{
"type": "Article",
"authors": [
{
"type": "Person",
"familyNames": [
"Chemla"
],
"givenNames": [
"Karine"
]
}
],
"description": "This article presents some of the theoretical issues that interest me in the\nhistory of mathematics. Each of them has its origin in the work I have done on\nmathematical sources in Chinese. However, they all have ramifications in other\nbodies of mathematical literature, and I have pursued them beyond Chinese\nsources.",
"identifiers": [],
"references": [
{
"type": "Article",
"id": "bib-bib1",
"authors": [],
"title": "\nI. G. Bašmakova, Diophant und diophantische Gleichungen.\nTranslated from the Russian 1972 original by L. Boll,\nBirkhäuser, Basel-Stuttgart (1974)\n"
},
{
"type": "Article",
"id": "bib-bib2",
"authors": [],
"title": "\nK. Chemla, Equations with general coefficients in the Ce Yuan Hai Jing.\nPublications de l’Institut de Recherche Mathématique de\nRennes, Fascicule II: Science, Histoire, Société, 23–30,\nwww.numdam.org/article/PSMIR_1985___2_23_0.pdf\n(1985)\n",
"url": "http://www.numdam.org/article/PSMIR_1985___2_23_0.pdf"
},
{
"type": "Article",
"id": "bib-bib3",
"authors": [],
"title": "\nK. Chemla, What is the content of this book? A plea for\ndeveloping history of science and history of text conjointly.\nPhilosophy and the History of Science: A Taiwanese Journal4, 1–46 (1995)\n[Republished in [5]]\n"
},
{
"type": "Article",
"id": "bib-bib4",
"authors": [],
"title": "\nK. Chemla, Euler’s Work in spherical trigonometry: Contributions and applications.\nIn Euler. Opera Omnia. Commentationes physicae ad theoriam caloris, electricitatis et\nmagnetismi pertinentes. Appendicem addidit Karine Chemla,\nedited by P. Radelet-de Grave and D. Speiser, CXXV–CLXXXVII, Birkhäuser, Basel (2004)\n"
},
{
"type": "Article",
"id": "bib-bib5",
"authors": [],
"title": "\nK. Chemla (ed.), History of science, history of text.\nBoston Studies in the Philosophy of Science 238, Springer, Dordrecht (2004)\n"
},
{
"type": "Article",
"id": "bib-bib6",
"authors": [],
"title": "\nK. Chemla, On mathematical problems as historically determined artifacts:\nreflections inspired by sources from ancient China.\nHistoria Math.36, 213–246 (2009)\n"
},
{
"type": "Article",
"id": "bib-bib7",
"authors": [],
"title": "\nK. Chemla, Une figure peut en cacher une autre. Reconstituer une\npratique des figures géométriques dans la Chine du XIIIe siècle.\nImages des mathématiques,\nimages.math.cnrs.fr/Une-figure-peut-en-cacher-une.html\n(2011)\n",
"url": "http://images.math.cnrs.fr/Une-figure-peut-en-cacher-une.html"
},
{
"type": "Article",
"id": "bib-bib8",
"authors": [],
"title": "\nK. Chemla (ed.), The history of mathematical proof in ancient\ntraditions. Cambridge University Press, Cambridge (2012)\n"
},
{
"type": "Article",
"id": "bib-bib9",
"authors": [],
"title": "\nK. Chemla, The Motley Practices of generality in various\nepistemological cultures, The Hans Rausing lecture 2017.\nSalvia Småskrifter, Uppsala,\nwww.idehist.uu.se/digitalAssets/775/c_775182-l_1-k_2019motley-practices-of-generality–final-versionrausinglecture2017originalcorrected.pdf\n(2019)\n",
"url": "https://www.idehist.uu.se/digitalAssets/775/c_775182-l_1-k_2019motley-practices-of-generality--final-versionrausinglecture2017originalcorrected.pdf"
},
{
"type": "Article",
"id": "bib-bib10",
"authors": [],
"title": "\nK. Chemla, From reading rules to reading algorithms.\nTextual anachronisms in the history of mathematics and their effects on interpretation.\nIn Anachronisms in the history of mathematics,\nedited by N. Guicciardini, Cambridge University Press, Cambridge (to appear)\n"
},
{
"type": "Article",
"id": "bib-bib11",
"authors": [],
"title": "\nK. Chemla, R. Chorlay and D. Rabouin (eds.),\nThe Oxford handbook of generality in mathematics and the\nsciences, Oxford Univ. Press, Oxford (2016)\n"
},
{
"type": "Article",
"id": "bib-bib12",
"authors": [],
"title": "\nK. Chemla and S. Guo, Les neuf chapitres. Dunod, Paris (2004)\n"
},
{
"type": "Article",
"id": "bib-bib13",
"authors": [],
"title": "\nK. Chemla and S. Pahaut,\nPréhistoires de la dualité: explorations algébriques en trigonométrie sphérique (1753–1825).\nIn Sciences à l’époque de la Révolution Française,\nedited by R. Rashed, Lib. Sci. Tech. Albert Blanchard, Paris, 151–201 (1988)\n"
},
{
"type": "Article",
"id": "bib-bib14",
"authors": [],
"title": "\nJ. D. Gergonne,\nConsidérations philosophiques sur les élémens de la science de l’étendue.\nAnn. Math. Pures Appl. [Ann. Gergonne]16, 209–231 (1825/26)\n"
},
{
"type": "Article",
"id": "bib-bib15",
"authors": [],
"title": "\nJ. Hudeček, Reviving ancient Chinese mathematics. Needham Research\nInstitute Studies, Routledge/Taylor & Francis Group, London (2014)\n"
},
{
"type": "Article",
"id": "bib-bib16",
"authors": [],
"title": "\nD. E. Knuth, Ancient Babylonian algorithms. Comm. ACM15,\n671–677 (1972)\n"
},
{
"type": "Article",
"id": "bib-bib17",
"authors": [],
"title": "\nR. Rashed, Diophante. Les Arithmétiques, Tome 3: Livre IV. Tome 4: Livre V–VII.\nLes Belles Lettres, Paris (1984)\n"
},
{
"type": "Article",
"id": "bib-bib18",
"authors": [],
"title": "\nA. Robadey, A work on the degree of generality revealed in the organization of\nenumerations: Poincaré’s classification of singular points of\ndifferential equations. In Texts, textual acts and the history of\nscience, edited by K. Chemla and J. Virbel, Springer, Dordrecht, 385–419 (2015)\n"
},
{
"type": "Article",
"id": "bib-bib19",
"authors": [],
"title": "\nW.-T. Wu, Recent studies of the history of Chinese mathematics. In\nProceedings of the International Congress of Mathematicians,\nVol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI,\n1657–1667 (1987)\n"
}
],
"title": "All roads come from China – For a theoretical approach to the history of mathematics",
"meta": {},
"content": [
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"id": "p1",
"content": [
"To an outside observer, I suppose I appear to be working on the history of\nmathematics in ancient and medieval China. To a certain extent, this is true.\nHowever, this is also partly wrong. By this (perhaps unexpected) statement, I\ndo not mean simply that I have also carried out research and published on the\nhistory of projective geometry and of duality more broadly, as well as on the\nhistory of medieval mathematics in Arabic, Greek, Hebrew and Sanskrit. I mean\nsomething deeper. Working on the history of mathematics in China is certainly\nmeaningful in and of itself. However, to my eyes, it becomes all the more\nmeaningful in that it confronts us with sources with which we are not used to\nthinking about mathematics, and these sources suggest interesting new issues,\nas well as new ways of addressing old issues. In other words, Chinese sources,\nlike in fact any mathematical document if treated appropriately, give us\nresources with which to nurture a theoretical approach to the history of\nmathematics. This, in the end, is my main goal. In what follows, I will\nillustrate how this has worked for me in practice, by discussing some of the\ntheoretical issues I have been led to address in the course of my research."
]
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"content": [
"1 History of science, history of text"
]
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"caption": [
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"Li Ye, ",
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"Measuring the Circle on the Sea-mirror"
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" To the right and to the left, resp., a polynomial (",
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") are written using a place-value notation."
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"content": [
"My first significant encounter with Chinese mathematical sources took\nplace in 1981, as I was studying in China at the Institute for the\nHistory of Natural Sciences (Chinese Academy of Sciences), and it\nconfronted me right away with striking phenomena, about which I still\nthink today."
]
},
{
"type": "Paragraph",
"id": "S1.p2",
"content": [
"Following a suggestion that had been made to me by Leuven sinologist\nUlrich Libbrecht, I started reading the book that Li Ye 李冶 (1192–1279)\nhad published in 1248 under the title ",
{
"type": "Emphasis",
"content": [
"Measuring the Circle on the\nSea-Mirror"
]
},
" (",
{
"type": "Emphasis",
"content": [
"Ceyuan haijing"
]
},
" 測圓海景, hereafter ",
{
"type": "Emphasis",
"content": [
"Measuring\nthe Circle"
]
},
"), which was to become the subject of my dissertation. For\nthis, I benefited from the guidance of the person in charge of\norganizing my study in Beijing at the time, Mei Rongzhao 梅榮照, who had\nalready worked on Li Ye’s book. I was also lucky to receive advice\nfrom the group of scholars who had been appointed to teach me during my\ntime in China, namely: Du Shiran 杜石然, Guo Shuchun 郭書春, He Shaogeng\n何紹庚 and Yan Dunjie 嚴敦傑."
]
},
{
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"id": "S1.p3",
"content": [
"Li Ye’s book opens with a diagram, to which the entire book is devoted (see\nFigure ",
{
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"2"
]
},
", and Chemla [",
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"] for an analysis). The diagram\nis followed by a set of about 700 formulas, stating relationships between its\nsegments, and then 170 problems, which basically all share the same structure.\nThey give two segments of the diagram and in general ask to determine the\ndiameter of the circle. The point of the problems is thus not the answer, since\nit is systematically the same, but rather the method. Li Ye begins with the\nchoice of an unknown (not always the diameter itself, but a magnitude that\ncould easily be related to it), to which he refers as “the celestial origin”.\nHe then brings into play polynomials, written using a place-value notation,\nalong with a geometrical reasoning that relies on the data and the unknown, in\norder to establish an algebraic equation, “the” root of which is the unknown\nsought (see Figure ",
{
"type": "Cite",
"target": "S1-F1",
"content": [
"1"
]
},
"). Indeed, at the time, in China, equations\nwere considered as having a single root. In brief, this was how the book had\nbeen understood up to then: it was the earliest extant book attesting to the\nalgebraic method known as “the procedure of the celestial origin ",
{
"type": "Emphasis",
"content": [
"tian\nyuan shu"
]
},
" 天元術.”"
]
},
{
"type": "Paragraph",
"id": "S1.p4",
"content": [
"However, something immediately struck me in the solutions Li Ye gave to the\nproblems. Every solution had the same structure, which consisted of two parts.\nEach of these parts described, in a different way, how to obtain the same\nequation that solved the problem. The first part, called “method” (",
{
"type": "Emphasis",
"content": [
"fa"
]
},
"\n法), described a sequence of algorithms that relied on the data to compute the\nsuccessive coefficients of the sought-for equation. In this part, there were no\nnumerical values, in contrast with the second part, called “detail of the\nprocedure” (",
{
"type": "Emphasis",
"content": [
"cao"
]
},
" 草), which, starting from the data and the chosen\nunknown, presented two ways of reasoning to obtain the same geometrical\nmagnitude and a numerical polynomial associated with it. The reasonings, along\nwith the related polynomial computations, systematically followed the same\npattern: each step consisted of an operation that took previously determined\nmagnitudes and the associated polynomial as its operands, and then yielded a\nresult in the form of another magnitude (the reasoning part) and the polynomial\nassociated with it (the computation part). At the end of a procedure of this\nkind, the equation was obtained numerically, by subtracting from each other the\ntwo polynomials that corresponded to the same magnitude. Why, I wondered,\nshould the author systematically tell the reader, twice and in two different\nways, how to get the same equation? This was my first question, soon followed\nby a second one: taking for granted that the “method” and the “detail of the\nprocedure” led to exactly the same equation, how were the algorithms given in\nthe “method” obtained?"
]
},
{
"type": "Paragraph",
"id": "S1.p5",
"content": [
"For each of the 170 problems, I made an experiment. I computed the sequences of\npolynomials leading to the final equation in the “details of the procedure”\nsymbolically, and not numerically as they were presented in Li Ye’s book.\nAlthough the text did not contain any computation of this kind, I established\nthat, in each case, my computations highlighted a missing link between the\nalgorithms of the “method” and the “details of the procedure”. Indeed,\nevery algorithm in the “method” actually described the sequence of operations\nthat, in the symbolic computations deriving from the “details of the\nprocedure”, had been applied to coefficients of successive polynomials to\nshape the corresponding coefficient of the final equation [",
{
"type": "Cite",
"target": "bib-bib2",
"content": [
"2"
]
},
"]. In\nbrief, using mathematical knowledge and practice that did not feature in the\nbook, and that appeared long after the book was completed (that is, algebraic\nsymbolism and algebraic computations), I could highlight a correlation between\nthe two parts of every solution. The correlation was so intimate that the\n“method” could not have been obtained independently from the “details of the\nprocedure”. Clearly, the systematic correlation yielded a clue indicating that\nthe description in the “method” derived from a work that Li Ye had carried\nout, but not recorded in his book. So the question became: what kind of\nmathematical work was that?"
]
},
{
"type": "Paragraph",
"id": "S1.p6",
"content": [
"Perhaps, in the future, someone will find clues in ",
{
"type": "Emphasis",
"content": [
"Measuring the Circle"
]
},
",\nor elsewhere, to answer this question with certainty. However, as far as we\nknow today, nothing in the book seems to indicate exactly how, for every single\nproblem, Li Ye produced the “method” part of the solution, relying on the\n“details of the procedure” part. I cannot attribute to him without further\nado the knowledge that I, as an observer, bring into play to establish the\ncorrelation between the “method” and the “details of the procedure”.\nNevertheless, my experiment sheds light on knowledge that Li Ye must have\npossessed, and practices that he must have used, in order to write\n",
{
"type": "Emphasis",
"content": [
"Measuring the Circle"
]
},
" as it stands, even though I cannot describe them\nprecisely since he did not expose them himself, even indirectly. As historians\nof mathematics, we cannot content ourselves with a superficial reading of the\nbook and offer a historical treatment that would ignore this new dimension that\nstudying ",
{
"type": "Emphasis",
"content": [
"Measuring the Circle"
]
},
" allows us to perceive. We are committed to\ntry to account for the knowledge the actors we observe possessed and the\npractices they put into play, even when these were not the objects of\ndiscursive exposition."
]
},
{
"type": "Paragraph",
"id": "S1.p7",
"content": [
"This example illustrates why, in order to fully accomplish their task,\nhistorians must look for clues and then strive to interpret those clues as best\nas they can. One might of course be tempted to consider this case as an\nexception and an outlier; however, since I began working as a historian, my\nexperience has convinced me of the contrary, not only because other similar\nphenomena occur in Li Ye’s book, but because they actually occur much more\nbroadly. In fact, as early as 1974, drawing on discussions with Igor\nShafarevich, Isabella Bashmakova once showed something quite similar about the\nfour books of Diophantus’ ",
{
"type": "Emphasis",
"content": [
"Arithmetics"
]
},
" that still exist in Greek [",
{
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"content": [
"1"
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},
"].\nTen years later, Roshdi Rashed fully developed this approach\nand observed the same phenomena in his publication of the four other books that\nhad just resurfaced in Arabic [",
{
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"target": "bib-bib17",
"content": [
"17"
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},
"]. These historians used insights\nfrom modern algebraic geometry to analyze the procedures Diophantus followed in\nthe ",
{
"type": "Emphasis",
"content": [
"Arithmetics"
]
},
" to solve Diophantine problems. This reading,\ninstrumented by a type of mathematical knowledge that Diophantus certainly did\nnot possess, revealed something that completely contradicted previous\ninterpretations, according to which Diophantus was fundamentally unpredictable\nin his approach to a problem, even after one has read dozens of his solutions\nto other problems. Indeed, the analysis of the ",
{
"type": "Emphasis",
"content": [
"Arithmetics"
]
},
" using\nalgebraic geometry showed that Diophantus’ solutions systematically made use of\nthe same methods. Exactly as was the case for Li Ye, we cannot attribute to\nDiophantus knowledge of the tool modern historians put into play to read the\n",
{
"type": "Emphasis",
"content": [
"Arithmetics"
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". However, this tool brings to light knowledge that\nDiophantus possessed and practices he used without recording them. How can we\napproach his knowledge and practices on the basis of these clues? This is the\ntheoretical problem raised by these phenomena [",
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"]."
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"content": [
"In the cases of Li Ye and Diophantus, the clues provided by a certain type of\nmathematical reading reveal facets of the knowledge and practice that these\nauthors have put into play in their approaches to specific problems, without,\nhowever, writing about them. In fact, clues can do more for historians, as Anne\nRobadey has illustrated in her work on Henri Poincaré. For instance,\nRobadey [",
{
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"content": [
"18"
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},
"] starts from the remark that Poincaré’s publications\nabound in enumerations, and she sets out to analyze what these textual\nphenomena can teach us about the way Poincaré carried out his mathematical\nwork. Robadey [",
{
"type": "Cite",
"target": "bib-bib18",
"content": [
"18"
]
},
"] establishes that these textual clues reveal an\nintellectual practice that Poincaré recurrently put into play in different\ncontexts and that left traces not only in his writings but also in the type of\nmathematical results he formulated. Indeed, faced with certain mathematical\nsituations, Poincaré regularly analyzes them, focusing first on the case that\npresents itself most often (in a sense of the latter expression for which\nPoincaré puts forward a definition and an assessment), then on the second\nmost frequent case, and so on, until reaching phenomena that he thinks he can\ndisregard, since they “almost never” occur (on the basis of an assessment of\nthe same type). The enumerations embody precisely this recurring intellectual\nprocedure that Poincaré follows. Moreover, they mesh with theorems of\nPoincaré’s in which he asserts that something holds true except for a set of\nsituations that can be neglected. In this example, textual phenomena,\nmathematical practice and mathematical results appear to be closely\nintertwined. How, as historians, we can find clues that allow us go deeper in\nour analysis, and how we can use them in historical research are precisely two\nof the main issues that I study, not least in the context of the project\n“History of science, history of text”, which I launched in 1995 and on which\nI have been working since then with a group of colleagues (see,\ne.g., [",
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"])."
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"content": [
"The result obtained by Robadey that I have summarized naturally leads us to\nanother interesting issue: why do practitioners of mathematics not always\npresent their knowledge and practices “explicitly” (as we would be tempted to\nsay, but I explain below why this term is inadequate), to the extent that\nhistorians need to rely on clues to uncover part of this knowledge and these\npractices? The example of Poincaré’s enumerations suggests a first answer to\nthis question. If his publications yield the clues I have mentioned, the\nreason seems to be that Poincaré carries his analysis forward while engaging\nwith his page. The page thereby keeps the trace of the procedure that\nrecurringly structures his mathematical exploration. This remark explains why\nthe writing gives us clues about his way of conducting mathematical work.\nPoincaré chooses to work with the textual structure of the enumeration, since\nit offers a support on which he can rely to unfold his reasoning."
]
},
{
"type": "Paragraph",
"id": "S1.p10",
"content": [
"We can observe a similar phenomenon in the prehistory and history of duality,\non which I have begun to work with Serge Pahaut [",
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"13"
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},
"]. If we\nconsidered that duality emerged in mathematics at the point when actors first\nexplicitly mentioned the phenomenon, we would set its beginnings in the 1820s.\nHowever, Pahaut and I noticed that starting from the 1750s, some mathematicians\nwho published on spherical trigonometry chose new notations, and shaped types\nof text, both of which were appropriate to highlight a phenomenon that they had\nobserved without thematizing it. Using new notation, Leonhard Euler, for\ninstance, presented in 1753 a memoir about spherical trigonometry that is\nremarkable for the following reason: its text displayed, without any comment, a\nsymmetry in a corpus of propositions asserted, and also in a corpus of proofs\nestablishing these propositions. Today, we associate this symmetry with the\nduality that affects spherical trigonometry. Euler did not address this\nphenomenon discursively. For him, as for several mathematicians who wrote about\nspherical trigonometry in the same way in the following decades, this was a\nphenomenon to be explored, and, instead of writing about it in a discursive\nway, they expressed what they observed using textual features of their\nwritings: they gave it to readers to read off from the structure of the text.\nShould we call such a way of expressing knowledge “not explicit” simply\nbecause it is not expressed with a subject, a verb and a complement? I don’t\nthink so. This would be quite a narrow interpretation of what “explicit”\nmight mean."
]
},
{
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"id": "S1.p11",
"content": [
"Indeed, we can establish that for these mathematicians, writing in this way was\na genuine choice. The reasoning goes as follows. In a second memoir on the\ntopic that Euler presented in 1781, Pahaut and I were able to show that he made\na mistake in a proof, which was then replicated in the dual proof. This clue\nthus indicates that Euler relied on the notation to produce the dual theorem\nand the dual proof by mere rewriting of the corresponding theorem and proof,\nwithout actually redoing the computation. In other words, Euler knew that a\ntheorem and a proof could automatically give rise to another theorem and\nanother proof, but he chose to present both systematically and to cast light on\nthe symmetry between them by means of the structure of his exposition. This\nremark allows us to establish another key point: the notation appropriate to\ninvestigate phenomena related to duality constituted a tool created by Euler to\nwork with and to produce a text that displayed the symmetry. More generally,\ntexts are not always merely discursive expositions of knowledge, as a modern\nreading all too often expects. This remark might seem obvious for rough\ndrafts, but Euler’s inquiry into duality and Poincaré’s enumerations show\nthat it also applies to texts intended for publication. We see mathematicians\nshaping notations and textual resources and developing practices using them, in\norder to work with them and explore new phenomena. As a result, these textual\nresources and practices present intimate correlations with the questions these\nactors pursue and the research they conduct. I take these textual innovations\nas a key dimension of their activity."
]
},
{
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"id": "S1.p12",
"content": [
"This observation highlights one of the reasons why, as a result, texts can give\nclues about the mathematical work that produced them and also about the\nknowledge that mathematicians acquired through working with them. In Euler’s\ncase, he met more than once with phenomena caused by duality, and regularly\nmade use of similar textual resources. Interestingly, when dealing with the\nsame topics, subsequent mathematicians used notation and textual resources that\nwere either identical or similar to Euler’s, which indicates that notation and\ntextual resources are, like mathematical theories and concepts, products of\nmathematical work that get picked up and used further by others [",
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"content": [
"4"
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"].\nThe joint production of knowledge and textual resources (in the broadest sense\nof this expression) is likewise one of the theoretical issues that interest me\nmost."
]
},
{
"type": "Paragraph",
"id": "S1.p13",
"content": [
"If we pursue this line of thought, we see that sometimes, in order to deal with\nspecific topics, new types of textual resources are introduced (like writing\npropositions and proofs in a symmetrical way), and that some among the\nsubsequent readers will not only grasp what is being given to read in this new\nmanner, but also then go on to reuse the new textual resources to carry on\nfurther research along the same lines. However, not all readers will notice\nwhat is given to read in this way. For instance, historians of mathematics had\nnot underline what the structures of these texts expressed, at least for works\nwritten before Joseph Diez Gergonne’s explicit introduction, in 1826, of a\ndouble-column device to display the symmetry between propositions and proofs\nelicited by duality [",
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]
},
"]. My purpose is not to blame these historians,\nbut to draw a conclusion from this observation. Obviously, we do not all read\nin the same way, particularly because we have not all been acculturated to\nreading mathematical writings in the same way. Reading (and reading\nmathematical texts is no exception) has a history that itself deserves to be\nstudied in the various contexts in which it has been carried out over time, in\norder to better account for what our sources convey in ways that are not always\nobvious to us. This latter issue, and more generally those brought to light in\nthis section, have turned out to be central in basically every single piece of\nresearch that I have conducted."
]
},
{
"type": "Heading",
"id": "S2",
"depth": 1,
"content": [
"2 ",
{
"type": "Emphasis",
"content": [
" The Nine Chapters"
]
},
": Algorithms, proofs, and epistemological values"
]
},
{
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"id": "S2-F2",
"caption": [
{
"type": "Paragraph",
"content": [
"The diagram in ",
{
"type": "Emphasis",
"content": [
"Measuring the Circle on the Sea-Mirror"
]
},
" (1248)",
{
"type": "Emphasis",
"content": [
"Tongwenguan"
]
},
" Edition 同文館, 1876"
]
}
],
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"content": [
"While working on ",
{
"type": "Emphasis",
"content": [
"Measuring the Circle"
]
},
", it appeared to me that this book\nwas deeply rooted in an ancient Chinese canonical work in mathematics to which\nLi Ye explicitly refers, namely the first-century classic ",
{
"type": "Emphasis",
"content": [
"The Nine\nChapters on Mathematical Procedures"
]
},
" (",
{
"type": "Emphasis",
"content": [
"Jiuzhang suanshu"
]
},
" 九章算術,\nhereafter ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
"). In fact, most mathematical writings\ncomposed in China before the fourteenth century referred to this work and to\nthe commentaries with which it has been handed down, that is, Liu Hui’s 劉徽\ncommentary, completed in 263, and the subcommentary published in 656 by a team\nworking under the supervision of Li Chunfeng 李淳風. When, as early as 1981,\nGuo Shuchun suggested that we could cooperate to translate ",
{
"type": "Emphasis",
"content": [
"The Nine\nChapters"
]
},
" and its commentaries into French, I thus found the project meaningful\nand accepted without hesitation. We agreed in 1983 that in addition to the\ntranslation, our joint book would offer a new critical edition of these texts\nas well as our own annotations, unaware that these tasks would take us over\ntwenty years to complete [",
{
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"]."
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{
"type": "Paragraph",
"id": "S2.p2",
"content": [
"The project was difficult not only because the Chinese text was hard and the\nestablishment of the critical edition challenging, but also because the\nendeavor raised many theoretical problems that I felt we needed to address to\ncomplete our task satisfactorily. I will illustrate some of these problems\nwhile outlining some of the research directions I have followed in my research\non ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" and the more theoretical projects that this work\nhas inspired me with."
]
},
{
"type": "Paragraph",
"id": "S2.p3",
"content": [
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" was composed of 246 mathematical problems along with\nprocedures to solve such problems. The “problem and procedure” form had led\nsome historians to read it either as an exercise book or as a manual for\nbureaucrats, who would simply need to pick up instructions and follow them\nblindly. It seemed to me that these interpretations, which derived from a\nmodern reading of an ancient text, could not explain why the book had been\nconsidered a classic for centuries. We thus needed to find ways of reading\n",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" that could be less anachronistic and for which we\ncould put forward arguments."
]
},
{
"type": "Paragraph",
"id": "S2.p4",
"content": [
"My efforts first bore on the procedures. In 1972, Donald Knuth, whose work on\nalgorithms needs no introduction, published an article that had a significant\nimpact on the history of ancient mathematics [",
{
"type": "Cite",
"target": "bib-bib16",
"content": [
"16"
]
},
"]. Indeed, his article\nwas proposing a completely new way of approaching cuneiform texts of the\nbeginning of the second millennium BCE, by reading them as they were written,\ni.e., as lists of operations, or “algorithms”, and not by rewriting them into\nmodern algebraic formulas, as had been the case until then. I became aware of\nthis breakthrough in 1981, thanks to discussions with Wu Wenjun, a topologist\nwho had turned to automated theorem proving and the history of mathematics in\nChina during the cultural revolution [",
{
"type": "Cite",
"target": "bib-bib15",
"content": [
"15"
]
},
"]. Wu immediately adopted\nKnuth’s perspective in his reading of the procedures contained in ancient\nChinese mathematical texts, since they too were written in the form of lists of\noperations. What mattered most to him was the emphasis that Chinese texts of\nthe past had placed on algorithms providing constructive means to solve\nproblems. A typical example is the algorithm described by Qin Jiushao 秦九韶 in\nhis 1247 book, in order to compute actual solutions to problems whose solution\nis known to exist by the Chinese remainder theorem [",
{
"type": "Cite",
"target": "bib-bib19",
"content": [
"19"
]
},
"]. I became more\ninterested in the work with lists of operations to which the procedures of\n",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" attest."
]
},
{
"type": "Paragraph",
"id": "S2.p5",
"content": [
"This work is eloquently illustrated by the procedures given in ",
{
"type": "Emphasis",
"content": [
"The Nine\nChapters"
]
},
" for square and cube root extraction, whose texts put into play three\noperations that Knuth identified as fundamental in the writing of algorithms:\nassignment of variables, iterations and conditionals. Before Knuth introduced\nthe idea of reading lists of operations as algorithms, historians knew which\ncomputations these texts referred to, but I claim they were unable to\nunderstand ",
{
"type": "Emphasis",
"content": [
"how"
]
},
" these texts in ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" actually referred\nto these computations. Thanks to the recent development of mathematical work on\nalgorithms, in relation to their implementation on computers, mathematicians\nhave shaped new types of texts to write procedures, and these textual resources\nhave given us new insights into how ancient procedural texts might have been\nwritten and consequently how we might interpret them [",
{
"type": "Cite",
"target": "bib-bib10",
"content": [
"10"
]
},
"].\nWe see again how, in different contexts, actors put into play different types\nof textual resources to carry out mathematical work and to write about it. In\nthe case of the procedures of ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
", we could not fully\ninterpret their texts and understand the work involved in producing them\nwithout first asking ourselves how these texts were expected to be read and how\nmathematical work put them into play. With respect to, e.g., the algorithms for\nroot extractions, this research brought to light two key points."
]
},
{
"type": "Paragraph",
"id": "S2.p6",
"content": [
"First, the way in which the author(s) had used conditionals and iterations to\nwrite a list of operations on the basis of which any square (resp. cube) root\nextraction could be performed highlighted important features of the work with\noperations in this context. To explain this point, let me make clear that the\nexecution relied on a decimal place-value system and that the roots were\ndetermined digit by digit. On this basis, the list of operations used for the\nfirst digit and the one used for each digit after the first one had been shaped\nin such a way that they could be integrated into a single text. Without\nentering into details (for which I refer to the 2004 book [",
{
"type": "Cite",
"target": "bib-bib12",
"content": [
"12"
]
},
"]), let\nme simply emphasize that the integration relied on the assignment of variables.\nIt also relied on the fact of treating operations formally and without taking\ntheir intention into account. More specifically, highlighting an operation\ncommon to the concluding part of a root extraction (when the digit of the units\nhas been dealt with) and to the preparation of the computation for the next\ndigit, if any, even though the purpose of using this operation differed in the\ntwo contexts, as well as placing the statement of this operation in the list of\noperations adequately with respect to the conditionals and the iteration,\nplayed a key part in allowing the authors to compose a single algorithm valid\nfor all cases. In ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" and their commentaries, we find more\ngenerally many indications that the authors worked on lists of operations\nformally, without taking into account the variety of reasons for bringing these\noperations into play in each of their respective contexts. As a result, we\nregularly see authors striving to unify lists of operations that performed\ndifferent tasks, but could be made formally identical. This highlights a form\nof algebra specific to the work with operations, to which I have devoted some\nresearch, but on which much remains to be done."
]
},
{
"type": "Paragraph",
"id": "S2.p7",
"content": [
"The latter remarks lead me to the second key point. A search of the kind just\ndescribed with respect to algorithms, that is, a search for lists of operations\nwhose efficacy would extend as broadly as possible, bespeaks actors’ valuing of\ngenerality. The fact of giving a single algorithm for square (resp. cube) root\nextraction points in the same direction. The text of the algorithm was general\nin the sense that for any number, an adequate circulation within it, guided by\nthe conditionals and the iteration, would yield the list of operations required\nto determine the desired square root. What is more, the text added this: should\nthe extraction not be completed when reaching the digit of the units, the\nresult should be given as the “side of the number”, i.e., as a quadratic\nirrational. If we considered this suggestion from the viewpoint of the\ndiscussions about irrationality by Greek authors of antiquity, we would\ncompletely miss the point – I return to this below. At stake in the\ninterpretation of the text of the square root algorithm is thus a better\nappreciation of how it reflects the importance actors in this context gave to\nthe epistemological value of generality."
]
},
{
"type": "Paragraph",
"id": "S2.p8",
"content": [
"In fact, generality proved to be a key value for these actors much more\nbroadly [",
{
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"target": "bib-bib9",
"content": [
"9"
]
},
"]. For instance, the way in which commentators read problems\nin ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" indicated that for them, a particular problem was\nto be interpreted as a paradigm. This might seem obvious: the problems from our\nchildhood about trains passing each other were not meant to stand only for\nthemselves, but expressed something more general. However, an observation of\nthe commentators’ way of reading problems in the classic shows that they meant\nsomething more specific. In a key case, when the procedure placed after a\nproblem correctly solves it, but lacks generality, the third century\ncommentator Liu Hui expresses dissatisfaction. After pointing out that the\nprocedure of ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" is based on the use of two singular\ncharacteristics of the problem, he proposes a first procedure that fixes the\nfailure of the original procedure to solve the most general problem,\ndistinguishing between two cases, and then a completely general and uniform\nprocedure. In other words, for Liu Hui, the fact that a mathematical problem\nwas not abstract did not affect the expectation he had with respect to its\ngenerality. This remark, inspired by this Chinese document, raised an important\ntheoretical issue: it was an invitation to dissociate the values of generality\nand abstraction in our reflection about mathematics and to see what a focus on\ngenerality alone might show. The emergence of projective geometry in France\nduring the first decades of the 19th century proved to be an ideal case for me\nto address this issue. Indeed, this new geometry took shape in the hands of\nCarnot, Poncelet, Chasles and others, on the basis of a comparative reflection\nabout the different types of generality brought about by analytic and geometric\napproaches to geometric problems. More broadly, this direction of research\nproved fruitful for a group of historians and philosophers of science, as is\nillustrated by the collective reflection we developed on this\nissue [",
{
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"target": "bib-bib11",
"content": [
"11"
]
},
"]."
]
},
{
"type": "Paragraph",
"id": "S2.p9",
"content": [
"The remarks that I have presented about the problems of ",
{
"type": "Emphasis",
"content": [
"The Nine\nChapters"
]
},
" illustrate a method that I have used more systematically. Indeed, if\nwe need to restore how ancient actors used and read the texts with which they\nperformed mathematical activity, or, in other words, if we need to develop a\nhistory of the reading and handling of ancient mathematical texts, observing\nhow ancient readers proceeded seems to be a method that has great potential.\nThis is precisely one of the reasons why it is so valuable to have early\ncommentaries on ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
". If, for instance, we continue to rely\non them to better understand how and why ancient actors used problems in their\nmathematical practice, we discover something quite unexpected, which\ndefinitively rules out the interpretation of ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" as an\nexercise book or as a manual for bureaucrats."
]
},
{
"type": "Paragraph",
"id": "S2.p10",
"content": [
"This facet of their practice also appears when we turn to another key point\nabout the commentaries on ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
", namely that they\nsystematically put forward proofs of the correctness of the algorithms\npresented in the classic. This is quite an important fact for a history of\nmathematical proof, to which I return shortly. What matters here is that the\ncommentators’ way of carrying out proving brings mathematical problems into\nplay. To put it differently, in their practice, mathematical problems appear to\nbe tools with which to conduct proofs, and not merely statements awaiting a\nsolution [",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
"]. If we think that the same fact held true for the\nauthor(s) of ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters,"
]
},
" this invites a radically new\ninterpretation of the work. The reading of problems and procedures that I have\nsuggested might help us understand better how ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" has been\nconsidered a canonical work over so many centuries. What is more, the fact that\n",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" was handed down with these commentaries might have\nalso played a part in giving the book its value in the eyes of its users. This\nremark brings us back to the proofs that Liu Hui as well as the team working\nwith Li Chunfeng formulated."
]
},
{
"type": "Paragraph",
"id": "S2.p11",
"content": [
"In contrast with what we read in Euclid’s ",
{
"type": "Emphasis",
"content": [
"Elements"
]
},
" and Archimedes’s\nwritings, these proofs aim at establishing the correctness of procedures.\nObserving them hence gives us source material to think about this other branch\nof the history of proof that has so far been almost completely neglected, and,\nmore broadly, about the various dimensions of the exercise of proving in\nmathematics [",
{
"type": "Cite",
"target": "bib-bib8",
"content": [
"8"
]
},
"]. What is essential here is that the\ncommentators use theoretical concepts to refer to key aspects of the conduct of\na proof."
]
},
{
"type": "Paragraph",
"id": "S2.p12",
"content": [
"To begin with, they devote a specific term, i.e., “meaning/intention” (",
{
"type": "Emphasis",
"content": [
"yi"
]
},
" 意), to designate the meaning of an operation that\ncorresponds to the interpretation of its result in the context in which the\noperation is used. Typically, for an operation, this is the kind of meaning\nthat the context of a problem enables a practitioner to make explicit. By\nextension, the term ",
{
"type": "Emphasis",
"content": [
"yi"
]
},
" also refers to a sequence of meanings of this\nkind, and in the end to the reasoning from which the sequence derives. As a\nrule, a reasoning of this type consists in making clear the “meaning” of the\nsuccessive steps of an algorithm, thereby showing why its end result\ncorresponds to what was expected. Interestingly, we find here an echo with the\ntype of reasoning Li Ye expounded in his “details of the procedure”, which we\nmentioned at the beginning of this article. The only difference lies in this:\nin Li Ye’s case, instead of yielding a meaning and a number, each operation\nyields a meaning and a polynomial. We nevertheless see that there might be\ntraditions of reasoning to which Chinese writings attest, but which were not\nyet studied."
]
},
{
"type": "Paragraph",
"id": "S2.p13",
"content": [
"The second term used by commentators of ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" in the context\nof their proofs, which I denote by ",
{
"type": "Emphasis",
"content": [
"yi’"
]
},
" 義, refers to another type of\n“meaning” for procedures. It designates a fundamental procedure that\nunderlies the procedure whose correctness must be established. As part of the\nproof of the correctness, this fundamental procedure highlights the strategy\nfollowed by the algorithm under consideration. At the same time, identifying it\nconnects this algorithm with others, which follow the same formal strategy,\neven though the reasons for using the same operations might differ, depending\non the context. The interest that commentators have for this kind of\n“meaning” thus appears to be connected with the formal work on operations to\nwhich the algorithms contained in ",
{
"type": "Emphasis",
"content": [
"The Nine Chapters"
]
},
" also attest. This\nfocus of their proofs is in fact more broadly connected with a research program\nfor which we have evidence between the first and the thirteenth century and\nwhich aimed at identifying the least number of algorithms from which all the\nothers derive [",
{
"type": "Cite",
"target": "bib-bib9",
"content": [
"9"
]
},
"]."
]
},
{
"type": "Paragraph",
"id": "S2.p14",
"content": [
"The last set of terms that commentators use for their proofs relates to what I\nhave called “algebraic proofs in an algorithmic context”. A proof of this\nkind consists in establishing a list of operations that starts from the same\ndata as the algorithm under consideration, and yields the desired result. The\ncommentator then takes the algorithm established as correct as a basis, and\noperates on its list of operations to transform it, ",
{
"type": "Emphasis",
"content": [
"qua"
]
},
" list of\noperations, into the algorithm whose correctness is to be established. In other\nwords, instead of rewriting equalities, as we do in an algebraic proof, this\ntype of proof rewrites algorithms. The meta-operations applied to the list of\noperations include swapping multiplication and division that follow each other,\nand cancelling a multiplication and a division inverse of one another. They\nalso include inverting an algorithm known to be correct. The essential point\nhere is that the commentators associate the correctness of these\nmeta-operations with the fact that divisions and square root extractions are\ngiven exact results, notably through the introduction of fractions and\nquadratic irrationals. They thus bring the set of numbers used and the\nmeta-operations applied to a list of operations in relation with each other.\nMoreover, here again, we see that these “algebraic proofs in an algorithmic\ncontext” also involve formal work on lists of operations."
]
},
{
"type": "Paragraph",
"id": "S2.p15",
"content": [
"What does all this tell us about the history of algebraic proof, of which we\nstill lack a proper account? What does it tell us about the history of algebra\nand the part played by operations in the history of mathematics? These are some\nof the theoretical questions that remain on my agenda."
]
},
{
"type": "Paragraph",
"id": "S2.p16",
"content": [
{
"type": "Emphasis",
"content": [
"Acknowledgements"
]
}
]
},
{
"type": "Paragraph",
"id": "S2.p17",
"content": [
"I have pleasure in expressing my gratitude to Fernando Manuel Pestana da Costa\nfor his invitation to write this article and Leila Schneps for her feedback,\nwhich greatly improved it."
]
},
{
"type": "Paragraph",
"id": "authorinfo",
"content": [
"Karine Chemla, Senior Researcher at the French National Center for\nScientific Research (CNRS), in the laboratory SPHERE (CNRS and Université de\nParis), focuses, from a historical anthropology viewpoint, on\nthe relationship between mathematics and the various cultures in the\ncontext of which it is practiced.\n\n",
{
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"target": "mailto:chemla@univ-paris-diderot.fr",
"content": [
"chemla@univ-paris-diderot.fr"
]
}
]
}
]
}