Corps de l’article

I. Intervista Impossibile[1]

Bertrand Russell : I have just finished writing my Introduction to Mathematical Philosophy while I was in prison. In the process of preparing a trip to China to give some lectures on mathematical logic,[2] I would like to learn more about the philosophy of mathematics in China from you, as a practitioner. Let us begin with a simple question inspired by Frege’s Grundlagen der Arithmetik : how would you define numbers ?

Zhu Shijie : I wouldn’t do that, it’s useless in calculations.

Russell : All right, I see. Maybe I shall reformulate my question, starting from the simple. Since I assume you have a certain concept in mind of what a number is when you do mathematics, what about unity ?

Zhu : The Classic of Changes (Yijing 易經) says it is heaven, the Supreme Ultimate (taiji 太極).

Russell : I am confident that as a literati you know the Confucian Classics by heart very well, but I wanted to learn more about the philosophy of mathematics.

Zhu : May I ask what you mean by “philosophy of mathematics” ?

Russell : No embarrassing questions,[3] please, I am the one who shall ask them. Let us rather proceed further with the interview. So, according to your quotation from the Changes, unity is something foundational, but how about the other numbers ?

Zhu : There are no other numbers, unity is the only one, ten thousand things originate from it.[4]

Russell : If I understand you correctly, beyond unity you consider integers only, that is, a series of natural numbers from one to ten thousand obtained by adding “1” repeatedly ?

Zhu : From one make two, from two make four, from four make eight, the process of generation never ends. How could this not be of itself naturally so ?

Russell : If I am not mistaken, this procedure does not produce numbers other than powers of two, thus not even ten thousand ?

Zhu : Ten thousand are so many that we cannot even count them.

Russell : A bit confusing…, so your “naturally so” numbers are an uncountable finite set ?

Zhu : If you don’t grasp my meaning, dear Sir Russell, I suggest you look at the River Chart (Hetu 河圖) and the Inscription of the River Luo (Luoshu 洛書), the principles will immediately become apparent.

Russell : But these diagrams only show numbers from one to ten, how should they convey a general philosophical meaning ? Maybe you could elaborate on the kind of mathematical principles you see in these diagrams, such as definitions or axioms. As you might have heard, there are many debates on foundations of mathematics in Europe these days.

Zhu : Principles (li 理) do not serve the purpose of justifying truth, they are rather immanent to mathematical objects, whose meaning can be elucidated by recognizing their structural properties and patterns.

Russell : That is futuristic, Hilary Putnam will love this, mathematics without foundations !

Zhu : How dare you say that, you moron ! Our mathematics is well grounded, august Emperors from Zhou dynasty have laid out their basis. I feel truly insulted by you, Sir. Goodbye.

The fictive interview between two historical actors who have never met, Bertrand Russell (1872-1970), a British mathematician, logician and philosopher, and Zhu Shijie 朱世傑 (active around 1300 ce), a Chinese Yuan 元 dynasty (1271-1368) author of two mathematical books, illustrates the absurdities of searching for an equivalent of what only in the 20th century became the field of philosophy of mathematics. The above dialogue’s aim is not to juxtapose a Western philosophy of mathematics to an inexistent counterpart in China, neither is its purpose to ridicule attempts of contemporary historians to describe mathematical practices in China from a philosophical point of view based on a few cherry-picked traces : they would all agree that there ARE philosophical aspects to be found in mathematical texts and even more so in their commentaries.[5] When I say “philosophical”, I mean traces of both kinds of reflections : the object of mathematical inquiry and the mathematician’s toolbox, which are his methods, practices of argumentation and classification and linguistic tools and standards developed to speak about mathematical concerns. All of these aspects we do find built into the corpus of pre-modern Chinese mathematics,[6] yet, nothing explicit is found outside of it in the form of a meta-discourse on mathematics itself. A rare exception is Hua Hengfang’s 華蘅芳 (1833-1902) Brush Talks on Mathematical Learning (Xue Suan Bitan 學算筆談), a collection of notes and short essays on mathematical concerns. In one short essay on the principles of mathematical methods (suanfa zhi li 算法之理), Hua assumes that at a certain stage of development, we have primary cognitive faculties for supporting our quantitative senses, which, when fine-tuned by learning mechanisms, allow for more efficient processing of numerical problems. Hua juxtaposes “methods” (fa 法) with “principles” (li 理), the latter being a well-known term from philosophical discourses and of particular interest in this article with respect to mathematics.[7] As I will show for a certain mathematical domain, “mathematical principles” there are the abstract foundations of mathematical objects and tools, including argumentative patterns among the latter. Contrary to some authors who interpret the term as referring to a specific kind of argumentative style in 17th and 18th century Chinese mathematics,[8] I do not believe that there is such unique interpretation given to the term by the actors themselves, but a variety of meanings dependent on the specific contexts to which li is applied.[9] Hua, for example, relates his discussion to mathematical cognition by assuming that children know the “principles” before they can measure and count :

If in a person’s mind there is indeed ignorance and no awareness, then there is no need to discuss the learning of mathematics. But if there is a little awareness, and the person is able to think and argue, then he already possesses the principles (li 理) for mathematical learning. It is innate, just try to observe children at play : when they see the fruit, they will necessarily strive for the biggest one, because their brain already has the capacity to perceive magnitudes. From this we know that the principles of mathematical learning (suanxue zhi li 算學之理) are inherent to our minds (ren xin suo zi you 人心所自有) and do not come from outside (zi wai 自外). Therefore, if we select some not very arduous problems from a mathematical book, those who were not trained in mathematics with words, can also, by bringing together their thoughts, find the number that is asked for. Yet, when the degree of difficulty is gradually picking up, then it is much easier for those who master mathematics. That is because if for the calculation one does not yet have a method, then all the numbers need to be entirely reckoned by applying one’s mind. Necessarily, this is truly difficult ! But if one knows the mathematical method (suanfa 算法), then no matter how the numbers are set up, everything can be solved by applying the method and it is not necessary to apply one’s mind to it in order to reduce the work and obtain twice the effect. I believe that any mathematical method at the outset entirely emerges according to mathematical principles (suanli 算理). It is only once that the method is obtained that principles do reside within the method (li ji yu yu fa zhi zhong 理即寓于法之中). One can, by engaging with a method, obtain the principles, and one can also by setting aside the principles apply a method. If the method does not fail, then the principles cannot be erroneous either ![10]

To clarify what “principles” concretely refer to in one specific field of mathematical inquiry, I will focus in the following on a branch of mathematics that evolved into a true discipline in the 19th century : “discrete accumulations” (duoji 垛積). Involving natural numbers only, my discussion will elaborate upon all aspects raised in the introductory dialogue. I will first explain what “discrete accumulations” are and show how through structural adjustments of text (i.e. by changing the order of certain problems) they turned into an object of mathematical inquiry. In a second step, I will analyze reflections upon a mathematician’s toolbox for dealing with these “discrete accumulations”. The main historiographical argument of this paper is to show that philosophical aspects of mathematics in China are sparse in their manifestation and dispersed between texts, paratexts and images, borrowing concepts from other Chinese contexts of philosophical nature. My argument builds upon two such concepts which play an important role in the field of “discrete accumulations” : “principles” (li 理) and “comparable categories” (bilei 比類).

General “principles” underlying mathematical objects and procedures there refer to common structural patterns in diagrams and procedures related to sequences of numbers. Recognizing these patterns, as some authors claim, allows to conjecture general procedures by analogy and (incomplete) induction. Although not explicitly spelled out as a valid mode of argumentation, by the 19th century, standard linguistic formulations and diagrammatic codes attest of an established set of discursive and visual elements in mathematical writings for expressing common patterns in mathematical objects and procedures of “comparable categories”. Such scholarly tools set forth to deal with “discrete accumulations”, as I will show, can be considered the reflection of philosophical considerations on the very nature of the underlying mathematical “principles” common to an entire set of mathematical objects and procedures.

II. “Discrete Accumulations” as Mathematical Objects

Li Shanlan 李善蘭 (1811-1882), the author of a 19th century treatise entitled Comparable Categories of Discrete Accumulations (Duoji Bilei 垛積比纇, 1867) sees his own work on the kind of mathematical object designated by “discrete accumulations” as the establishment of a true field of mathematical knowledge outside the canonical tradition :

I want those who learn mathematics to know that the procedures for discrete accumulations erect another flag beyond the Nine Chapters. Their theory has begun with no other than myself (yu ling xi suanjia zhi duoji zhi shu yu Jiuzhang wai bie li yi zhi, qi shuo zi Shanlan shi 欲令習算家知垛積之術於九章外別立一幟,其說自善蘭始) ![11]

Li’s book relates to summations of finite series, but research on this mathematical subject was far from being a novelty in his time. Since the canonical Nine Chapters, it has been approached in different contexts, within geometry first under the Han 漢 (202 bc-220 ad), but also in astronomical and algebraic contexts under the Yuan. The very expression “discrete accumulations”, or literally “accumulated heaps”, gives a name to a real field of mathematical research in pre-modern China. The choice to translate the expression by “discrete accumulations” reflects the strong links that this field has with geometry, and in particular with figurate numbers. The idea is to accumulate unitary — and therefore countable — elements by forming certain geometric objects whose volume (ji 積) is known for the case of a continuous space.

1. Shared patterns underlying structural principles

The first seeds of this type of rapprochement of the arithmetically discrete and the geometrically continuous can be found in one chapter of the Nine Chapters. A series of twenty-two problems asking to calculate the volume of certain geometric shapes is followed by three problems where grains are piled up in a corner or against a wall. One asks for the volume of the heap — not for the number of grains piled up. In the solution the same procedure as for a circular cone given earlier in the chapter is applied. This fact is mentioned in the commentary by Liu Hui 劉徽 (263). What seems like a simple side remark turns into a central conceptual stance in light of the later history. Looking for example at Song 宋 dynasty (960-1279) commentaries one is rather inclined to read Liu Hui’s statement as a testimony of classificatory reflections upon mathematical objects : although a heap of grains and a continuous volume is constituted differently and thus has a different inner structure, yet both mathematical objects share the same procedure for calculating their volume.

Here is the later history of “discrete accumulations” in more detail to confirm my point : It is actually under the Song that a more systematic elaboration by Yang Hui 楊輝 (ca. 1238-1298)[12] is attested in the transmitted sources. Like Li Shanlan later on, Yang proceeds by “comparable categories” (bilei 比類),[13] for example, when he juxtaposes the calculation of the volume of a pyramid with a square base with the summation of the square numbers. In Yang Hui’s work, objects are classified and explained by being compared or assimilated to each other, and the related problems are then solved by similar or even identical procedures. His book Detailed Explanations of the Nine Chapters on Mathematical Methods (Xiangjie Jiuzhang Suanfa 詳解九章算法), printed in 1261, therefore contains mathematical problems from the Nine Chapters which are rearranged and supplemented by problems of “comparable categories”. The following example illustrates how the procedural similarity between the new analogous problem and its counterpart in the canonical book is established.

While the Nine Chapters provided the procedure for a problem asking to calculate the volume of a truncated pyramid with a rectangular base, called chutong 芻童 (lit. a haystack), Yang Hui added a problem of “comparable category” in which the same geometrical shape (except that it is mirrored horizontally) is not a solid volume but constructed from discrete elements. The question of the problem is thus no longer what would be the volume of the resulting solid, but rather what is the total number of unitary objects that are stacked in the form of a truncated pyramid with a rectangular base.[14] Yang Hui even explains in the solution procedure, which sequences of operations remain unchanged, and which operations modify the original method[15] :

  • Suppose we have a chutong. Its lower width [a] is 2 zhang, its length [b] is 3 zhang, its upper width [c] is 3 zhang, its length [d] is 4 zhang, its height [h] is 3 zhang. The question is : what is its volume (ji 積) ?

    The answer says : 26500 chi.[16]

    Explanation of the problem : [the shape] is similar to an observation platform, lengthened in length.

    The calculation says : Doubling the upper length of 4 zhang makes 80 chi. Adding the lower length of 30 chi makes 110 chi. Multiplying this by the upper width of 30 chi, we get 3300 chi. Doubling the lower length of 3 zhang makes 60 chi. Adding the upper length of 40 chi makes 100 chi. By multiplying this by the lower width of 20 chi, we get 2000 chi. By adding the two positions together, we get 5300 chi. Multiplying this by the height of 30 chi gives 159000 chi. Dividing this by 6 gives 26500 chi. This corresponds to what has been asked for.

    The comparable category (bilei 比類) : a pile of seeds. An upper length of 4, a width of 2, a lower length of 8, a width of 6, a height of 5. The question is : how many in total ? The answer is : 130. The method says : Double the upper length, add the lower length. Multiply this by the upper width and you get 32. Also, double the lower length, add the upper length. Multiplying the lower width by this value gives 120. When the two positions are added together, the result is 152. This is the original method that governs the volume of a chutong. With the upper length, we reduce the lower length, we also add the remaining 4. A stack of seeds is therefore not equal to a circular object, nor to a rectilinear volume. This is why this term must be added. By the height, we multiply this and [we get] 780. The division by 6 is also part of the original method of a chutong.

The procedures indicated in the text above correspond to the following calculations for the volume of the chutong in the continuous case :


and to :


for the accumulation in the “comparable” discrete case. Due to the parallel formulation of the operations to be carried out in both cases, the corrective term (b − d) for the discrete case with respect to the continuous can easily be recognized.[17] This type of textual organization, where old mathematical problems are confronted with new mathematical objects and rearranged according to procedural similarities and structural analogies, is more systematically applied by Yang Hui in his book, Rapid Methods of Multiplication and Division Compared to Categories of Fields and [their] Measurements (Tianmu Bilei Chengchu Jiefa 田畝比類乘除捷法) of 1275.[18] When Zhu Shijie included in 1303 in his Jade Mirror of the Four Unknowns (Siyuan Yujian 四元玉鑑) four chapters entirely devoted to the problems of “discrete accumulations”, the mathematical domain under the same name took on well-defined contours.[19] Even more so under the Qing 清 (1644-1912), when an important number of commentaries were written following the rediscovery of Zhu Shijie’s treatise. The field of “discrete accumulations” became one of the major themes of mathematical research, partly influenced by Western mathematics from the end of the 19th century on.[20]

Table 1

Chronological list of books placed in the category “Discrete Accumulations” of the Combined Commentary on the Bibliography of Mathematical Books, Ancient and Recent and the Preliminary Compilation of Mathematical Studies (Gujin Suanxue Shulu Suanxue Kao Chubian Hezhu古今算學書錄算學考初編合注), 1957.[22]

-> Voir la liste des tableaux

The above list gives some titles of works which, by the actors themselves or in the Biographies of Astronomers and Mathematicians, are declared to be part of the field of research on “discrete accumulations” (see table 1). But besides having in common from a pragmatic point of view the same object of study, what was it that provided unity to this field ? In absence of an explicit answer provided by the compilers of bibliographies, it is best to turn again to the mathematical practitioners.

2. Interconceptuality between mathematics and philosophical traditions

Li Shanlan himself is said to have made a distinction between numbers (shu 數) and principles (li 理), claiming that “although numbers have a myriad transformations, their principle alone is the fundamental procedure !” (shu you wan bian li wei yuan shu 數有萬變理惟元術).[23] Confronting this paradigm with Li’s work on Discrete Accumulations, it seems indeed to be underlying his conception of finding a unique, general algorithm for the different kinds of number sequences in the diagonals of the kind of arithmetic triangles which he presents : based on the first four or five summation procedures, Li Shanlan systematically asks the reader to induce the general method by analogy on the basis of “comparable categories” (bilei 比類),[24] thus assuming that there is a single pattern underlying all of them. As mentioned above, the expression bilei has already been at the core of Yang Hui’s way of organizing continuous and discrete geometric objects into procedurally connected pairs.[25] Li Shanlan goes a step further by looking at more than two objects that share a common algorithmic pattern and inductively seeks a general procedure for an entire sequence of objects.

That classes of mathematical objects do share a common principle, is not a philosophical topos restricted to “discrete accumulations”, neither is the habit of relying on terms and concepts borrowed from other fields of intellectual inquiry. Combinatorics, indeterminate analysis, or, as the example of conic sections below will show, were all domains that fell for example back on notions from the Changes.[26] When Xia Luanxiang 夏鸞翔 (1823-1864), a contemporary of Li Shanlan, gives a summary of his 1861 work on conic sections, he introduces the subject matter of his book as follows :

Heaven is big and round. When describing the objects of heaven, there is none that does not relate to the circle. Although “circle” is only a unique name, there are a myriad of species. When following the circle for one round, curves are generated. Westerners divide [the different kinds of] lines according to the order of generation into the following categories : lines of the first order, [this class consists of] the straight line only. Lines of the second order, [this class comprehends] four species : the circle, the ellipse, the parabola and the hyperbola. Lines of the third order have eighty different kinds. Lines of the fourth order have more than five thousand kinds. Beyond the lines of the fifth order, it is such that it is impossible to investigate them.[27] Here, I explore the four kinds of lines of the second order and trace their origins, and additionally, I add explanations to all the parabolas of higher order.[28] Although their forms have a myriad variations, their principles are from a single strain (xing sui wan shu li shi yi guan 形雖萬殊理實一貫). All the equations of conics are entirely provided for on the solid [i.e. the surface] of a cone. That is the reason why the cone is the mother of the curves of the second order.[29] For the ellipse one uses congregation, for the parabola one uses extension, for the hyperbola one uses dispersion, but their principles all stem from the plane circle. If indeed, we unite what they have in common, then we “build tools to imitate the cosmos” (zhiqi shangxiang 制器尚象).[30] By “bowing [to the earth] and looking upwards [to the heaven], by observing [the sky] and investigating [the ground]” (fu yang guang cha 俯仰觀察),[31] their applications are without limitations (weiyong wuqiong yi 為用無窮矣).[32]

It is perhaps surprising that a Chinese mid-nineteenth-century mathematician who was familiar with translations of English works on analytic geometry, differential and integral calculus, as well as with the works of his Chinese predecessors, frames philosophically a complex mathematical topic that he deals with using a technically original and novel approach. That Xia wanted to promote his work among Confucian scholars with a preface that proved that he was versed in the classics and familiar with the Classic of Changes, is not unusual in paratextual discourse. But Xia certainly also wanted to understand mathematics as a numerical science that studies change. Analogous to the transformations of broken into unbroken lines in the divinatory hexagrams, conics, as he underlines in the above-quoted passage, can be obtained from a single origin, the cone, and transformed one into another : the ellipse can be obtained by joining together the two extremities of the parabola, and if one extends one endpoint of the major axis of the ellipse to infinity, it is transformed into a hyperbola.[33] Yet, this was clearly a philosophical thought experiment, since strictly speaking mathematically, in order to perform a mapping from an ellipse to a hyperbola or parabola a projective transformation is needed. No affine transformation can change a bounded curve into an unbounded one.[34]

Xia’s statement that “although their forms have a myriad variations, their principles are from a single strain” (xing sui wan shu li shi yi guan 形雖萬殊理實一貫) closely relates to the similar phrase by Li Shanlan quoted at the beginning of this section, “although numbers have a myriad transformations, their principle alone is the fundamental procedure” (shu you wan bian, li wei yuan shu 數有萬變理惟元術), yet on a more visual note. The myriad manifestations of a single entity are a common trope in both, Confucian and Daoist philosophy,[35] it is thus difficult to link these statements to any specific school of thought. The kind of intertextuality observed for both authors nevertheless confirms that mathematics was not an isolated knowledge system but participated in processes of conceptual transfers, and even possibly influenced the formation of technical terms in the late Qing which translated foreign concepts. Induction, as a 1913 Dictionary of Philosophical Terms shows, was then translated as wan shu yi ben 萬殊一本, literally meaning “a myriad transformations, a single origin”.[36]

III. A Mathematician’s Toolbox for “Discrete Accumulations”

To understand better the philosophical implications of “principles” (li 理) specifically in mathematical writings, one needs to go beyond narratives and include their visual elements, too. In the above impossible interview with Russell, Zhu Shijie’s statement about the visibility of “principles” in diagrams (tu 圖), and thus about a cognitive foundation for understanding mathematical knowledge, was not entirely fictional. In the text that accompanies a diagram showing the geometric configuration obtained from squaring the magnitudes associated to four unknowns at the beginning of Zhu’s Jade Mirror (1303), he concludes by saying that “if one studies the diagram, one will recognize this.[37] Its principle is of evident nature” (kao tu ren zhi qi li xian ran 考圖認之, 其理顯然).[38] Turning more specifically to figured numbers in China and their representations, one can thus ask what visualized “principles” signify in a number theoretical context.

Visual representations of numbers are found abundantly since the Song period in philosophical writings, most prominently in relation to The Chart of the Yellow River (Hetu 河圖) and The Writ of the Luo River (Luoshu 洛書) diagrams,[39] in which numbers are shown as composed of black or white pebbles, each corresponding to a unit.[40] Beginning with Yang Hui, figured numbers, so-called “heaps” (duo 垛), appear in diagrams as “comparable categories” to continuous plane geometric objects, so-called “fields” (tian 田) in the shape of squares, circles, triangles and trapezoids.[41] Next to the diagrams in figure 1 we are told that “for all [the surfaces shown] the method for the trapezoidal field can be applied. One does not need to establish yet another problem with a calculation sketch”. In the absence of further explanations, one might ask what justifies, for example, that a topsy-turfy trapezoid or the number of pebbles in the triangle composed of lines with 1 to 7 pebbles are calculated by the same procedure as the surface of the trapezoid.[42] That Yang Hui has grouped together all four diagrams seems indeed to imply that he intends to show a structure and properties common to all of them.

Figure 1

Simple Methods for Multiplication and Division with Comparable Categories in Field Measurement, in Yang Hui’s Mathematical Methods 楊輝算法 (1275)

Simple Methods for Multiplication and Division with Comparable Categories in Field Measurement, in Yang Hui’s Mathematical Methods 楊輝算法 (1275)

-> Voir la liste des figures

As for three-dimensional figured numbers, algorithms for calculating the number of wine jars piled up in the shape of a truncated pyramid with a rectangular base were first stated by Shen Gua in the 11th century.[43] Although such algorithms circulated widely in Ming 明 (1368-1644) dynasty mathematical manuals, even in versified form for easy memorization,[44] I have not found a single illustration of such three-dimensional forms before the Essence of Numbers and their Principles Imperially Composed (Yuzhi Shuli Jingyun 御製數理精蘊, 1723), a compilation integrating Chinese and Western mathematical writings and discursive forms.[45] Naturally, representing integer sequences of higher order by geometrical shapes has its cognitive limits. Hua Hengfang admitted this, but doomed diagrams as unnecessary tools for making apparent mathematical “principles” once algebra as an all-encompassing technique is used to express and manipulate symbols and formulas :

Among the ancient mathematical books, none of them draws many charts. In the Nine Chapters, there is only the diagram of base and height [in a right-angled triangle]. By eliminating the blue [surfaces] and inserting the red ones it is explained that the sum of the two squares of base and height is equal to the square on the hypotenuse. For all the other procedures no [other ancient mathematical book] resorts to a diagram to elucidate it. […] The problem is that if one uses diagrams to make calculation principles apparent, one can only do so up to solids, yet, this is the end. For higher dimensions, one cannot draw diagrams. For that which diagrams cannot make evident, mathematicians have additionally invented algebra. Among all pieces of segments there is none that cannot be elucidated by a mathematical formula. Therefore, what in ancient times could not be elucidated without a diagram, today can be done so without necessarily using a diagram. What in ancient times no diagram could be drawn for, today can be expressed by algebraic formulas, replacing the function of diagrams. In my mathematical books I never draw diagrams, the reason being also that I use algebraic formulas.[46]

Besides the potential possibility or impossibility to express “principles” verbally or visually, there is another important connection between diagrams and text with respect to philosophical aspects to be found in mathematical writings. It relates to textual and visual practices of inductive argumentation. I have shown that pebble diagrams, by depicting a geometric interpretation of sequences of natural numbers, form an epistemological unity with the accompanying text in Wang Lai’s essay The Mathematical Principles of Sequential Combinations (Dijian Shuli 遞兼數理).[47] Both are persuasive representations of a procedural aspect that establishes a recursive relationship between the mathematical objects that are either depicted or constructed rhetorically.[48] Each item is produced out of the previous result through specific operations. By decomposing geometric figures into layers of unitary elements, the structure of the successive diagrams refers to the patterns of recursive algorithms and inductive arguments.

In contrast to Wang Lai, explicit meta-discursive elements are entirely absent in the case of Li Shanlan. The analogy between diagram and text rests entirely on structures : Li shows the first few numbers in the first few diagonals of a triangular table with unitary pebbles geometrically arranged, and, for each of these first diagonals, gives the procedure that calculates the sum of the first n terms. As for the latter, the reader is asked to infer inductively by analogy (lei tui 類推) the general summation algorithm valid for all diagonals, whereas for the former, the invitation is visually extended : may the reader continue the pattern by piling up further unitary building blocks. Figure 2 (left), for example shows in the right column the squares of 1, 2, 3 and 4, the left column the squares of 1, 1+2, 1+2+3 and 1+2+3+4. These are the first numbers in the second and third diagonal (from top) in the corresponding triangular table (see figure 2 on the right, cells colored in red and blue). The text which follows then gives the corresponding algorithms for calculating the sums of 12+22+32+42+… +n2, 1+(1+2)2+(1+2+3)2+(1+2+3+4)2+…+(1+2+…+n)2 as well as the algorithms for the next two series before stating that “beyond these one can proceed by analogical extension” the general procedure for the sums of the first n cells in any diagonal of the triangle.

Figure 2

Piles of squared triangular [numbers] (Li Shanlan,Duoji Bilei, 1867, juan 3, p. 1b) and corresponding triangle

Piles of squared triangular [numbers] (Li Shanlan,Duoji Bilei, 1867, juan 3, p. 1b) and corresponding triangle

-> Voir la liste des figures

A slightly earlier author and friend of Li Shanlan, Dai Xu 戴煦 (1805-1850), in his 1844 commentary to the Jade Mirror of Four Unknowns has equally contributed to the field of “discrete accumulations” by providing diagrams of the mathematical objects involved. Yet, he does neither figure in the above list (table 1) nor is he referred to by Li Shanlan in spite of the fact that Dai might have been a source of inspiration for him.[49] Dai indeed proposes an approach to Zhu Shijie’s series by diagrams similar to the ones found in Li Shanlan’s work. An example from Dai’s manuscript allows to illustrate the role which diagrams could play in depicting justifications of procedural patterns even beyond an inductive scheme.

-> Voir la liste des figures

Considering the sequence of even numbers 2i (i = 1, … n), each taken four times : , Dai visualizes the sequence of these terms for by “empty” squares with each side composed of an odd number of pebbles. The sum of these empty squares, arranged around an initial unity at its center, visually gives a “full” square where each side is equal to . One might thus conjecture a first formula of summation :

equation: 5059829.jpg

or[50] :

equation: 5059830.jpg

By observing in detail the “full” square on top, one recognizes a second “formula” : the square is composed of six triangles with , a big triangle with and a small triangle with units at its base and as its height. One thus also recognizes the following identity :

equation: 5059831.jpg

It is precisely this last type of “formula” that Li Shanlan establishes (in procedural language) for the sums in the diagonals in his generalized arithmetic triangles. The “principles” of mathematical procedures, provided without a discourse of justification, are thus visualized in the diagrams. If extended to the general case, i.e. when the pattern has been recognized, these diagrams can play the role of a proof without words. That such was indeed the intension of the authors, we can only argue on the ground of statements such as the one quoted above by Zhu Shijie, claiming that “the principles are of evident nature” if one studies the diagrams. The massive presence of pebble diagrams in Dai Xu’s and Li Shanlan’s work would certainly confirm the importance attached to the epistemological function of visualization as a tool for justification (and discovery).

Another kind of stacked numbers systematically presented and strategically placed within the mathematical writings relevant to “discrete accumulations” are the numerical tables in triangular shape (cf. fig. 3). These triangular diagrams display a high degree of freedom as concerns the direction with which they were to be read : the lines connect the cells with the neighboring cells above, below and on their sides (if there are any). Yet, what seems like an interpretative ambiguity is productive mathematically.[51] Read horizontally or diagonally, the tables do not allow to produce the same mathematical meaning : binomial coefficients in one way, arithmetic series in the other, the manifold “uses” of arithmetic triangles have been developed systematically in the West by Pascal in his Traité du triangle arithmétique (1667). Pascal, as well as Zhu Shijie and Li Shanlan, place the diagram before or right at the beginning of their book, thereby making manifest its crucial necessity as a foundation for the methods to follow.

Figure 3

The “Pascal Triangle” in Zhu Shijie, Siyuan Yujian, plate 1 of the preliminary diagrams (left) and p. 1b following Li Shanlan’s preface (right)

The “Pascal Triangle” in Zhu Shijie, Siyuan Yujian, plate 1 of the preliminary diagrams (left) and p. 1b following Li Shanlan’s preface (right)

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Other arrangements of such triangular tables are also to be found in other Chinese mathematical texts. Among the authors listed in table 1 and mentioned by Li Shanlan in his preface as one of his predecessors, for example, is Dong Youcheng. Compared to Zhu Shijie, Dong does not provide any new summation “formulas” but he does change the layout of integer sequences in a diagram, arranging them into rectangular tables.[52] Others, like Fu Jiuyuan, even integrate verbally the combinatorial interpretation of arithmetic series into the diagram of the arithmetic triangle. By arranging differently the cells for each term (cf. fig. 4), Fu indicates horizontally and to the right of each cell the combinatorial meaning of each numerical value “k of n” (n k), which is equivalent to the binomial coefficients equation: 5059923.jpg.

As in the pebble diagrams, a general pattern of generation of the content of the represented cells can be induced from the limited number of cells depicted. The structure of and the relation between the numbers involved are therefore made apparent, even if they are not accompanied by any conceptual discourse.

Figure 4

Fu Jiuyuan 傅九淵, Illustrated Explanation of Heaps of Piles (Duiduo Tushuo 堆垛圖說), 1888/1889.[53]

Fu Jiuyuan 傅九淵, Illustrated Explanation of Heaps of Piles (Duiduo Tushuo 堆垛圖說), 1888/1889.53

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Coherent traces of epistemological assumptions about the “principles” of mathematical objects and procedures have been shown to exist in the so-called field of “discrete accumulations”, or what in the West is known as figured numbers. A system of classification and of reasoning in textual and diagrammatic form evolved historically for these mathematical objects until the turn of the 20th century, influenced by concepts that were common in philosophical discussions about ways of knowing outside of the mathematical domain of inquiry. While in ancient Greece questions on foundations of mathematics were treated by Plato and Aristotle within the framework of metaphysical reflections, no explicit discussion of the nature of mathematics, its objects and tools, is recorded in the transmitted sources from China. Yet, a philosophical interest in cognition, representation and justification is evidenced in normative narratives about mathematical objects and their visualization in the mathematical literature itself. The imagined dialogue between Russell and Zhu Shijie reflects these two kinds of philosophical narrative with respect to numbers : while Russell discusses explicitly and spontaneously his own ideas about the nature of numbers, Zhu Shijie responds through quotations from mathematical texts, refers to commonplace diagrams or phrases, and thus implies philosophical content by letting the reader engage with these elements.