traité de logique algoritmique

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Publications des Archives Henri-PoincaréPublications of the Henri Poincaré Archives

Textes et Travaux, Approches Philosophiques en Logique, Mathématiques

et Physique autour de 1900Texts, Studies and Philosophical Insights in Logic, Mathematics andPhysics around 1900

Éditeur/Editor: Gerhard Heinzmann, Nancy, France


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Birkhäuser

Edited by

Oliver Schlaudt and Mohsen Sakhri,

with an introduction and annotations by

Oliver Schlaudt

Louis Couturat Traité deLogique algorithmique

−


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Dr. Oliver Schlaudt Universit ̈at Heidelberg Philosophisches SeminarSchulgasse 669117 Heidelberg Germany

www.birkhauser.ch

Editors:

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© Springer Basel AG 2010

Library of Congress Control Number: 2010927519

ISBN 978-3-0346-0410-9 e-ISBN 978-3-0346-0411-6

978-3-0346-0411-6DOI 10.1007/

Dr. Mohsen Sakhri

Philosophie - Archives Henri Poincar

Universit Nancy 291 avenue de la Lib ration - BP 45454001 Nancy CedexFrance

é

éé

’ des Sciences et deLaboratoire d Histoire

CNRS - Nancy-Universit ́eUMR 7117


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Preface

Louis Couturat (1868–1914) was an outstanding intellectual of the turn of the nineteenthto the twentieth century. He is known for his work in the philosophy of mathematics,for his critical and editorial work on Leibniz, for his attempt to popularise modernlogic in France, for his commitment to an international auxiliary language, as well asfor his extended correspondence with scholars and mathematicians from Great Britain,the United States, Italy, and Germany. From his correspondence we know of fourunpublished manuscripts on logic and its history, which were largely complete andsome of which must have been of considerable size. We publish here for the first time ina critical edition the only one of these manuscripts that has been rediscovered: the Traité de Logique algorithmique , presumably written in the years 1899–1901. It is a highlyinteresting document of the academic reception and popularisation of symbolic logic inFrance. It provides evidence of the discussions and controversies which accompanied thecreation of logic as a new branch of science. At the same time it completes the pictureof Couturat’s work, which has been opened up to systematic study by the publicationof important parts of his correspondence during the last decade. We append the articleon Symbolic Logic of 1902 which Couturat wrote in collaboration with Christine Ladd-

Franklin for Baldwin’s Dictionary of Philosophy and Psychology. This article, as nowbecomes evident, is a sort of résumé of the Traité ; at the same time it points the way toCouturat’s Algèbre de la Logique of 1905. It thus helps to situate the Traité in Couturat’sœuvre . The same purpose is served by the second document appended, a short part of Couturat’s report of the first International Congress of Philosophy, which took place inParis in 1900. This report documents Couturat’s reception of Platon Poretsky, whosework was of considerable importance for the outline of L’Algèbre de la Logique andmarks the main difference between this later work and the Traité . – Since history of modern logic already attracts a lot of attention, the introduction focusses on Couturatand his perspective on modern logic in order to provide information the reader may

lack. Finally a critical apparatus should help the reader to find his way through theTraité and to understand its genesis.

Acknowledgements

We are much obliged to the Laboratoire d’Histoire des Sciences et de Philosophie – Archives Henri Poincaré (UMR 7117 CNRS / Nancy-Université), especially to Gerhard Heinz-mann, for the generous intellectual and material support which rendered possible therealisation of this project, as well as to the CDELI ( Centre de documentation et d’étude

sur la langue internationale ) at the municipal library of La Chaux-de-Fonds (Switzerland)for granting the printing licence for the manuscript. We are furthermore indebted to


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a number of colleagues for their help and advice which were essential for our project.We would like to thank Paolo Mancuso (Berkeley), Peter McLaughlin (Heidelberg),Philippe de Rouilhan (Paris), Fabien Schang (Nancy/Dresden), Anne-Françoise Schmid

(Lyon/Paris), Christian Thiel (Erlangen), and Paul Ziche (Utrecht).

Preface


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Contents

I Introduction 1

I. Presentation of the Manuscript 3

II. Biographical Note 3

III. Dating of the Manuscript 5

IV. Origin and Meaning of the Term “Logique algorithmique” 7

V. Characterisation of the Manuscript 9

VI. Couturat’s Interest in Algebraic Logic 11

VII. Editorial Policy 32

VIII. Editorial Symbols in the Presentation of the Text 33

II Transcription of the Manuscript 35Tome I 37

I. Définitions et notations : A. Logique des concepts 37

II. Définitions et notations : B. Logique des propositions 49

III. Principes 63

IV. Lois de la multiplication et de l’addition 75

V. Lois de la négation 93

VI. Développement des fonctions 107

VII. Théorie des équations 121

Appendice II. Sur les opérations inverses : Soustraction et division 145

Tome II 157

VIII. Théorie des inégalités 157

IX. Calcul des propositions constantes 185


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viii Contents

X. Calcul des jugements variables (ou des probabilités) 205

XI. Comparaison avec la Logique classique 225

XII. Conclusions 241

Editor’s Appendix: 251

A: Louis Couturat and Ladd-Franklin: Symbolic Logic 251

B: Couturat on Schröder and Poretsky, on the Ist International congress 258

Christine , 1902

III Critical Apparatus 261Variants and Annotations 263

Table of Correspondence with L’Algèbre de la Logique 294

List of Signs and Abbreviations 295

Table of Figures 296

Bibliography 297

Index Nominum 311

Index Rerum 312

1900 of Philosophy, Paris


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Part I

Introduction


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I. Presentation of the Manuscript

The present text is a transcription of Louis Couturat’s manuscript entitled “Traité de

Logique algorithmique ”. Of this manuscript, only the tenth chapter has been published –posthumously in 1917. The present edition of the entire text is based on the only knowncopy, an undated handwritten version preserved at the CDELI (Centre de documentationet d’étude sur la langue internationale) at the library of La Chaux-de-Fonds in Switzerland.It was found there by Mohsen Sakhri in 2003 in the course of his research on Couturat’swork on international auxiliary languages. With dismay we must note that according tothe CDELI the remaining papers of Couturat, including the manuscript of the Traité ,have in the meantime been seriously damaged and partly destroyed by water duringconstruction work. The only version available for study thus is a photocopy of themanuscript kept at the Archives Henri Poincaré at Nancy University (France).

The manuscript consisted of two volumes of about 200 sheets each. The sheetsare mostly used on one side; if used, the versos contain additional notes or corrections.Three handwritings can be distinguished: The rather uniform main text is in Couturat’sclear and easily legible handwriting; additional notes in different ink were presumablyadded by Couturat himself; and some notes that correspond to the 1917 edition of chapter X of the manuscript are in a different hand. On the last two pages of the secondvolume are placed a table of contents and a table of figures. The table of contentsadditionally indicates three appendices: I. Sur les signes adoptés, II. Soustraction et division(15 p.) , and III. Solutions générales symétriques. However only one of them – on thelogical operations of division and substraction – is given at the end of the first volume.

II. Biographical Note

Alexandre-Louis Couturat was an outstanding intellectual of the French Third Republic.He was born on January 17, 1868 in Ris-Orangis near Paris and died on August 3, 1914,on the eve of the First World War, when on the way from Paris to his country house inBois le Roi he was involved in a fatal traffic accident with an army transporter duringthe mobilisation of the French army.1 Couturat is today still known as a Leibniz scholarand as an important figure in the propagation of modern logic in France. Couturat’smost creative period as a philosophical writer as well as a tireless propagator of theinternational auxiliary language Ido in the name of peace and internationalism coincidedwith the impact of the Dreyfus Affair on the French society. He expressed his pacifistconvictions – coupled with profound knowledge in the history of philosophy – in a

1The main sources of his biography are André Lalande’s synopsis L’Œuvre de Louis Couturat , publishedin 1914 in the Revue de métapysique et de morale , and the obituary Couturat’s friend Louis Benaerts publishedin 1915 in L’annuaire de l’Association amicale de secours des Anciens Elèves de l’Ecole Normale Supérieure . Thesearticles have been reprinted together with Arnold Reymond’s short obituary from 1915, but this bookletis difficult to find today. There are two monographs on Couturat, one from the Argentinean historian of mathematics Claro C. Dassen (1873–1941) from 1939, and a more recent one from Ubaldo Sanzo, publishedin 1991. Dassen’s book, reviewed by Quine in 1940, benefits from the author’s correspondence with Couturatfrom January 1902 until at least 1911 on the subject of Ido.

Publications des Archives Henri Poincaré, DOI 10.1007/978-3-0346-0411-6_1,

© Springer Basel AG 2010

O. Schlaudt, M. Sakhri (eds.), , Louis Couturat – Traité de Logique algorithmique


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4 Introduction

polemic over Kant’s notion of war with the conservative writer Ferdinand Brunetière, akey figure of the Dreyfus Affair.

After a very successful academic education in philosophy at the Ecole Normale

Supérieure (ENS) 1887–1890 and then in mathematics at the ENS as well as at the Faculté des sciences where he attended lectures given by Jules Tannery, Picard, Jordanand Poincaré in 1890–1892, he was appointed as a lecturer in 1894 at the Universityof Toulouse where he stayed for one year. 1896 saw the defence of his two thesesprepared during an academic leave of absence of two years: the principle thesis De l’infini mathématique which marks an important – though not uncriticised1 – step inthe philosophical reception of Cantor in France, and the secondary thesis in Latin De

Platonis mythicis.2 After a further year of studies in physics at the Sorbonne , Couturataccepted a position as a lecturer at the University of Caen. In Toulouse, Couturat hadtaught Plato, and still in 1900 at the First International Congress of Philosophy, he

presented a paper, Le système de Platon exposé dans son développement historique . Butwhen he went to Caen in October 1897 he devoted his lectures and research activitiesalmost exclusively to philosophy of mathematics and to logic. About this time he alsostarted his important exchange of letters with Giuseppe Peano (1896) and BertrandRussell (1897).3 After two years in Caen he successfully applied for a leave of absenceand went back to Paris, where he organised the logic section of the first InternationalCongress of Philosophy, which took place in August 1900. Aside from his unexpectedreplacement of Bergson at the Collège de France in 1905/06, this leave marked the endof his teaching activity.4 In 1900 and 1901 he spent some time in Hanover where hesifted through Leibniz’s unpublished manuscripts on logic, the existence of which hadbeen indicated to him by the Italian mathematician Giovanni Vacca. This work yieldedthe book La Logique de Leibniz d’après des documents inédits (1901) and the Opusculeset fragments inédits de Leibniz (1903). The following years saw the publications of hispopularisation of Russell’s Principles of Mathematics, Les principes des mathématiques(1904), and of his short textbook L’Algèbre de la Logique (1905) which earned him areputation as an important representative of modern logic in France. During the years1905/06 he had a famous quarrel with his former teacher Henri Poincaré on the rôle of intuition in logic, which underlines Couturat’s turning away from the then dominantKantian philosophy. Already at the first International Congress of Philosophy there

occured for the first time in Couturat’s writings the idea of an international auxiliarylanguage. His commitment to Ido increasingly absorbed Couturat. It gave rise to dozensof articles and works. The most important of them is probably the Histoire de la langue

1Cf. Tannery 1897 and Dugac 1983.2A comprehensive report of Couturat’s defence before a jury composed of his director Boutroux, Jules

Tannery, Evellin, Séailles and Egger for De l’infini, and of Brochard, Croiset, Bouché-Leclercq and Decharmefor De Platonis mythicis was published in RMM, Supplément du numéro de septembre 1896, pp. 13-20.

3These important correspondences have meanwhile been edited, cf. Schmid 2001 as well as Luciano andRoero 2005; in the following we refer to the former by indicating the date of the letter prefixed with a C forCouturat and a R for Russell, the volume and page number, and to the latter by indicating the date and pagenumber. For a survey of the published part of Couturat’s letters see part B of the bibliography.

4In 1902 he applied unsuccessfully for the chair of general history of science at the Collège de France , cf.Paul 1976, p. 393.


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Introduction 5

universelle (1903) written together with Léopold Leau. They emphasised the necessityof a universal language as a linkage of nations and of the scientific community. From1900 to 1908, Couturat held the office of treasurer of the Délégation pour l’adoption d’une

langue auxiliaire internationale ; in 1908 he was secretary of the delegation’s committee.In the same year he founded the journal Progreso, and from 1910 on he was secretaryof the Akademio di la linguo internacino Ido. This engagement led from 1908 on to amore serious interest in linguistics and also resulted in some articles on the relation of language and logic which are of distinctive interest.

III. Dating of the Manuscript

The Traité de Logique algorithmique is, besides Logique mathématique , the Manuel de logistique and the Histoire de la Logistique , one of four unpublished manuscripts Couturatmentioned in his correspondence.1 It is the only one that has been rediscovered. It musthowever have been accessible during the first years after Couturat’s unexpected death inAugust 1914, since its tenth chapter was published posthumously in 1917 in the Revue de Métaphysique et de Morale . The anonymous editor – presumably André Lalande, withwhom Couturat collaborated for the Vocabulaire technique et critique de la philosophie and who in his 1915 study L’Œuvre de Couturat had already proved to be acquaintedwith Couturat’s remaining papers, including his correspondence with Russell – adjoinedthe following annotation:

The present article is extracted from an unfinished treatise on algorithmiclogic written by Couturat perhaps a long time ago, definitely before 1902,perhaps a long time before that. [. . . ] After this period he abandonedpublication; he completely revised this first version in order to transformit into a Manual of mathematical logic which, as we hope, will be publishedsoon. (1917b, p. 291)

1For references to the Traité see the information given in this section. In C 30.08.04 Couturat reports thecompletion of his Logique mathématique for the publisher Naud, with whom he had agreed upon this projectin January 1904 (cf. letter to Peano, p. 62, and C 11.02.1904; Couturat sent a résumé of this book to Peano in

July 1904, p.69. In October 1904 he still waited for page proofs, which shows that he indeed had deliveredthe completed manuscript, cf. p. 77). It purported to present Peano’s system completed by Russell’s logic of relations and some methodological considerations. For reasons not mentioned in the correspondence the bookwas never published. In December 1904, after having finished L’Algébre de la Logique , Couturat announced hisproject of the Manuel de Logistique for the publisher Alcan (C 18.12.1904, II/453). In January he reported hisongoing work to Russell (C 22.01.1905) and to Peano (05.01.1905, p. 85). In July 1905 he announced to Peano(p. 89) the upcoming completion of the book, containing his two articles already published in the journal Enseignement mathématique (1900f and 1900g). One chapter of the Manuel has been published posthumouslyin 1917 (1917a), which gives altogether a quite precise idea of the book. Couturat, having made good progresswith this project, stopped it because of his unexpected call to the Collège de France as a substitute for HenriBergson ( cf. his letters to Russell C 10.11.1905, II/546, and to Peano, p. 93). At the Collège de France he lecturedon the history of modern logic and took up again a book on this subject, the Histoire de la Logistique , thefourth manuscript, which he mentioned for the last time in his correspondence with Russell in C 22.07.1906II/614, and in his correspondence with Peano in October 1906 (p.120).


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The available information, in particular Couturat’s correspondence with Bertrand Rus-sell and Giuseppe Peano published in 2001 and in 2005 respectively, confirms this dating.Indeed Couturat was charged with giving lectures at the university of Caen (Normandy)

from November 1897 on.1

For his second academic year in Caen he announced a course“Studies of the diverse systems of algorithmic logic: Boole, de Morgan, Stanley Jevons,Delbœuf, Peirce, MacColl, Schröder, Peano, etc. On the relations of mathematics andlogic; on the scope of the mathematical method. The Idea of universal algebra (White-head).”2 This course, mainly inspired by a preliminary reading of Whitehead’s Treatise on Universal Algebra3, lies at the origin of the present manuscript. In September 1899,one year later, Couturat wrote to Russell: “I concentrate on writing my course on

algorithmic logic which I intend to publish next year” (C 05.09.1899, I/135). A littlemore than one year later he confirms: “I still intend to write an Algorithmic Logic in twovolumes” (C 03.01.1901, I/219). The consequences to be drawn from this last utterance

are not unambiguous: On the one hand, the specification of the two volumes seems torefer to our manuscript; on the other hand the expressed intention to write the treatisein the future contradicts the letter of September 1899 reporting the already initiatedproject of writing up the lecture notes. Most probably the production of the Traité consisted of two major parts, the relatively homogeneous collection of his lecture notesand, as the annotations in the manuscript also suggest, a later rough proof-reading. Thefirst part may even just predate the more careful reading of Whitehead, since one findshis name subsequently added four times, while the original version does not mention it.Of course it is possible, too, that the additional annotations originate in the preparatorywork for L’Algèbre de la Logique in 1904. There is also another curious detail whichconfirms the dating of the manuscript: In the Traité Couturat still used the spelling“Leibnitz”4, abandoned in his La Logique de Leibniz (1901a, p. vii, note 1). In his cor-respondence with Russell, one can localise this change in the spelling quite accuratelybetween November 1900 and January 1901 (cf. C 05.11.1900 and C 03.01.1901).

For reasons that are not evident from (the published parts of) his correspondence,Couturat abandoned the plan to publish the Traité de logique algorithmique sometimeafter January 1901. He in any case interrupted his work on modern logic in order tocomplete his study La Logique de Leibniz, published in 1901, as well as his Opuscules et

fragments inédits de Leibniz, finally published in 1903.5 It seems that the only work on

logic he published in the meantime is the article Symbolic Logic for Baldwin’s Dictionaryof Philosophy and Psychology, written in collaboration with Christine Ladd-Franklin (cf.Appendix A, p. 251). This article, as can be seen now, is in fact a précis of our Traité ,

1Letter to Peano, p. 7; letter to Russell, C 09.11.1897, I/69.2RMM, Supplément Septembre 1898, p. 2; cf the letters to Peano and to Bettazzi, p.9 and p.190.3In C 08.07.1898 he reports having read the introduction only; the complete reading for his review in

RMM followed in 1899 (C 03.02.1899 and C 05.09.1899).4In a letter to Peano, Couturat explains in June 1899, p. 24: «Pour Leibnitz, j’ai adopté l’orthographe

que préconise M. Boutroux, pour des raisons d’analogie, tout en sachant que l’auteur et les éditeurs écrivaient Leibniz. Mais je n’y tiens pas autrement.»

5In a letter to Peano he later confirmed this break during 1901, 1902 and 1903 (p. 85). – It should beremarked that this work on Leibniz again was partly inspired by the reading of Whitehead (and thereafter of Grassmann); cf. C 13.05.1900.


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differing from it only in some minor details, like the propositional interpretation of “o”and “1” and the explicit discussion of the Ladd-Franklin-formula. In 1902 Couturat waslooking forward to returning to his studies on logic after having finished the Opuscules

(C 08.07.1902, I/282). In the middle of 1903 we find Couturat again working on logic,however no longer on Boole and Schröder, whose works were crucial to the Traité , buton Peano and Russell – a “complete change” in his own words (C 10.06.1903). In 1904he announces his popularisation of Russell’s Principles of Mathematics of 1903, and at thesame time also a booklet on mathematical logic, i. e. in particular on Peano’s symbolism(C 11.02.1904). Nevertheless he returned in 1904 to algorithmic logic and wrote his bookon the Algebra of Logic . At this time algebraic logic was of course no longer an end initself for Couturat, but a didactical means: the book was supposed first of all to facilitatethe book on mathematical logic (C 15.11.1904, C 22.01.1905). Mathematical logic,i.e. Peano’s symbolism, was then considered by Couturat as more fundamental than

algebraic logic, though admittedly less practicable (C 07.05.1905). Couturat reported thecompletion of L’Algèbre de la logique in December 1904. It contains a considerable part of the Traité in a compressed form (cf. the table of correspondence, p. 294). There are only afew points which cannot be found in the Traité . These mainly concern issues with whichCouturat may have become acquainted at the International Congress of Philosophy in Parisin 1900, where Russell, Schröder, MacColl, Peano, Johnson, and in particular PlatonPoretsky were present. Primarily the writings of the latter had a considerable influenceon L’Algèbre de la logique (cf. Appendix B, p.258). Couturat however suppressed theexposition of the calculus of variable propositions, i. e. of probabilities (ch. X), as well asthe detailed comparison to classical logic (ch.XI). In the conclusion of L’Algèbre de lalogique he declared:

The foregoing exposition is far from being exhaustive; it does not pretend tobe a complete treatise on the algebra of logic, but only undertakes to makeknown the elementary principles and theories of that science. (1905a, § 60,p.94)

Since he seems to allude to the present manuscript in speaking of a “traité completd’Algèbre de la Logique”, we feel all the more justified in publishing it, though thecompletion of L’Algèbre de la Logique might finally have contributed to Couturat’s

decision to abandon the Traité .

IV. Origin and Meaning of the Term “Logique algorith-mique”

Algebraic or algorithmic logic resulted from the application of the mathematical methodto logic, achieved particularly in the work of Boole and Schröder.1 By the term “logic”must be understood first and foremost classical logic, i. e. the theory of the syllogism

1For a short account of the history of algebraic logic see Jourdain’s introduction to the English translationof L’Algèbre de la Logique , published in 1914; for a detailed study cf. Grattan-Guinness 2000.


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or the logic of concepts. The application of mathematical method was made possibleby the discovery of structural similarities between classical logic and algebra. Insofaras algebraic logic applies a method borrowed from mathematics, it subsumes logic

under mathematics. This becomes notably clear in Alfred North Whitehead’s Trea-tise on Universal Algebra from 1898, a comparative study of Boole’s symbolic logic,Hamilton’s calculus of quaternions and Hermann Grassmann’s calculus of extension( Ausdehnungslehre ), all subsumed under a general notion of calculus. Due to this mathe-matical treatment, logical propositions figure in algebraic logic mainly in an uncommonform, namely as equations. Its main subject hence is the theory of equations and thedevelopment of functions. In the Traité these subjects are completed by a chapter on thetheory of inequalities and on probabilities, interpreted – following Boole and MacColl– as truth coefficients. After these first successful steps were taken in algebraic logicauthors like Peirce and again Schröder started working on relational logic. For Coutu-

rat, though acquainted with these efforts, relational logic became important only in1901, due to Russell’s work on this topic. Relational logic was essential for the logicistprogramme of reducing mathematics to logic, i. e. reducing mathematical reasoning tological inference and introducing the basic concepts of mathematics by means of thelogical primitive terms. In so far, the rise of relational logic turned the relation of logicand mathematics upside down.

In the years from 1900 onward Couturat became an ardent advocate of the logicistprogramme. The Traité however predates these developments and originates in anepoch where Couturat was still sceptical with regard to relational logic (see my remarksbelow). It is an interesting fact that Couturat, notwithstanding his changing attitudestowards logic, in particular towards relational logic, always preferred the notion of “algorithmic logic”, that Church traced back to Castillon (1803) and Delbœuf (1877).1 Intwo articles (1904e and 1912a), where Couturat discussed the different occurring namesfor the new logics, he always opts in favour of “logique algorithmique” (or “logistique”as was proposed independently by Couturat, Lalande, and Itelson). He rejected theterm “symbolic logic” because it overemphasises the use of symbols, the term “Algebraof logic” because it presents the subject as a branch of mathematics, as a special algebrainstead of logic, and finally also the easily misunderstood term “mathematical logic”(1904e, p. 1042, 1905e, p. 706, 1912a, p. 138). In our manuscript, “algorithmic logic”

however still stands simply for algebraic logic, i.e. a logic providing rules for solvinglogical problems by means of standardised calculating procedures (cf. ch. VII.4). TheTraité thus documents the completion of a development originating in the nineteenthcentury, rather than the beginning of the logicist programme that was of fundamentalimportance in the early twentieth century. The study of algebraic logic was neverthelessof philosophical importance for Couturat, as I will show in the following sections.

1Church 1956, p.56-7, note 124, and 1936, p. 126-7 and p. 129. Delbœuf’s treatise was indeed part of thecourse Couturat held at Caen in 1899; Castillon is mentioned by Couturat in connection with his lecture atCollège de France , C 24.11.1905, II/555.


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V. Characterisation of the Manuscript

As the information used for dating the manuscript has already revealed, the Traité is

essentially based on lecture notes, which suggests the character of a compendium or asurvey. A cursory reading indeed confirms the character of a popularisation or synopsis:A popularisation in that it compiles recent developments in algebraic logic (Boole,Peirce, Schröder, MacColl, Venn, Whitehead) and presents them in French; a synopsisin that it clearly shortens long and fatiguing algebraic procedures (e. g. of equationsolving) in order to show the reader the central ideas, and in so far as it neglects –knowingly or due to lack of familiarity – more subtle problems connected with conceptformations in algebraic logic1. The Traité is in this sense an interesting historicaldocument on the propagation of symbolic logic in France at the turn of the twentiethcentury. This characterisation in addition corresponds to how Couturat understood

his own work and the rôle he played in the intellectual life of his epoch. Thus, heprogrammatically declared in a letter to Russell: “In my review I will leave aside alllogical difficulties and subtleness in order to bring out the mathematical principles thatconstitute, from the point of view of general philosophy, the most important part of your work.” (C 19.11.1903, I/338) In a different letter to Russell he willingly admitted:“the debate surpasses my competency and rather concerns the inventive authors (likeyou, Whitehead, Peano) . . . ” (C 27.07.05). Schmid concludes from that, that Couturatdid not conceive of himself as an “inventive author” (Schmid 2001, I/ 37) and comes tothe following general characterisation:

[. . . ] the intellectual rôle of Couturat: correspondent of scholars and philoso-phers, intermediary in the circulation of ideas between several countries(particularly England, Italy, Germany, and the United States). His concernwas not to invent, but to identify the results of recent research in order tokeep up a vivid thinking against the conservatism of the institutions. (2001,I/341)

One is tempted to say that Schmid sketches the image of a modern Marin Mersenne,whose merit consists rather in detecting, transmitting, and connecting with a keen senserecent trends in science than in contributing to them in an inventive manner. This

image indeed seems to be fair in a number of its characterisations. The last point inparticular, concerning Couturat’s undogmatic activity in academia, is confirmed by theTraité itself, which, as we have seen, emerged from a course on symbolic logic which

1Cf. for example the issues raised by Husserl’s and Frege’s critique of Schröder (analysed in Heinzmann1992), published several years before the composition of the Traité . The reader will note that Couturatunderestimated in particular the importance of the copula ε (ch.II.28, cf. also 1899b, p. 628) as well asthe problems that come along with the concept of a “universe of discourse”. Couturat avoided these latterproblems simply by abandoning any absolute meaning of the notion of universe. According to him theuniverse is always to be understood relatively to a given problem (ch. I.10). In ch. VII.2 he even speaks of the“universe of the problem“ instead of the “universe of discourse”. The necessity of the distinction betweeninclusion < and membership ε was acknowledged by Couturat in 1901 (cf. the letter to Russell C 27.01.1901,I/232), i.e. shortly after the termination of his work on the Traité . Couturat anyhow did not go into theproblems connected with the concept of the “universe of discourse”, indicated by Russell, R 20.10.1903.


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must have been quite exceptional at that time1. In the winter 1905/06, when Couturattaught history of mathematical logic at the Collège de France , he still complained of thelack of approval.2 As regards the first point – concerning Couturat’s lack of originality –

one should nevertheless be careful when jumping to the suggested conclusion. SimilarlyRussell judged Couturat’s Principes des mathématiques to be a too dogmatic synopsis,presenting his theses as apodictic, although they were rather intended as problematic.In a letter to Ottoline Morrell he expressed his rather severe opinion of Couturat:

Then when my “Principles of Math’cs” came out, he wrote a short bookprofessing to explain its doctrines simply to the French public. He left outall the doubts and difficulties, all the places where consistency had led meinto paradox, and everything which he imagined calculated to shock, and atthe same time put the thing forward as a dogmatic doctrine finally solving ahost of difficulties. In consequence he made me appear absurd, and took tothe international language, which now occupies him wholly.3

But in fact there are good reasons for doubting Russell’s judgement, at least in thisrigour. Couturat’s Principes des mathématiques are in fact the result of an autonomousand critical work. This finds its most obvious expression in the theory of extensive mag-nitudes presented in Chapter V, essentially based on the works of Otto Hölder, EdwardV. Huntington and Cesare Burali-Forti. This supplement in effect answers a failing of Russell’s “absolute” theory of magnitude, the inability to grasp the additive propertyof extensive magnitudes, which however is constitutive for the numerical expressionof magnitudes and hence for the essential purpose of measurement.4 Surely Couturatdidn’t dare to confront Russell directly. Instead he tried to reconcile the opposing posi-

1I infer this judgement from Couturat’s own reports, since there is unfortunately no survey of logicteaching activities in France of the 19th and the early 20th century. In C 22.01.1905, II/467, Couturat reportsthe abandoning of the idea to teach logic at the Sorbonne because of the low expected effect in view of thetraditional program and the traditional requirements for passing examinations. Having begun his lecture onthe history of logic at the Collège de France , he confirms: «C’est peut-être la seule fois que depuis cent anson aura fait un cours de Logique formelle au Collège de France.» (C 10.11.1905). In his inaugural lecture, heemphasises that it was indeed in 1838, when Barthélemy Saint-Hilaire (1805–1895) lectured on Aristotle’slogic, that anyone was concerned with logic at the Collège de France . A remarkable exception to this absenceof modern logic is presented by Louis Liard (1846–1917) who published a book on contemporary Englishlogicians in 1878. We have no information about Liard’s teaching activities, but we know at least that he wasimportant for Couturat’s career: It was Liard, then director of higher education, who supported Couturat’s journey to Hanover where he studied the manuscripts of Leibniz at the Royal Library (cf. Couturat 1903a,p. ii), and it was Liard too, who encouraged Couturat to accept the position at the university of Caen in 1897(cf. Dassen 1939, p. 97).

2At the beginning, Couturat was quite pleased about the large audience (C 12.12.1905, II/565); but someweeks later he already had cause for complaint: «Je dois vous dire que l’auditoire se fait de plus en plus rare:le ‹public› n’y vient naturellement pas, et les étudiants se trouvent obligés d’assister à des conférences dephilosophie qui ont lieu aux mêmes heures.» (C 06.02.1906, II/593)

3Letter to Ottoline Morrell, March 25 1912, Russell Archive no. 400, quoted with permission of theBertrand Russell Archives Copyright Permissions Commitee, McMaster University, Hamilton. I am gratefulto Kenneth Blackwell and Carl Spadoni for transcribing the quotation and for granting the permission topublish. For a French translation cf. Schmid 2001, I/339.

4Cf. Schlaudt 2009b, p. 288-293 and Nagel 1931, p.326-7, who criticised Russell in the same sense butdid not notice the ambiguity in Couturat’s text which he also quoted.


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tions (cf. C 06.06.1904, C 26.06.1904) and ended up in his Principes des mathématiqueswith a presentation rightly characterised by Schmid as a rather awkward compromise(2001, II/416-7, ann. 3). Be that as it may, Couturat’s Principes des mathématiques give

good evidence of an awareness of existing problems as well as of a certain originality. If Couturat’s philosophical originality did not result in contributions to the developmentof logic, it nevertheless found expression in the presentation of the subject matter. Thesehints should serve as a guideline when looking for the originality of the Traité , as I willdo in the following section.

VI. Couturat’s Interest in Algebraic Logic

The Traité owes its content to the writings of the founders of symbolic logic, i. e. first

and foremost to George Boole, Charles Sanders Peirce, Ernst Schröder, Hugh MacColl,Alfred North Whitehead and others. Since Couturat himself constantly spells out thesedebts, itemising them here in detail does not promise to yield unforeseen insights. Thepresentation and composition of the Traité however is of interest, as I pointed out inthe last section. It is governed by several particular preferences or decisions, whichare held together by a general aspect. The particular preferences are (1.) calculus vs.ordinary language, (2.) analytical vs. synthetical reasoning, (3.) algebra of logic vs. logicof algebra, (4.) equivalence vs. implication, (5.) extension vs. comprehension, and (6.)structure vs. interpretation. The common general motivation behind these preferencescan be found in Couturat’s interest in formalisation of thought, more precisely in his

discovery (or rather recovery of Leibniz’ discovery) of non-quantitative formalisation asa part of algebra in a more general sense, the characteristica universalis. This frameworkallowed Couturat, as we will see, to conceive the study of logic as a way to study themind itself.

Quantitative formalisation, or briefly quantification, was already one of the majortopics of Couturat’s De l’infini mathématique of 1896. Measurement theory had becomean important issue during the nineteenth century, primarily in consequence of twoimportant developments in science: on the one hand the emergence of quantitativepsychology due to Herbart and Fechner, and on the other hand the growing interest in

the foundations of mathematics that led, among other things, to an axiomatization of quantity. Both developments are reflected in Hermann von Helmholtz’s epoch-makingessay Messen und Zählen. Erkenntnistheoretisch betrachtet from 1887 (cf. Schlaudt 2009a,pp. 173 et seqq.). This essay was for Couturat, too, indeed one of the most importantreferences of his De l’infini. In this book Couturat studied the concept of numberon the one hand and the concept of quantity or magnitude on the other hand. Hedeveloped thereby a highly differentiated and refined conceptual framework, whoseeffectiveness he proved anew two years later in his reply to Russell’s On the Relationsof Number and Quantity (1897). In this still neo-hegelian essay, later judged by itsauthor to be “unmitigated rubbish” (1959, p. 41), Russell claimed to show that each

alternative in the conception of quantity leads to contradictions, thus coming finallyto the conclusion of an “inadequacy of thought to sense, or, if we prefer it, of the


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fundamental irrationality of sense” (p. 341). Such a conclusion was not acceptable forCouturat who held reality to be for us “what we think is true, and the true is that whichreason understands and affirms as existing” (Bowne 1966, p.30). In his reply (1898b)

Couturat showed that the alleged antinomies can in large part be resolved simply byattentively distinguishing between (kinds of) quantities, determinate states of quantity,numerical expressions of quantities, and concrete instantiations of abstract quantities.This is the vocabulary he had developed in his De l’infini mathématique . The basicidea of this book is, that the concepts of number and quantity are independent of oneanother and that both are independent of experience. This conception led him to aconsideration of number and quantity as isomorphic systems and thus to a conceptionof measurement which anticipates its currently prevailing definition as “assignment of numbers according to rules” (Stevens 1946), which in effect underlies the formal studyof numerical representations as it was accomplished in the second half of the twentieth

century (Krantz et al. 1971–1990). Couturat’s De l’infini already shows an interest in,and a familiarity with, algebra in the narrow sense, i. e. as a calculus of number andquantity. His main source seems to have been Henri Padé’s Leçons d’algèbre published in1892. The acquaintance with Whitehead’s Universal Algebra, the first volume of whichwas published in 1898 and which Couturat read shortly thereafter (see above), seemsto be at the origin of Couturat’s wider idea of formalisation. This comparative studyoffered to Couturat the idea of alternative ways of formalisation which, though theydo not proceed in a quantitative manner, permit rigorous reasoning. These alternativeways of formalisation of thought found for Couturat their most striking expression inthe two formulae:

a + a = 2a and a + a = a.

These formulae characterise two different algebras proceeding according to differentrules: The first characterises the quantitative calculus, in which taking a two times gives2a, thus leading to numerical expressions; the second characterises the logical calculus,in which considering a a second time does not add anything to a (ch. IV.5 of the Traité as well as 1901a, p. 365-6, and 1900d, p. 331). Classical logic can therefore rightly beconsidered as a qualitative algebra. In turn traditional algebra could be considered asthe logic of quantity. Both points of view lead to the same result: Logic and algebra aresubsumed under a general or universal algebra which conversely is specified with thespecial sense given to the relation designated by the sign of equality “=”: quantitative equality leads to ordinary algebra as the calculus of number and quantity, especiallyGrassmann’s extensional calculus; identity and mutual inclusion lead to the calculus of concepts and classes (intensional and extensional respectively); equivalence leads to thecalculus of propositions; similitude , congruence , and equipollence finally lead to specialgeometric calculi.1 The reading of Whitehead, and then of Grassmann, led Couturat toa systematic study of Leibniz, where he found that the project of formalising thoughtwithout first quantifying it had been anticipated. In Leibniz this endeavour was coupled

11900d, p. 331, 1901a, p. 318 et seqq. and p. 410. – As regards the idea of the universal algebra in thework of Schröder, cf. Peckhaus 1994.


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with a critique of Spinoza and Descartes, who, according to Leibniz, had failed toextend the geometrical method to metaphysics (1901a, p. 94, and p. 280). The subjectof his critique was nota bene not the attempt to subject other topics to a rigorous and

secure method, but the failure to examine the preconditions of these efforts. This lackof methodology resulted in a procedure more geometrico, which merely imitated themathematical method without establishing a real calculus in metaphysics. The latterhowever would have resulted in a revision and broadening of the concept of mathematicsitself, transcending the narrow borders of the “science of quantity” (1901a, p. 290-1).This is the path leading from Leibniz’ efforts to establish a metaphysical calculus tomodern algebraic logic, as I will discuss in more detail below.

It is noteworthy that Couturat not only appropriated Whitehead’s ideas as a gener-alisation of his own former studies, but conversely also applied a fundamental standpointof his De l’infini mathématique to universal algebra. One of the noticeable traits of this

work is its distinction between a logical and a rational point of view: According toCouturat, concept formation is essentially underdetermined by logic; there are alwaysvarious ways to form concepts and to extend theories which are equally possible fromthe point of view of logic. They thus demand a supplementary justification from a ra-tional point of view (cf. Bowne 1966, p. 29 and Schlaudt 2009b, p. 220-2). In arithmetic,the topic of De l’infini, the issue in question is the broadening of the number concept:Although the extension of number from integer number to whole number, fractionalnumber and so on is logically possible, Couturat still demands a rational justificationfor it. This justification is provided by the criterion of applicability to quantity. Wholenumbers, for example, logically introduced as sets of pairs of integers, can be regardedas numbers in so far as there are quantities adequately representable by them. The samereasoning holds, following Couturat, with regard to the logical calculus. In his reviewof Whitehead (1900d, p. 328), Couturat explains that, from a purely logical point of view, the calculus can be performed without regarding the meaning of the symbols(this is even one of the major features of symbolic logic); from a philosophical point of view however, the raison d’être of the symbols remains to be explored. This is the taskSpinoza and Descartes missed. Couturat’s account of this specific value of symbolicrepresentations, allowing the application of a calculus, will be examined below. How-ever, we can already get a first idea of how Couturat adopted the more general point of

view provided by Whitehead’s Universal Algebra and how he also kept at once his ownoriginal philosophical standpoint. In addition we can observe that the intermixture of the Traité with measurement-theoretic issues (the discussion of the measure of extensionand comprehension, of the measure of probability, of the notion of homogeneity, of Fechner’s psychophysical law, of mathematical variables, etc.) is not merely due to theaccidental intellectual biography of its author, but is indicative of the fact that algebraiclogic and measurement are two sides of one and the same coin, namely, formalisation of thought. I will now examine the particular consequences that Couturat drew from thisgeneral idea for the conception of his treatise on symbolic logic.

1. Calculus vs. ordinary language . – The calculus of symbolic logic provides the

advantages of absolute rigour together with purely mechanical performance. It evenpermits the construction of logical machines, “mills into which the premises are fed


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and which turn out the conclusions by the revolution of a crank”, as Peirce put it(1887, p. 165). This mechanical reasoning not only rules out error, but also permitsus to save human intellectual power or, as one may put it, permits us to effectuate

complicated logical inferences which surpass our limited natural powers (cf. ch.XII).Ordinary languages on the contrary are afflicted with equivocality, inexactness, andambiguity. This is the opposition as Couturat sees it. It is this last advantage of the calculus, the advantage of saving intellectual power, which permits Couturat tounderstand the logical calculus as a realisation of Descartes’ idea of method .1 This traithas been emphasised by MacColl, who began his Calculus of Equivalent Statements withthe following considerations:

Symbolical reasoning may be said to have pretty much the same relation toordinary reasoning that machine-labour has to manual labour. In the case of

machine-labour we see some ingeniously contrived arrangement of wheels,levers, &c., producing with speed and facility results which the hands of man without such aid could only accomplish slowly and with difficulty, orwhich they would be utterly powerless to accomplish at all. In the case of symbolical reasoning we find in an analogous manner some regular systemof rules and formulæ, easy to retain in the memory from their generalsymmetry and interdependence, economising or superseding the labour of the brain, and enabling any ordinary mind to obtain by simple mechanicalprocesses results which would be beyond the reach of the strongest intellectif left entirely to its own resources. (1880, p. 45)

We find the same rhetoric in W. E. Johnson:

As a material machine is an instrument for economising the exertion of force, so a symbolic calculus is an instrument for economising the exertion of intelligence. And, employing the same analogy, the more perfect the calculus,the smaller would be the amount of intelligence applied as compared withthe results produced. (1892, p. 3)

It is also on this intuitive level that Couturat could establish the equivalence of mechan-

1Grossmann argued in his sociological study on Descartes’ method from 1946 that Descartes’ ideas aroseby a kind of generalisation from experiences with early machines in manufactural production; he thus linkedthe birth of rationality in the scholastic world with the emergence of automatic machines, (2009, p. 163, p. 188,p. 197 and p.215). With his algebraic method, Descartes created an “intellectual auxiliary means” (p. 163),“a kind of intellectual machine operating automatically, quickly, and surely” (p. 223-4). This conception isof particular interest, as will become clear, for Couturat’s idea of logic that was also closely linked to theconcepts of method and mechanization. According to Grossmann, concepts gained from technique can inturn be applied to nature, as the editors point out (p. 23), by taking the machine as a model of (the relevantpart of) nature. Nature itself is transformed by that into a huge machine, the “machina mundi” (p. 10 andp. 190). As regards our case, we find a similar situation: The logical calculus as well as logic machines thusprovide a model for the human intellect as regards its faculty to reason logically. We will see the resourcesof Couturat’s philosophy for such a conception of the mind in what follows. This leads to Leibniz’ famousstatement that “omne opus mentis nostrae esse computationem” (cf. Couturat 1901, p. 458, and Marciszewskiand Murawski 1995, p. 76). As Couturat pointed out, in Leibnizian methaphysics finally the whole natureappears as an “admirable calculating machine” (1901, p. 256).


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ical and formal reasoning.1 As regards machines, the connection between mechanicaland formal reasoning is immediately evident: Once a machine has been built (once thecalculus has been established), it can be used without the slightest idea of how it works

and what it is good for, and even an untrained worker is able to effectuate work of thehighest accuracy. This idea becomes even more evident in regard to early clocks andin regard to machines from antiquity whose inventors did not succeed in mechanisingthe whole procedure to be effectuated. In this case of partial mechanisation, a repeatedintervention of a keeper is necessary, i.e. intervention of someone who understandswhat purpose the machine should serve and who consequently knows how the gapsin its performance can be filled by hand.2 Couturat’s critique of the early calculusof relations (as well as of Whitehead’s notation) will consist exactly in the existenceof such gaps which cannot be filled without considering the sense of the utterances inquestion (see below, point 3). Here a lack of mechanical performability is interpreted

as indicating an imperfect formalisation. Mechanical and formal reasoning are thusequated.As regards the advantages of the calculus compared to ordinary language – me-

chanical rigour vs. ambiguity –, Couturat at first sight only repeats commonplaces.His distinction between ordinary language and calculus however rests on a remarkablephilosophical conception. This conception essentially follows from Couturat’s anti-psychologism. This conviction marks a constant element of Couturat’s philosophy. Afirst outline is already found in the preface of his thesis De l’infini from 1896 (pp. viii-xii).Couturat later spelled out this critique of psychologism, accompanied by “sociologism”and “moralism”, in his inaugural lecture at Collège de France in 1905. His argumentsagainst psychologism are at first sight common too: Logic is the normative science of correct reasoning whereas psychology can at best be the natural history of the soul.3But Couturat went one decisive step further. He argued that introspective psychologyis not even competent to investigate reasoning, the latter being a “non- or sub-consciousprocess”. Introspection has no privileged access to reasoning, being the action of the“esprit ” (in the following translated as mind), not of the “conscience ” (consciousness). Wefind hence in Couturat the very same “extrusion of thoughts from the mind” which

1This equivalence was later questioned by the specification of the idea of the algorithm, particularlyin the works of Alonzo Church and Alan Turing on undecidability of arithmetic and first-order logic in1936 (cf. Krämer 1988, p. 138 et seqq.). Semidecidability of predicate logic, formulated by Thoralf Skolemand Jacques Herbrand in the 1920s, nevertheless provided a basis for further research on the mechanizationof deductive reasoning. The appearance of computer technology in the 1950s additionally stimulated thisresearch and efforts towards automated theorem provers (cf. Marciszewski and Murawski 1995, p. 209 et seqq.and p. 228-30).

2Grossmann 2009, p. 197. Freudenthal 1982 analysed the momentous ideological differences resultingfrom taking either a perfect scientist’s watch or an artisan’s clock as a model of the physical universe, as wasdone by Leibniz and by Newton and Clarke respectively in their famous controversy. As the artisan’s clockdemands repeated interventions of a keeper, it suggests, taken as a model, a universe that is not causally closedand leaves space for divine interventions.

3For the critique of psychologism, in particular the arguments of Husserl and Frege, cf. Kusch 1994.As Kusch points out, Husserl observed that the normative-descriptive-distinction alone does not provide aconclusive argument against psychologism, since thought as it ought is a special case of thought as it in fact occurs (p. 44). In Couturat too this distinction is a mere preliminary to the key argument.


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Dummett pointed out in Frege.1 But unlike Frege, Couturat did not move thoughts toa “third realm” of objective but non-causal entities. Couturat drew the conclusion thatreasoning cannot be studied empirically unless it has been externalised, for example in

the form of language, i.e. written or spoken utterances:In order to get an accurate picture of reason’s subliminal operations we haveto compare it to an invisible machine whose products only can be conceivedand whose structure can only be discovered in terms of the traces it producesand the forms it impresses upon them. (1896b, p. xi; cf. also 1906b, p. 321 etseqq.)

This point of view suprisingly resembles the approach of analytic philosophy in that thelatter, too, rejected Frege’s “third realm” and accounted for the objectivity of thoughtsby localising them in language, understood as an externalisation essentially shared by

a community of competent users. It is however a difficult question whether Couturatperformed a linguistic turn, i. e. attempted to analyse thought by means of the analysisof language, which would make him a kind of analytic philosopher avant la lettre (according to the definition in Dummett 1994, p. 25). The concrete significance of Couturat’s approach essentially depends on (1) what counts as a trace of the invisiblemind and (2) how the relation between the mind and its empirical traces is conceived.

As regards the first question, in 1896 it was nothing else than science in its mostrecent developments which Couturat considered as the mind’s proper work and henceas the subject to be studied. This conception is reflected in his thesis De l’infini. Someyears later, at the time of the Traité , science is replaced by symbolic logic. This changewas justified, as we have seen, by the discovery that logic, in its algebraic form, can beconceived as a “calculus of quality” that complements the quantitative calculus of science.We can thus state a first result concerning the philosophical significance of algorithmiclogic for Couturat: Algorithmic logic was indeed a promising candidate for studying theobjective structure of the mind. This highlights the philosophical importance that thestudy of algorithmic logic had for Couturat when he wrote the Traité . However, thisapproach does not fit the idea of a linguistic turn. Remember that at that time Couturatconceived logic as opposed to language. Couturat considered natural language rather asan obstacle to precise reasoning, and it is the aim of his analysis to show the advantages

of the calculus as compared to ordinary language. In his inaugural lecture at Collège de France he still holds that logic does not arise from the study of languages, but on thecontrary permits one to judge and criticise the different languages, which are nothingbut coarse and imperfect instruments of the mind (1906b, p. 328). It is in 1911/12,i. e. after a considerable gap in Couturat’s philosophical bibliography, that we find himto be much more liberal on this point, probably as a result of his linguistic studies,which came along with his increasing engagement for Ido.2 Couturat now emphasisesthat language is among the manifestations of thought ( pensée ) the most universal and

1Dummett 1994, p. 22 et seqq.; Couturat, by the way, studied Frege’s works not before 1903, cf. hisletter to Frege of February 11, 1904, in Gabriel et al., 1976, p. 25.

2Couturat mentions the lectures the linguist Antoine Meillet (1866–1936) held at the Collège de France in1911. Cf. Dassen 1939, p. 176.


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– for all that – the most adequate. He expresses the conviction that the mind ( esprit )indeed impresses its pattern on language by continuously reshaping it. Linguistic formstherefore mirror patterns of thought. They are part of a “universal grammar” which has

been partially detected, and partially is still to be detected, in linguistic (i. e. empirical)research (1912b, p. 1-2).1 Logic is thereby nevertheless still considered as a regulativeelement: Logic defines the “ideal language”, helps to identify the essential elements of language, and to eliminate interfering factors like the impact of tradition.2 NonethelessCouturat finally professes that “theoretical logic” and “logic inherent to our languages”are basically the same (1911a, p. 510-6). Here, finally, language takes over the placethat science and logic had occupied before. Thus, although Couturat never explicitlyexpressed the conviction that “philosophical problems are problems which may besolved (or dissolved) either by reforming language, or by understanding more about thelanguage we presently use” (Rorty 1997, p. 3), we indeed find in his late writings the idea

of an analysis of thought by means of an analysis of language, i. e. a kind of linguisticturn.3 This idea is in addition based on anti-psychologism in so far as the latter demandsthat thought be studied not by means of introspection, but by means of an analysis of itsexternalisations. This linguistic turn nevertheless does not rest on a principle somehowequivalent to Frege’s Context Principle, but rather on the (empirical) assumption of a “universal grammar”. But this is the later Couturat, where logic has been reducedto a kind of regulative element; around 1900, on the contrary, algorithmic logic wasstill considered as the sole way to grasp the mind, whereas language was associated withambiguity and error.

Before turning to a more detailed analysis of the relation of calculus and ordinarylanguage, I will shortly discuss the second question, i.e. the question how to conceivethe relation between the mind and its traces that are the proper subject of philosophicalinvestigation. In the introduction of L’Algèbre de la Logique , Couturat wrote:

It belongs to the realm of philosophy to decide whether, and in what measure,this calculus corresponds to the actual operations of the mind, and is adaptedto translate or even to replace argument; we cannot discuss this point here.(1905a, § 1)

Couturat himself is hence not explicit on this point. There is certainly a lot that

1Giuculescu 1983, p. 122, pointed out the resemblance to the the idea of a general grammar in NoamChomsky. But since Chomsky himself (1968, p.12 et seqq.) dates back the origin of this idea to the Port-Royal Grammar the fact that it appears in Couturat should not be overemphasised. As Savatovsky 1992, p. 99,suggests, the idea of universal grammar was at that time even rather outdated.

2As regards the relation of psychology and logic in Couturat as well as the rôle of logic in the creationof an international auxiliary language cf. Ziche 2009. As Lalande reported (1913, p. 373), Couturat’s 1912article provoked an animated discussion in the Société de philosophie in which notably Meillet, Vendryes andLévy-Bruhl took part.

3This story does not lack irony: It is known that Russell was disappointed by the fact that Couturatdevoted himself from 1900 on increasingly to the propagation of the international auxiliary language Ido. Itwas however this engagement for Ido that led Couturat via his linguistic studies to a point of view quite closeto what Russell himself had written in his Principles of Mathematics: “The study of grammar, in my opinion,is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. . . and in what follows, grammar, though not our master, will yet be taken as our guide.” (1903, p.42).


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can be inferred from his works. Instead of tackling a historical analysis I will restrictmyself to sketching the possible answers in order to appreciate the systematic interestof Couturat’s approach. A plausible model for the study of the mind-logic-connection

would be mechanics in so far as it, too, studies causes that are known solely by theireffects (like the moving force of a body and gravitation). But one notes that thisapproach does not guarantee the scientific character of the investigation (as is showne. g. by the “entelechia” introduced by Couturat’s contemporary Driesch, explicitly inreference to Newton’s gravitation). In this approach, the mind risks being nothing morethan a qualitas occulta incorporating human beings’ faculty of reason. In addition, thisapproach links the mind more closely to its traces than Couturat’s anti-psychologismprobably permits. To remove the mind too far from its traces risks on the other handturning Couturat’s whole project into a kind of physico-theology, i.e. the attempt todemonstrate the existence and properties of God from his works, simply replacing

“God” by “mind”. Couturat’s intention unquestionably has to be located somewherein between these two approaches. In order to avoid the problems that accompany bothof them I propose a third reading that surely does not fit Couturat’s own intentions,but instead points out the promising resources in his philosophy. The point is: If algorithmic logic is to be understood as the work of the mind, and if in addition it isimpossible to compare the material manifestations of the mind to the mind itself, andif thirdly it is an essential feature of the logical algorithm, as we will see, to replacethe mind, it is quite plausible to say that the mind can only be grasped by taking thelogical algorithm as its model . When Leibniz declared “omne opus mentis nostrae essecomputationem”, we now can interpret this as taking computation as a model of thereasoning process and thus conceiving the latter as computation rather than, e. g., as“inner speech”, that is, reasoning modeled by language. This approach seems at firstsight to come along with a loss, i. e. the loss of an explanation of language and logic interms of reasoning. But as we have already remarked, the ability to reason looks quitesimilar to a qualitas occulta. What counts is thus what we gain within this framework.And this gain is that our approach suggests a genetic theory of the mind that accountsfor the gnoseological primacy of algorithmic logic: It is viewed from the perspectiveof the primacy of the effect vis-à-vis the cause (the working mind), of the materialrepresentation vis-à-vis the entity represented (the laws of the mind), and of the means

vis-à-vis the end (sound reasoning). This approach resembles Sellars’ idea that “thesemantical characterisation of overt verbal episodes is the primary use of semanticalterms, and that overt linguistic events as semantically characterised are the model for theinner episodes [i. e. for thoughts] introduced by the theory” (1956, § 58, p. 319). In ourcase we are concerned not with single thoughts, but with the process of reasoning, andthe model is not provided by language, but by the logical algorithm. The real difficultynow consists in this, that we need a model that really does what the mind is intended to do, i.e. reasoning. This brings us back to Couturat’s main point, the analysis of the logical algorithm as compared to language. Couturat’s analysis indeed yields whatwe are looking for and hence is even reevaluated by putting it into a framework that

admittedly does not correspond to Couturat’s original intentions, given his rejection of “sociologism”.


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I thus complete the preceding considerations and turn now to a closer investiga-tion of how Couturat conceived the supremacy of the calculus as compared to naturallanguage in the epoch of the Traité . He does so in two steps: Firstly, his anti-psychologist

conception has the remarkable consequence of equalising the different forms of reason-ing, – spoken, diagrammatical, algebraical, and even mechanical – as different exter-nalisations of reasoning which are situated within a continuum.1 The peculiarity of Couturat’s view becomes evident as opposed to Husserl’s, who distinguished categor-ically between language as a “method of symbolico-systematical utterance of mentalphenomena”, and calculus as a “method of symbolico-systematical reasoning” (1891b,p. 258). Accordingly language can be a calculus, but a mere calculus could never serveas a language. In Couturat on the contrary, language is no longer the proper mediumof thought, replaced by symbolic logic as an artificial representation. Both are properor artificial to the same extent. This permits Couturat on the one hand to legitimise

the calculus with respect to language, and on the other hand to discuss its advantagescompared to language. This is what the second step consists in. Both language and thelogical calculus are symbolic representations. The answer to the question in what thevalue of the latter consists, is taken from, or at least inspired by, Leibniz’ characteristicauniversalis, i. e. the attempt to extend mathematics to metaphysical subjects.2 As regardsnumbers and quantities, the work is already done by ordinary algebra. As regardsgeometry, in so far as it treats of spatial quantities with both direction and magnitude,the application of algebra had been achieved only by the detour of analytic geometry,where the algebraic symbols stand for numbers which themselves encode quantities.Leibniz sought a way to represent geometric entities by symbols without the interme-diary of number. But this was not accomplished until the work of Grassmann andHamilton. The last and most ambitious extension of the mathematical method consistsin establishing a calculus of metaphysical subjects. Here too Leibniz sought to establisha calculus based on symbols representing ideas. Thereby these symbols should not onlyprovide a “stenography” or a “tachygraphy”, i. e. a abridged notation which facilitatesreasoning, but a “logical pasigraphy” or “ideography” which permits one to establisha calculus which replaces reasoning (1901a, p. 89 and p. 101). He experimented, forexample, with methods inspired by integer factorisation. Denoting the primitive ideasby prime numbers should lead to a unique numerical representation of complex ideas:

human (6) = rational (2) × animal (3). Although none of his efforts were successful,they are among the few constructive ones in an epoch which has seen mainly precipi-tate ad-hoc-applications of mathematical methods to philosophical subjects and surveysof those approaches which, while sometimes of considerable measurement-theoreticinterest, are purely critical in purpose.3

1Logic machines, as Couturat himself mentions them in L’Algèbre de la Logique , § 49, and in La Logique de Leibniz, p. 116 note 2, are well situated within this continuum too: On the one hand early logic machineswere nothing else than logical diagrams manipulable in a reversible way (cf. Gardner 1958, Leibniz p. 113note 2); on the other hand there is, as regards the purely formal nature of the operations, no real differencebetween setting into operation a machine’s clockwork by pulling a lever and applying mechanical rules e. g.of adding or multiplying numbers in decimal notation, as we will explain in detail.

2Compare also the description of the characteristica given by Peckhaus 1996, p. 31.3As an example we mention Thomas Reid’s Essay on Quantity (1748), a critique of Hutcheson’s quanti-


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20 Introduction

Couturat’s project was of course much less ambitious than Leibniz’. He didnot seek to establish a metaphysical calculus, he was simply looking for a conceptualframework that would permit him to understand algorithmic logic. The analysis of

successfully working algorithms should yield important heuristics. The basic arithmeticoperations of addition, substraction, multiplication, and division provide a simplebut instructive example. Couturat knew it from Cournot’s careful study De l’origine et des limites de la correspondance entre l’algèbre et la géométrie (1847).1 As Cournotpointed out, these operations depend on the decimal notation for numbers. It is afeature of this notational system that it permits the application of simple transformationrules which correspond to the basic arithmetic operations of addition, substraction,multiplication, and division. These rules are applied in a purely mechanical way andpermit computations with numbers of any size which surpass one’s ability to performthem mentally. This is an advantage of the decimal notation compared to roman

numerals.2 In establishing a metaphysical calculus, the notational system thus plays acrucial rôle. The symbols are characterised by the feature of not merely representingobjects (as hieroglyphs, chemical symbols or roman numerals do, too) but also providingreasoning (1901a, p. 81). Couturat speaks of “manageable” or “expressive” symbols(1901a, p. 87, and C 11.02.1904, II/351). In short one can say that algebraic symbolsare characterised by an operative value , using a notion borrowed from Krämer (1991,p. 88 et seqq). By operative value I mean that the symbols are constructed in such a waythat the application of transformation-rules results in signs which symbolise, accordingto the very same construction rules, the result of the corresponding operations on thesymbolised objects. To establish such a system of operative symbols in all fields of intellectual interest – mathematics, geometry, and metaphysics – is the problem of thecharacteristica universalis. The ideas that Couturat found in Cournot indeed enabledhim in his interpretation of Leibniz’ characteristica universalis to go far beyond what,e. g., Trendelenburg had presented in 1856, and to highlight a point that even Cassirermissed two years later in his study on Leibniz (1902, pp. 135-138). The story howeverdoes not end with the notational system. Once the desired ideographical symbols aregiven for the domain in question, their combination and manipulation is governedby a set of rules – as Couturat emphasises repeatedly. These rules form the calculusratiocinator , which serves to draw in a mechanical way all conclusions from given

tative approach to moral philosophy. Cf. also the introduction to the German translation, Schlaudt 2009a,pp. 21-4. As regards practical and moral questions, it can be stated that the theory of probability came closerto a formal theory of rational decision than any other approach (cf. Daston 1995). Understood as the calculusof variable judgments, the theory of probability becomes a part of algorithmic logic and hence can also befound in Couturat’s Traité (ch.X).

1Couturat 1901a, p. 106, and Cournot 1847p. 3 et seqq. Couturat added a second example, the Leibnizeansymbolism for the infinitesimal calculus (1901a, p.84-5, 1906a, p. 213) which illustrates Cournot’s point too,but in a less manifest way.

2Detlefsen et al. 1976 have shown that for roman numerals, too, algorithmic procedures for arithmeticaloperations can be established. This demands a reevaluation of roman numerals, but does not affect the mainpoint, that algorithmic procedures essentially depend on appropriate notational systems. – A. von Humboldtin his study on systems of numerals offers a quite natural explanation of how the decimal notation wasestablished. According to him (1829, p.217), the early notational systems arose themselves from the use of material calculation devices. It is hence not surprising that they in turn support algorithms.


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premisses.1 In this regard it is of special interest that Couturat in the Traité used solelythe word “rule” whereas he prefered later, in L’Algèbre de Logique (1905), Principes desmathématiques (1905) and Prinzipien der Logik (1912), where the idea of the calculus

became less prominent, to speak of “principles”. Obviously in our Traité “rule” is notmerely a different name for principles or axioms, but has really to be understood as aninstruction about how to manipulate material symbols. All these aspects are subsumedin Couturat’s notion of calculus, which I provide here since it is missing in the moretechnical Traité :

A calculus is the art of manipulating and combining certain substitutionalsigns according to a set of rules in such a manner that the result of the op-erations, having been interpreted, expresses a proposition about the objectsdesignated. (1900d, p. 324)

I recapitulate the important consequences which follow from this conception of thecalculus. First we state that the calculus applies to signs tout court (i.e. not to ideasthrough the medium of signs, but to the signs themselves). The subject matter of algorithmic logic is hence not concepts, but letters (1901a, p. 284). Couturat furthermoreemphasises the physical or material nature of the calculus (1900d, p. 327). From thisfollows, that the calculus replaces reasoning.2 Alternatively we can say that it is thecalculus itself that reasons. This may seem astonishing, but in fact in the mechanicalexecution of the rules of the arithmetic algorithm, too, arithmetic operations actuallyare carried out though nobody intentionally calculates. I stressed this point in themodel theory of the mind. These points taken altogether justify attributing to Couturatan operative notion of logic, “operative” being understood as the claim that logicconsists, basically, in manipulating material signs according to rules (Lorenzen 1955,p. 4). One should however not overemphasise this point, since it is not certain whetherthe operative aspect really has its roots in Couturat’s philosophical convictions or if itrather resulted from an accentuation of the operative nature of algebra, which can beadmitted even by someone who has no operative notion of logic and mathematics atall. The peculiarity of Couturat’s concept of algorithmic logic indeed does not lie in hisnotion of the calculus which was not uncommon at his time3, but rather in his analysisof the operative characteristic of the symbols the calculus is based on.4

1As regards the characteristica as the basis for the calculus ratiocinator , cf. 1901a, p. 96. In so far as thecalculus is based on the characteristica, Frege’s opposition of both in his critique of Boole is misconceived( Booles rechnende Logik und die Begriffsschrift , published posthumously in Frege 1969, pp. 9-52; cf. in particularp. 13). Cf. the clarifying discussion in Peckhaus 2004, pp. 7-8. I will discuss van Heijenoort’s conception of logic, which took up Frege’s critique of Boole, below, point 3, p. 26.

2Couturat 1899b, p. 617, letter to Peano, p. 6, 1901a, p. 55, p. 88 et seqq, and p. 101. The complementarythesis, that language too is a – admittedly imperfect – kind of reasoning machine that replaces the mind, hasbeen explored by Dugas 1896, esp. ch. IV, § 1.

3Cf. for example Whitehead 1898, p. 4-5, and Husserl 1891b, p. 246 and p. 259, but of course also thequotations from MacColl and Johnson given above. Lorenz 1971, p. 47, refers to the following definitionsfrom Leibniz: “nihil aliud enim est calculus quam operatio per characteres” and “Calculus vel operatio consistit inrelationum productione facta per transmutationes formularum, secundum leges quasdam praescriptas facto”. For

more historical details cf. also Lorenz 1976.4A comparison to Husserl’s critique of Schröder elucidates this point. In his review of Schröder’s


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22 Introduction

There is a last point which yet needs to be clarified. I emphasised that, on the onehand, the calculus applies to mere letters independent of the interpretation given to them.On the other hand, in the definition of the calculus quoted above, Couturat refers to the

interpretation of the calculus. And elsewhere he explains that in algorithmic logic the“intuitive rules of manipulation of signs” correspond to the abstract laws of logic (1901a,p. 89). One should note that Couturat, though he defines the calculus as a mere play withsymbols1, is far from being a nominalist. Quite the contrary, militant anti-nominalisticattacks are a common characteristic of his work. By “nominalism” Couturat obviouslydid not understand the denial of the existence of abstract objects. He used the termrather in the sense of conventionalism which tends to degrade rational justification tomere utilitarian considerations.2 Couturat’s attack consequently consists in showing thatthe calculus is not merely conventional. The main argument is provided by his theoryof representation in so far as it enabled him to refer to the correspondence between

the manipulations of symbols and manipulations of things represented symbolically(1900d, p. 325, 1901a, p. 89, and still in 1905b, p. 276-7). This is the raison d’être of thecalculus. It is thus a consequence of his “rational” point of view that Couturat alwaysrefers to an interpreted calculus (see the definition quoted above) though he regards as itsmain feature the abstraction from all content. This kind of “anti-nominalism” of coursehas nothing to do with realism with respect to universals. On the contrary, it permitsCouturat to maintain a critical attitude towards premature reifications of ideas – anattitude present in the Traité too (ch.XII). – I have elaborated on Couturat’s distinctionbetween calculus and language. The remaining points can now however be explainedquite easily.

2. Analytical vs. synthetical reasoning . There is a straightforward consequenceof the mechanical aspect, which is of considerable philosophical importance: In so faras the process of reasoning can be replaced or performed by a machine, it cannot reston intuition, at least not in a somehow constitutive manner (1906a, p. 214). Couturatreserves for this characteristic the notion of analyticity (cf. C 18.12.1903, II/456).

Algebra der Logik, Husserl emphasises the difference between symbolic language ( lingua characteristica) and thealgebraic procedure ( calculus rationcinator ). He asserts that the latter is not necessarily given with the former(1891b, p. 258). But this is only true if the symbols are considered as mere and arbitrary representationswithout further analysis of their features. Couturat, who provides such an analysis, thus is obviously notaffected by this critique.

11901a, p. 96: «jeu d’écriture», and p. 101: «jeu de symboles».2This reading is suggested by Loi 1976. Couturat indeed labeled the position of his former teacher

Poincaré as nominalism (1898a, p. 361; cf. also Sanzo 1975, p. 405 and p.414-5). One may be surprisedthat nonetheless both Couturat and Poincaré rejected Edouard Le Roy’s (1870–1954) position as nominalist(Poincaré 1902, General Conclusions of part III; Couturat 1900b). The difference consists in this: Poincarérejected the claim that science is entirely conventional, whereas he still argued for the weaker thesis that sciencecontains, or partially rests on, irreducible conventional elements. Couturat did not even accept this weakerclaim. What he had in mind obviously was a rational justification of each single term of a scientific theory,whereas the existence of conventional elements allows at best a justification of larger parts of theories, if notonly of whole theories. That both Couturat and Poincaré share anti-nominalism has been well remarked inLalande 1928, II/518. Couturat there added: «Il est vrai que le point de départ du nominalisme scientifiquen’est pas l’ancien nominalisme des logiciens; mais il reste un caractère commun entre les deux doctrines,qui justifie la communauté d’appellation: refuser toute valeur objective à nos concepts, et par suite aux loisscientifiques.» ( op.cit., p. 517-8).


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Introduction 23

In the Traité , Couturat is not unambiguous on that point: In establishing the fulldistributive law he indeed refers to intuition (ch. IV.2), whereas he completely rejects itfrom abstract theory on the occasion of discussing geometrical illustrations (ch. II.31).

Finally however he indeed draws the conclusion that reasoning is purely analytical (ch.XII). This immediately leads to the question whether mathematics rests on intuitionas it was raised in the famous controversy with Henri Poincaré in 1905 and 1906.1 Toanswer this question demands clarifying the relations between logic and mathematics.In his article La philosophie des mathématiques de Kant , annexed to the Principes desmathématiques de 1905, Couturat, arguing for logicism, i. e. the claim that mathematicsrests entirely on logic, and identifying this with the claim that mathematics is purelyanalytic, rejected the synthetic character of mathematics.2 In the Traité he still argues thecontrary: Here Couturat argues for the synthetical character of mathematics, perhapseven more explicitly than in his published works such as his article on the definition of

number (1900a). He emphasises that the very method of mathematics, deduction, mustbe synthetic in character, since it leads to more general theorems, whereas logic alwaysdescends from the general to the particular. Here, by the way, Couturat is not far fromPoincaré (1902, ch. I). If the very method of deduction is regarded as synthetic, it is inconsequence not even sufficient to admit that the mathematical theorems follow in ananalytic manner from synthetic principles and axioms. This latter way was chosen laterby Cassirer in order to reconcile Kant with logicism.3 Couturat nota bene distinguishedin the Traité between logical and mathematical reasoning. Bowne has already remarkedin her 1966 study that this distinction still seemed natural to both Russell and Couturatin 1896, whereas Peano already had abandoned it (Bowne 1966, p. 46). All in all one willstill find a rather Kantian idea of mathematics in the Traité ; but nevertheless Couturatabsolutely advocates the analyticity of reasoning as a consequence of its algebraisation.

3. Algebra of logic vs. logic of algebra. Couturat distinguished between two maincurrents of modern logic, the system Boole-Schröder on the one hand and the systemPeano-Russell on the other hand. He later adopted – referring to a bon mot of GregorItelson – the labellings “algebra of logic” and “logic of algebra”, the first being a logicin algebraic form and hence an application of mathematical methods, the latter on thecontrary being logic as concerned with the foundations of mathematics.4 Initially, i. e. in

1Cf. Poincaré’s articles, reprinted in Heinzmann 1986, Couturat 1906a, and also Heinzmann 1985, p. 16et seqq. as well as Sanzo 1975.

2Cf. C 16.11.1903, II/332. Russell did not agree, neither with this identific

traité de logique algoritmique

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