Abstract
This paper aims at analyzing the ways in which the description of an algorithm can refer to the reasons for its correctness. I rely on ancient Chinese mathematical sources: the Book of Mathematical Procedures (ca. 186 B.C.E.), The Nine Chapters on Mathematical Procedures, (probably first century C.E.), commentaries on the latter by Liu Hui (completed in 263 C.E.) and by a group of scholars working under Li Chunfeng’s supervision (presented to the throne in 656 C.E.). These sources show two fundamental ways in which an algorithm can indicate the reasons for its correctness. First, the algorithm can be decomposed into steps and sequences of steps, the meaning of which can be formulated with respect to the problem by reference to which the algorithm was formulated; second, the algorithm can use indirect speech acts to prescribe the operations to be executed: Instead of directly prescribing the operation(s), it refers to them either by the material effect they will have in the situation (first case) or by a term indicating both their material effect and their formal intention (second case). The latter description goes along with prescribing not one operation but several at a time, since it is their combination that achieves the aim intended and indicated by the term used. Such modes of indirect prescription occur in both ancient books. However, the second case occurs only in The Nine Chapters and is abundantly discussed by the commentators. This may indicate an evolution in the modes of approaching the correctness of algorithms between the dates when the two books were composed.
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- 1.
Compare the critical edition with annotations in Peng Hao (彭浩 2001).
- 2.
Below, I shall abbreviate the title into The Nine Chapters. For a critical edition and a French translation of this book and its earliest commentaries, compare Chemla and Guo Shuchun 2004. Chapter B, by Guo Shuchun, discusses the opinions of several scholars regarding the time period when The Nine Chapters was compiled. In my introduction to chapter 6 in the same book, I argue for dating the end of the compilation to the first century C.E. (Chemla and Guo Shuchun 2004: 475-481).
- 3.
Below, we refer to this layer of the text as “Li Chunfeng’s commentary.” Two other supra-commentaries, composed during the Song dynasty, respectively in the eleventh and the thirteenth century, survived only partially. They were not handed down systematically with the collection, by that time coherent, that The Nine Chapters and the two earlier commentaries formed.
- 4.
I introduced this distinction in Chemla 1996. I shall come back to it below.
- 5.
More precisely, when such proofs were analyzed, their analysis seldom aimed at determining the specificities of proofs, whose goal is to establish the correctness of algorithms. I have suggested elsewhere that once we understand better the history of such proofs, we might be in a position to formulate hypotheses regarding the part they played in a world history of mathematical proof and, more specifically, in a history of algebraic proof. However, in my view, we have not yet reached that point.
- 6.
It would be impossible to mention here the many papers and books that in the last decades were devoted to the proofs contained in the commentaries. Let me simply evoke: Li Yan (李儼 1958: 40-54); Qian Baocong (錢寶琮 1964: 62-72); Wu Wenjun (吳文俊 1982), Li Jimin (李繼閔 1990); Guo Shuchun (郭書春 1992); Wu Wenjun (吳文俊), Bai Shangshu (白尚恕), Shen Kangshen (沈康身) and Li Di (李迪 1993). For a fuller bibliography, refer to Chemla and Guo Shuchun 2004. In general, the publications seldom analyze the proofs from the viewpoint that they establish the correctness of algorithms. I have attempted to identify the main operations involved in the proof of the correctness of algorithms to which these commentaries bear witness in Chap. A of Chemla and Guo Shuchun 2004: 27-39.
- 7.
The first synthetical article that I devoted to this issue is Chemla 1991.
- 8.
- 9.
The working seminar “History of science, history of text,” organized with Jacques Virbel since 2002, and especially Agathe Keller’s contribution, helped me clarify this dual dimension of an algorithm. It is my pleasure to express my gratitude to the group gathered around this seminar.
- 10.
See below for some concrete examples.
- 11.
I owe this element of description of an algorithm, that is, the “action,” to the presentation of the project “Histoire de la calculabilité” by M. van Atten, M. Bourdeau, and J. Mosconi (Final Conference of the Program of the CNRS and MESR: “Histoire des savoirs,” November 29-December 1, 2007). The proceedings of the Program can be found at http://www.cnrs.fr/prg/PIR/programmes-termines/histsavoirs/synth2003-2007Histoiredessavoirs.pdf.
- 12.
I describe a text of that kind for an algorithm as well as Liu Hui’s proof of the correctness of the algorithm in Chemla 1991.
- 13.
- 14.
These are problems 1.31 and 1.32. The pair of numbers I attach to a given problem in The Nine Chapters refers, first, to the chapter in which it is placed (here, Chap. 1) and, then, to the order in which the problems are arranged in this chapter (here, 31st and 32nd problems). Note that these numbers are not part of the source material.
- 15.
Note that the diagram is restored on the basis of the references Liu Hui makes to its structure. However, I do not attempt to produce a figure conforming to the features known to be specific of the diagrams Liu Hui used. For instance, to conform to modern usage, I name some of the points. Before the thirteenth century C.E., we have no evidence in China of such ways of marking figures.
- 16.
Evidence supporting this claim is given in Chemla 2003.
- 17.
This remark is important only because there are other modes of prescribing an operation that constitute another family of cases, in which the text of an algorithm refers to the reasons for its correctness (see below).
- 18.
I composed a glossary of technical expressions used in The Nine Chapters and its early commentaries (Chemla and Guo Shuchun 2004: 895-1042). In what follows, I shall refer to it as Glossary. It provides evidence for the meanings and facts regarding technical terms. For yi (“meaning, intention”) see Glossary: 1018-1022.
- 19.
The terms I translate here by “base” and “height” are in fact technical terms referring, respectively, to the shorter and the longer sides of the right angle in a right-angled triangle.
- 20.
I follow the structure of the Chinese term for prescribing a square root extraction and underline, as the Chinese does, the link of that operation to division.
- 21.
See the term “look for 求 qiu,” in Glossary: 971. The corresponding problem and procedure in Chap. 9 appear in Chemla and Guo Shuchun 2004: 704-707.
- 22.
Such transformations constitute parts of proofs to which I referred as “algebraic proofs in an algorithmic context.” On this set of transformations and how their correctness was approached in ancient China, see Chemla 1997/1998.
- 23.
For those algorithms in The Nine Chapters the text of which does not have a transparent structure, the commentators regularly argue that the reason lies precisely in such rewriting. They compose, in the way just outlined, an algorithm carrying out the task expected from the algorithm commented upon. They further bring to light the cumbersome character of the algorithm they have composed, when it comes to computations, to account for the fact that the algorithm recorded in The Nine Chapters differs from the one they just composed. The transformations they describe in order to transform the latter algorithm into the former, thereby proving its correctness and accounting for its shape, constitute the part of the proof to which I refer by the expression of “algebraic proofs in an algorithmic context.”
- 24.
See for example the texts for algorithms computing the volumes of solids recorded in bamboo slips 142-145 (Peng Hao (彭浩) 2001: 101-105). They share common features with texts for algorithms in The Nine Chapters and the structure of which the commentators interpret as transparent (Chemla 1991). Cullen 2004: 90-99 developed this idea of mine.
- 25.
See the new critical edition and French translation in Rashed 2007: 100 ff.
- 26.
In the only case in al-Khwarizmi’s book when the algorithm proved differs in its structure from the algorithm to be proved, we find two hints indicating that al-Khwarizmi’s intention is to prove the algorithm with the structure with which its text is formulated. First, at the end of his proof, he addresses the differences between the two algorithms. Second, this is the only time when al-Khwarizmi develops a second proof, which in fact establishes the correctness of the algorithm, on the very basis of the structure of its formulation (see Rashed 2007: 108-113). Incidentally this remark shows that the structure of the text is not transparent in and of itself: It is made transparent by an interpretation.
- 27.
In both cases, the proof consists in making the meanings of the successive results explicit. However, the two authors carried out this operation differently. In the Liu Hui excerpt analyzed here, the meanings are made explicit in the text itself. However, al-Khwarizmi’s book presents the proof as a separate text, the structure of which follows the structure of the text for the algorithm. Moreover, the dispositifs within which the meanings are expressed differ. Liu Hui makes use of diagrams as well as of problems and procedures attached to them. These are precisely the elements with which Liu Hui claims to have made the yi (意, “meaning”) explicit (see yi in Glossary). Al-Khwarizmi uses only diagrams, the nature of which differs from Liu Hui’s.
- 28.
I have devoted several publications to this text. I shall strictly limit myself here to what is essential to deal with the topic of this article. For greater detail, compare, for instance, Chemla 1997.
- 29.
I argued for an interpretation of this text in Chemla 1992. In a forthcoming paper, I examine how the text covers the various cases in greater detail. This paper will be published in the volume edited by J. Virbel and myself, as the outcome of the seminar “History of science, history of text.” Here, I rely on my 1992 publication without repeating its argument, my main focus being to analyze the text of the algorithm from the perspective of how it refers to reasons for correctness.
- 30.
In fact, the general case meant here corresponds to the second formula, the first corresponding to e equal to 0.
- 31.
My forthcoming article points out that such types of text, organizing cases in exactly the same way, recur in Chinese sources from the second century B.C.E. till at least the seventh century. The next section of this article will show another example of this phenomenon. The way in which the practitioner used the text to derive lists of actions requires clarification. It illustrates how, behind what appears to be a list of operations, complex structures may be hidden. However, I cannot dwell on this issue here.
- 32.
- 33.
I am grateful to Professor Ma Biao, who has established that the reading of the character 石, when it designates a unit of measure for capacities, should be dan, and not shi as occurs in most Western sinological literature. I refer the reader to his forthcoming article on the topic. When the Book of Mathematical Procedures was composed, this character designated both the highest unit of capacity and the highest unit of weight used. In both cases, it read dan. There are reasons to believe that both units of measures are meant in the title of this operation and that they paradigmatically refer to the highest unit in a given series of units. The critical edition of the part of the Book of Mathematical Procedures that I analyze here can be found in Peng Hao (彭浩 2001: 73-75). Note that the manuscript found in a tomb was written on bamboo slips, which were discovered unbound. In such cases, the operations of the critical edition include suggesting an order of the bamboo slips. The order for the slips to which I refer is the one suggested by Professor Peng Hao. Below, we shall refer to two series of units. For the units of weight, the relationships between them are given in slip 47, as follows: 24 zhu for 1 liang, 384 zhu for 1 jin, (…), 46080 zhu for 1 dan. We can deduce the relationships between the units of capacity used in the Book of Mathematical Procedures from its text. They are, respectively, 10 sheng for 1 dou, 100 sheng for 1 dan. These values correspond to what contemporary sources attest to.
- 34.
To support my reconstruction of the use of the surface for computing, see my description in Chemla and Guo Shuchun 2004. Simply, I use Arabic figures in place of the configurations of counting rods with which in ancient China figures were written down on the surface. Moreover, for a more detailed discussion of the interpretation provided, see Chemla 2006.
- 35.
Note that the same term “divisor” designates different values at different points in the flow of computations. This is one of the many examples of the use of the “assignment of variables” in ancient Chinese texts of algorithms.
- 36.
The 1 by which the amount of cash was supposed to be multiplied will now be modified. This explains why I initially suggested not executing the multiplication immediately. This recalls how the text for division is formulated in The Nine Chapters.
- 37.
I owe this notion to Jacques Virbel, who took part in research in cognitive psychology on texts of instructions (private communication). Compare also J. Virbel, J.M. Cellier, J.L. Nespoulous (éds.), Cognition, discours procédural, action. Pôle Universitaire Européen de Toulouse & PRESCOT, Novembre 1997, p. 163; Cognition, discours procédural, action. Volume II. PRESCOT, Mai 1999, p. 308.
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Acknowledgments
It is my pleasure to express my deepest gratitude to John Holt, who had the difficult task of taming my English, and to Sarah-Jane Patterson who helped me in a crucial way to implement these changes. Without them, the paper would not be as readable as it has become. Nevertheless, I remain responsible for all remaining shortcomings. My most sincere thanks to Gila Hanna and Niels Jahnke, for their support and their patience in all circumstances!
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Chemla, K. (2010). Proof in the Wording: Two Modalities from Ancient Chinese Algorithms. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_17
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