Overlooking mathematical justifications in the Sanskrit tradition: the nuanced case of G. F. Thibaut Agathe Keller To cite this version: Agathe Keller. Overlooking mathematical justifications in the Sanskrit tradition: the nuanced case of G. F. Thibaut. History of Mathematical Proof in Ancient Traditions, Cambridge University Press, pp.260-273, 2012. ￿halshs-00150736v2￿ HAL Id: halshs-00150736 https://shs.hal.science/halshs-00150736v2 Submitted on 21 Jan 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - ShareAlike| 4.0 International License Overlooking mathematical justifications in the Sanskrit tradition: the nuanced case of G. F. Thibaut Agathe Keller Abstract How did the narratives of the history of Indian mathematics ex- plain the tradition of mathematical justifications that existed in me- dievalSanskritcommentaries? WhentheGermanphilologerG.F.Thibaut published a translation of a set of Vedic geometrical texts in 1874 and 1875, he established that India had known other mathematical ac- tivities than ‘practical calculations’. Thibaut’s philological work and historiographical values determined his approach to these texts and provided a bias for understanding the reasonings and efforts which establish the validity of algorithms in this set of texts. Introduction 1 Untilthe1990s, thehistoriographyofIndianmathematicslargelyheldthat Indians did not use “proofs”2 in their mathematical texts. Dhruv Raina has shown that this interpretation arose partly from the fact that during the sec- ond half of the nineteenth century, the French mathematicians who analyzed Indian astronomical and mathematical texts considered geometry to be the 1I would like to thank K. Chemla and M. Ross for their close reading of this article. They have considerably helped in improving it. 2Srinivas 1990, Hayashi 1995. 1 measure of mathematical activity3. The French mathematicians relied on the work of the English philologers of the previous generation, who considered the computational reasonings and algorithmic verifications merely ‘practical’ and devoid of the rigor and prestige of a real logical and geometrical demon- stration. Against this historiographical backdrop, the German philologer Georg Friedrich Wilhem Thibaut (1848-1914) published the oldest known mathematical texts in Sanskrit, which are devoted only to geometry. These texts, ´sulbasu¯tras4 (sometimes called the sulvasu¯tras) contain trea- ¯ tisesbydifferentauthors(Baudha¯yana,Apastamba,Ka¯tya¯yanaandMa¯nava) and consider the geometry of the Vedic altar. These texts were written in the style typical of aphoristic su¯tras between 600 and 200 BCE. They were sometimes accompanied by later commentaries, the earliest of which may be assigned to roughly the thirteenth century. In order to understand the methods which he openly employed for this corpus of texts, Thibaut must be situated as a scholar. This analysis will focus on Thibaut’s historiography of mathematics, especially on his perception of mathematical justifications. 1 Thibaut’s intellectual background G. F. Thibaut’s approach to the ´sulbasu¯tras combines what half a century before him had been two conflicting traditions. As described by D. Raina and F. Charette, Thibaut was equal parts acute philologer and scientist in- vestigating the history of mathematics. 3See Raina 1999: chapter VI. 4We will adopt the usual transliteration of Sanskrit words which will be marked in italics, except for the word Veda, which belongs also to English dictionaries. 2 1.1 A philologer Thibaut trained according to the German model of a Sanskritist5. Born in 1848 in Heidelberg, he studied Indology in Germany. His European career culminated when he left for England in 1870 to work as an assistant for Max Mu¨ller’s edition of the Vedas. In 1875, he became Sanskrit professor at Benares Sanskrit College. At this time, he produced his edition and stud- ies of the ´sulbasu¯tras, the focus of the present article6. Afterwards, Thibaut spent the following 20 years in India, teaching Sanskrit, publishing trans- lations and editing numerous texts. With P. Griffith, he was responsible for the Benares Sanskrit Series, from 1880 onwards. As a specialist in the study of the ritualistic mim¯am. sa school of philosophy and Sanskrit scholarly grammar, Thibaut made regular incursions in the history of mathematics and astronomy. Thibaut’s interest in mathematics and astronomy in part derives from his interest in mim¯am. sa. The authors of this school commented upon the ancillary parts of the Vedas (ved¯an˙ga) devoted to ritual. The ´sulbasu¯tras can be found in this auxiliary literature on the Vedas. As a result of hav- ing studied these texts, between 1875 and 18787, Thibaut published several articles on vedic mathematics and astronomy. These studies sparked his cu- riosityaboutthelatertraditionsofastronomyandmathematicsintheIndian subcontinent and the first volume of the Benares Sanskrit Series, of which Thibaut was the general scientific editor, was the Siddh¯antatattvaviveka of Bhat.t.a Kamala¯kara. This astronomical treatise written in the seventeenth 5The following paragraph rests mainly on Stachen-Rose 1990. 6See Thibaut 1874, Thibaut 1875, Thibaut 1877. 7The last being a study of the jyoti.saved¯an˙ga, in Thibaut 1878. 3 century in Benares attempts to synthesize the re-workings of theoretical as- tronomy made by the astronomers under the patronage of Ulug Begh with the traditional Hindu siddh¯antas8. Thibaut’s next direct contribution to the history of mathematics and as- tronomy in India was a study on the medieval astronomical treatise, the Pan˜casiddh¯anta of Var¯ahamihira. In 1888, he also edited and translated this treatise with S. Dvivedi and consequently entered in a heated debate with H. Jacobi on the latter’s attempt to date the Veda on the basis of descriptions of heavenly bodies in ancient texts. At the end of his life, Thibaut published several syntheses of ancient Indian mathematics and astronomy9. His main oeuvre, was not in the field of history of science but a three volume transla- ´ tion of one of the main mim¯am. sa texts: San˙kar¯ac¯arya’s commentary on the Ved¯antasu¯tras, published in the Sacred Books of the East, the series initiated by his teacher Max Mu¨ller 10. Thibaut died in Berlin at the beginning of the first world war, in October 1914. Amongthe´sulbasu¯tras,ThibautfocussedonBaudha¯yana(ca. 600BCE)11 ¯ and Apastamba’s texts, occasionally examining Ka¯ty¯ayana’s ´sulbapari´si.s.ta. Thibaut noted the existence of the M¯anavasulbasu¯tra but seems not to have had access to it12. For his discussion of the text, Thibaut used Dv¯araka¯na¯tha 8See Minkowski 2001 and CEES, Vol 2: 21. 9Thibaut 1899, Thibaut 1907. 10Thibaut 1904. 11Unlessstatedotherwise,alldatesrefertotheCESS.Whennodateisgiven,theCESS likewise gives no date. 12For general comments on these texts, see Bag & Sen 1983, CESS, Vol 1: 50; Vol 2: 30; Vol 4: 252. For the portions of Dv¯arak¯an¯atha’s and Venkate´svara’s commentaries on Baudh¯ayana’s treatise, see Delire 2002 (in French). 4 Yajvan’scommentary13 ontheBaudha¯yanasulbasu¯traandRa¯ma’s(fl. 1447/1449) commentary on Ka¯tya¯yana’s text. Thibaut also occasionally quotes Kapar- disva¯min’s (fl. before 1250) commentary of A¯pastamba14. Thibaut’s intro- ductory study of these texts shows that he was familiar with the extant philological and historical literature on the subject of Indian mathematics and astronomy. However, Thibaut does not refer directly to any other schol- ars. The only work he acknowledges directly is A. C. Burnell’s catalogue of manuscripts15. For instance, Thibaut quotes Colebrooke’s translation of L¯ıl¯avat¯ı16 but does not refer to the work explicitly. Thibaut also reveals some general reading on the the history of mathematics. For example, he implic- itly refers to a large history of attempts to square the circle, but Thibaut’s sources are unknown. Hisapproachtothetextsshowstheimportanceheascribedtoacutephilo- logical studies17. Thibaut often emphasizes how important commentaries are for reading the treatises18: the su¯tra-s themselves are of an enigmatical shortness (...) but the commentaries leave no doubt about the real meaning The importance of the commentary is also underlined in his introduction of the Pan˜casiddh¯anta19: 13Thibaut 1875: 3. 14Thibaut 1877: 75. 15Thibaut 1875: 3. 16Thibaut 1875: 61. 17See for instance Thibaut 1874: 75-76 and his long discussions on the translations of vr.ddha. 18op. cit. : 18. 19Thibaut 1888: v. 5 Commentaries can be hardly done without in the case of any Sanskrit astronomical work... However, Thibaut also remarks that because they were composed much later than the treatises, such commentaries should be taken with critical distance20: Trustworthy guides as they are in the greater number of cases, their tendency of sacrificing geometrical constructions to numer- ical calculation, their excessive fondness, as it might be styled, of doing sums renders them sometimes entirely misleading. Indeed, Thibaut illustrated some of the commentaries’ ‘mis-readings’ and de- voted an entire paragraph of his 1875 article to this topic. Thibaut explained that he had focussed on commentaries to read the treatises but disregarded what was evidently their own input into the texts. Thibaut’s method of openly discarding the specific mathematical contents of commentaries is cru- cial here. Indeed, according to the best evidence, the tradition of ‘discussions onthevalidityofprocedures’21 appearinonlythemedievalandmoderncom- mentaries. True, the commentaries described mathematics of a period differ- ent than the texts upon which they commented. However, Thibaut valued his own reconstructions of the ´sulbasu¯tras proofs more than the ones given by commentaries. The quote given above shows how Thibaut implicitly values geometrical reasoning over arithmetical arguments, a fact to which we will return later. 20Thibaut 1875: 61-62. 21Thesearediscussed,inaspecificcase,intheotherarticleinthisvolumeIhavewritten, Keller same volume. 6 It is also possible that the omission of mathematical justifications from the narrativeofthehistoryofmathematicsinIndiaconcernsnotonlytheconcep- tion of what counts as proof but also concerns the conception of what counts as a mathematical text. For Thibaut, the only real mathematical text was the treatise, and consequently commentaries were read for clarification but not considered for the mathematics they put forward. In contradiction to what has been underlined here, the same 1875 article sometimesincludedcommentator’sprocedures, preciselybecausethemethod they give is ‘purely geometrical and perfectly satisfactory’22. Thus there was a discrepancy in between Thibaut’s statements concerning his methodology and his philological practice. Thibaut’s conception of the Sanskrit scholarly tradition and texts is also contradictory. Healternatesbetweenavisionofahomogenousanda-historical Indian society and culture and the subtleties demanded by the philological study of Sanskrit texts. In 1884, as Principal of Benares Sanskrit College (a position to which he had been appointed in 1879), Thibaut entered a heated debate with Bapu Pramadadas Mitra, one of the Sanskrit tutors of the college, on the ques- tion of the methodology of scholarly Sanskrit pandits. Always respectful to the pandits who helped him in his work, Thibaut always mentioned their contributions in his publications. Nonetheless, Thibaut openly advocated a ‘Europeanization’ of Sanskrit Studies in Benares and sparked a controversy about the need for Pandits to learn English and history of linguistics and 22Thisconcludesadescriptionofhowtotransformasquareintoarectangleasdescribed by Dv¯arakan˙tha in Thibaut 1875: 27-28. 7 literature. Thibaut despaired of an absence of historical perspective in Pan- dits reasonings–an absence which led them often to be too reverent towards the past23. Indeed, he often criticized commentators for reading their own methodsandpracticesintothetext, regardlessofthetreatises’originalinten- tions. His concern for history then ought to have led him lead to consider the different mathematical and astronomical texts as evidence of an evolution. However, although he was a promoter of history, this did not prevent him from making his own sweeping generalizations on all the texts of the Hindu tradition in astronomy and mathematics. He writes in the introduction of the Pan˜casiddh¯anta24: (...) these works [astronomical treatises by Brahmagupta and Bha¯skara¯carya]25 claim for themselves direct or derived infalli- bility, propound their doctrines in a calmly dogmatic tone, and either pay no attention whatever to views diverging from their own or else refer to such only occasionally, and mostly in the tone of contemptuous depreciation. Throughhisbeliefinacontemptuousarroganceonthepartofthewriters, Thibautimplicitlydeniesthetreatisesanyclaimforreasonablemathematical justifications, as we will see later. Thibaut attributed part of the clumsiness which he criticized to their old age26: 23See Dalmia 1996: 328 sqq. 24Thibaut 1888: vii. I am setting aside here the fact that he argues in this introduction for a Greek origin of Indian astronomy. 25[] indicate the author’s addenda for the sake of clarity. 26Thibaut 1875: 60. 8 Besides the quaint and clumsy terminology often employed for the expression of very simple operations (...) is another proof for the high antiquity of these rules of the cord, and separates them by a wide gulf from the products of later Indian science with their abstract and refined terms. After claiming that the treatises had a dogmatic nature, Thibaut extends this to the whole of “Hindu literature”27: The astronomical writers (...) therein only exemplify a general mental tendency which displays itself in almost every department of Hindu Literature; but mere dogmatic assertion appears more than ordinarily misplaced in an exact science like astronomy... Thibaut does not seem to struggle with definitions of science, mathemat- ics or astronomy, nor does he does discuss his competency as a philologer in undertaking such a study. In fact, Thibaut clearly states that subtle philol- ogy is not required for mathematical texts. He thus writes at the beginning of the Pan˜casiddh¯anta28: ...texts of purely mathematical or astronomical contents may, withoutgreatdisadvantages, besubmittedtoamuchrougherand bolder treatment than texts of other kinds. What interests us in these works, is almost exclusively their matter, not either their general style or the particular words employed, and the peculiar 27Thibaut 1888: vii. 28Thibaut 1888: v. 9