First of all, thank you Professor Lackner and the consortium for letting me be part of this lecture series.

And I hope to make a worthwhile contribution to it.

This presentation has developed over the course of three or four earlier talks I gave.

And I have substantially redesigned and extended it for today.

The material I draw on is the result of slightly more than three years during which I had the opportunity to work on my doctoral dissertation,

including the last year here at the IKGF that provided such a fabulous environment for research, which I was very privileged to enjoy.

I study forms of arguments in cosmological speculation from traditional China.

That is, from the classical period to late medieval and early modern times.

And today I want to talk about argumentation through diagrams.

An article in a German weekly magazine I read recently made me think about the timelessness of certain developments that we often assume are hitherto unknown or without comparison.

A new software has been described in that article, which interpreted the human voice of a person being interviewed for a job to predict that person's qualification and suitability for a certain project or task at hand.

However, the rather distressing thing about this new technology to me was the fact that the algorithm was not used to interpret what the human interlocutor said in any literal sense.

Instead, it merely focused on the voice itself, its pitch, rhythm, volume, modulation, and so on.

And then, based on a large dataset of anonymous profiles, it gave a prognosis of that person's motivation, persistence, flexibility, courage, charisma, and so on.

Recruitment and effectiveness of officialdom were also among the goals of reform projects projected by Wang Anshi, 1021 to 1086.

And the reason I begin with this anecdote is that my talk includes certain applications of, broadly speaking, technology and computation.

One fundamental way of looking at these, therefore, is to ask ourselves whether what is achieved with the help of computation qualifies as mathematics.

In a sense, that for there to be mathematics, it is necessary, so goes the saying, that the person doing it knows what he or she is doing, regardless of the practical outcome.

Or otherwise, we have to modify our notion of the concept of mathematics.

When I say forms of arguments, think about the differences in knowledge representation between linear text and visual images.

How what is not said in any text seems to be implicit in visual images.

How charts and graphs can communicate ideas that are impossible or only very impractical to put into words. To use a metaphor, think about how solving a puzzle on paper structures the way the resulting complete image is discovered.

And how, more importantly, an existing picture can be broken down into a puzzle that should only have one correct way of assembling parts.

Or, more concrete, think about how logical operators such as conjunction, disjunction, alternation, causation, and so on, in linear exposition are translated in a sequence of diagrams to illustrate and prove at the same time the correctness of the sequence of one model of cosmogenesis over another, as it is the case in my study.

Also, think about how conventionally established associations in symbolism work very differently in writing as opposed to visual statements.

And how an author might leave a holy scripture, one that must not be altered, untouched by paraphrasing parts of it with graphs, while at the same time new ideas are injected.

The material used in this paper dates from the early Song and is an adaptation of classical cosmological texts that revolve around the Book of Changes, the ET, and the role that numbers and diagrams play in this adaptation.

And what I am interested in primarily is the tension between symbolic and computational uses of numbers.

My talk will have three parts. First, I give you a historical background that includes the graphical notes and textual problems of my sources.

This source is first of all the issue going into by a scholar official named Liu Mu, as well as the texts that are the required reading, so to say, for the issue going into.

Second, we will look in detail at a series of diagrammatic problems of which each will be contextualized with its discursive background, a technical explanation and some tentative methodological remarks.

And third, I try to give a conclusion that will take up and answer some of the questions that were posed at the outset and also broaden the scope of the study and suggest a few further lines of inquiry.

A few words on the School of Charts and Books.

Beginning in the Northern Song dynasty, several classical scholars began to explore new visual modes of argumentation, such as charts, diagrams, maps, and so on.

These scholars were proponents of what will later be called the School of Charts and Books, Tu Shu Xie Pai.

This branch of learning represents a temporary flourishing of intermediate discourse and had its own specific relation to the two established schools, meaning pattern Li and image and number Xiang Shu.

In the conventional view of lines of transmission, which goes back to the editors of the Silkwood Transchroom, the School of Charts and Books is seen as a subbranch of the image and number tradition, and I'd like to comment on some problematic implications of this view in my conclusion.

The School of Charts and Books had lasting influence on later thinkers who were faced with similar difficulties in making sense of classical texts. Among the early proponents of Tu Shu were Liu Mu and Shao Yong, and also later Zhu Zhen, Zhang Li, as well as many others in varying degrees.

The innovative method of these thinkers was the invention of diagrams alongside verbal explication.

However, instead of tracing lines of transmissions and identifying schools, it is, I think, equally illuminating to see how the ideas themselves acquire a dynamic of their own when they become transplanted across time and translated across media and how the media and the practice of diagrammatics shape these translations.

During the early decades of the 11th century, Liu Mu invented numerically motivated cosmological diagrams that were, followed by the contemporary Shao Yong, 1011 to 1077, the work that initiated a temporary flourishing of diagrammatic speculation in the Northern Song.

According to a close contemporary, Li Yigo, 1009 to 1050, who wrote a harsh critique on Liu Mu's work, Liu Mu was even responsible for initiating the revival of the studies of Yijing in his time.

Liu Mu can therefore be called the founder of the School of Charts and Books, or at least of its cosmological wings, long before that name was coined.

In retrospect, Shao Yong became the famous and influential numeratus and cosmologist of his time, as well as the topic of many contemporary books and articles, while Liu Mu's contributions, on the other hand, have never been systematically examined.

Liu Mu's biography and life dates are a matter of debate. We cannot go into this for today's talk.

Suffice it to say that very early, already a few generations after his death, two Liu Mus appeared on the screen, and it was argued that Liu Mu, with the second name Liu Min Shi, was the true author of the issue going to, and not the other one Liu Jiang Min, who had been credited with it up to then.

There were two advantages of this claim. First, the person making this claim was a later relative of Liu Min Shi, and second, that Liu Min Shi had lived even slightly earlier than Liu Jiang Min, and therefore would have been clearly prior to, or for example, Shao Yong, and also prior to Zhou Dun.

The beginnings of the School of Charts and Books are therefore vague, and the name Liu Mu may, for the time being, serve more as a label for a diverse and partly self-contradictory collection of materials that stem from different periods and probably from different authors.

So, for the second part, let us first look at two classic examples of diagrams of this time. Here, we can identify, so these are the famous Hurtu and Luoshu, which spawned a large debate in the Song Dynasty, which I will have more to say in a few minutes.

Let us first look at them. We can identify several assumptions that qualify a valid diagrammatic statement, and how these translate into linear linguistic propositions and vice versa. The diagram on the left represents simply a so-called magic square of the order three.