this difference between the “conceptual” and the “stylistic”
level is expressed trough our second difficulty: there was a circulation of Descartes’
“style”, made explicit by the actors themselves, but based on profound disagreements as
regards the basic concepts involved in this new kind of Geometry and the legitimacy of a
reduction of Geometry to Algebra. What these authors took up, we claim, was not some
Cartesian conceptions, but a certain way of doing mathematics, which was embodied in
a new way of writing it.
Conceptual plasticity
I would like to put a particular emphasis on this aspect, because it is the most striking
one when confronted not only to Granger’s, but more generally to the usual description
of what “Descartes’ style of Geometry” is supposed to be. If we follow the description
centered on an internal analysis of concepts, it is indeed very striking that Descartes’
first heirs were not… “Cartesian”. If we focus on the analysis of external factors, like
institutions, communities, and more generally “cultural” backgrounds, it is also very
striking that we encounter, depending on the context, many different ways of
interpreting one and the same book (not to say “theory” since it is unclear that
Descartes’ Géométrie presents us with a theory properly speaking) 56. Nonetheless,
Incidentally it is very important not to confuse a “way of writing” with “writing” per se: a way of writing
can involve diagrams juxtaposed or mixed with writings, exactly in the same way that literary styles such
as Apollinaire’s one in the Calligrammes or Stendhal’s one in La vie d’Henry Brulard involve diagrammatic
elements.
56 One could call them, following Ehrhardt (in this volume), different “cultures”. This is indeed one of my
main reasons not to hold that “styles” can always be interpreted as “cultural” process. Or better that style
55
18
something circulated – to the point that it changed the way of doing mathematics and
determined its overall aspect for at least one century, if not more.
Let me develop quickly that aspect since it is of great importance in our argument. It is
very well known that famous readers of Descartes’ Géométrie, such as Newton and
Leibniz did not accept the delimitation of what Geometry was supposed to be according
to him. But it is less well known that this phenomenon was not an heresy, but the rule. In
fact, the first cracks in the wall appeared already with Descartes’ first editor and
commentator: Franz van Schooten! Already in his notes on Descartes’ Géométrie
published in 1649 (when Descartes was still alive), Van Schooten did not hesitate to
include in his commentary (which was accompanying the first Latin translation of the
treatise) a development on the tangent to the Cycloid: a curve which was not, according
to Descartes, admissible in Geometry 57. More than that, in the second edition of his notes,
accompanying the first volume of the great Latin edition of La Géométrie (1659-1661),
Van Schooten presented Descartes’ method of normal on the same foot as that of…
Fermat! 58 When one remembers the vivid quarrel over the respective value of their
methods which occurred between the two men in the year 1638, this is quite a startling
situation 59.
This last fact is also very important if one wants to understand an intriguing feature of
the circulation of the “Cartesian” style of Geometry. If the name of Descartes is almost all
the time mentioned, it is often so in connection with others, like that of Fermat, Vieta,
Oughtred or Harriot. Hence the name “Cartesian” or even the proper name “Descartes”
has to be understood here in a generic way. More than that, what appears to us as
different methods (and even, for some readers, as different “styles” 60), which were
opposed very strongly by their creators, were rapidly intertwined in such a way that one
could use one in the name of the other (this is typically the case for John Wallis who
refers to Descartes, but uses Fermat’s method of tangents; James Gregory is another
name which comes to mind here). This is a good reason not to confuse “style” and
“method”: A same “style”, in the sense of “way of writing”, can involve different (and
quite opposite) “methods”. Last important thing to be noticed: the “way of writing”
cannot be here a simple matter of notations: Fermat did not use “Cartesian” exponential
notations (but Vieta’s) and his method was made public by Hérigone, who used a
symbolic writing of his own 61; later, mathematicians such as Sluse, when developing
other methods of tangents, could still use Vieta’s notations; others, like Leibniz, could
as ways of writing are not the same thing as “cultural styles” (even if they may coincide on some specific
corpus).
57 Reproduced in (Descartes 1659, 264-270). The treatment of the Cycloid was truly Cartesian in the sense
that it is inspired by one of Descartes’ letter to Mersenne from April 1638. It is nonetheless noteworthy
that what Descartes was dealing with in letters, rejecting it explicitly from the realm of his Géométrie, was
included by Van Schooten in the course of his commentary to this book.
58 (Descartes 1659, 252-253).
59 On this quarrel, see (Mahoney 1994, chap. IV § IV).
60 On Fermat’s style, as opposed to Descartes’, see (Mahoney, 1994, 57).
61 (Hérigone 1642).
19
jump with no problem from one symbolic system of representations to another. We
hence need to give a better characterization of “ways of writing” than limiting them to
the invention of new means of expression.
Next to Van Schooten, one could also mention in the first generation of Descartes’ heirs
the work of John Wallis. His treatise on conic sections, dated from 1655, was indeed the
first treatise of “Algebrized geometry” produced outside of the “Cartesian school” (i.e.
people assembled around Van Schooten and participating to the great Latin edition, the
first volume of which appears only four years later after Wallis’s treatise, in 1659) 62. Its
quite simple and clear structure is very interesting: the first part is a summary of certain
properties of conic sections known since Apollonius and demonstrated in the ancient
manner (so much for the introduction of “new objects”!)63; the second part is a
repetition of the same propositions in the setting of the new algebraic analysis of curves.
The purpose of the book is hence twofold: showing, at first, that nothing was lost by
introducing the new algebraic tools; but, in the same move, showing that something was
gained in terms of simplicity of proofs (and also in the possibility of their unification, the
algebraic setting making it obvious that some structural identity holds between
properties which were treated separately in the ancient setting). This constitutes
therefore a beautiful example of a treatise confronting two “styles” (at least on the
restricted part of the theory of curves in which they coincide, namely the study of conic
sections).
In the Dedicatio of his Book, Wallis put his use of symbolical algebra under the patronage
of Oughtred, Harriot and Descartes 64, but, as I have already mentioned, the method he
used for finding tangents was that of… Fermat (who he did not mention). This is not
however the only surprise since Wallis also mentioned another mathematician who
deeply influenced his “modern” presentation of conic sections: Cavalieri. When one
opens the treatise, one realizes that Wallis presents curve in a quite opposite way to that
of Descartes and Fermat. He considers them not as generated by the regular composition
of some continuous magnitudes (i.e. through algebraic manipulations on magnitudes),
but as “fluxions” of “indivisible” quantities. This led him to develop in parallel a new field
of research, which he coined the “arithmetic of the infinite” (the title of another of his
treatise published in the same period), but also to enlarge the realm of the algebraic
analysis of curves to the finding of quadrature for algebraic and “mechanical” curves 65.
We have here a beautiful example of a circulation in the way of writing which goes along
with an opposition in the ontological interpretations associated with it, in this case, what
is the nature of the curve. Wallis’s views on the subject were instrumental to the way in
which Newton and Leibniz conceived of “geometrical” curves as opposed to Descartes.
(Wallis 1655).
One representative of this “style” at that time, quoted by Wallis, was typically Mydorge who also wrote a
treatise on conic sections.
64 (Wallis 1655, Introduction)
65 (Wallis 1659).
62
63
20
The case of Newton, writing in the margin of the Latin edition from 1659-1661 non est
geometria !, but whose “method of fluxions” was at the same time so deeply influenced
by Descartes’ method for normals is well known 66 - as is Leibniz early refusal of the
limitation of Geometry to the case of “algebraic curves” (he is indeed the one who
invented the expression “transcendental curves” and began his mathematical career by
the resolution of a problem which was considered by Descartes as an impossible one:
“the arithmetical quadrature of the circle” 67). One could add to the list: Huygens,
Tschirnhaus, Gregory, amongst others. All of these great mathematicians, who were so
deeply influenced by Descartes’ Geométrie and contributed to change the face of
Geometry in the second half of the XVIIth Century, disagreed on fundamental aspects of
Descartes’ initial conceptual framework (be it the limitation of admissible “curves” to
algebraic ones, the characterization of what a geometrical object is, the superiority of the
“analytic” method over the “synthetic” one, the notation, etc., etc.). This conceptual
plasticity of the “Cartesian Style” is a crucial reason why the “Algebrizing of Geometry”
evolved so smoothly and so rapidly into “Differential Calculus” (invented between 1665
and 1675, but made public only in the 1680’s). It is therefore no surprise to realize that
the stabilization of “analytical Geometry” (even if it was not yet called this way) can be
found at the beginning of the XVIIIth in handbooks combining the two approaches (by
authors such as L’Hospital, Reynaud or Guisnée, and later the great Euler).
This evolution is also very important in order to mark the difference between “styles”
and “theories”. The development of Descartes’ style led indeed to (at least) two
“theories”: “analytical geometry” on the one hand (including differential calculus and the
treatment of transcendental curves) and “algebraic geometry” on the other (limited to
the treatment of algebraic curves). This is one reason why teleological characterizations
of “Descartes’ Geometry” are often embarrassed as what to consider as its essential core:
on the one hand, Descartes himself was very clear about the fact that Geometry should
limit itself to the treatment of algebraic curves (which he called, for this very reason,
“geometrical”); but on the other hand, his style of geometry, mixed with “indivisible”
methods and concepts, was at the core of the development of differential techniques and
evolves naturally to the creation of a new branch of mathematics, soon to be called
“analytical geometry”.
My point in this paper is not to enter into the details of the conceptual divergences
occurring amongst the first generation of Descartes’ heirs, but to emphasize the
following claim: the only fact that these authors disagreed on deep epistemological
issues makes it impossible to characterize this new “style” of Geometry in terms of
“shared epistemology”, “conceptual settings”, new “domains of objectivity”, etc., which is
indeed the variable part in this story. This is a reason why historians are often so
skeptical when confronted with philosophical studies which project a uniform
On Newton’s Cartesian heritage, see (Panza 2005).
(Probst 2008). The problem was supposed impossible by Descartes, because he adequated
« arithmetic » with « algebraic », where Leibniz, coming after Wallis’s Arithmetica Infinitorum, was ready
to enlarge the « arithmetic » procedures to the manipulation of infinite series.
66
67
21
conceptual analysis on historical developments. This is not to say that there is never an
harmonious development between these two aspects, but this parallel development has
to be documented and cannot be derived a priori from the analysis of the conceptual
setting related to some mathematical texts 68. In the case of Descartes, as in many other
cases, it so happens that there was no harmonious development between these two
lines. This is an historical fact, of which I produced the sketch of an historical
demonstration above. Anybody claiming the contrary should at least produce documents
supporting this claim.
***
Coupling systems of inferences
Now the next question is: what did these authors have in common, if it was not a “shared
epistemology”? How are we to characterize a “Cartesian style” if not in terms of
“concepts”, “domain of objectivity”, new “theories” or new notations? Was this
circulation more than a series of divergent readings, based on local misunderstandings?
As was pointed out by Hacking and George Smith (on the famous case of the quarrel over
the air pump studied by Shapin and Schaffer), the answer to this question is strongly
influenced by the time scale under consideration. When taking this kind of development
in the middle of the controversies, one could have the impression that science is nothing
but a perpetual battlefield. Nevertheless it so happens that some parts of scientific
activities may sometimes stabilize and that this stabilization is no less interesting than
the controversies preceding it.
Considering our case study at a middle size scale, there are certainly good reasons not to
think that this history was a simple succession of local misunderstandings. First of all
stands the fact that there certainly was a “stabilization” of the Cartesian style in the
beginning of the XVIIIth Century. It took the form, amongst other expression, of a series
of standard textbooks 69. This presentation, which is not that of a particular “theory”,
involves features, which were rarely present in the authors from the previous
generation and are, to our modern eyes, typical of “Cartesian Geometry”: first of all, the
initial setting of some “axes of coordinates” and the direct identification of curves with
equation. Since the equation of the curve depends a priori on the choice of a coordinate
system, this kind of presentation implies to have a way of identifying different forms of
equations corresponding to a given curve, or in other words to determine a “general
form” corresponding to the variety of algebraic expressions representing one and the
same curve. By tackling this issue, authors from the beginning of the XVIIIth Century
raised incidentally a very interesting claim: the real core of the “algebrizing of
Geometry” per se, according to them, was not the use of algebraic techniques in
68
69
This is certainly a problem in the descriptions given by (Schubring 2005).
Such as the one which I briefly mentioned before: Guisnée (1705), L’Hospital (1707), Reynaud (1708).
22
geometry 70; it was more likely to be found in a particular technique, which allows
finding these general algebraic “forms”: the method of “indeterminate coefficients” 71.
This method, which is at the center of Descartes’ “method of normals”, consists in
comparing two equations of the same degree (expressing some equivalent geometrical
relations) and solving the problem in play by identifying their coefficients 72. When
describing this method, Reynaud identifies it as a key feature of the resolution of Pappus
Problem and the construction of equations in Descartes’ Géométrie (this last fact is in
fact reconstructed by Van Schooten in his commentary). He even blames “Descartes’
commentators” for not having emphasized that aspect (with the exception of Craig in
England and L’Hospital in France). In the article “méthode des coefficients
indéterminées” of the Encyclopédie méthodique, the author (presumably Condorcet)
goes as far as crediting Descartes for the invention of this technique 73. By contrast, it is
highly interesting that no one of the modern commentators or philosophers, who we
encounter so far trying to give a characterization of something like a “Cartesian” style of
Geometry did even mention this technique as a key element. When one revisits the
history of the circulation of the “Cartesian” Geometry with this indication in mind, one
sees immediately that this method played a central role and can indeed be considered as
a kind of leitfaden surviving the divergence of “readings” 74.
Under this consideration, it could be thought that “styles” as “ways of writing”
designates simply what can be better characterized as the circulation of certain
“methods”. But as I pointed out before, this is not true: many “Cartesian” mathematicians
did not follow Descartes’ method for normals and preferred Fermat’s one. In Fermat’s
method, the core procedure is not the identification of the coefficients of two equations,
but the introduction of a quantity e, which acts as an increment of the variable and
which can be, after identification of some points on the curve, considered as “becoming
zero”. At that stage it can therefore be eliminated from the equations and produces the
Even if they do not know about Arabic mathematicians, or remain silencious about them, they all know
that it already occurred in the works of mathematicians before Descartes such as Vieta.
71 See (Reynaud 1708, t. II, Préface, ix), which refers to (L’Hospital 1707, 213).
72 Here is Descartes’ exact formulation: « Mais je veux bien en passant vous avertir que l'invention de
supposer deux équations de même forme, pour comparer séparément tous les termes de l'une à ceux de
l'autre, et ainsi en faire naître plusieurs d'une seule, dont vous avez vu ici un exemple, peut servir à une
infinité d'autres Problèmes, et n'est pas l'une des moindres de la méthode dont je me sers ». (La Géométrie
AT VI, 423).
73 Since there are testimonies of the use of this method in an arithmetical context before Descartes, he
should more likely be credited for its introduction in Geometry.
74 Leibniz and Tschirnhaus spent, for example, a lot of time and energy trying to enter into the realm of
“transcendental” Geometry by generalizing the use of this method. Unfortunately, and for reasons which
may be linked to the way in which the “Cartesian style” was identified, this material (now published in the
Akademie Edition) awaits to be studied (in the manner as (Panza 2005) has studied the importance of the
method of indeterminate coefficients for Newton). At some point, Leibniz convinced himself that one
cannot tackle all the problems in play with this method, whereas Tschirnhaus considered the contrary and
remained all of his life a convinced “Cartesian”. This played a very important role in the public controversy
occurring between the two men in the beginning of the 1680 around the invention of the methods of
quadrature.
70
23
expected result. The question is then: what did our authors consider as being new and
equally powerful in both these methods, if they did?
A possible answer to this question is the following one: a striking feature of these
methods is the fact that they display inferences which are at the same time coupled to a
geometrical setting (and more precisely to the resolution of geometrical problems), but
have no geometrical interpretation whatsoever. By contrast, one main goal of authors
having brought algebra into geometry since the Arabic mathematicians was to secure
algebraic inferences by grounding them either in substitutions operating on basic
equalities (typically the kind of “common notions” expressed by Euclid at the beginning
of the Elements) or in geometrical relations (typically the kind of identities between
squares and rectangles obtained in book II of the Elements). Although they may have
used inferences that they were not able to justify through classical means, they would
certainly not have posited these kinds of inferences as the core of their “methods”.
The identification of the coefficients of two equations of the same degree is something
which belongs to the general “form” of an algebraic equation, i.e. it relies on a law of
identity operating on the symbolic writings themselves; the identification of two
equations by elimination of a quantity which becomes zero is also a “rule” of
computation which has no geometric interpretation since the same symbol e will
sometimes represent a given magnitude (and be manipulated as such) and sometimes
“nothing” (and be eliminated as such); here also the symbolical writing gain a form of
autonomy in the “blind” process of the algorithm. In a word, both Descartes and
Fermat’s methods rely on a kind of inferential “black box” coupled with geometrical
reasoning. This allows us to give a more precise characterization of the “Cartesian style”
(at least for one important aspect): the core of the new algebraic style is not here the use
of algebra in and of itself (which existed way before Descartes and Fermat), but the
coupling of specific kinds of computational inferences with geometrical ones. In this
sense, one can say that the Cartesian style of Geometry, even if it did not suddenly
disappear, took a dramatic turn around 1750 with the first formulations which were, as
was later emphasized by Lagrange, free from any diagrammatic inferences: (Euler 1748)
can be considered as a starting point here. It would give rise to a new “style” which
Chevalley described in his paper (along with many other actors and commentators) as
that of “Algebraic Analysis” 75.
To conclude this quick description, let me emphasize the following points:
-
75
First, “styles-as-ways-of-writing” appear here as much more than a descriptive
category. They provide a heuristic tool allowing us to detect invariants which
remain hidden in other notions of “styles” (be they tied to the delimitation of
“domain of objectivity” or to “shared epistemologies” characteristic of some
“cultural” settings or of some “local” determinations). In this sense, they may be
useful tools for philosophers and for historians.
On this style, see (Panza 2005), (Fraiser 1989) et (Jahnke 1993).
24
-
-
Second, it is possible to give a contentful meaning to the category of “way of
writing”, so that it would not reduce to the trivial fact that people do not write
mathematics in the same way in different periods. This is deeply linked to the
attention paid to inferences (more than to objects and/or concepts) and the
specification of type of inferences attached to a given style 76.
It is clear that the “conceptual analysis” in terms of objects/concepts and theories
is limited, when we are confronted with the historical development of scientific
activity. This fact is now well established in the case of natural science, but it is
less in the study of mathematics where the role of the “practice” and of the
“material setting” is not so easy to characterize. The study of ways of writing as
proxies for specific kind of inferences offers a very interesting entry to this
problem.
4. Other Examples and epistemological program
It is, of course, difficult to draw general conclusions from a single case study. It seems to
me, however, that what I have presented on the example of the Cartesian Geometry is a
very general phenomenon corresponding to what Chevalley described in his paper as
the development of a “style” in mathematics. The reasons to call it a “style” are first that
it is a natural way to designate a “way of writing”; second, as I have tried to show, that it
is clearly transversal to different “cultures”, “practices” or “methods” (it crosses different
cultures and gathers different practices and methods) – so that these classical terms
would not be well fitted to describe the kind of phenomenon which we are interested in.
Finally, as I have recalled following Chevalley’s motivations, there are good strategic
reasons to invest the use of the category of “style” and not leave it either to purely
individual determinations or to cultural ones. In order to convince the reader that the
case under study is not isolated, I would like to conclude this paper by mentioning some
other examples taken from existing studies in history and philosophy of mathematics.
The Leibnizian style of Differential Calculus
The first example which comes to mind is that of Leibniz’s new “style” of
(“transcentandal”) Geometry which presents many analogies with the description of the
Cartesian Style. It is first very intriguing indeed that the Leibnizian way of writing
differential calculus developed so quickly in the beginning of XVIIIth century Europe
with such deep disagreements on the nature of its fundamental concepts and objects. To
establish this fact in detail would necessitate a study of its own, but let me just give a
beautiful quote from its main actor:
76
When our friends were disputing in France with the Abbé Gallois, Father Gouye
[sic] and others, I told them that I did not believe at all that there were actually
infinite or actually infinitesimal quantities; the latter, like the imaginary roots of
Note that this is precisely what Chevalley is doing when indicating the role of inequalities in modern analysis
as opposed to algebraic analysis.
25
algebra were only fictions, which however could be used for the sake of brevity
or in order to speak universally . . . But as the Marquis de l'Hôpital thought that
by this I should betray the cause, they asked me to say nothing about it, except
what I already had said in the Leipzig Acta (Leibniz to Dangicourt, 11
September 1716 77).
In other words, when the main supporters of the “Leibnizian” Calculus in the Académie
des Sciences asked their hero to take a public stance on the nature of the fundamental
entities involved in his calculus, they realized that he… disagreed with them (and asked
him politely to keep silence in order not to ruin their whole strategy!). This phenomenon
is not limited to the first generation of supporters of the Differential Calculus. A deep
evolution of the conception involved in Differential Calculus, as already emphasized by
D’Alembert, was the progressive introduction of a new interpretation of its basic
constituents in terms of “limit processes” – an interpretation which is absent from the
practice of Leibniz and his first followers, but present in the Newtonian tradition 78.
This is the occasion to recall another striking analogy with the development of
“Cartesian” Geometry: the Leibnizian Calculus developed indeed quite rapidly as a
mixture of features coming from Leibniz’ and Newton’s techniques, whereas the two
men were taken at the end of their lives in an important controversy about their
respective originality and launched two apparently incompatible traditions (“culturally”
speaking 79).
Set Theory
It would not be difficult to find other similar examples. Take, closer to us, the case of Set
Theory. It is very well known that Set Theory was taken into important debates and
disagreement about conceptual issues. Nonetheless it did stabilize rapidly in a quite
neutral “way of writing” to the point that Brouwer, who was such a vivid opponent to
Cantor’s ideas about transfinite, could write in the 1918-1920 an “intuitionistic” Set
Theory (an idea which is now quite well accepted 80). The distinction between “Set
Theory” as a way of writing and as an axiomatic theory (the most successful version of it
being the famous “ZFC”) is a very important historical fact if one wants to understand
the incredible success of the first one notwithstanding the quarrel over what was
involved in the second 81. But it is also an important philosophical fact since one can then
wonders, as did Solomon Feferman, if the adoption of a set-theoretical way of writing
As quoted and translated in (Jesseph 2008, 230).
See the seminal paper by (Bos 1974).
79 On the tradition of the Newtonian calculus, see (Guicciardini 1989).
80 (L.E.J. Brouwer 1918). Harvey Friedman even showed in 1973 that one can develop an « intuitionistic »
Set Theory (called IZF) with the same logical strength than the usual formalization of the “classical” one
(called ZF for “Zermelo-Fraenkel”).
81 On that aspect, see (Ferreiros 1999), who will designate the first one more likely as a “language”: “A
second distinction (…) is that between set theory as an autonomous branch of mathematics- as in
transfinite set theory or abstract set theory- and set theory as a basic tool or language for mathematics:
the set-theoretical approach or the language of sets. As indicated above, abstract set theory came about
after the set-theoretical approach began to develop, not the other way around” (p. xix). Although Ferreiros
does not dwell upon this aspect in his book, this distinction was also important in the following
developments, as is clearly described by (Gispert 1995).
77
78
26
implies a real commitment to the so-called “theory” axiomatized by mathematicians
such as Zermelo and Fraenkel in the first half of the XXth Century 82. Once again “style”
could here be a useful category for historians and for philosophers.
The Euclidean style of Geometry
To insist on the epistemological issues at stake in this research program, I will take as a
third example the characterization of the “Euclidean” practice of Geometry proposed by
Ken Manders in his article on “the Euclidean Diagram” 83. As is well known, the use of
geometric intuition in Ancient Geometry was harshly criticized by modern
mathematicians, such as Pasch and Hilbert, because it involved hidden (and sometimes
erroneous) assumptions. Without denying the strength of these criticisms, Manders
claims that they paradoxically render more urgent to understand the success of
Euclidean Geometry. No modern mathematician would indeed claim that Euclid’s
theorems are false. In fact, it is very rare to find false statements in Ancient Greek
Geometry, although some of them may lack the generality we would expect them to
have. It is hence strange, especially for modern mathematicians claiming that
mathematics needs “rigorous” demonstrations, that a Geometry relying on misleading
intuitions did not lead to false results 84. But there is another mystery: since Euclid relies
heavily on reductio ad absurdum, his geometry is full of impossible diagrams. It is hence
not possible to maintain a “semantic” reading of them, i.e. a view according to which
they should be interpreted in terms of corresponding mathematical “objects” 85.
Manders’ proposal is that both mysteries can be solved at the same time by looking at
the way in which diagrams are used not as representing some objects (what he calls the
“semantic” view on diagrams), but as proxies for carrying specific inferences (which he
coins “co-exact”). His interpretation is that Greek mathematicians lacked resources to
express some relations through discourse, whereas other relations, such as equality,
could not be expressed by diagrams. They therefore elaborated a style of discourse
coupling these two types of complementary resources. Typical examples of “co-exact”
relations given by the diagram (and by the diagram only) are: intersections, inclusion of
a region in another, contiguity of two regions (Manders 2008, 93).
Even if this is not the terminology employed by Manders, what he described in terms of
correlation between different kinds of inferences purported by discourse and diagrams
(“exact” and “co-exact”) corresponds well to what we have proposed to designate,
following Chevalley, as a “style” – even if more emphasis could be put, following (Netz
2003), on a precise study of the material display involved in this correlation. This
approach allows giving an account of a very important historical fact: as testified by
ancient commentators such as Pappus or Proclus, disagreements about the nature of
The surprising result is that the answer given by Feferman and others seems to be “no”. See: “Infinity in
mathematics: Is Cantor necessary?” in (Feferman 1998).
83 (Manders 2008).
84 Note that exactly the same point can be made concerning the first century of Differential Calculus.
85 (Manders 2008, 84 sq.).
82
27
geometrical objects were common in Antiquity. These disagreements largely exceeded
the philosophical debate mentioned by Hacking on the ontological status of
mathematical entities: they also involved mathematical questions about how to
manipulate these objects and what standards should be accepted for the
demonstrations. These debates continued amongst the Arabic authors, where they took
more and more importance – to the point that they contributed to introduce in geometry
new concepts, such as that of geometrical transformations. This general evolution led to
quite a radical change in term of conceptual foundation (especially in the recourse to
movement in geometry) 86. Once again, our approach allows acknowledging these
conceptual changes without losing the continuity of a style: Manders’ description of the
“Euclidean diagram” points indeed to features which are stable in this history without
preventing important conceptual changes to occur within it.
*
Material Anchors and Conceptual Blends
To conclude this section, I would like to sketch the kind of epistemological framework
which could be adapted to Chevalley’s description of mathematical “styles” and make
more explicit what I have alluded at several occasions as regards the “material” aspect of
mathematics and the way in which some material setting can serve as “proxies” for
reasoning. As was noticed by the anthropologist Edwin Hutchins, the study of the
phenomenon of stabilization in human reasoning confronts us with at least two very
different strategies. The first one is linked to the way in which we stabilize reasoning by
interpreting their components (for example, when we interpret a reasoning in a familiar
context, easier to handle) and could hence be coined “semantic” 87. The other strategy
does not appeal to interpretations and meanings, but to the fact that one can use
material settings (which can be perceived or imagined) to, so to say, “reason for us”. This
is what happens, for example, when one computes with a slide-rule or even with his
fingers 88. Parts of the inferences are not tied, in this case, to the interpretation of the
operations involved, but to the very functioning of the material setting. For this kind of
cognitive strategy, Hutchins has coined the word “material anchor” of reasoning. He has
provided a series of examples going from simple perception (seeing people attending in
a queue) to much more sophisticated reasoning used in Micronesian navigation.
As noticed by Hutchins, this second strategy is not well studied yet and opens a very
interesting program for history and philosophy of mathematics. There seems indeed to
be a form of continuity between this quite widespread use of “material anchors” in
everyday reasoning and the use of symbolical artifacts in mathematics (continuity which
remained hidden because of the modern myth claiming that “formal languages” are
nothing “material” and resemble more like a “transparent” picture of “concepts”):
(Crozet 2010).
Interestingly enough Hutchins designates this type of stabilization as “culturally based” (Hutchins 2005,
1558).
88 (Hutchins 2005, 1665 and 1571).
86
87
28
In some cases it is possible to do the cognitive work by imagining the manipulation of a
physical structure. In others, the case of the sliderule for example, it is necessary to
manipulate the physical device itself because it is not possible to imagine it accurately
enough to be of use. The cultural process of crystallizing conceptual models in material
structure and saving those up through time puts modern humans in a world where thinking
depends in significant measure on the availability of a set of physical structures that can be
manipulated in this way.
A final turn on this path is that when a material structure becomes very familiar, it may be
possible to imagine the material structure when it is not present in the environment. It is even
possible to imagine systematic transformations applied to such a representation. This happened
historically with the development of mathematical and logical symbol systems in our own
cultural tradition. Beginning as external representations physically embodied and operated
upon with manual skills, we learn to imagine them and to operate on the imagined structures
(Rumelhart et al., 1986). Unfortunately, much of cognitive science is based on the mistaken
view that this relatively recent cultural invention is the fundamental architecture of
cognition (Hutchins, 1995). The very idea of rationality, as held by gametheorists,
economists, and political scientists, is a cultural construct that owes its existence to the
ability to create a certain class of materially anchored conceptual blends. It is a mistake to
assume that thinking is, in general, a symbol manipulation process. (Hutchins 2005, 1575.
My emphasis).
This picture is in accordance with what I have tried to develop in this paper: style as
ways of writing can indeed be considered as some kind of blend 89 in which some
material anchor (the way of writing in its material aspect) is coupled with a conceptual
space by some kind of a mapping (note that a material anchor can enter into quite
complex processes and involve different layers of conceptual mappings, as is
exemplified by Hutchins in the case of the ars memoriae 90). What is important in this
picture, and mostly emphasized by Hutchins, is that the blend then acquires a form of
autonomy so that it can be used for its own sake and be inserted in other conceptual
frameworks than the one in which it was initially forged – provided that the inferences
in play are preserved in this new interpretation 91. This will suffice, I hope, to indicate
that “styles” considered as “ways of writing” provide not only an important phenomenon
to study, but also an interesting entrance into the question of mathematical knowledge.
Conclusion
My claim in this paper is certainly not that “ways of writing” offer the one and only way
to understand the phenomenon of circulation and stabilization in history of
mathematics, nor that it is the only legitimate use of the category of “style”. But I would
claim that this use is well fitted and fruitful in order to understand some phenomena of
stabilization provided by the history of mathematics – precisely the one in which the
conceptual variability is important. More than that, I think that it purports a very
important view on the nature of mathematical knowledge, in which “objects”, “concepts”
and “theory” are brought to the back, whereas material displays and the kind of
89
90
91
On this notion, see (Hutchins 2005, 1556).
(Hutchins 2005, 1563-1565).
(Hutchins 2005, 1556).
29
inferences which they carry are brought to the fore. This could be read in accordance
with one of Hacking’s requisite emphasizing the turn occurring in what he calls
“metaphysics” after Putnam. But the emphasis on material displays opens, according to
me, a much broader horizon, on which Hutchins called attention with his notion of
“material anchors”.
To conclude this study, I would also like to emphasize another stake in play in it. The
approach in terms of “material anchors”, because it leaves the representational aspects
entering into a given style to the back, is also precious if one wishes to understand how
mathematical “theories”, “concepts” and “objects” can stabilize in history, without
projecting on their development a teleological view in which they are supposed to be
given before hand as already constituted (denying in this sense the very idea of an
historical process of their stabilization). A study of “mathematical styles” could therefore
help in sketching an answer to one of the main challenges posed nowadays in the
integration of history and philosophy of science: how to give an account of the
constitution of theories and domains of objectivity by accepting that this constitution is
an historical process, i.e. that it may happen (and indeed happens all the time) that the
theory and its intended domain of objectivity are stabilized (if they stabilize) only as a
result, and not as a condition, of an historical evolution.
30
References
Arboleda, L. C. And Recalde, L. C., 2003, “Fréchet and the logic of the constitution of abstract
spaces from concrete reality”, Synthese 2003, 134/1-2: 245-272.
Bieberbach, L., 1934, “Stilarten mathematischen Schaffens”, Sitzungsbericht der preußischen
Akademie der Wissenschaften: 351–360.
Bos, H. J. M. 1974, “Differentials, higher order differentials and the derivative in the Leibnizian
calculus”, Archives for the History of Exact Sciences, 14: 1-90.
Bos, H. J. M. 1984, “Arguments on Motivation in the Rise and Decline of a Mathematical Theory:
The 'Construction of Equations,' 1637-ca. 1750”, Archive for History of Exact Sciences, 30:331-80.
Bos, H. J. M., 2001, Redefining Geometrical Exactness. Descartes’ Transformation of the Early
Modern Concept of Construction, New York: Springer, Sources and Studies in the History of
Mathematics and Physical Sciences.
Bourbaki, N., 1950, “The Architecture of Mathematics”, The American Mathematical Monthly,
57/4: 221-232.
Brechenmacher, F., 2010, “Une histoire de l'universalité des matrices mathématiques“, Revue de
synthèse, 131/4: 569-603.
Brouwer, L.E.J., 1918, Begründung der Mengenlehre unabhängig vom logischen Satz vom
ausgeschlossenen Dritten. Erster Teil: Allgemeine Mengenlehre, Koninklijke Nederlandsche
Akademie van Wetenschappen, Verhandelingen, 1e sectie 12 no.5.
Chemla, K., and Keller, A., 2008, "When Nations Shape History of science" in Disquisitions on the
Past & Present, 18 : 169-210.
Chevalley, C., 1935, “Variations du style mathématique”, Revue de Metaphysique et de Morale, 3:
375–384.
Corry, L., 2004, Modern Algebra and the Rise of Mathematical Structure, Basel: Birkhäuser; 2nd
edition.
Crombie, A., 1994, Styles of Scientific Thinking in the European Tradition, London: Duckworth.
Crozet, P., 2010, “De l'Usage Des Transformations Géométriques à la Notion D'Invariant: La
Contribution d'Al-Sijzī“, Arabic Sciences and Philosophy 20 (1): 53-91.
Daston, L and Otte, M., 1991, “Style in Science”, special issue of Science in Context, 4/.2.
De Lorenzo, J., 1971, Introducción al estilo matematico, Madrid: Editorial Tecnos.
Descartes, R., 1964-1974 [AT], Œuvres de Descartes, publiées par C. Adam et P. Tannery, 11 vol.,
nouvelle présentation en coédition avec le CNRS, Paris: Vrin.
31
Descartes, R., 1659-1661, Geometria à Renato Descartes anno 1637 Gallice edita ; postea autem
Unà cum Notis Florimondi De Beaune [. . .]. Operâ atque Studio Francisci a Schooten [. . .],
Amsterdam: Apud Ludovicum & Danielem Elzevirios.
Euler, L., 1748, Introductio in analysin infinitorum, Lausanne: Bousquet
Ferreiros, J., 1999, Labyrinth of Thought: A History of Set Theory and its Role in Modern
Mathematics, Basel and Boston: Birkhäuser.
Fleck, L., 1935, Entstehung und Entwicklung einer wissenschaftlichen Tatsache. Einführung in die
Lehre vom Denkstil und Denkkollektiv, Basel: Schwabe (English translation by F. Bradley: Genesis
and Development of a Scientific Fact, Chicago: University of Chicago Press, 1979).
Fraiser, G., 1989, “The calculus as algebraic analysis: some observations on mathematical
analysis in the 18th century”, Archive for History of Exact Sciences, 39: 317–331.
Feferman, S., 1998, In the Light of Logic, Oxford: Oxford University Press (Logic and Computation
in Philosophy series).
Galuzzi, M. and Rovelli, D., Nouveauté et modernité dans les mathématiques de Descartes,
(forthcoming).
Gayon, J., 1996, “De la catégorie de style en histoire des sciences”, Alliage, 26: 13–25. (english
translation: Gayon, 1999).
Gayon, J., 1999, “On the Uses of the Category of Style in the History of Science”, Philosophy and
Rhetoric 32 (3):233 - 246
Geertz, C., 1973, The Interpretation of Cultures, New York: Basic Books.
Geertz, C., 1983, Local Knowledge Further Essays in Interpretive Anthropology, New York: Basic
Books.
Gispert, H., 1995, "La théorie des ensembles en France avant la crise de 1905: Baire, Borel,
Lebesgue et tous les autres", Revue d´histoire des mathématiques 1 (1): 39-81
Giusti, E., 2000, La naissance des objets mathématiques, Paris: Ellipses.
Goldstein, C., 1995, Un théorème de Fermat et ses lecteurs, Saint-Denis: PUV (Histoires de
science).
Granger, G.G., 1968, Essai d'une philosophie du style, Paris: Armand Colin.
Guicciardini, N., 1989, The Development of Newtonian Calculus in Britain, 1700-1800, Cambrige:
Cambridge University Press.
Guisnée, N., Application de l'algèbre à la géometrie, ou Methode de démontrer par l'algebre, les
Theorêmes de Geometrie, Paris : Boudot & Quillau.
Hacking, I., 1991, “Statistical Language, Statistical Truth and Statistical Reason” in McMullen E.
(ed.), Social Dimensions of Sciences, University of Notre-Dame Press.
32
Hacking, I., 2002, Historical Ontology, Cambridge, MA: Harvard University Press (chap. 12 is a
new version of Hacking, I., 1992, “‘Style’ for historians and philosophers”, Studies in History and
Philosophy of Science, 23: 1-20).
Hérigone, P., 1642, Supplementum Cursus Mathematici: Continens Geometricas Aequationum
Cubicarum Purarum, Atque Affectarum Effectiones, Paris : Henri le Gras.
Hutchins, E, 2005, “Material Anchors for Conceptual Blends”, Journal of Pragmatics 37, 1555–
1577.
Jahnke, H.-N., 1993, "Algebraic analysis in Germany, 1780-1840: some mathematical and
philosophical issues", Historia Mathematica, 20(3), 265–284.
Jesseph, D., 2008, “Truth in Fiction: Origins and Consequences of Leibniz’s Doctrine of
Infinitesimal Magnitudes” in Infinitesimal Differences: Controversies between Leibniz and this
Contemporaries, edited by Ursula Goldenbaum and Douglas Jesseph, Berlin and New York: De
Gruyter, 215-233.
Latour, B., 2012, Enquête sur les modes d’existence. Une anthropologie des modernes, Paris : La
Découverte.
L’Hospital, G., 1707, Traité Analytique des Sections Coniques & de leur Usage, par Mr. le Marquis de
l'Hospital, Paris: Boudot & Boudot.
Mahoney, M. S., 1994, The Mathematical Career of Pierre de Fermat (1601-1665), Princeton:
Princeton University Press, 2nd ed.
Mancosu, P., 2010, “Mathematical style” in E. N. Zalta (Ed.), The Stanford Encyclopedia of
Philosophy, http://plato.stanford.edu/archives/spr2010/entries/mathematical-style.
Manders, K., 2008, “The Euclidean diagram” in P. Mancosu (Ed.), The Philosophy of Mathematical
Practice, Oxford: Oxford University Press, 80-133.
Netz, R., 2003, The Shaping of Deduction in Greek Mathematics : A Study in Cognitive
History, Cambridge: Cambridge University Press.
Otte, M., 1991, “Style as a Historical Category” in Daston and Otte (1991), 233-264.
Panza, M., 2005, Newton et les Origines de l'analyse, 1664-1666, Paris: Blanchard.
Panza, M., 2007, “Euler’s Introductio in analysin infinitorum and the program of algebraic
analysis: quantities, functions and numerical partitions”, in R. Backer (Ed.) Euler Reconsidered.
Tercentenary essays, Heber City (Utah): Kendrick Press, 119-166.
Poincaré, H., 1883, “Sur un théorème de la théorie générale des fonctions”, Bulletin de la Société
Mathématique de France, 11: 112-125.
Probst, S., 2008, “Neues über Leibniz’ Abhandlung zur Kreisquadratur”, in Hecht, H. and al., eds.,
Kosmos und Zahl. Beiträge zur Mathematik- und Astronomiegeschichte, zu Alexander von
Humboldt und Leibniz, Stuttgart: Franz Steiner Verlag, 171-176.
33
Putnam, H., 1975, "Language and Reality" in Philosophical Papers, Vol. 2: Mind, Language and
Reality. Cambridge: Cambridge University Press.
Raina, Dhruv, 1999, “Nationalism, Institutional Science and the Politics of Knowledge”, PhD
Thesis, Göteborgs Universitet.
Rashed, R., 2005, « La modernité mathématique : Descartes et Fermat » in Rashed, R. and
Pellegrin, P. (ed.), Philosophie des mathématiques et théorie de la connaissance. L’Œuvre de Jules
Vuillemin, Paris: Albert Blanchard, 239-252.
Reyneau, C.-R., 1708, Analyse demontrée ou La méthode de résoudre les problêmes des
mathématiques, expliquée et d'apprendre facilement ces sciences, Paris: J. Quillau.
Rowe, D., 2003, “Mathematical schools, communities, and networks”, in Cambridge History of
Science, vol. 5, Modern Physical and Mathematical Sciences, Mary Jo Nye (ed.), Cambridge:
Cambridge University Press, 113–132.
Smith, G. E, 2010, “Revisiting Accepted Science: The Indispensability of the History of Science”,
The Monist 93 (4), 545-579.
Schubring, G., 1996, “Changing cultural and epistemological views on mathematics and different
institutional contexts in nineteenth-century Europe” in C. Goldstein Mathematical Europe. Myth,
History, Identity (Edited by Goldstein et al.), Paris: Editions de la Maison des sciences de
l’homme, 363- 388.
Schubring, G., 2005, Conflicts between Generalisation, Rigor and Intuition, New York: Springer.
Segal, S., 2003, Mathematicians under the Nazis, Princeton: Princeton University Press.
Wallis, J., 1655, De sectionibus conicis nova methodo expositis, Oxford, ex typis Leon Lichfield.
Wallis, J., 1659, Tractatus duo, de Cycloide et Epistolaris de Cissoide, Oxford, ex typis Leon
Lichfield.
34