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Justifier en Mathematiquesby Dominique Flament and Philippe Nabonnand (eds.)

PARIS: EDITIONS DE LA MAISON DES SCIENCES DE L’HOMME, 2011, 372

PP., 29€, ISBN 978-2-7351-1414-6

REVIEWED BY PAOLA CANTU

TThis collection of essays shows how the investigationof mathematical practice extends to justification aswell as proof. Investigations on mathematical prac-

tice are based on the idea that a deep understanding ofmathematical knowledge cannot be obtained only by ananalysis of the logical structure of mathematical theories,but requires a detailed study of the activities of mathema-ticians including, for example, discovery, explanation,justification, and representation. The investigation of theseactivities is taken to be, along with the examination ofproofs to establish the truth of a mathematical fact, arelevant part of the analysis of the foundations of mathe-matics, given that the latter also concerns epistemologicalaspects.

In this recent trend of research in philosophy of mathe-matics, devoted to the investigationof interesting case studiesrather than to the development of a unifying theory, it issometimes difficult to trace a clear distinction between thenotions of explanatory proof, justification, and explanation.The essays contained in this volume explain what mathe-matical justification may include, and also why and when itoccurs inmathematical practice, even if anultimatedefinitionis nowhere to be found in the book. Justification includes theintroduction of new notions or the extension of previouslyused concepts (definitions), the choice of demonstrativeprocedures, a change of notation, and the emergence of anew theory or point of view. The need for justification arisesboth in revolutionary phases of the history of a science, whennew ideas and methods are developed, and in normal pha-ses, when objections, counterexamples, problems, andparadoxes induce either a revision or a justification of certainprocedures and concepts. For example, some argumentativestrategies, basedongenerality and rigor, are used to convincethe reader of the merits of a particular method with respect toalternative procedures. The editors thus suggest that thejustificatory discourse has a normative value, because it canbe used to distinguish good from bad mathematics, but thisaspect is not specifically developed in the book, and justifi-cation is not specifically analyzed as an argumentativepractice.

The editors consider justification to be a rhetorical aspectof the mathematical discourse that is intrinsic to

� 2014 Springer Science+Business Media New York

DOI 10.1007/s00283-014-9452-6

epistemological and ontological issues, and which cannot beseparated from the debate on the foundations of mathemat-ics. To focus on foundational problems and on the relationsbetween mathematics and logic, the editors have restrictedthe domain of investigation to the 19th century, althoughseveral essays inevitably take the early 20th century intoaccount. This might explain why the book’s philosophicaldiscussion concerning ontology and epistemology prevailsover the interest in specific rhetorical strategies, in the kindsof justificatory arguments or mathematical styles, and in theanalysis of the targeted readers and the means ofcommunication.

One of the most interesting results of this book consists—according to our reading—precisely in showing, althoughonly implicitly, a different understanding of the origin andevolution of metatheoretical considerations and of meta-mathematics, which are related to practices of justification.First, metatheoretical investigations should not only be basedon the study of proofs and demonstrations and on the 20th-century history of fully developed axiomatic systems, but canbe traced back to the 19th century. Second, justification iscrucial in the understanding of the distinction betweentheory and metatheory, which is usually taken to be a cor-nerstone of modern axiomatics.

All contributors share the belief, quite common in inves-tigations on mathematical practice, that justification dis-courses should be investigated in a historical perspective.They suggest that justification styles are particularly useful tolearn (1) how mathematicians themselves perceived theirownwork anddiscipline, and (2) how the justificatory debateinfluenced their own practice: two aspects that, if duly con-sidered, might seriously affect the traditional historiographyof mathematics. It is thus a pity that the introduction does notexplicitly summarize all the historiographical changes orreadjustments concerning the foundations of mathematicssuggested by the various essays in the book.

The case investigated by Philippe Nabonnand concernsjustification in a revolutionary situation of science. In theworks by Carnot, Poncelet, and Chasles the search for ageneral treatment of different problems is also a way toreshape geometry as a theory based on general principlesrather than on the analysis of geometric figures. The argu-mentby generality is usednot only todefend the introductionof new methods, but also as a rhetorical way to justify theemergence of a different conception of geometry. This essayintroduces a leitmotiv of the book: justification is often pre-liminary to the moment when a new, well-established theorybecomes grounded in philosophicalmeans and canbe foundin different but essential steps of a revolutionary mathemat-ical transformation. It is not a matter of finding the best ormost definitive formulation of a theory: justification is used toinvestigate alternatives, and even impasses and limits of newmethods.

The attention to mathematical practice—one of the greatmerits of the book—provides original results once applied tothe analysis of the theory of quaternions by W. R. Hamilton.According toDominiqueFlament’s reconstruction,quaternionsoriginated from a sort of mathematical and philosophicalimpasse concerning the geometrical representation of imagi-nary quantities. They were developed by an extension of the

concept of algebraic couples and triplets, which implied theabandonment of a conception of mathematics as based onseparate disciplines: algebra and geometry, science of time,and science of space. The justification required by quater-nions concerns: (1) the emergence of a unitary conception ofmathematics, (2) the need to account for a revolutionarychange such as the abandonment of commutativity and ofthe principle of permanence in algebra, and (3) the intro-duction of a new notation based on the extension ofpreviously used concepts.

The distinction between separate mathematical disci-plines and the possibility of transferring resources from onedomain to another domain is central to the justification of theuse of geometrical representations in the development ofnumber theory made by Gauss and Minkowski. SebastienGauthier investigates several arguments introduced to justifythe use of mixed methods: from the applicative and episte-mological function suggested by Gauss to the heuristic,pedagogic, and foundational function advanced by Min-kowski. The study of this rhetoric of justification togetherwith the associated mathematical practices is—according tothe author—the best way to understand the foundationalviews of mathematicians who did not make explicit foun-dational remarks, such as Minkowski.

Catherine Goldstein describes the alternative philosoph-ical view implicitly expressed by Hermite, as he discusses hisown preferences for a certain method or point of view. Thisimplicit philosophical conception is presented as an exampleof antifoundationalism based on a defense of the interrela-tions between different branches of mathematics. Byquestioning the foundational program of arithmetization ofanalysis, Hermite condemns the idea of an a priori andphilosophical adoration of rigor that might limit the freedomof mathematics to create links between distinct domains. Andat the same time he defends an organicistic conception ofmathematics, where nothing is created ex nihilo, but newproperties emerge from a different approach, as in the case ofKummer’s ideals. The article by Goldstein offers anothergood example of how the attention to mathematical practicesmight alter the standard view on the history of the foundationof mathematics. On the one hand she shows that this historyneeds to take into account authors who have not explicitlydiscussed classical foundational questions, but nonethelesshave a specific philosophical point of view on the topic. Onthe other hand she shows, by a well chosen and less knownexample, why the question of the foundations of mathe-matics cannot be focused only on explicit polemics about theobjects of mathematics or investigate only the standard‘‘foundational’’ authors, but should include a larger numberof mathematicians and their mathematical practices.

The analogy between different domains (mathematicsand physiology) is not only investigated but also used rhe-torically by Klaus Volkert to emphasize the justificatory roleof exceptions and counterexamples in the discourse onmathematical generality. In physiology the appearance ofmonsters (1) first stimulated a horrified reaction to what wastaken to be outside the course of nature, (2) then suggestedan understanding of monsters as borderline cases in whichsomething is missing or added, and (3) finally led to theinclusion of them in the natural order to preserve the validity

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of general laws and a principle of continuity in nature. Sim-ilarly, in mathematics the interest in monsters was associatedwith an interest in generality, because atypical cases emergedwhen mathematicians began to analyze not just paradigmaticcases but all possible cases. Investigating the cases ofDirichlet, Riemann, and Hankel, Volkert claims that mathe-matical monsters not only made their appearance by chance,but were also explicitly built and used as a mathematical wayto investigate mathematics itself. Teratology had thus such arevolutionary impact because it was used to study metathe-oretical properties of theories. The acceptance of monsters—evident from the efforts to classify them and to find generalprinciples for their generation—was thus identified with astep toward the modern conception of mathematics. Thefurther reference to the case of Du Bois-Reymond offers anoriginal contribution to the historical investigation of thefoundations of mathematics. The easy acceptance of mon-sters is associated with the philosophical conception of theidealist, who accepts monsters as well as infinitesimalquantities,whereas the refusal to acceptmonsters—or at leastthe idea that future developments of mathematics will letthem disappear entirely—is associated with the point of viewof the empiricist, who refuses to accept anything that goesbeyond the limits of representation.

One of the polemic targets of the paper by Javier Legris—but I would say that this was already the case of Goldstein’spaper, even if only implicitly—is van Heijenoort’s recon-struction of the history of the foundations of mathematicsbased on the well-known distinction between logic as cal-culus and logic as language. Contrary to the standardhistoriographical view that pits Frege against Schroder, Legrisrecalls the similarities between the two authors—especiallythe idea of a universal scientific language that is common totheir programs. The main differences concern: (1) theirconception of semantics (internal for Frege and external forSchroder), (2) their justificatory strategies and aims (based onthe reduction of arithmetic to logic for the former and prag-matically oriented to the application of general algebra todifferent branches of mathematics for the latter), and (3) theirontological or non-ontological understanding of the prop-erties of numbers. As a result, the opposition between Fregeand Schroder suggested by the standard historiography is notreally abandoned, but rather reconstructed and justifiedbased on their mathematical practice rather than on the basisof van Heijenoort’s distinction, which fails to apply, becauseboth Schroder and Frege developed logic both as a calculusand as a language.

Jose Ferreiros insists on the essential role played by justi-fication in the process of formation and development ofmathematical theories. The prehistory of set theory—in par-ticular the justifications of Dirichlet’s notion of arbitraryfunction and of the extendeduse of the notion of number—istaken to be an essential element of the revolutionary stage ofset theory resulting from the works by Cantor and Dedekind.This kind of justification of terms and conceptual domains isdistinguished from the foundational justification to be foundin Hilbert, who saw ‘‘logic as thought’’ before he moved on toa formal and metamathematical kind of justification. It is alsofound in Zermelo, whose justification is based on axioms.This justificatory phase is characterized by an investigation of

alternative axiomatic systems, and it is opposed to the kind ofjustification developed by Godel, that is, a conceptual,metaphysical, and pragmatic justification of a well-estab-lished theory based on the notion of iterative set, onPlatonistic intuition, and on the role of the continuum inscience. Interestingly enough, some recent analysis ofunpublished manuscripts by Godel developed by GabriellaCrocco and her research group shows that even in the case ofGodel’s philosophical groundingof set theory, thequestion isoften that of finding alternative solutions to specific prob-lems, rather than that of furnishing a general justification ofthe best solution. Thehistorically detailed investigationof therelation between the notion of arbitrary function and theaxiom of choice, and of the evolutions and different justifi-catory strategies ofmathematiciansdevelopedbyFerreiros, isvery helpful for putting things into perspective: the founda-tional questions concerning set theory cannot be easilyreduced to the emergence of a couple of paradoxes at thebeginning of the century, or only to different philosophicalapproaches to the relations between logic and mathematics, butare part of a complex history extending from Dirichlet to Godel.

The evolutionary and historical approach followed bymost articles seems to be overturned by the last, morephilosophical, contribution, which suggests that the passagefrom a logicist to a formalist justification of mathematics is nota result of some evolution of mathematical practices but ofopposed philosophical beliefs. Yet, these beliefs are notidentified with the standard alternative ontological concep-tions of mathematical objects (realism, Platonism, idealism),but with two approaches to the philosophy of history. On theone hand is the logicist belief in the absolute truth of scienceand on the other hand the formalist (pragmatist?) belief in therationality of history. The originality of this approach withrespect to the standard treatment of the foundations ofmathematics consists (1) in the reduction of the foundationaltrilemma between intuitionism, logicism, and formalism to adilemma—the intuitionist perspective is taken to be a merelycritical rather than justificatory perspective—and (2) in theidea that there are no winners or losers in the foundationalmatchplayed in the 1930s, but rather that there are alternativestrategies in the same justificatory game.

The book would have profited from the addition of aname index and subject index, and of an introductory sectiondevoted to a more detailed description of the content of thevolume that would have brought out more clearly the varietyof approaches to the topic of justification. Yet the essays arethemselves so clear, even when difficult and technicalmathematical examples are examined, that the relevance andrichness of the content emerges easily from an attentivereading, useful both to historians and to philosophers ofmathematics, because this volume has the rare quality ofshowing how historical investigation might forge and modifyour philosophical understanding of crucial mathematicalissues.

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