## 1. Introduction



The goal of this essay is to survey and analyze the literature on
style in history and philosophy of mathematics. In particular, the
problem of how one can philosophically approach the notion of
'style' in mathematics will be addressed towards the
end. Although this is not one of the canonical topics in philosophy of
mathematics, the presentation will avail itself of relevant
discussions on style in the history and philosophy of science.



Speaking about mathematics in terms of style is a common enough
phenomenon. One encounters such appeals to stylistic features in
mathematics already early in the seventeenth century. Bonaventura
Cavalieri, for instance, as early as 1635 contrasts his indivisibilist
techniques with the Archimedean style:



> 
> I know in fact that all the things mentioned above [Cavalieri's
> own theorems obtained by indivisibilist proofs] can be reduced to
> Archimedean style. (In the original Latin: "*Scio autem
> praefata omnia ad stylum Archimedeum reduci posse.*"
> (Cavalieri 1635, 235)).
> 



Later in the century it is easier to find examples. For instance
Leibniz (1701, 270-71) writes: "Analysis does not differ
from Archimedes' style except for the expressions that are more
direct and more appropriate to the art of discovery" (French:
"*L'analyse ne differe du style
d'Archimede que dans les expressions, qui sont plus
directes et plus conformes a l'art
d'inventer*"). It is an interesting fact that such
occurrences predate the generalized use of the notion of style in
painting, which only dates from the 1660s (sporadic occurrences, as
pointed out in Sauerlander 1983, are also found in the sixteenth
century). Earlier in the seventeenth century the word of choice in
painting was "*maniere*" (see Panofsky 1924;
English translation (1968, 240)). Here are a couple of additional
examples from the nineteenth and the twentieth centuries. Chasles in
his *Apercu historique* (1837) speaking about Monge
says:



> 
> He initiated a new way of writing and talking about this science.
> Style, in fact, is so intimately welded to the spirit of a methodology
> that it must advance in step with it; likewise, if it has anticipated
> it, style must of necessity play a powerful influence over it and over
> the general progress of science. (Chasles, 1837, SS18, 207)
> 



Another example comes from Edward's evaluation of
Dedekind's approach to mathematics:



> 
> Kronecker's brilliance cannot be doubted. Had he had a tenth of
> Dedekind's ability to formulate and express his ideas clearly,
> his contribution to mathematics might have been even greater than
> Dedekind's. As it is however, his brilliance, for the most part,
> died with him. Dedekind's legacy, on the other hand, consisted
> not only of important theorems, examples, and concepts, but of a whole
> style of mathematics that has been an inspiration to each successive
> generation. (Edwards 1980, 20)
> 



Obviously, one could pile up quotations of the same kind (see, among
others, Cohen 1992, de Gandt 1986, Dhombres 1993, Epple 1997,
Ferreiros and Reck 2020, Fleckenstein 1955, Granger 2003,
Hoyrup 2005, Laugwitz 1993, Novy 1981, Reck 2009, Tappenden
2005, Weiss 1939, Wisan 1981) but that would not be very
interesting. Even in mathematics style ranges from 'individual
styles' to 'national styles' to 'epistemic
styles', among others. What is needed is first of all an
understanding of the major contexts in which appeal to
'style' in mathematics occurs, although this essay will
not contain much discussion of 'individual styles'
(examples of such would include, to follow a suggestion by Enrico
Bombieri, the "*very* personal" styles of Euler,
Ramanujan, Riemann, Serre and A. Weil).



In many cases the appeal to the notion of style is conceived as
borrowed from the fine arts and some cases will be discussed
forthwith. Harwood 1993 claims that "the concept of style was
devised in order to classify cultural patterns observed in the study
of fine arts". Wessely 1991 talks about "transferring that
concept [of style] to the history of science" (265). While this
might perhaps be true for the twentieth century (see also Kwa 2012),
one should keep in mind, as was pointed out above, that this claim
must be qualified for the seventeenth century.


## 2. Style as a central concept in comparative cultural histories



Notwithstanding the previous caveats, it is a fact that some major
twentieth century appeals to the category of style in mathematics have
done so in reference to the arts. This is especially true of those
authors who were motivated by accounting in a unified way for the
cultural production of mankind and who saw thus a uniformity in the
processes of scientific and artistic production. It was in such
context that Oswald Spengler in *The Decline of the West*
(1919, 1921) attempted a morphology of world history and claimed that
the history of mathematics was characterized by different stylistic
epochs which depended on the culture that produced it:



> 
> The Style of any mathematics which comes into being, depends wholly on
> the Culture in which it is rooted, the sort of mankind it is that
> ponders it. The soul can bring its inherent possibilities to
> scientific development, can manage them practically, can attain the
> highest levels in its treatment of them--but is quite impotent to
> alter them. The idea of the Euclidean geometry is actualized in the
> earliest forms of Classical ornament, and that of the Infinitesimal
> Calculus in the earliest forms of Gothic architecture, centuries
> before the first learned mathematicians of the respective Cultures
> were born. (Spengler 1919, 59)
> 



Not only are there parallels between mathematics and other artistic
productions of a culture. Relying on Goethe's statement that the
complete mathematician "feels within himself the beauty of the
true" and on Weierstrass's pronouncement that "who
is not at the same time a bit of a poet will never be a true
mathematician", Spengler went on to characterize mathematics
itself as an art:



> 
> Mathematics, then, is an art. As such it has its styles and style
> periods. It is not, as the layman and the philosopher (who is in this
> matter a layman too) imagine, substantially unalterable, but subject
> like every art to unnoticed changes from epoch to epoch. (Spengler
> 1919, 62)
> 



The most extensive treatment that builds on the parallel between art
and mathematics and exploits the notion of style as a central category
for an analysis of the history of mathematics is that of Max Bense. In
a book appropriately entitled *Konturen einer Geistesgeschichte der
Mathematik* (1946), Bense devoted a whole chapter (ch. 2) to
articulating how the notion of style applies to mathematics. For Bense
style is form:



> 
> For style is form, essential form, and we designate this form as the
> "Aesthetic", if it controls categorially the sensible, a
> material. (Bense 1946, 118)
> 



Bense saw the history of art and the history of mathematics as aspects
of the history of mind [*Geistesgeschichte*]. In fact
"style is given wherever the human imagination and the capacity
of expression arrive to creation". Bense was certainly prone to
draw parallels between styles in the history of art and styles in
mathematics (he especially treated the baroque and the romantic styles
in his book) but he kept, in opposition to Spengler, the nature of art
and mathematics separate. Indeed he recognized that a stylistic
history of mathematics could not be reduced "to a coincidence
between certain mathematical formal tendencies and the great
artistic-worldviews-spiritual styles of single epochs such as the
Renaissance, Classicism, the Baroque or Romanticism" (p.132; see
Fleckenstein 1955 and Wisan 1981 for more recent parallels between the
baroque in art and the mathematics of the seventeenth century). He
referred to Felix Klein's "*Elementarmathematik vom
hoheren Standpunkte aus*" to point out that certain
lines of development characterized by Klein could be seen as pointing
to styles in the history of the development of mathematics (see Klein
1924, 91).



Attempts such as Spengler's and Bense's appeal certainly
to those theoreticians who would like to use the category of style as
a tool for describing, and perhaps accounting for, cultural patterns.
However, they leave the reader who is knowledgeable in mathematics
and/or history of art skeptical on account of the usually far-fetched
parallels that are supposed to provide evidence for the account. Of
course, this is not to reject ultimately the approach or the
usefulness of the appropriateness of the category of style in
mathematics but one would like its use to be more directly related to
aspects of mathematical practice. A recent attempt at connecting the notions of style in literature and in mathematics is van Bengedem and van Kerkhove 2014.



In general, one can distinguish two types of theorizing that can be
associated with such attempts. The first is purely descriptive, or
taxonomic, and satisfies itself with showing certain common patterns
between a certain area of thought, such as mathematics, and other
cultural products of a certain society. The second approach
presupposes the first but also enquires after the causes that account
for the presence of a certain style of thought or production and
normally tries to ascribe it to psychological or sociological factors.
In Spengler and Bense there are elements of both although the emphasis
is more on the parallels than on the causes underlying or explaining
the parallels.



Attempts to extend the use of the concept of style in art to other
domains of human endeavors abound in the early twentieth century. A
well-known case is Mannheim's sociological attempt to
characterize styles of thought within different social groups
(Mannheim 1928). While Mannheim had not excluded scientific thought
from the realm of sociological analysis of knowledge, he did not
actively pursue such an analysis. By contrast, Ludwik Fleck practiced
a sociological analysis of science in which "styles of
thought" played a central role. Fleck focused however on
medicine (Fleck 1935).



Here it is important to point out that the notion of thought style has
received, by and large, two different developments in contemporary
research, which also affect mathematics. First, there is the notion
encountered in Fleck. Depending on how generous one wants to be in
drawing connections, one could see this approach to styles of thought
to be related to the later work by Kuhn, Foucault and Hacking (see
below for a discussion of Hacking). There is however a different way
of thinking about styles of thought, which usually goes under the name
of cognitive styles. This is an area of interest to cognitive
psychologists and mathematical educators (for an overview of the
psychological research in this area see Riding 2000 and Stenberg and
Grigorenko 2001). Here the focus is on the psychological make up of
the individual who displays preference for a certain cognitive style
either in learning, understanding or thinking about mathematics (i.e.,
processing and organizing mathematical information). The old
distinction between visual and analytic mathematicians emphasized by
Poincare (see Poincare 1905) is still part of the
picture although there are a great variety of models and
classifications. For an historical overview and a theoretical proposal
centered on mathematics see Borromeo Ferri 2005.



In the area of history and philosophy of mathematics there are no book
length accounts of mathematical styles that explain the emergence of a
certain style with sociological or psychological categories (although
Netz 1999 has been of interest to theoreticians of style as an attempt
at a cognitive history of an important segment of Greek mathematics).
This is in contrast to books in history of the natural sciences such
as Harwood 1993, whose goal is to explain the emergence of the thought
style of the German genetics community through sociological arguments.
The closest one comes to such an account is Bieberbach's
conception of style in mathematics as dependent on psychological and
racial factors. He will be discussed in the next section on national
styles.


## 3. National styles in mathematics



Something less ambitious than the previous attempts at a general
history of human cultural productions or far-reaching parallels
between art and mathematics consists in a use of the notion of style
as an historiographical category in the history of mathematics without
particular reference to art or other human cultural activities. If one
goes back to the beginning of the twentieth century, one finds that
"national styles" were often referred to for categorizing
certain features characterizing mathematical production that seemed to
fall squarely within national lines. In the history of science such
cases of "national styles" have often been studied. One
should recall here J. Harwood's book *Styles of Scientific
Thought* (1993) and the contributions Nye 1986, Maienschein 1991,
and Elwick 2007. A case of interest for mathematics is the opposition
between French and German style in mathematics studied by Herbert
Mehrtens.



Mehrtens (1990a, 1990b, 1996) describes, in terms of styles, the
conflict in mathematics between "formalists" and
"logicists" on the one hand and
"intuitionists" on the other hand as a battle between two
conceptions of mathematics (see also Gray 2008 for a critical take on
of Mehrtens's approach while emphasizing the
"modernist" transformation of mathematics). Hilbert and
Poincare are used as paradigms for the sources of the
opposition that later led to the Hilbert-Brouwer foundational debate
in the 1920s (on the history of the Brouwer-Hilbert debate see Mancosu
1998). Mehrtens also points out that this opposition did not
necessarily run along national lines as, for instance, Klein could be
seen as close to Poincare. Indeed, a certain internationalism
in mathematics was dominant at the end of the nineteenth century and
the early part of the twentieth century. However, WWI was to change
the situation and gave rise to strong nationalistic conflicts. A
central player in 'nationalizing' the opposition was
Pierre Duhem who opposed the *esprit de finesse* of the French
to the *esprit de geometrie* of the Germans:



> 
> To start from clear principles...then to make progress step by
> step, patiently, painstakingly, at a pace that the rules of deductive
> logic discipline with extreme severity: this is what German genius
> excels at; the German *esprit* is essentially *esprit de
> geometrie*...The Germans are geometers, they
> are not subtle [*fin*]; the Germans completely lack *esprit
> de finesse*. (Duhem 1915, 31-32)
> 



Duhem intended his model to apply to the natural sciences but also to
mathematics. Kleinert 1978 showed that Duhem's book was only
part of a reaction by French scientists to the 1914 declaration
"*Aufruf an die Kulturwelt*" signed by 93 prominent
German intellectuals. This led to the so called "*Krieg der
Geister*" in which the polarization between Germany and
France reached the point not only of criticizing the specific ways of
making use of science (say practicing science with military aims) but
also led to a characterization of scientific knowledge as essentially
determined by national characteristics. In fact this strategy was
basically used by the French in criticizing "*La Science
Allemande*" but it will, twenty years later, be used by the
Germans with the replacement of "national" by
"*rassisch*". The best known case is that of
"*Deutsche Physik*" but here the focus will be on
"*Deutsche Mathematik*" (see also Segal 2003 and
Peckhaus 2005).



The most extreme form of this ideological confrontation, which
ironically reversed the role of Germans and French in the comparison
used by Duhem, is found in the writings of Ludwig Bieberbach, the
founder of the so-called "*Deutsche Mathematik*".
Taking his start from the dismissal of Landau from the mathematical
Faculty in Gottingen, Bieberbach tried to rationalize why the
students had forced Landau's dismissal. In a
*Kurzreferat* for his talk he summarized his aims as
follows:



> 
> My considerations aim at describing the influence for my own science,
> mathematics, of the people [*Volkstum*], of blood and race, on
> the style of creation by using several examples. For a
> national-socialist this requires of course no proof at all. It is
> rather an insight of great obviousness. For all our actions and
> thoughts are rooted in blood and race and receives from them their
> specificity. That there are such styles is also familiar to every
> mathematician. (Bieberbach, 1934a, 235)
> 



In his two papers 1934b and 1934c, he claimed that the mathematics
practiced by Landau was foreign to the German spirit. He compared
Erhard Schmidt and Landau and claimed that in the first case



> 
> The system is directed towards the objects, the construction is
> organic. By contrast, Landau's style is foreign to reality,
> antagonistic to life, inorganic. The style of Erhard Schmidt is
> concrete, intuitive and at the same time it satisfies all the logical
> demands. (Bieberbach 1934b, 237)
> 



Other important oppositions brought forward by Bieberbach as
"evidence" for his claims were Gauss vs. Cauchy-Goursat on
complex numbers; Poincare vs. Maxwell in mathematical physics;
Landau vs. Schmidt; and Jacobi vs. Klein.



By relying on the psychology of types by the notorious Marburg
psychologist Jaensch he then went on to oppose Jewish/Latin and German
psychological types. The fault line, so to speak, was between a
mathematics driven by intuition, typical of German mathematics, and
the formalism allegedly espoused by the Jewish/Latin mathematicians.
Obviously, Bieberbach was forced to do a lot of gerrymandering to make
sure that important German mathematicians did not end up on the wrong
side of the equation (see what he says of Weierstrass, Euler and
Hilbert). The basis of these mathematical differences was to be found
in racial characteristics:



> 
> In my considerations I have tried to show that in mathematical
> activity there are issues of style and that therefore blood and race
> are influential in the way of mathematical creation. (Bieberbach
> 1934c, 358-359)
> 



The reason for discussing Bieberbach in this context is that his case
exemplifies an attempt at rooting the notion of style in something
more fundamental, such as national characteristics interpreted in
terms of psychology and racial features. Moreover, his case is also of
interest as his approach to style shows how such theorizing can be put
in the service of a twisted political program.



Fortunately, talk of national styles in mathematics does not have to
carry with it all the implications that were found in Bieberbach.
Indeed, when historians today refer to national styles they do so
without the nationalism that motivated the older contributions.
Rather, they are concerned with describing how "local"
cultures play a role in the constitution of knowledge (see also Larvor
2016). While increased mobility and email communications make it
harder for national styles to thrive, special political conditions
might also favor the persistence of such a style. This is the case,
for instance, of the Russian style in algebraic geometry and
representation theory. As Robert MacPherson has pointed out to the
author, this case of national style would deserve a more extensive
investigation and it would be interesting to study how the fall of the
Soviet Union impacted this style. By contrast, an instance of a
national style that has been extensively studied is that of Italian
style algebraic geometry. This case has been studied with care by a
number of historians of mathematics and in particular by Aldo
Brigaglia (see also Casnati et al. 2016). For instance in a recent
article, Brigaglia writes:



> 
> Moreover, the Italian school was not strictly a *national*
> 'school,' but rather a working style and a methodology,
> principally based in Italy, but with representatives to be found
> elsewhere in the world. (Brigaglia 2001, 189)
> 



The scare quotes highlight the problem of trying to grasp the
difference between 'schools', 'styles',
'methodologies' etc. (see Rowe 2003) There has been no
attempt to discuss analytically the notion of 'national
style' for the history of mathematics--in any case, nothing
comparable to what Harwood 1993 does in the first chapter of his book.
The situation is also complicated by the fact that different authors
use different terminologies while perhaps referring to the same issue.
For instance, there has been much talk recently of 'images of
mathematics' (Corry 2004a, 2004b, Bottazzini and Dahan
Dalmedico, 2001). In the last section, we shall return to reflect on
these different usages of style in the historiographical literature on
mathematics and how they compare with those in the natural
sciences.


## 4. Mathematicians on style



So far the discussion has focused on style as a tool for philosophers
of culture and for historians of mathematics. But do mathematicians
recognize the existence of styles in mathematics? Once again, it would
not be difficult to give isolated quotes where mathematicians might
talk about the style of the ancients or the abstract algebraic style
or categorial style. In logical work one finds occurrences of style in
such denominations as 'Bishop-style constructive
mathematics'. What are difficult to find are systematic
discussions by mathematicians of the notion of style. The case of
Bieberbach was mentioned above but no detailed discussion of the
examples he adduced as evidence of differences in style was given
there, partly because they are so twisted by his desire to provide
support for his ideological point of view that there are reasons to
doubt that one would gain much by way of an analysis of his case
studies.



An interesting contribution is an article by Claude Chevalley from
1935 titled "*Variations du style
mathematique*". Chevalley takes the existence of
style for granted. He begins as follows:



> 
> 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)
> 



However, Chevalley did not try to reflect on the notion of style here
involved. Rather he was concerned to show by means of an important
example the features of the transition between two styles of doing
mathematics that had characterized the passage from nineteenth century
mathematics to twentieth century approaches. The first style described
by Chevalley is the Weierstrassian style, 'the style of
e'. It finds its '*raison
d'etre*' in the need to rigorize the calculus
moving away from the unclarities related to such notions as
"infinitely small quantity" etc. The development of
analysis in the nineteenth century (analytic functions, Fourier
series, Gauss' theories of surfaces, Lagrangian equations in
mechanics etc.) led to a critical analysis



> 
> of the algebraic-analytical framework in front of which they found
> themselves; and it is from this critical examination that a completely
> new mathematical style was to emerge. (Chevalley 1935, 377)
> 



Chevalley went on to single out the discovery of a continuous nowhere
differentiable function, due to Weierstrass, as the most important
element of this revolution. As Weierstrass' function can be
given in terms of a Fourier expansion with a quite normal appearance,
it became obvious that many demonstrations in mathematics assumed
closure condition that needed to be rigorously established. The
concept of limit, as defined by Weierstrass, was the powerful tool
that allowed such investigations. The reconstruction of analysis
pursued by Weierstrass and his followers turned out to be not only
foundationally successful but also mathematically fruitful. Here is
how close Chevalley comes to characterizing this style:



> 
> The use on the part of the mathematicians of this school of the
> definition of limit due to Weierstrass can be noticed in the external
> appearance of their writings. First of all, in the intensive, and at
> times immoderate, usage of the "e" equipped with
> various indexes (this is the reason why we have spoken above of a
> style of the "e"s). Secondly, in the progressive
> replacement of equality for inequality in the demonstrations as well
> as in the results (approximation theorems; upper bound theorems;
> theory of increase, etc.). This last aspect will occupy us for it will
> make us understand the reasons that forced the overcoming of the
> Weierstrassian style of thinking. Indeed, while equality is a relation
> that is meaningful for mathematical beings whatsoever, inequality can
> only be applied to objects equipped with an order relation,
> practically only on the real numbers. In this way one was led, in
> order to embrace all of analysis, to reconstruct it entirely from the
> real numbers and from functions of real numbers. (Chevalley 1935,
> 378-379)
> 



Out of this approach one could also build the system of complex
numbers as pair of reals and the points of spaces in *n*
dimensions as *n*-tuples of reals. This gave the impression
that mathematics could be unified by means of constructive definitions
starting from the real numbers. However, things went differently and
Chevalley tries to account for the reasons that led one to give up
this "constructive" approach in favor of an axiomatic
approach. Various algebraic theories, such as group theory gave rise
to relationships that could not be constructed starting from the real
numbers. Moreover, the constructive definition of complex numbers was
equivalent to fixing an arbitrary reference system and thus endowing
these objects with properties that hid their real nature. On the other
hand one was familiar with Hilbert's axiomatization of geometry
which, although rigorous, did not have the character of artificiality
of the constructive theories. In this case the entities are not
constructed but rather defined through the axioms. This approach
developed to influence analysis itself. Chevalley mentioned the theory
of Lebesgue integral that was obtained by first setting down what
properties the integral had to satisfy and then showing that a domain
of objects satisfying those properties existed. The same idea was used
by Frechet by setting down the properties that were to characterize
the operation of limit thereby arriving at a general theory of
topological spaces. Another example mentioned by Chevalley is the
axiomatization of field theory given by Steinitz in 1910. Chevalley
concluded that



> 
> The axiomatization of theories has modified very deeply the style of
> contemporary mathematical writings. First of all, for every result
> obtained, one always needs to find out which ones are the strictly
> indispensable properties needed to establish it. One will seriously
> address the problem of giving a minimal demonstration of such result
> and to that effect one will need to delimit exactly in which domain of
> mathematics one is operating in such a way as to reject methods that
> are foreign to this domain since the latter are likely to bring about
> the introduction of useless hypotheses. (Chevalley 1935, 382)
> 



Moreover, the constitution of domains that are perfectly suited to
certain operations allows one to establish general theorems on the
objects under consideration. In this way one can characterize the
operations of infinitesimal analysis algebraically but without any of
the naivete which had characterized the previous algebraic
approaches.



Chevalley's article is a precious source from a contemporary
mathematician on the topic of style. He forcefully shows the
difference between the late nineteenth century arithmetization of
analysis and the early twentieth century axiomatic-algebraic approach.
However, it has its limitations. The notion of style is not thematized
as such and it is not clear that the features adduced to explain the
particular historical events might provide the general tools for
analyzing other transitions in mathematical style. But perhaps that
should, if anything, be the task of a philosopher of mathematics (for
a detailed analysis of Chevalley's approach to style see Rabouin
2017; for further developments, focused on Bourbaki style, see Marquis 2021).


## 5. The locus of style



In a book entitled "*Introduccion al estilo
matematico*" (1971) the Spanish philosopher Javier de Lorenzo
attempted to write a history of mathematics (admittedly partial) in
terms of style. Although by 1971 Granger's work, to be discussed
in section 5, had already appeared, de Lorenzo was not aware of it and
the only source on style he uses is Chevalley's article. Indeed
this book is merely an extension of Chevalley's study to include
many more 'styles' that have appeared in the history of
mathematics. The list of mathematical styles studied by de Lorenzo is
the following:


* Geometric style;
* Poetic style;
* Cossic style;
* Cartesian-algebraic style;
* The style of indivisibles;
* Operational style;
* Epsilon style;
* Synthetic vs analytical styles in geometry;
* Axiomatic style;
* Formal style.



The general set up reminds one much of Chevalley's approach and
one would look in vain in de Lorenzo's book for a satisfactory
account of what style is. It is true that there are some interesting
observations about the role of language in determining a style but a
general philosophical analysis is missing. There is however an
important point to be emphasized concerning the treatment given by
Chevalley and de Lorenzo, which seems to point to an important feature
of the use of 'style' in mathematics.



In his paper "*De la categorie de style en histoire
des sciences*" (Gayon 1996), and in the later Gayon 1999,
Jean Gayon presents the different usages of 'style' in the
historiography of science as falling between two camps (in a way he
follows Hacking 1992 here). First, there is the usage of
'scientific style' on the part of those who pursue a
'local history of science'. Usually this type of analysis
focuses on 'local groups or schools' or on
'nations'. For instance, this type of history deemphasizes
the universal component of knowledge and stresses the difficulties
involved in translating experiments from one setting to another. Such
difficulties are shown to depend on the 'local'
traditions, which include specific technical and theoretical know-how
which is "fundamental to setting-up, realizing, and analyzing
the outcome of those experiments" (Corry 2004b) Secondly, there
is the use of 'scientific style' exemplified in works such
as Crombie's 1994 'Styles of Scientific Thinking in the
European Tradition'. Crombie enumerates the following scientific
styles:


1. postulation in the axiomatic mathematical sciences
2. experimental exploration and measurement of complex detectable
relations
3. hypothetical modelling
4. ordering a variety by comparison and taxonomy
5. statistical analysis of populations, and
6. historical derivation of genetic development (quoted from Hacking
1996, 65)



Gayon remarks that this latter notion of 'style' could be
replaced by 'method' and that 'the styles discussed
here have nothing to do with local styles'. He also remarks that
when it comes to local styles the groups that act as sociological
support for such analyses are either 'research groups' or
'nations'. There has been much emphasis in recent history
of the experimental sciences on such local factors (see, for instance,
Gavroglu 1990 for the 'styles of reasoning' of two low
temperature laboratories, that of Dewar (London) and that of
Kamerlingh Onnes (Leiden)).



Historians of mathematics are now attempting to apply such
historiographical approaches also to pure mathematics. A recent
attempt in this direction is the work of Epple in terms of
'epistemic configurations' such as his recent article on
Alexander and Reidemeister's early work in knot theory (Epple
2004; but see also Rowe 2003 and 2004, and Epple 2011). The support
groups for such investigations are not referred to as
'schools' but rather as 'mathematical
traditions' or 'mathematical cultures'.



What about the 'methodological' notion of style a
la Crombie? Have historians of mathematics made much use of this?
Apart from numerous treatments of the first style (axiomatic method),
there is not much in this area but an interesting historical
contribution is Goldstein's work on Frenicle de Bessy (2001).
She argues that the pure mathematics as practiced by Frenicle de Bessy
had much in common with the Baconian style of experimental science.
Perhaps one should mention here that experimental mathematics is now a
blossoming field which might soon find its historian (see Baker 2008
for a philosophical account of experimental mathematics and
Sorensen 2016 for an analysis in terms of mathematical
cultures). This tends to be a topic of high interest for philosophers,
as it impinges on issues of mathematical method. The problem can be
simply put as follows: in addition to what Crombie lists as
methodological style (a) [axiomatic], what other styles are pursued in
mathematical practice? Corfield 2003 touches upon the problem in the
introduction to his book "Towards a philosophy of
'real' mathematics" when he, referring to the
Crombie list above, says:



> 
> Hacking applauds Crombie's inclusion of (a) as 'restoring
> mathematics to the sciences' (Hacking 1996) after the logical
> positivists' separation, and extends the number of its styles to
> two by admitting the algorithmic style of Indian and Arabic
> mathematics. I am happy with this line of argument, especially if it
> prevents mathematics been seen as an activity totally unlike any
> other. Indeed, mathematicians also engage in styles (b) (see chapter
> 3), (c) and (d) [7] and along the lines of (e) mathematicians are
> currently analyzing the statistics of the zeros of the Riemann zeta
> function. (Corfield 2003, 19)
> 



In note 7 Corfield mentions John Thompson's comment to the
effect that the classification of finite simple groups is an exercise
in taxonomy.



It is not the goal of this essay to address squarely the vast set of
issues that emerge from the previous quotes. But it should be pointed
out that these issues represent a fresh and stimulating territory for
a descriptive epistemology of mathematics and that some work has
already been carried out in this direction (see Etcheverria
1996; van Bendegem 1998; Baker 2008).



Finally, how to put together the 'local' and
'methodological' styles with what is found in Chevalley
and de Lorenzo? 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. Here are some
possible examples, where the names attached to each position should
not mislead the reader into thinking that one is merely dealing with
'individual' styles.


1. Direct vs. indirect techniques in geometry (Cavalieri and
Torricelli vs. Archimedes)
2. Algebraic vs. geometrical approaches in analysis in the
seventeenth and the eighteenth century (Euler vs. McLaurin)
3. Geometrical vs. analytical approaches in complex analysis in the
nineteenth century (Riemann vs. Weierstrass)
4. Conceptual vs. computational approaches in algebraic number theory
(Dedekind vs. Kronecker)
5. structural vs intuitive styles in algebraic geometry (German
school vs. Italian school)



Of course, it might just be the case that also in history and
philosophy of science there are 'intermediate' levels of
style such as the ones being described here (one example that comes to
mind is 'Newtonian style' in mathematical physics) but the
fact that Jean Gayon did not detect them as central seems to point to
the fact that the situation in history and philosophy of mathematics
is quite different, as these 'intermediate' styles are
those that have been more thoroughly discussed and that correspond to
the styles analyzed by Chevalley and de Lorenzo. Moreover, discussions
of local mathematical cultures tend to do without the concept of
style.


## 6. Towards an epistemology of style



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? The notion of coloring here comes from Frege
who distinguished in "The thought" between the truth
condition of a statement and those aspects of the statement which
might provide information about the state of mind of the speaker or
hearer but do not contribute to its truth conditions. In natural
language, typical elements of coloring are expressions of regret such
as "unfortunately". "Unfortunately, it's
snowing" has the same truth conditions as "it's
snowing" and "unfortunately", in the first sentence,
is only part of the coloring. Jacques and Monique Dubucs have
generalized this distinction to proofs in "La couleur des
preuves" (Dubucs and Dubucs 1994) where they deal with the
problem of a 'rhetoric of mathematics', a problem quite
close to that of an analysis of style. Dubbing traditional rhetoric as
'residualist', for it takes into account only phenomena of
non-cognitive significance such as ornamentation etc. of the
mathematical text but leaves the object (such as the content of a
demonstration) untouched, they explored the options for a more
ambitious "rhetoric of mathematics".



One can thus begin to articulate the first position that can be
defended with respect to the epistemological significance of style. It
is a position that denies style any essential cognitive role and
reduces it to a phenomenon of subjective coloring. According to this
position, stylistic variations would only reveal superficial
differences of expression that leave the content of discourse
untouched.



Two more ambitious positions have been defended in the literature
concerning the cognitive content of style. The first seems to be
compatible with a form of Platonism or realism in mathematics whereas
the second is definitely opposed to it. What is being alluded to are
the two main proposals available in the literature, namely those of
Granger 1968 and Hacking 1992, which will now be briefly
described.



Granger's *Essay of a philosophy of style* (*Essai
d'une philosophie du style* 1968) is the most systematic and
worked-out effort to develop a theory of style for mathematics.
Granger's program is so ambitious and rich that a thorough
discussion of the structure of his book and of his detailed analyses
would need a paper by itself (see also Benis-Sinaceur 2010). Due to
limitation of space, the aim here is to give just a rough idea of what
the project consists in and to show that the epistemological role of
style defended by Ganger is compatible with a realism about
mathematical entities or structures.




Granger's aim is to provide an analysis of 'scientific
practice'. He defines practice as "an activity considered
with its complex context, and in particular the social conditions
which give it meaning in a world effectively experienced
(*vecu*)" (1968, 6). Science he defines as
"construction of abstract models, consistent and effective, of
the phenomena" (13). Thus, a scientific practice has both
'universal' or 'general' components and
'individual' components. The analysis of scientific
practice requires at least three types of investigations:


1. There are many ways of structuring, by means of models, a certain
phenomenon; and the same models can be applied to different phenomena.
Moreover, scientific constructions, including mathematical ones,
reveal a certain "structural unity". Both of these aspects
will be the theme of a stylistic analysis.
2. The second investigation concerns a 'scientific
characteriology', aimed a studying the psychological components
which are relevant in the individuation of scientific practice;
3. The third investigation concerns the study of the
'contingency' of scientific creation, always located in
space and time.



All three aspects would be necessary for an analysis of
'scientific practice' but in his book Granger only focuses
on 1. Where do style and mathematics come in? Mathematics comes in as
one of the areas of investigation that can be subjected to a stylistic
analysis of science (Granger's book provides applications not
only to mathematics but also to linguistics and the social sciences).
What about style? Every social practice, according to Granger, can be
studied from the point of view of style. This includes political
action, artistic creation and scientific activity. There is thus a
general stylistics that will try to capture the most general stylistic
features of such activities and then more 'local'
stylistic analyses such as the one provided by Granger for scientific
activities. Obviously, the concept of style here invoked must be one
that is much more encompassing than the one usually associated with
this term and indeed one that would make application to such areas as
political activity or scientific activity not just metaphorical but
rather 'connaturate' to such activities.



Granger's analysis of mathematical style takes up chapters 2, 3,
and 4 of his book. Chapter 2 deals with Euclidean style and the notion
of magnitude; chapter 3 with the opposition between 'Cartesian
style and Desarguian style' (on the Cartesian style see also
Rabouin 2017); finally, chapter 4 concerns the 'birth of
vectorial style'. All of these analyses center around the
concept of "geometrical magnitude".



One gets a good sense of what Granger is after by simply looking at an
example that he describes in his preface. This is an example
concerning the complex numbers.



Style, according to Granger, is a way of imposing structure to an
experience. Experience must be taken here to go beyond empirical
experience. In general the kind of experience the mathematician
appeals to is not empirical. From this experience come the
"intuitive" components that are structured in mathematical
activity. But one should not think that there is an
"intuition" to which, as it were externally, one then
applies a form. The mathematical activity gives rise at the same time
to form and content within the background of a certain experience.



> 
> Style appears to us here 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. (Granger 1968, 20)
> 



As an example Granger gives three ways of introducing the complex
numbers; all three ways account for the structural properties which
characterize the algebraic structure in question. The first way
introduces the complex numbers by trigonometric representation using
angles and directions. The second introduces them as operators applied
to vectors. In the first case, one defines a complex number as a pair
of real numbers and the additive properties are then immediate. By
contrast, in the second case, it is the multiplicative properties that
are immediately seized. But, and this is the third way, one can also
introduce complex numbers by regular square matrices. This leads to
seeing the complex numbers as a system of polynomials in *x*
modulo *x*2+1.



> 
> These different ways of grasping a concept, of integrating it in an
> operative system and of associating to it some intuitive
> implications--of which one will have to delimit the exact
> extent--constitute what we call aspects of style. It is evident
> that the structural content of the notion is not here affected, that
> the concept qua mathematical object subsists identically through these
> effects of style. It is however not always so and we will encounter
> stylistic positions which demand true conceptual variations. What
> changes always, in any case, is the orientation of the concept towards
> this or that usage, this or that extension. Thus, style plays a role
> that is perhaps essential both with respect to the dialectic of the
> internal development of mathematics and to that of its relation to
> worlds of more concrete objects. (Granger 1968, 21).
> 



Thus, in Granger's theory mathematical styles are modes of
presentation, or modes of grasping the mathematical structures. At
least in some cases these effects of style leave the mathematical
objects or structures unaffected although they will affect the
cognitive mode in which they are apprehended, therefore affecting how
they might be subjected to extension, applied in various areas etc.
Even though Granger might have sympathized with a Kantianism without a
transcendental subject, and thus think of style as constitutive, it
seems that his position is at least compatible with a form of realism
about mathematical entities. This does not seem to be the case for the
third and final epistemological position to be discussed, which is due
to Ian Hacking.



As pointed out earlier, Hacking, following Crombie, has proposed
investigating the notion of style as a "new analytical
tool" for history and philosophy of science. His preference is
to speak of styles of reasoning (see also Mancosu 2005) as opposed to
Fleck's thought styles or Crombie's styles of thinking
(his most recent preference is to speak of 'Styles of scientific
thinking & doing'; for the most recent discussion of
Hacking's program at the time of writing see Kusch 2010 and the
special issue of *Studies in History and Philosophy of Science*
(issue 43, 2012), including Hacking 2012 and several other
contributions). The reason is that Hacking wants to move away from the
psychological level of reasoning and work with the more
'objective' level of arguments. He explicitly defines his
project as a continuation of Kant's project aimed at explaining
how objectivity is possible. And indeed, Hacking's position
rejects realism and embraces a strongly constitutive role for style.
According to Hacking, styles are defined by a set of necessary
conditions (he does not attempt, wisely, to provide sufficient
conditions):



> 
> There are neither sentences that are candidates for truth, nor
> independently identified objects to be correct about, prior to the
> development of a style of reasoning. 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. One should
> also notice, on occasion, new types of classification and new types of
> explanation. (Hacking 1992, 11)
> 



It should be clear that this notion of style, just like
Granger's, attributes a very important role to style as
grounding the objectivity of an entire area of scientific activity but
that, unlike Granger's, it is committed ontologically to a
rejection of realism. Styles are essential in the constitution of
mathematical objects and the latter do not have a form of existence
independent of them. Hacking has not extensively discussed case
studies from the history of mathematics although one of his papers
(Hacking 1995) deals with four constructionalist images of mathematics
(the word "constructionalism" is borrowed from Nelson
Goodman) and shows how well they fit with his picture of 'styles
of thinking'. By implication, it is also clear that more
robustly committed realistic positions will not fit well with
Hacking's account of reasoning styles.



Thus, three possible models for explicating the epistemological role
of 'styles' in mathematics have been considered. There are
surely many more possible positions waiting to be articulated but so
far this is all that can be found in the literature.


## 7. Conclusion



As pointed out at the outset, the topic of mathematical style is not
one of the canonical areas of investigation in philosophy of
mathematics. Indeed, this entry is the first attempt to encompass in a
single paper the multifarious contributions to this topic.
Nonetheless, it should be clear by now that reflection on mathematical
style is present in contemporary philosophical activity and deserves
to be taken seriously. But the work is just beginning. One needs many
more case studies of mathematical styles and a clearer articulation of
the epistemological and ontological consequences yielded by different
conceptualizations of style. In addition, one would like to see a
better integration of all this work with the work on cognitive styles
that is found in cognitive psychology and mathematical education.
Finally, standard philosophical chestnuts, such as the relationship of
form and content to style, and the relation of style to normativity
and intentionality would also have to be addressed (for a very good
discussion of such topics in aesthetics see Meskin 2005).