The essay begins with a taxonomy of the major contexts in which thenotion of ‘style’ in mathematics has been appealed tosince the early twentieth century. These include the use of the notionof style in comparative cultural histories of mathematics, incharacterizing national styles, and in describing mathematicalpractice. These developments are then related to the more familiartreatment of style in history and philosophy of the natural scienceswhere one distinguishes ‘local’ and‘methodological’ styles. It is argued that the naturallocus of ‘style’ in mathematics falls between the‘local’ and the ‘methodological’ stylesdescribed by historians and philosophers of science. Finally, the lastpart of the essay reviews some of the major accounts of style inmathematics, due to Hacking and Granger, and probes theirepistemological and ontological implications.

• 1. Introduction
• 2. Style as a central concept in comparative cultural histories
• 3. National styles in mathematics
• 4. Mathematicians on style
• 5. The locus of style
• 6. Towards an epistemology of style
• 7. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

The goal of this essay is to survey and analyze the literature onstyle in history and philosophy of mathematics. In particular, theproblem of how one can philosophically approach the notion of‘style’ in mathematics will be addressed towards theend. Although this is not one of the canonical topics in philosophy ofmathematics, the presentation will avail itself of relevantdiscussions on style in the history and philosophy of science.

Speaking about mathematics in terms of style is a common enoughphenomenon. One encounters such appeals to stylistic features inmathematics already early in the seventeenth century. BonaventuraCavalieri, for instance, as early as 1635 contrasts his indivisibilisttechniques with the Archimedean style:

I know in fact that all the things mentioned above [Cavalieri’sown theorems obtained by indivisibilist proofs] can be reduced toArchimedean style. (In the original Latin: “Scio autempraefata omnia ad stylum Archimedeum reduci posse.”(Cavalieri 1635, 235)).

Later in the century it is easier to find examples. For instanceLeibniz (1701, 270–71) writes: “Analysis does not differfrom Archimedes’ style except for the expressions that are moredirect and more appropriate to the art of discovery” (French:“L’analyse ne diffère du styled’Archimède que dans les expressions, qui sont plusdirectes et plus conformes à l’artd’inventer”). It is an interesting fact that suchoccurrences predate the generalized use of the notion of style inpainting, which only dates from the 1660s (sporadic occurrences, aspointed out in Sauerländer 1983, are also found in the sixteenthcentury). Earlier in the seventeenth century the word of choice inpainting was “manière” (see Panofsky 1924;English translation (1968, 240)). Here are a couple of additionalexamples from the nineteenth and the twentieth centuries. Chasles inhisAperçu historique (1837) speaking about Mongesays:

He initiated a new way of writing and talking about this science.Style, in fact, is so intimately welded to the spirit of a methodologythat it must advance in step with it; likewise, if it has anticipatedit, style must of necessity play a powerful influence over it and overthe general progress of science. (Chasles, 1837, §18, 207)

Another example comes from Edward’s evaluation ofDedekind’s approach to mathematics:

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

Obviously, one could pile up quotations of the same kind (see, amongothers, Cohen 1992, de Gandt 1986, Dhombres 1993, Epple 1997,Ferreirós and Reck 2020, Fleckenstein 1955, Granger 2003,Høyrup 2005, Laugwitz 1993, Novy 1981, Reck 2009, Tappenden2005, Weiss 1939, Wisan 1981) but that would not be veryinteresting. Even in mathematics style ranges from ‘individualstyles’ to ‘national styles’ to ‘epistemicstyles’, among others. What is needed is first of all anunderstanding of the major contexts in which appeal to‘style’ in mathematics occurs, although this essay willnot contain much discussion of ‘individual styles’(examples of such would include, to follow a suggestion by EnricoBombieri, the “very personal” styles of Euler,Ramanujan, Riemann, Serre and A. Weil).

In many cases the appeal to the notion of style is conceived asborrowed from the fine arts and some cases will be discussedforthwith. Harwood 1993 claims that “the concept of style wasdevised in order to classify cultural patterns observed in the studyof fine arts”. Wessely 1991 talks about “transferring thatconcept [of style] to the history of science” (265). While thismight perhaps be true for the twentieth century (see also Kwa 2012),one should keep in mind, as was pointed out above, that this claimmust 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 majortwentieth century appeals to the category of style in mathematics havedone so in reference to the arts. This is especially true of thoseauthors who were motivated by accounting in a unified way for thecultural production of mankind and who saw thus a uniformity in theprocesses of scientific and artistic production. It was in suchcontext that Oswald Spengler inThe Decline of the West(1919, 1921) attempted a morphology of world history and claimed thatthe history of mathematics was characterized by different stylisticepochs which depended on the culture that produced it:

The Style of any mathematics which comes into being, depends wholly onthe Culture in which it is rooted, the sort of mankind it is thatponders it. The soul can bring its inherent possibilities toscientific development, can manage them practically, can attain thehighest levels in its treatment of them—but is quite impotent toalter them. The idea of the Euclidean geometry is actualized in theearliest forms of Classical ornament, and that of the InfinitesimalCalculus in the earliest forms of Gothic architecture, centuriesbefore the first learned mathematicians of the respective Cultureswere born. (Spengler 1919, 59)

Not only are there parallels between mathematics and other artisticproductions of a culture. Relying on Goethe’s statement that thecomplete mathematician “feels within himself the beauty of thetrue” and on Weierstrass’s pronouncement that “whois not at the same time a bit of a poet will never be a truemathematician”, Spengler went on to characterize mathematicsitself as an art:

Mathematics, then, is an art. As such it has its styles and styleperiods. It is not, as the layman and the philosopher (who is in thismatter a layman too) imagine, substantially unalterable, but subjectlike every art to unnoticed changes from epoch to epoch. (Spengler1919, 62)

The most extensive treatment that builds on the parallel between artand mathematics and exploits the notion of style as a central categoryfor an analysis of the history of mathematics is that of Max Bense. Ina book appropriately entitledKonturen einer Geistesgeschichte derMathematik (1946), Bense devoted a whole chapter (ch. 2) toarticulating how the notion of style applies to mathematics. For Bensestyle is form:

For style is form, essential form, and we designate this form as the“Aesthetic”, if it controls categorially the sensible, amaterial. (Bense 1946, 118)

Bense saw the history of art and the history of mathematics as aspectsof the history of mind [Geistesgeschichte]. In fact“style is given wherever the human imagination and the capacityof expression arrive to creation”. Bense was certainly prone todraw parallels between styles in the history of art and styles inmathematics (he especially treated the baroque and the romantic stylesin his book) but he kept, in opposition to Spengler, the nature of artand mathematics separate. Indeed he recognized that a stylistichistory of mathematics could not be reduced “to a coincidencebetween certain mathematical formal tendencies and the greatartistic-worldviews-spiritual styles of single epochs such as theRenaissance, Classicism, the Baroque or Romanticism” (p.132; seeFleckenstein 1955 and Wisan 1981 for more recent parallels between thebaroque in art and the mathematics of the seventeenth century). Hereferred to Felix Klein’s “Elementarmathematik vomhöheren Standpunkte aus” to point out that certainlines of development characterized by Klein could be seen as pointingto styles in the history of the development of mathematics (see Klein1924, 91).

Attempts such as Spengler’s and Bense’s appeal certainlyto those theoreticians who would like to use the category of style asa tool for describing, and perhaps accounting for, cultural patterns.However, they leave the reader who is knowledgeable in mathematicsand/or history of art skeptical on account of the usually far-fetchedparallels that are supposed to provide evidence for the account. Ofcourse, this is not to reject ultimately the approach or theusefulness of the appropriateness of the category of style inmathematics but one would like its use to be more directly related toaspects 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 beassociated with such attempts. The first is purely descriptive, ortaxonomic, and satisfies itself with showing certain common patternsbetween a certain area of thought, such as mathematics, and othercultural products of a certain society. The second approachpresupposes the first but also enquires after the causes that accountfor the presence of a certain style of thought or production andnormally tries to ascribe it to psychological or sociological factors.In Spengler and Bense there are elements of both although the emphasisis more on the parallels than on the causes underlying or explainingthe parallels.

Attempts to extend the use of the concept of style in art to otherdomains of human endeavors abound in the early twentieth century. Awell-known case is Mannheim’s sociological attempt tocharacterize styles of thought within different social groups(Mannheim 1928). While Mannheim had not excluded scientific thoughtfrom the realm of sociological analysis of knowledge, he did notactively pursue such an analysis. By contrast, Ludwik Fleck practiceda sociological analysis of science in which “styles ofthought” played a central role. Fleck focused however onmedicine (Fleck 1935).

Here it is important to point out that the notion of thought style hasreceived, by and large, two different developments in contemporaryresearch, which also affect mathematics. First, there is the notionencountered in Fleck. Depending on how generous one wants to be indrawing connections, one could see this approach to styles of thoughtto be related to the later work by Kuhn, Foucault and Hacking (seebelow for a discussion of Hacking). There is however a different wayof thinking about styles of thought, which usually goes under the nameof cognitive styles. This is an area of interest to cognitivepsychologists and mathematical educators (for an overview of thepsychological research in this area see Riding 2000 and Stenberg andGrigorenko 2001). Here the focus is on the psychological make up ofthe individual who displays preference for a certain cognitive styleeither in learning, understanding or thinking about mathematics (i.e.,processing and organizing mathematical information). The olddistinction between visual and analytic mathematicians emphasized byPoincaré (see Poincaré 1905) is still part of thepicture although there are a great variety of models andclassifications. For an historical overview and a theoretical proposalcentered on mathematics see Borromeo Ferri 2005.

In the area of history and philosophy of mathematics there are no booklength accounts of mathematical styles that explain the emergence of acertain style with sociological or psychological categories (althoughNetz 1999 has been of interest to theoreticians of style as an attemptat a cognitive history of an important segment of Greek mathematics).This is in contrast to books in history of the natural sciences suchas Harwood 1993, whose goal is to explain the emergence of the thoughtstyle of the German genetics community through sociological arguments.The closest one comes to such an account is Bieberbach’sconception of style in mathematics as dependent on psychological andracial factors. He will be discussed in the next section on nationalstyles.

3. National styles in mathematics

Something less ambitious than the previous attempts at a generalhistory of human cultural productions or far-reaching parallelsbetween art and mathematics consists in a use of the notion of styleas an historiographical category in the history of mathematics withoutparticular reference to art or other human cultural activities. If onegoes back to the beginning of the twentieth century, one finds that“national styles” were often referred to for categorizingcertain features characterizing mathematical production that seemed tofall squarely within national lines. In the history of science suchcases of “national styles” have often been studied. Oneshould recall here J. Harwood’s bookStyles of ScientificThought (1993) and the contributions Nye 1986, Maienschein 1991,and Elwick 2007. A case of interest for mathematics is the oppositionbetween French and German style in mathematics studied by HerbertMehrtens.

Mehrtens (1990a, 1990b, 1996) describes, in terms of styles, theconflict in mathematics between “formalists” and“logicists” on the one hand and“intuitionists” on the other hand as a battle between twoconceptions of mathematics (see also Gray 2008 for a critical take onof Mehrtens’s approach while emphasizing the“modernist” transformation of mathematics). Hilbert andPoincaré are used as paradigms for the sources of theopposition that later led to the Hilbert-Brouwer foundational debatein the 1920s (on the history of the Brouwer-Hilbert debate see Mancosu1998). Mehrtens also points out that this opposition did notnecessarily run along national lines as, for instance, Klein could beseen as close to Poincaré. Indeed, a certain internationalismin mathematics was dominant at the end of the nineteenth century andthe early part of the twentieth century. However, WWI was to changethe situation and gave rise to strong nationalistic conflicts. Acentral player in ‘nationalizing’ the opposition wasPierre Duhem who opposed theesprit de finesse of the Frenchto theesprit de géométrie of the Germans:

To start from clear principles…then to make progress step bystep, patiently, painstakingly, at a pace that the rules of deductivelogic discipline with extreme severity: this is what German geniusexcels at; the Germanesprit is essentiallyesprit degéométrie…The Germans are geometers, theyare not subtle [fin]; the Germans completely lackespritde finesse. (Duhem 1915, 31–32)

Duhem intended his model to apply to the natural sciences but also tomathematics. Kleinert 1978 showed that Duhem’s book was onlypart of a reaction by French scientists to the 1914 declaration“Aufruf an die Kulturwelt” signed by 93 prominentGerman intellectuals. This led to the so called “Krieg derGeister” in which the polarization between Germany andFrance reached the point not only of criticizing the specific ways ofmaking use of science (say practicing science with military aims) butalso led to a characterization of scientific knowledge as essentiallydetermined by national characteristics. In fact this strategy wasbasically used by the French in criticizing “La ScienceAllemande” but it will, twenty years later, be used by theGermans 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 andPeckhaus 2005).

The most extreme form of this ideological confrontation, whichironically reversed the role of Germans and French in the comparisonused by Duhem, is found in the writings of Ludwig Bieberbach, thefounder of the so-called “Deutsche Mathematik”.Taking his start from the dismissal of Landau from the mathematicalFaculty in Göttingen, Bieberbach tried to rationalize why thestudents had forced Landau’s dismissal. In aKurzreferat for his talk he summarized his aims asfollows:

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

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

The system is directed towards the objects, the construction isorganic. By contrast, Landau’s style is foreign to reality,antagonistic to life, inorganic. The style of Erhard Schmidt isconcrete, intuitive and at the same time it satisfies all the logicaldemands. (Bieberbach 1934b, 237)

Other important oppositions brought forward by Bieberbach as“evidence” for his claims were Gauss vs. Cauchy-Goursat oncomplex numbers; Poincaré vs. Maxwell in mathematical physics;Landau vs. Schmidt; and Jacobi vs. Klein.

By relying on the psychology of types by the notorious Marburgpsychologist Jaensch he then went on to oppose Jewish/Latin and Germanpsychological types. The fault line, so to speak, was between amathematics driven by intuition, typical of German mathematics, andthe formalism allegedly espoused by the Jewish/Latin mathematicians.Obviously, Bieberbach was forced to do a lot of gerrymandering to makesure that important German mathematicians did not end up on the wrongside of the equation (see what he says of Weierstrass, Euler andHilbert). The basis of these mathematical differences was to be foundin racial characteristics:

In my considerations I have tried to show that in mathematicalactivity there are issues of style and that therefore blood and raceare influential in the way of mathematical creation. (Bieberbach1934c, 358–359)

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

Fortunately, talk of national styles in mathematics does not have tocarry with it all the implications that were found in Bieberbach.Indeed, when historians today refer to national styles they do sowithout 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 Larvor2016). While increased mobility and email communications make itharder for national styles to thrive, special political conditionsmight also favor the persistence of such a style. This is the case,for instance, of the Russian style in algebraic geometry andrepresentation theory. As Robert MacPherson has pointed out to theauthor, this case of national style would deserve a more extensiveinvestigation and it would be interesting to study how the fall of theSoviet Union impacted this style. By contrast, an instance of anational style that has been extensively studied is that of Italianstyle algebraic geometry. This case has been studied with care by anumber of historians of mathematics and in particular by AldoBrigaglia (see also Casnati et al. 2016). For instance in a recentarticle, Brigaglia writes:

Moreover, the Italian school was not strictly anational‘school,’ but rather a working style and a methodology,principally based in Italy, but with representatives to be foundelsewhere in the world. (Brigaglia 2001, 189)

The scare quotes highlight the problem of trying to grasp thedifference between ‘schools’, ‘styles’,‘methodologies’ etc. (see Rowe 2003) There has been noattempt to discuss analytically the notion of ‘nationalstyle’ for the history of mathematics—in any case, nothingcomparable to what Harwood 1993 does in the first chapter of his book.The situation is also complicated by the fact that different authorsuse different terminologies while perhaps referring to the same issue.For instance, there has been much talk recently of ‘images ofmathematics’ (Corry 2004a, 2004b, Bottazzini and DahanDalmedico, 2001). In the last section, we shall return to reflect onthese different usages of style in the historiographical literature onmathematics and how they compare with those in the naturalsciences.

4. Mathematicians on style

So far the discussion has focused on style as a tool for philosophersof culture and for historians of mathematics. But do mathematiciansrecognize the existence of styles in mathematics? Once again, it wouldnot be difficult to give isolated quotes where mathematicians mighttalk about the style of the ancients or the abstract algebraic styleor categorial style. In logical work one finds occurrences of style insuch denominations as ‘Bishop-style constructivemathematics’. What are difficult to find are systematicdiscussions by mathematicians of the notion of style. The case ofBieberbach was mentioned above but no detailed discussion of theexamples he adduced as evidence of differences in style was giventhere, partly because they are so twisted by his desire to providesupport for his ideological point of view that there are reasons todoubt that one would gain much by way of an analysis of his casestudies.

An interesting contribution is an article by Claude Chevalley from1935 titled “Variations du stylemathématique”. Chevalley takes the existence ofstyle for granted. He begins as follows:

Mathematical style, just like literary style, is subject to importantfluctuations in passing from one historical age to another. Withoutdoubt, every author possesses an individual style; but one can alsonotice in each historical age a general tendency that is quite wellrecognizable. This style, under the influence of powerful mathematicalpersonalities, is subject every once in a while to revolutions thatinflect writing, and thus thought, for the following periods.(Chevalley 1935, 375)

However, Chevalley did not try to reflect on the notion of style hereinvolved. Rather he was concerned to show by means of an importantexample the features of the transition between two styles of doingmathematics that had characterized the passage from nineteenth centurymathematics to twentieth century approaches. The first style describedby Chevalley is the Weierstrassian style, ‘the style ofε’. It finds its ‘raisond’être’ in the need to rigorize the calculusmoving away from the unclarities related to such notions as“infinitely small quantity” etc. The development ofanalysis in the nineteenth century (analytic functions, Fourierseries, Gauss’ theories of surfaces, Lagrangian equations inmechanics etc.) led to a critical analysis

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

Chevalley went on to single out the discovery of a continuous nowheredifferentiable function, due to Weierstrass, as the most importantelement of this revolution. As Weierstrass’ function can begiven in terms of a Fourier expansion with a quite normal appearance,it became obvious that many demonstrations in mathematics assumedclosure condition that needed to be rigorously established. Theconcept of limit, as defined by Weierstrass, was the powerful toolthat allowed such investigations. The reconstruction of analysispursued by Weierstrass and his followers turned out to be not onlyfoundationally successful but also mathematically fruitful. Here ishow close Chevalley comes to characterizing this style:

The use on the part of the mathematicians of this school of thedefinition of limit due to Weierstrass can be noticed in the externalappearance of their writings. First of all, in the intensive, and attimes immoderate, usage of the “ε” equipped withvarious indexes (this is the reason why we have spoken above of astyle of the “ε”s). Secondly, in the progressivereplacement of equality for inequality in the demonstrations as wellas in the results (approximation theorems; upper bound theorems;theory of increase, etc.). This last aspect will occupy us for it willmake us understand the reasons that forced the overcoming of theWeierstrassian style of thinking. Indeed, while equality is a relationthat is meaningful for mathematical beings whatsoever, inequality canonly be applied to objects equipped with an order relation,practically only on the real numbers. In this way one was led, inorder to embrace all of analysis, to reconstruct it entirely from thereal numbers and from functions of real numbers. (Chevalley 1935,378–379)

Out of this approach one could also build the system of complexnumbers as pair of reals and the points of spaces inndimensions asn-tuples of reals. This gave the impressionthat mathematics could be unified by means of constructive definitionsstarting from the real numbers. However, things went differently andChevalley tries to account for the reasons that led one to give upthis “constructive” approach in favor of an axiomaticapproach. Various algebraic theories, such as group theory gave riseto relationships that could not be constructed starting from the realnumbers. Moreover, the constructive definition of complex numbers wasequivalent to fixing an arbitrary reference system and thus endowingthese objects with properties that hid their real nature. On the otherhand one was familiar with Hilbert’s axiomatization of geometrywhich, although rigorous, did not have the character of artificialityof the constructive theories. In this case the entities are notconstructed but rather defined through the axioms. This approachdeveloped to influence analysis itself. Chevalley mentioned the theoryof Lebesgue integral that was obtained by first setting down whatproperties the integral had to satisfy and then showing that a domainof objects satisfying those properties existed. The same idea was usedby Frechet by setting down the properties that were to characterizethe operation of limit thereby arriving at a general theory oftopological spaces. Another example mentioned by Chevalley is theaxiomatization of field theory given by Steinitz in 1910. Chevalleyconcluded that

The axiomatization of theories has modified very deeply the style ofcontemporary mathematical writings. First of all, for every resultobtained, one always needs to find out which ones are the strictlyindispensable properties needed to establish it. One will seriouslyaddress the problem of giving a minimal demonstration of such resultand to that effect one will need to delimit exactly in which domain ofmathematics one is operating in such a way as to reject methods thatare foreign to this domain since the latter are likely to bring aboutthe introduction of useless hypotheses. (Chevalley 1935, 382)

Moreover, the constitution of domains that are perfectly suited tocertain operations allows one to establish general theorems on theobjects under consideration. In this way one can characterize theoperations of infinitesimal analysis algebraically but without any ofthe naïveté which had characterized the previous algebraicapproaches.

Chevalley’s article is a precious source from a contemporarymathematician on the topic of style. He forcefully shows thedifference between the late nineteenth century arithmetization ofanalysis and the early twentieth century axiomatic-algebraic approach.However, it has its limitations. The notion of style is not thematizedas such and it is not clear that the features adduced to explain theparticular historical events might provide the general tools foranalyzing other transitions in mathematical style. But perhaps thatshould, if anything, be the task of a philosopher of mathematics (fora detailed analysis of Chevalley’s approach to style see Rabouin2017; for further developments, focused on Bourbaki style, see Marquis 2021).

5. The locus of style

In a book entitled “Introducción al estilomatematico” (1971) the Spanish philosopher Javier de Lorenzoattempted to write a history of mathematics (admittedly partial) interms of style. Although by 1971 Granger’s work, to be discussedin section 5, had already appeared, de Lorenzo was not aware of it andthe only source on style he uses is Chevalley’s article. Indeedthis book is merely an extension of Chevalley’s study to includemany more ‘styles’ that have appeared in the history ofmathematics. The list of mathematical styles studied by de Lorenzo isthe 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 andone would look in vain in de Lorenzo’s book for a satisfactoryaccount of what style is. It is true that there are some interestingobservations about the role of language in determining a style but ageneral philosophical analysis is missing. There is however animportant point to be emphasized concerning the treatment given byChevalley and de Lorenzo, which seems to point to an important featureof the use of ‘style’ in mathematics.

In his paper “De la catégorie de style en histoiredes sciences” (Gayon 1996), and in the later Gayon 1999,Jean Gayon presents the different usages of ‘style’ in thehistoriography of science as falling between two camps (in a way hefollows 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 analysisfocuses on ‘local groups or schools’ or on‘nations’. For instance, this type of history deemphasizesthe universal component of knowledge and stresses the difficultiesinvolved in translating experiments from one setting to another. Suchdifficulties are shown to depend on the ‘local’traditions, which include specific technical and theoretical know-howwhich is “fundamental to setting-up, realizing, and analyzingthe outcome of those experiments” (Corry 2004b) Secondly, thereis the use of ‘scientific style’ exemplified in works suchas Crombie’s 1994 ‘Styles of Scientific Thinking in theEuropean Tradition’. Crombie enumerates the following scientificstyles:

1. postulation in the axiomatic mathematical sciences
2. experimental exploration and measurement of complex detectablerelations
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 Hacking1996, 65)

Gayon remarks that this latter notion of ‘style’ could bereplaced by ‘method’ and that ‘the styles discussedhere have nothing to do with local styles’. He also remarks thatwhen it comes to local styles the groups that act as sociologicalsupport for such analyses are either ‘research groups’ or‘nations’. There has been much emphasis in recent historyof the experimental sciences on such local factors (see, for instance,Gavroglu 1990 for the ‘styles of reasoning’ of two lowtemperature laboratories, that of Dewar (London) and that ofKamerlingh Onnes (Leiden)).

Historians of mathematics are now attempting to apply suchhistoriographical approaches also to pure mathematics. A recentattempt in this direction is the work of Epple in terms of‘epistemic configurations’ such as his recent article onAlexander and Reidemeister’s early work in knot theory (Epple2004; but see also Rowe 2003 and 2004, and Epple 2011). The supportgroups for such investigations are not referred to as‘schools’ but rather as ‘mathematicaltraditions’ or ‘mathematical cultures’.

What about the ‘methodological’ notion of style àla 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 historicalcontribution is Goldstein’s work on Frenicle de Bessy (2001).She argues that the pure mathematics as practiced by Frenicle de Bessyhad much in common with the Baconian style of experimental science.Perhaps one should mention here that experimental mathematics is now ablossoming field which might soon find its historian (see Baker 2008for a philosophical account of experimental mathematics andSørensen 2016 for an analysis in terms of mathematicalcultures). This tends to be a topic of high interest for philosophers,as it impinges on issues of mathematical method. The problem can besimply put as follows: in addition to what Crombie lists asmethodological style (a) [axiomatic], what other styles are pursued inmathematical practice? Corfield 2003 touches upon the problem in theintroduction to his book “Towards a philosophy of‘real’ mathematics” when he, referring to theCrombie list above, says:

Hacking applauds Crombie’s inclusion of (a) as ‘restoringmathematics to the sciences’ (Hacking 1996) after the logicalpositivists’ separation, and extends the number of its styles totwo by admitting the algorithmic style of Indian and Arabicmathematics. I am happy with this line of argument, especially if itprevents mathematics been seen as an activity totally unlike anyother. Indeed, mathematicians also engage in styles (b) (see chapter3), (c) and (d) [7] and along the lines of (e) mathematicians arecurrently analyzing the statistics of the zeros of the Riemann zetafunction. (Corfield 2003, 19)

In note 7 Corfield mentions John Thompson’s comment to theeffect that the classification of finite simple groups is an exercisein taxonomy.

It is not the goal of this essay to address squarely the vast set ofissues that emerge from the previous quotes. But it should be pointedout that these issues represent a fresh and stimulating territory fora descriptive epistemology of mathematics and that some work hasalready been carried out in this direction (see Etcheverría1996; van Bendegem 1998; Baker 2008).

Finally, how to put together the ‘local’ and‘methodological’ styles with what is found in Chevalleyand de Lorenzo? In the case of mathematics there is good evidence thatthe most natural locus for ‘styles’ falls, so to speak, inbetween these two categories. Indeed, by and large, mathematicalstyles go beyond any local community defined in simpler sociologicalterms (nationality, direct membership in a school etc.) and are suchthat the support group can only be characterized by the specificmethod of enquiry pursued. On the other hand, the method is not souniversal as to be identifiable as one of the six methods described byCrombie or in the extended list given by Hacking. Here are somepossible examples, where the names attached to each position shouldnot mislead the reader into thinking that one is merely dealing with‘individual’ styles.

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

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

6. Towards an epistemology of style

The problem of an epistemology of style can perhaps be roughly put asfollows. Are the stylistic elements present in mathematical discoursedevoid of cognitive value and so only part of the coloring ofmathematical discourse or can they be seen as more intimately relatedto its cognitive content? The notion of coloring here comes from Fregewho distinguished in “The thought” between the truthcondition of a statement and those aspects of the statement whichmight provide information about the state of mind of the speaker orhearer but do not contribute to its truth conditions. In naturallanguage, typical elements of coloring are expressions of regret suchas “unfortunately”. “Unfortunately, it’ssnowing” has the same truth conditions as “it’ssnowing” and “unfortunately”, in the first sentence,is only part of the coloring. Jacques and Monique Dubucs havegeneralized this distinction to proofs in “La couleur despreuves” (Dubucs and Dubucs 1994) where they deal with theproblem of a ‘rhetoric of mathematics’, a problem quiteclose to that of an analysis of style. Dubbing traditional rhetoric as‘residualist’, for it takes into account only phenomena ofnon-cognitive significance such as ornamentation etc. of themathematical text but leaves the object (such as the content of ademonstration) untouched, they explored the options for a moreambitious “rhetoric of mathematics”.

One can thus begin to articulate the first position that can bedefended with respect to the epistemological significance of style. Itis a position that denies style any essential cognitive role andreduces it to a phenomenon of subjective coloring. According to thisposition, stylistic variations would only reveal superficialdifferences of expression that leave the content of discourseuntouched.

Two more ambitious positions have been defended in the literatureconcerning the cognitive content of style. The first seems to becompatible with a form of Platonism or realism in mathematics whereasthe second is definitely opposed to it. What is being alluded to arethe two main proposals available in the literature, namely those ofGranger 1968 and Hacking 1992, which will now be brieflydescribed.

Granger’sEssay of a philosophy of style (Essaid’une philosophie du style 1968) is the most systematic andworked-out effort to develop a theory of style for mathematics.Granger’s program is so ambitious and rich that a thoroughdiscussion of the structure of his book and of his detailed analyseswould need a paper by itself (see also Benis-Sinaceur 2010). Due tolimitation of space, the aim here is to give just a rough idea of whatthe project consists in and to show that the epistemological role ofstyle defended by Ganger is compatible with a realism aboutmathematical entities or structures.

Granger’s aim is to provide an analysis of ‘scientificpractice’. He defines practice as “an activity consideredwith its complex context, and in particular the social conditionswhich give it meaning in a world effectively experienced(vécu)” (1968, 6). Science he defines as“construction of abstract models, consistent and effective, ofthe phenomena” (13). Thus, a scientific practice has both‘universal’ or ‘general’ components and‘individual’ components. The analysis of scientificpractice requires at least three types of investigations:

1. There are many ways of structuring, by means of models, a certainphenomenon; and the same models can be applied to different phenomena.Moreover, scientific constructions, including mathematical ones,reveal a certain “structural unity”. Both of these aspectswill be the theme of a stylistic analysis.
2. The second investigation concerns a ‘scientificcharacteriology’, aimed a studying the psychological componentswhich are relevant in the individuation of scientific practice;
3. The third investigation concerns the study of the‘contingency’ of scientific creation, always located inspace and time.

All three aspects would be necessary for an analysis of‘scientific practice’ but in his book Granger only focuseson 1. Where do style and mathematics come in? Mathematics comes in asone of the areas of investigation that can be subjected to a stylisticanalysis of science (Granger’s book provides applications notonly to mathematics but also to linguistics and the social sciences).What about style? Every social practice, according to Granger, can bestudied from the point of view of style. This includes politicalaction, artistic creation and scientific activity. There is thus ageneral stylistics that will try to capture the most general stylisticfeatures of such activities and then more ‘local’stylistic analyses such as the one provided by Granger for scientificactivities. Obviously, the concept of style here invoked must be onethat is much more encompassing than the one usually associated withthis term and indeed one that would make application to such areas aspolitical activity or scientific activity not just metaphorical butrather ‘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 notionof magnitude; chapter 3 with the opposition between ‘Cartesianstyle and Desarguian style’ (on the Cartesian style see alsoRabouin 2017); finally, chapter 4 concerns the ‘birth ofvectorial style’. All of these analyses center around theconcept of “geometrical magnitude”.

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

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

Style appears to us here on the one hand as a way of introducing theconcepts of a theory, of connecting them, of unifying them; and on theother hand, as a way of delimiting the what intuition contributes tothe determination of these concepts. (Granger 1968, 20)

As an example Granger gives three ways of introducing the complexnumbers; all three ways account for the structural properties whichcharacterize the algebraic structure in question. The first wayintroduces the complex numbers by trigonometric representation usingangles and directions. The second introduces them as operators appliedto vectors. In the first case, one defines a complex number as a pairof real numbers and the additive properties are then immediate. Bycontrast, in the second case, it is the multiplicative properties thatare immediately seized. But, and this is the third way, one can alsointroduce complex numbers by regular square matrices. This leads toseeing the complex numbers as a system of polynomials inxmodulox²+1.

These different ways of grasping a concept, of integrating it in anoperative system and of associating to it some intuitiveimplications—of which one will have to delimit the exactextent—constitute what we call aspects of style. It is evidentthat the structural content of the notion is not here affected, thatthe concept qua mathematical object subsists identically through theseeffects of style. It is however not always so and we will encounterstylistic positions which demand true conceptual variations. Whatchanges always, in any case, is the orientation of the concept towardsthis or that usage, this or that extension. Thus, style plays a rolethat is perhaps essential both with respect to the dialectic of theinternal development of mathematics and to that of its relation toworlds of more concrete objects. (Granger 1968, 21).

Thus, in Granger’s theory mathematical styles are modes ofpresentation, or modes of grasping the mathematical structures. Atleast in some cases these effects of style leave the mathematicalobjects or structures unaffected although they will affect thecognitive mode in which they are apprehended, therefore affecting howthey might be subjected to extension, applied in various areas etc.Even though Granger might have sympathized with a Kantianism without atranscendental subject, and thus think of style as constitutive, itseems that his position is at least compatible with a form of realismabout mathematical entities. This does not seem to be the case for thethird and final epistemological position to be discussed, which is dueto Ian Hacking.

As pointed out earlier, Hacking, following Crombie, has proposedinvestigating the notion of style as a “new analyticaltool” for history and philosophy of science. His preference isto speak of styles of reasoning (see also Mancosu 2005) as opposed toFleck’s thought styles or Crombie’s styles of thinking(his most recent preference is to speak of ‘Styles of scientificthinking & doing’; for the most recent discussion ofHacking’s program at the time of writing see Kusch 2010 and thespecial issue ofStudies in History and Philosophy of Science(issue 43, 2012), including Hacking 2012 and several othercontributions). The reason is that Hacking wants to move away from thepsychological level of reasoning and work with the more‘objective’ level of arguments. He explicitly defines hisproject as a continuation of Kant’s project aimed at explaininghow objectivity is possible. And indeed, Hacking’s positionrejects realism and embraces a strongly constitutive role for style.According to Hacking, styles are defined by a set of necessaryconditions (he does not attempt, wisely, to provide sufficientconditions):

There are neither sentences that are candidates for truth, norindependently identified objects to be correct about, prior to thedevelopment of a style of reasoning. Every style of reasoningintroduces a great many novelties including new types of: Objects;evidence; sentences, new ways of being a candidate for truth orfalsehood; laws, or at any rate modalities; possibilities. One shouldalso notice, on occasion, new types of classification and new types ofexplanation. (Hacking 1992, 11)

It should be clear that this notion of style, just likeGranger’s, attributes a very important role to style asgrounding the objectivity of an entire area of scientific activity butthat, unlike Granger’s, it is committed ontologically to arejection of realism. Styles are essential in the constitution ofmathematical objects and the latter do not have a form of existenceindependent of them. Hacking has not extensively discussed casestudies 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 NelsonGoodman) and shows how well they fit with his picture of ‘stylesof thinking’. By implication, it is also clear that morerobustly committed realistic positions will not fit well withHacking’s account of reasoning styles.

Thus, three possible models for explicating the epistemological roleof ‘styles’ in mathematics have been considered. There aresurely many more possible positions waiting to be articulated but sofar this is all that can be found in the literature.

7. Conclusion

As pointed out at the outset, the topic of mathematical style is notone of the canonical areas of investigation in philosophy ofmathematics. Indeed, this entry is the first attempt to encompass in asingle paper the multifarious contributions to this topic.Nonetheless, it should be clear by now that reflection on mathematicalstyle is present in contemporary philosophical activity and deservesto be taken seriously. But the work is just beginning. One needs manymore case studies of mathematical styles and a clearer articulation ofthe epistemological and ontological consequences yielded by differentconceptualizations of style. In addition, one would like to see abetter integration of all this work with the work on cognitive stylesthat is found in cognitive psychology and mathematical education.Finally, standard philosophical chestnuts, such as the relationship ofform and content to style, and the relation of style to normativityand intentionality would also have to be addressed (for a very gooddiscussion of such topics in aesthetics see Meskin 2005).

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Acknowledgments

I would like to thank Karine Chemla for having encouraged me to thinkabout this topic and Andrea Albrecht, Enrico Bombieri, Leo Corry,Jacques Dubucs, Jean Gayon, James Hamilton, Robert MacPherson, MarcoPanza, Chris Pincock, Martin Powell, Erich Reck, and Jamie Tappendenfor useful comments on a previous draft and for their help in trackingdown relevant literature. This essay is dedicated to AlessandraSchiaffonati, in memory of her inimitable style.