11.1. Mathematics: specific use of an operating mode

Speaking of the difference between a physicist and a mathematician: “A mathematician also deals with these things, spheres, but not each as the limit of a natural body. Nor does he study their attributes in natural bodies. Nor does he study their attributes as they are attributed to such ‘natural’ beings. This is also why it separates them, for they are separable from movement of thought, it makes no difference and no error is even produced by separating them” (Aristotle, Physics II, 2, 193 b).

“A mathematician is someone who applies, teaches or creates mathematics. It is a science in which one deals with describing, understanding and acting on the world; in which the object of study is abstract, and for which knowledge accumulates through the tool of evidence obtained through deductive reasoning” – Cédric Villani.¹

“In the past, there was Baoxi, who, firstly, drew the eight trigrams to communicate with the abilities of clairvoyance and illumination in order to classify (lei) the situations (qing) of all the existing ones, then created the procedure of the multiplication table so that it was in concordance with the mutations of the six lines (hexagrams)” – Liu Hui, Classic Chinese Mathematics.

Like the linguistic and musical MO.O.N., we will not take the path of what mathematics is or what a mathematician is. Primarily because it would lead us to slide down the explanatory slope and, consequently, the exploration of knowledge relative to the word, but also, because we would open a door behind which the objects of thought have no limit (useless for our purpose). Nor is it a question of amalgamating the operating mode called “logical-mathematical” and the mathematician’s name referring to an activity or a trade. However, because of the importance of mathematics as the central (or almost central) “point” of the so-called Western scientific thought, it seems difficult to us to economize on a few detours. Moreover, it would be restricting to only think of mathematics from the Western point of view and to oust the Chinese mathematical mind, hence our investigation of “that side” of thinking. The operator of the company, whoever they are, can bear in mind that the treatment given to this MO.O.N. (like the others before) has the sole intention of apprehending the way in which dynamic model principles shape, mold, drive, tilt, obstruct and accelerate “performance”. No MO.O.N. is “good or bad”; as by definition, they are all sometimes positive and sometimes negative, and one or the other of these tendencies is always linked to the C.U.P., in other words, following the trend in relation to the expected result (positive) or deviating from it, leading to a risk of embedding and rigidity, and consequently a decline (negative). Natural operating modes could be understood as tea in its natural state. They are neither able to judge themselves, to appreciate themselves or to define themselves. It takes the Other and usage in order to “qualify” a MO.O.N. (tea): for... is it useful all the time? Thus, the figure is worth something as if someone thinks it, figures it or sees it (idein). Who remembers that minute “minutae” is called “vulgar fraction” for Babylonians? Is it useful to know? Who remembers that Galileo (Galileo Galilei), at the dawn of his twentieth birthday, used his knowledge of mathematics and physics to translate the duration of his pulse beats, while he observed the oscillation of the lamp over the cathedral of Pisa and placed the foundations of a universal measurement? Competency no longer signifies knowing what “it is” or what it “means”, or even whether it meet the standard (or the law), but how “I” think of the question, that is, how and from where my language and its principles led me to “believe” and think what I wish to elaborate?

Western mathematics, a discipline of thought, but also a method of cultural thinking, has spread for more than 2000 years in the West and in China. However, the operating mode has continued on a distinct incline for both. Indeed, contrary to our “thought”, Chinese logic has never taken the path of a pure activity of the mind (nous), but rather, as Marcel Granet says, it describes reality (functional logic):

“The numbers are remarkable, like the emblems, for their versatility conducive to efficient handling. […] Numbers are used to classify because they can be used to situate and represent in a concrete way. These are emblems. First of all, they are given real descriptive power” [GRA 34].

The study of many authors and mathematicians shows that there are as many mathematical concepts developed as there are mathematicians. Cédric Villani, for example, presents mathematics (see two interviews²) by the following enumeration: describe, understand and act on the world. He clarifies notions of the “appropriable” as well as the reconstitution of the entirety of a reasoning of the mind, that is, one can verify everything said and appropriate the mathematical object. During his Tout est mathématique (Everything is mathematical) lecture at HEC, he emphasizes the gap between the mathematician (which he specifies: surveyor) and the physicist:

“On the one hand, the abstract and disembodied character of mathematical science which goes hand in hand with a formidable efficiency which, among the brightest minds throughout the centuries, has amazed many; precisely linked to the fact that something abstract will be able to be embodied in many different realizations and situations. And bringing a non-intuitive glance on certain objects [...] this new glance makes it possible to have access to phenomena qualitatively finer and richer than that which you would have simply by being satisfied with qualitative and phenomenological observations”.³

For André Lichnerowicz, the mathematician is “a craftsman who learns by knocking their own mind”.⁴ As for Alain Connes, he says that in order to illustrate the dynamics of a mathematician, the latter chooses a theorem in a book and must absolutely not read the work so as to not remain passive in front of the work done by others, and then must ask themself: “How the hell can I prove that?”. After checking the result in the book, the mathematician detects a small piece of information (about the whole) causing a change in mental image⁵. They then finish by saying “the true way to understand is to see the statement bringing a mental image to mind. You think about it for a while, then you come back and look”⁶.

We stress, if it is necessary, that the “abstract-general” mode of operation is expressed by the word “mathematical” (mathesis), which originally means “what can be taught-learned”. The mathematical word (signifier), as a semantic-consonant invention (concept), cannot (therefore) be elevated to the rank of reality (physicality). This widespread and reckless assertion with regard to social implications says that “all” the world “possesses” this language as a universal constant, or that “everything is mathematical”. This statement is false because not everything is “mathematical”; everything is, at best, mathematical from the point of view of the “pure” surveyor. Before being the manifestation of an operating mode of the mind (a skill), mathesis is organized into an abstract symbolic system (similar to ideograms); it translates objects of thought into “absolute” and/or functional principles. Perhaps it is worth remembering that this discipline of the mind (nous) accumulates nearly 2600 years of “geometrical-mathematical” trial-and-error; it is acceptable (or even obvious) that this “way” of thinking can produce such a variety of mental complexity that it may come to believe that “everything is mathematical”. “Lastly, is it necessary to impose” the principle of “wise caution” on oneself (phronesis), as indeed, the history of mathematics, like all stories, has a “beginning” (a primer) for which there was only “an” idea, then a hypothesis, then an application, then the beginning of a design, then principles, then rules and laws, etc. This “idea” began around 650 BC, at a time when Hoplites, both a warrior model developed on a geometric principle (phalanx) and a social model where every man in the city, whatever his function and privileges, took his place in the phalanx when there was a battle (citizen-soldiers). The “Hoplitic reform” [VER 99, p. 10] brought about a profound change in the use of words, on the one hand, because it became an “autonomous object subject to its own laws⁷,” and on the other hand, because its use was organized into two main functions. The first was a social function and was developed through a tool, promoting social relations (logos) through debate (rhetoric, sophisticated). It was also an instrument to define reality; the reality which mathematicians and geometers discovered and for which language invented the language of mathematics (as a system of homogeneous and delineated signs). Thus, speech and numbers were organized in languages that shared the same intention: to know (participating in a universal order).

The history of Chinese mathematics through the main works, the Book of Chinese Mathematical Procedures (Suanshushu), the Gnomon of the Zhou (Zhoubi) or The Nine Chapters, seems to follow a similar principle, but whose mental path is something “other”; here, “to know” (xing) is to be taken in the sense of experience and the consecutive teaching (jia) of reality. The Mathematical Classic (Jiuzhang suanshu) reminds us of this opposite whose “abstract-general” practice is just as real, although distanced from its common ground of putting forward a problem (wen, 問) or agreeing on the method of treatment (dao, 道). This “abstract-general” form of operation has many applications, and the “pure” use of Western mathematics is one of its most “visible” manifestations. Lastly, let us keep in mind that any statement is, by definition, “false”, especially if it claims to be “true” (alètheia, power of efficiency).

IMPORTANT.– From reality, we only have our human interpretations.

11.2. Mathesis: pure logic

At the risk of exaggerating the feeling of caution that should be cultivated, we specify that we are not “comparing” two cultures in order to show something that one would do better than the other, but how the common ground of each generates an “abstract-general” type of operating mode that millennia settles for. Our focus remains on the modus operandi, in other words, the process by which a specific observable-assessable result is called “logical-mathematical”. Let us keep in mind that our Greek culture has led our thinking (paradigm) to lean towards the incline of ideas, of the mind, of the Truth and the demonstrable. Plato, as the “first” perfect geometer (mathematician), participated in founding a method (methodos), a “right and fixed opinion” of which it is possible to think of science by demonstration, that is, the chain of reasoning by which every “thing” is what it is, by what it has determined and fixed. Thought is therefore objective, and objectivity defines knowledge (epistemus). Knowledge, thanks to reason (doxa), can then produce a true reasoning, that is, it is supported-constructed by reasoning and calculation (logismos). Reasoning makes it possible to both think in a precise order (coherent and logical), and to also know by proceeding according to a rigorous principle or deduction, as deduction-knowledge implies “knowing the causes of things” (rerum cognoscere causas) and consequently explaining (explicare). The idea is the “starting point”, and the starting point is the “cause”. Those who know causes, possess “knowledge”.

The mathematician, as a subject who thinks, is the one who “sees” the decorated objects of reality in his/her mind. Euclid, who is part of an ancient tradition initiated – among others – by people such as Proclus or Pythagoras, elaborates a so-called demonstrative science whose functioning is organized into two principles: the axiom and proof:

“Any demonstration of mathematics is either problem, or theorem. What mathematicians call a problem, is a demonstration that instructs on making or constructing something: and theorem is a demonstration that only seeks some properties of one or more quantities together. In short, problems instruct on finding and constructing something and theorems demonstrate the affections and properties of things already constructed. Both the problem and the theorem are called proposition, all the more so since both of them propose something to us”⁸.

His subject evokes the intellectual “conflict” between physicists and mathematicians (which is still “present”, as it seems, among purists):

“Physicists say that the point is the slightest object of sight and that it can be described in some way or other. But mathematicians who reject this definition say that the point is such a subtle object of the thinker, that it cannot be divided into any part, and it cannot be written; only heard and imagined. Although it is true that in order to represent it to our external senses, we use the physical point”⁹.

In order to “save time” and economize a long and unnecessary development here, let us examine the process that led to geometric thought.

From the point of view of the “mathematical” operator, everything that is advanced must be verified through reasoning and proof (logos). What begins, with Plato, will be updated in the “theory of forms”. Platonic thought elaborates an ideal thought through “ideal” objects ([VER 09, foreword]); it renders visible a specific skill of what is called mathematical: the skill to mentally transform objects. From an operational point of view, this process involves memorizing long chains of reasoning, themselves organized into objects decorated with a real (physical) form; this is what is called “pure intellection” (noesis). Thus, the chief (the geometer) is the one who knows not of the sensitive experience (knowledge) specific to Homer and consequently to Odysseus, the polûtropos (the cunning, the sprawling); Plato, in the line of his ancestor Solon, crystallizes the virtues of morality and truth by making geometry and mathematics the one and only discourse (true logos). Anything that does not correspond to these “principles”, such as the inherent ability of the rhetorician (the one by whom the artifice of language induces lies in speech) to be sophistically polymathic or polytechnic, deviates from it.

To understand the “first” committed mathematician (Plato), not in terms of practice, but in the implementation of this form of thought resulting from an operating mode, all you must do is remember the contempt that the latter has for Odysseus, the “wanderer”: “All wanderers [alêtai] invent lies, each in their own way; truth is the least of their worries”¹¹.

Thus, rather than wandering (not Being), he prefers straight and fixed knowledge (Being), to the approximate knowledge of situations (metis), to the method of discourse, to the technique inherent in the confident glance (eustochia) and to dexterity (euchéreia), to the technique (technê) of the tool mobilizing mathematics, to the effect of illusion (apátā), he prefers Truth to what’s correct (orthón), he prefers perfection to fugitive measurement (oligokairos) and he prefers the exact measurement.

11.3. The abstract and the general within a company

It wouldn’t be cautious to think that these polemics of another time are anachronistic with our reality; not at all, on the contrary. Out of more than 800 managers and top managers assessed, all (without exception) of those whose logical-mathematical mode of operation was “dominant” – in other words, whose operating principle was implemented in any (intellectual-situational) situation where it was considered a “problem” (problema) – refuse, or, even for some, generate a repulsion leading to a micro-expression of disgust or contempt with regard to the idea of developing one of the skills of the metis (extrapersonal MO.O.N.), the “feint”. For them, it symbolizes a lack of morale, a form of mediocrity, deceit, a lack of legitimacy, an unacceptable imperfection, a decay in the nobility of the mind (nous). A person deploying the mathematical MO.O.N. in a significant way will be nearly unable to train this specific skill. Now, if the feint (fingere) historically designates the ability to “shape” (the earth), it is then attributed to a lack of courage, to laziness or lies. The operating principle (the meaning) of pretending is to rely on the existing thing or to exploit the resource, in order to take advantage of it with the least effort (efficiency), or, again, for the search of a lesser risk (caution), that is, to divert attention by adopting-imitating an attitude, a form, a sound, a movement that the other translates in a “way” of their own, and for which they adopt the “expected” attitude. On the one hand, there is an amalgam – as is often the case – between the operating method and its use, and on the other hand, there is ignorance of the principle of the “unnatural”, in other words, that a person cannot (Lorentz) learn to apply what is not part of their “nature”, or at the cost of mental and physical effort considered by the latter as “painful”.

From the company’s point of view, this can be observed by looking (in the original sense) at how many top managers explain their “strategy” with numbers at the starting point: “here is where we are in terms of turnover. Our R.O.I. (return on investment) is as such and this is what we have to achieve in the next six months in order to meet our performance targets”. Then, they present the plan developed through reflection (logos-logismos) and calculation (logizesthai), that is, the course of action to be followed by the operational teams. The line of conduct (tactics) is conceived starting from the Euclidean type of principle of “cause-effect” (if one does this, then one inevitably arrives at that) translated by the principle that results from it: “end-means”.

This activity of the mind, which emancipates itself from an observation of reality (and its tendencies), must lead to the “result”, in other words (an abstract sign) to the defined “point of view”. Let us bear in mind for a moment what we mentioned earlier in the book, concerning the belief of the “everything in us” (autos-kratos); the one who operates according to this mode of operation “thinks” – in all good faith – that the mental object is worth reality. Consequently, it is necessary to define primary actions (causes) for the expected effect to be achieved. Neuroscience research helps us understand how the brain can create internal images without interacting with the outside world [CHA 83, p. 169].

Our experience and our study of the results of the I.LAB competition (2014, 2015, 2016) relating to the detection of “talents” of innovative entrepreneurs show mathematical “logics”, in the way of envisaging the direction of the future company.

So, to answer the questions “How do you decide? How do you take action?”, one of the leaders (case 934¹²) replies:

“Finally, for many strategic decisions (which markets first, which customers, which prices, etc.), I build Excel models based on publications and real figures in order to properly analyze the data and draw conclusions”.

This example illustrates the Platonic principle, artithmos, referring only to the science of numbers [JUL 09, p. 114]. For the study of this candidate, whose operating principle can be observed with more than 60% of the applicants (approximately 400), the use of analysis and deduction “are enough” (actual figures) in order to then know what makes it possible to act, in other words, “draw conclusions” with a view of implementing a strategy.

Information, as an object, can be “analyzed” by a tool (Excel); the resulting conclusion does not (therefore) need to be “validated” by testing reality, as long as rational analysis produces logical knowledge (by pure abstraction). This is one of the mathematical-type skills: abstracting data (objects, data and figures) by a logical formalization whose real validation has no (or little) importance, as long as it is valid through demonstration. This specific way of “seeing” the strategy, and also the motivation of the teams, the decisions or the way of “reacting” to the spontaneous, finds a similarity with Clausewitzian thinking. Indeed, he who is considered as one of the great war strategists of the 19th Century, theorizes war as an object of knowledge (an art) for which he mobilizes a true reasoning (doxa): “The object of an art is the using of means available for an end that is proposed” [ARO 76, p. 82].

The so-called mathematical mode of operation is deployed in the world of objects (ideas, signs), a world settled by more than 2500 years of geometric and then mathematical thought. These two millennia have made this natural mode of operation a central paradigm (from which everything is thought) of the Western paradigm. It “activated” itself and “interwove” itself through its own complexity by elaborating a “virtual/abstract world” cohabiting (often with indifference) with a real physicality. Generations of management books illustrate this: knowing and doing in order to “be” a “good” manager, a perfect leader. As Gardner points out, this form of intelligence is “highly centralized” and can “function” without developing links (or few of them) with other modes of operation (which he calls “forms of intelligence”). His principle is primarily “abstract”, as can be found in “abstract groups”, that is, “a series of operations subject to very specific conditions” [HOD 14, ndp. 1, p. 134], resulting from a rigorous explanation process. Regarding this, there is a form of proximity with the signifier in linguistics and, in particular, the idea of an abstract idea behind the word. This is what the mathematician and logician Gottlob Frege achieves through his Ideography project. The intention of the latter is to demonstrate that pure thought is capable of performing arithmetic demonstrations in order to produce knowledge without mobilizing conceptual intuition [WAG 07, p. 16].

Frege’s ideography, which is close to the Chinese ideogram, is considered an “auxiliary language” and seeks to clarify “the logical relationships between the contents of thought”¹³. The latter evokes an analogy (operating principle of the mathematical skill): “Ideography is to the current language what the microscope is to the eye”¹⁴. Each graphic signifier has a meaning, which leaves no room for ambiguity. Here, we find the mathematician’s “common theme”: clarity, precision, logic, perfection and rigor in thinking about mental objects; it must be said that such “perfection” is only (it seems to us) valid in the field of “ideas”. Thus, a symbol (graphic signifier) is worth a word or expression. The challenge of this conceptual language is to universally quantify statements inducing approximation or subjectivity; in doing so, it becomes possible to generalize (quantify) elements whose intuition and intellectual risk cause the risk and inadequacy of classical phonic signifiers and ideographic signifiers favoring logical relations between “contents of thought”. Thus, to express the logical idea that “Christophe is older than Martine” in the same way (in the identical sense) as all “people are mortal”, Frege’s conceptual writing would say it as follows: ∀x(Hx→Mx) and Rab¹⁵. This ideographic writing makes it possible to expel the ontological assertion, since the concepts of age and death are associated with the subject (Christophe, Martine, people are subjects); it is therefore not necessary to explain the conceptual nuances that the classical language implies. This avoids entering into metaphysical considerations that are considered useless.

Daniel Tammet, considered a “number genius”, for whom numbers are colors, explains with precision how, while he was in primary school, he mentally solved an exercise: there are 27 people in a room and each person shakes hands with the other. How many handshakes are there? Daniel Tammet explains how he solves the problem:

“Imagine two men in a big bubble, then half a bubble stuck in the side of that first bubble with one person in it [...] then I imagined a second half of a bubble with a fourth person [...]. I continued in the same way, imagining two more men, each man in his half of the bubble, so as to make six people and fifteen handshakes. The handshake sequence looked like this: 1, 3, 6, 10, 15. And I realized they were triangular numbers” [TAM 07, pp. 79–80].

At this stage, it is possible to formulate several “mathematical” operating abilities, or should we say, whose observable results and operating principles are called “logical-mathematical”, from the Western (Greek) point of view.

From the point of view of the “logical” meaning, we note an abstract logical type of skill, whose result has no vocation to be verified by experimentation or reality. It is a thought organizing objects of thought in an “ideal” way out of reality. In doing so, it can rigorously explain according to a deductive and cascading principle, that is, by associating ideas by their coherence of form and their plausibility in an “infinite” way (this explains that, which is why that leads to this). This mode of operation mobilizes doubt, in other words, the mental path aimed at seeking truth by refusing approximation or likelihood. In the world of ideas, only truth “exists”. Doubt means refusing what cannot be correlated with the so-called “perfect” forms, or what cannot be proved either by logical demonstration or by exact measurement. In companies, proving performance through figures (objectives “achieved”, R.O.I., income statements, fundraising, increases in net income after tax, etc.) remains a questionable indicator of the company’s performance. Doubt implies the operating principle aimed at cutting off current thinking, in order to trace “the” source of thought, of the problem. We will see that the so-called Western doubt deviates from the Chinese (Confucian) doubt. To be able to develop models by the use of symbols and objects of thought, you must manage long chains of reasoning, but also mentally associate “ideas”.

11.4. Classical Chinese mathematical thinking, a general and functional principle

“As a child, I studied The Nine Chapters; as an adult, I examined it again in detail. I observed the sharing of Yin and Yang, and I synthesized the source of mathematical procedures. In a moment of leisure when I explored its depths, I came to understand its meaning (yi). It is the reason why I dare to use all my weak intellectual resources, and gather (the materials) that which I have seen to make comments about them” – Liu Hui, Foreword to The Nine Chapters on the Mathematical Art [CHE 04, p. 127].

The point of similarity between Western mathematical thinking and Chinese mathematical thinking seems to be the ability to generalize a principle. But although, on the Greek side, generality draws from its language and consequently uses the resource of pure abstraction offered by the alphasyllabary language, the Chinese language proposes general principles, without “locking itself in”, that is, without closing thought on itself by resorting to a truth relative to the said principles. Language guides and constructs use, but an operating mode remains an operating mode. This general “rule” is found in strategy, painting and mathematics. Whether it is in the Chinese Mathematical Book of Procedures (Suanshushu), the Zhou Gnomon (Zhoubi) or in The Nine Chapters on the Mathematical Art (Jiuzhang suanshu), mathematics as much as geometry, is developed with an approach of homogenization (qi) and equalization (tong). Liu Hui, the author of Jiuzhang suanshu, says: “Multiply to disintegrate them, simplify to unite them, homogenize, equalize to make them communicate, how could that not be the point (gangji) of mathematics? The term ganghi is interesting because it helps to “identify the fundamental principles that link the different fields of study”.¹⁶

On the one hand, mathematics is organized and based on the will to understand, but also allows us to emancipate ourselves from the ancient gods of mythology. The Cosmos, located in a mathematized and geometrized space, proves to be one of the bases from which the great principles that are favorable to the emergence of mathematical rules are elaborated, but also the Laws, Morality and Method. On the other hand, the Chinese common ground leads Chinese mathematical thinking to remain anchored in the use of algebra, more than in that of geometry, which, for example, allowed it to discover decimal fractions, the zero (white space [NEE 73, p. 11]), but which conversely did not allow it to progress, because, as Needham says, the Chinese did not know Euclidean geometry¹⁷. Thus, instead of deploying mathematics oriented by a geometric materialism, they deployed organic materialism (philosophia perenis). On the side of the Chinese, common ground is something “other”. On one side, Heaven (qian), on the other, Earth (kun), and in between, the Ten thousand beings. We believe that there is no comparison in the most analogical sense of the term (identical), nor even similarity, “just” an “opposite”, deploying similar operating modes in order to understand, analyze, organize, generalize, “model” (concept to be taken with care), describe and abstract. Liu Hui says that mathematics (suan) was used by elders to select talented people¹⁸, as well as to teach the children of high dignitaries. Mathematics favors the transmission of methods, but also of common knowledge; the Chinese mathematician “highlights” that mathematics “gives the capacity to exhaust what is subtle and to penetrate the minute, to explore without limits”¹⁹, or even, “that which has no place (wufang)”. The Chinese mathematical operating principle allows us to deepen and explore what the yin and yang cannot “probe”; this is what we call shen.

Although Cartesian doubt, in its principle of breaking with the “obvious”, aims to revisit as much as to re-explore one’s thought (one’s ideas) and one’s thinking (one’s common ground), in order to re-found new resources, Chinese doubt (yi) refers to a state of confusion: “doubt appears when the original simplicity disappears, and with it, according to the Taoist thought, the harmony from early times” [JUL 09, p. 28]. From the point of view of Chinese thinking, it would be possible to say that doubt begins where any activity or thought leaves reality (hence confusion). On our side, confusion is linked to an undermining of the knowledge carried on the “self” or on the “object” of thought, and on the other, confusion comes from the gap with the Dao, that is, what deploys effortlessly and for which no action (in the effective sense, “end-means/cause-effect”) is necessary. Contrarily, “by going back up from the branches to the stump (ben/mo), we will coherently renew, from which reality and its “withdrawal”, whatever the register, does not cease to deploy”²⁰.

The gap in mathematical practice between Chinese and Greek thought may stem from the fact that the great enigma put forward by the world has not adhered to the “mind” of the Chinese. Indeed, the mind has not been an “object” of thought (without epistemology, without science of the mind). The mind was (therefore) not thought of as Plato, Socrates, Parmenides and Aristotle could think of it: if “being” and “one” exist, then “two” exist (Aristotle, 143 c-d); “Parmenides: So what, when I said being and one, did I not designate a couple?”. It is from these logical discussions that the foundations of Greek addition and multiplication began.

Although numbers become abstract (objects of thought) on the Western side, numbers remain attached to what Granet calls Emblems, images, on the Chinese side:

“Wayward to mechanical explanations, Chinese thought does not seek to exercise itself in a field, that of movement and quantity. It stubbornly confines itself to a world of emblems that it does not want to distinguish from the real universe” [GRA 34, p. 334].

This functional logic would allow us to understand how “naturally” the Chinese perceived the wave theory in the period of Qin and Han. This theory is linked to the polarized movement of Yin and Yang, in other words, the way in which reality, by incitement (gan), according to Chinese thought, generates a propensity, a process whose influx is, in appearance, uncertain and yet not deviant. Thus, Heaven (qian, initiating capacity) and Earth (kun, receiving capacity) generate a flow-reflux and have allowed China to “conceive” nuclear theories since the 2nd Century of the common era, without however supplanting scientific culture. There is a reluctance among the Chinese, as Needham points out, to engage in theory, and in particular geometric theory (ibidem, p. 16). However, it brings an important (even fundamental) point, that of the Chinese scientific mind, although it is inclined towards a practical sense, it is not easy to satisfy. In other words, they are interested in an in-depth exploration of the principles governing reality, but not from the same point of view as Westerners: the “purely abstract”.

The Chinese abstract (kong) means the will to associate thought with events in reality (by assessment, ji, 计), in addition to the detail of operations and their understanding in the context of a specific problem (wen). This abstract form is part of a maturation process, where the Western abstract form is part of a modeling principle (model form, eidos).

Given that the “foundations” of Chinese mathematical thought have been succinctly laid out, we propose to explore six operating modes (represented by six characters) extracted from the 331 characters that can be found in classical mathematics (glossary of technical expressions (ibidem, pp. 899–1035)). These six characters have been useful to us in our work. Indeed, our motivation to use the ancient Chinese mathematical system of thought, without going through an “equation”, put forward the challenge of its applicability to complex human problems whose treatment in pure abstraction (in line with Frege) did not seem appropriate to us. The complexity of human, animal and plant operations cannot be reduced to an equation, however “beautiful and perfect” it may be. The intention is neither to understand nor to “demonstrate” by explanation, but to understand by explanation so that it is possible to develop competencies (intentional operating modes). We kept the will of Liu Hui in mind, a mathematician of another time, in order to associate mathematics with events of reality. In this principle of thought, there isn’t sensitivity on one side, and reason on the other. There is a process. We wanted to be part of Liu Hui’s dynamic:

“If we are able to make distinctions in a clutter of different quantities (shu), to make them communicate when they are closed to one another, by relying on things, to make their lü²¹ happen, to carefully distinguish their positions and their relations with each other, to smooth out the disparities between them and to homogenize the differences in size between them, then, in the end, there is nothing that does not come back to this procedure”²².

For example, the character 化(hua) means transformation-metamorphosis from the point of view of the “thing”, as perceived by the observer. This notion implies “observation” in the latter without directly “observing” the process in progress of said transformation. The observer can certainly “perceive” the external transformation, but not the internal transformation, nor even the influence of external “factors”. Thus, from the point of view of Chinese mathematics, hua refers to the transformations undergone by the nature of the sizes concerning the quantities in procedures: “hua, it is the distinctive marks of one thing changing into the distinctive marks of another”²³. The character 化 used to think of mathematics is the one from which poetry or strategy is thought; it is its use and consequently the mathematical writing that will differ. Hua, if necessary, will combine with 默 化 (mo hua, silence transformation), 變 (bian, modification) or with 变 化 (bian hua, transform).

Piyin and character	Mathematical principle	Understanding/application
hua 化	Transform, metamorphose	The term refers to the transformation that the nature of sizes undergoes by the action of the operations that concern their quantities in the procedures. “Hua is the distinctive marks of one thing changing into the distinctive marks of another”24. The transformation to which hua refers to is the change made from the point of view of the thing, and such as it appears to an external observer, who notices a metamorphosis at the end of a process that is not spoken of. We emphasize the proximity with the concept developed by Jullien, Mo hua 默化 (silence transformation) and its proximity with bian (modification).
bian 變	Transformation, mutation	Refers to the transformations that a problem undergoes, because of the continuous variation to which its data are likely to be subjected to. It is not a matter of a transformation/metamorphosis (hua) relative to a context, but of maintaining continuity between the states it links. It is possible to follow these variations (bian) by relying on the variation of these visible criteria, which are the data of the problem.
bianhua 变化	Transform	General term, commonly refers to transformation processes and highlights that the procedure is conceived in terms of the change it brings about, “what they are transformed by” is like [the situation of] (7.6). see hua, “metamorphosing”.
bianyi 變易	Transformation	Given by Liu Hui as treating “exchanges and transformations”; refers to transformations from one state to another (metamorphoses of grains which are the object of the first problems); can also evoke transformations from one problem to another.
dong 動	Modification, transformation	“By such a transformation, coming to harmonize”bian “mutation”, bianhua “transform”, hua “metamorphose”. Movement of numbers on surfaces to be calculated (which progress, bu, or downgrading, tui), as opposed to “numbers that do not move”. Assuming that the procedures are read, among other things, as a ballet coordinating the movements of numbers on the surface to be calculated, a continuity would occur between the two senses of dong.
shi (shi’) 勢	Situation, procedure/situation, mechanism, configuration (influence, condition)	Generally, a device with effect or an effective arrangement [CHE 04, p. 979]. It is a concept specific to scholars and absent from the classics. The term shi’ intervenes in comparisons, when it is a question of noting the identity presented by the “situations” of certain geometric entities. It is to the extent that the confrontation between several (different) of its occurrences suggests that shi’ captures this pair of aspects, which we have chosen to add to the translation of “situation”, to specify: the “procedure” mode.

Our short focus on the notions of transformation (hua) or the minute (wei) allows us to understand this mode of operation that is inherent to the observation-perception of subtle principles. If the notion of abstraction necessary to perceive principles that escape the eye is solicited, it is not to enter into a principle of “pure abstraction” (geometric), but a dynamic one (how it operates to arrive at such a result). It is also through this example that we can find the “abstract-general” core component that is inherent in the mathematical MO.O.N., understood in both Chinese and Western thought, without sharing the same common ground (eidos-mathesis/bian hua 变化, shi 勢).

The assessment of hundreds of entrepreneurs in the field of innovation, particularly in the field of medicine and high technologies (between 2014–2015), shows how the mathematical operating mode (abstract-general) conditions the way of thinking of “strategy”, but also business situations. For the following questions: how do they decide and act? How do they deal with adversity, a crisis? How do they go about it if a competitor comes to thwart a product launch close to or identical to theirs? Out of eighteen entrepreneurs studied and selected of the 700 for our research, 94%²⁵ of the operating principles were centered on analysis (model), figures (statistics), opinion (discussion, knowledge, expertise) and the Being (what “I am”, what “I know”). 1% represented skills to operate by following the course of things (moment-position, shi wei), another 1% represented skills in understanding reality and favoring the principles of actualization, 3% showed skills to operate (metis-sing²⁶) and 1% represented an efficient type of mode of operation (in the sense of non-acting, wuwei).

For example, let us take three remarks by Western entrepreneurs and underline the way in which the “abstract-general” principle is effective in the field of decision and action:

“I turn to people who can help me in my analysis when subjects are outside my area of expertise” (Case 792 [RIC 16, p. 264]).

“Maintaining a high level of quality in all aspects: product, sales, marketing and customer relations. However, I will insist on investing in Research and Development to raise the quality of our products and services in order to maintain a competitive edge and customer satisfaction” (Case 897).

“To make a decision, I follow several steps:

• – define the problem in detail;
• – define some choices with, if possible, a SWOT diagram;
• – if the elements are not sufficient to make the decision, I move on to the phase of recovering more information by exchanging with external people (team, partners, internet, etc.). If this is not possible, I postpone the decision as much as possible;
• – I take action by first defining an action plan if there are multiple actions. I delegate everything possible and I manage teams and operators. I use software tools using the Getting things done approach that optimizes my execution efficiency” (Case 1125).

For the case of a Chinese leader²⁷ met in Shanghai, with identical themes discussed with Westerners, the latter said, while talking about major political decisions:

“So, if he says that, I’ll do that. I wait, it’s always passive. It’s like wuwei means, I’m not going to do anything, it’s not that he’s not doing anything, he’s actually trying to linger in the wind. As long as time moves, in fact, it turns like so, it’s Tai Chi, he doesn’t want to attack, he actually seeks, follows, adapts”.

NOTE.– The “intelligible” aspects, in the Western sense, are almost non-existent in Asia, that is, “pure abstraction-generalization” is not an operating principle that can be considered systematic. The case of this leader in this short excerpt shows the implicit common ground put forward by the Chinese Wen. Liu Hui’s mathematical precepts, through the “general” principles relate to the transformation proposed by the notions of wuwei, of time that moves (hua, 化), but also to the potential of situation (shi, 勢), evoked with the wind (feng, 风: which influences), for which we should linger, are also “abstract” and “general”.