Go as hierarchically ordered

    Keywords: Theory

This is an edited and amplified version of a thread I started on rec.games.go in March 2002, prompted by a remark of Bill Spight about the perceived relation of go and mathematics.

Bill: Charles, what remark was that? One might get the impression that we have quite different views on the relation between go and mathematics, but I think that our views are rather close.
Charles 'Twas on 25 March 2002 that you wrote this (it was a propos a discussion about A Beautiful Mind):

As one who has written and lectured on mathematical aspects of go, I have always found it curious that there seems to be an association between go and mathematics in the West. Other, more popular games, such as craps and poker, seem much more mathematical to me. :-)

Bill: Thanks, Charles. :-)

Others who posted in the thread were gowan, Richard Brown, Michael Alford, Robert Jasiek and Morten Ofstad. This summary uses my own words and shortly paraphrases those other comments.

This page can for example be read as a contrast with DieterVerhofstadt/IdeasOnGoTheory.

Charles Matthews

Why (in what way? whether?) go is connected with maths

There seems to be an association between go and mathematics in the West. My exposure to Korean go would suggest that there are very different roots. But it isn't really worth pursuing the 'why' of the association.

I do wonder about the 'what'. We know go has many levels - and it is sometimes argued that they are quite strictly hierarchical (you'll never understand X until you have mastered P, Q and R). Now that really is like mathematics, particularly in secondary school, as a practical comment. It is very tempting to say "you can't become 10 kyu without spotting snapbacks".

That's anyway not so interesting. Take a better example: opening theory. Saying you can't understand global openings without the corner openings is instructively wrong - obviously something to it. Saying you can understand openings without life-and-death knowledge is also instructively wrong. I'd say that you can't possibly understand openings without a feeling for things at the level of open-skirted territory; but I fear Takemiya might show me up there.

It might be helpful to discuss how this sort of brick-on-brick picture of go breaks down. One suggestion is that as a learner, one may become stuck at a local 'peak'.

Lower levels

Here there is a detailed discussion. What is the next step up from the rules?

Some very early matters are

  • recognising atari on chains of all sizes
  • recognising cutting points, a crosscut formation
  • don't play inside your own territory for no reason
  • simple blocking plays to defend territory.

These are low down in the hierarchy as self-organising, e.g. human playing a computer program without much information other than rules defining legal play, a young child playing an experienced player, or a spectator watching games without asking questions. Clearly matters to do with eyes are fundamental knowledge but would take a little longer to discover experimentally.

On the other hand it is reasonably argued in the case of eyes, that there are concepts behind or beneath this one: it isn't a primitive one, anyway. Inside and outside matter greatly.

Territory comes at what level?

The territory concept depends on understanding why invasions fail: that is, it certainly is at a higher level than life and death.

Your territory consists of the empty points, such that any stone of the opponent's played there can be captured.

That is a little superficial, even supposing your own group is definitely safe, and we say it to include stones already dead.


It doesn't apply straightforwardly to this position, for example. As things stand there may be an exchange of the white+circle and black+square stones, in a ko fight.[1]

There is the semedori concept to worry about, too.

A point is your territory if any stone your opponent plays there can be captured at a net cost of one point or less.

But sometimes the cost is greater, for tactical reasons, in faulty shapes that have no easy characterisation.

Therefore this is already a complex concept. Knowledge is required. An operational definition of territory is usually assumed, like this.

All invasions fail to do any of the below:

  1. kill all or part of the walls;
  2. make life with two eyes;
  3. make seki;
  4. make ko for life.

In the worst case, though, there might be even be a choice of various semedori tactics (eg sente or gote sequences).

There are good reasons why go isn't ever taught this way, however intellectually honest it might be to do so.[2]

Against the hierarchical model

My argument from the start was that there are problems with the hierarchical model. I was arguing that the mathematical component of go is overstated.

Reducing the territory concept to simpler elements illustrates my point. If you have an intuitive idea of territory (adequate for some cases) you can understand the principle of not filling it in. You don't need a complete understanding, embodied in a definition of territory.

This is the type of reasoning that scales up to more serious strategic concepts. For example, you can play in a way that takes account of influence without being able to say exactly what influence is.

The overall picture I see is that the hierarchical model isn't too bad up to about 1 dan level. From a teaching point of view you can say 'you fail to understand X and it seems to be because you are missing the Y concept', or 'you could add these extra modules (openings, life-and-death patterns) to the way you play without changing the rest'. That is, you don't expect as a teacher to analyse a big capturing race, show a way of gaining a liberty by throwing in to create a false eye, and get the query 'what's a false eye?' - this really can be taken for granted.

What interests me is how and why it breaks down.

Intuition, foundations and talking about them

Does anyone have a "complete" knowledge of any major go concept? For example an abstract understanding of territory as a working concept in the middle game.

And if one does have deep knowledge, can it be articulated in the expected way? That is, as discussed here?

In fact there is evidence that well-structured learning does help in go. It counters bad habits, not just by prohibitions but with explanations. Sometimes though the explanation does refer to conventional wisdom as if it were normative. Which it certainly isn't.[3]

In some qualified sense the hierarchical picture therefore is far from being wrong. On the other hand foundations (as for example set out in Kageyama's book Lessons in the Fundamentals) are probably arranged in a flatter pyramid than would be expected from the detailed comments in the second section. And there are many more exceptions than are tolerated in mathematics.

The challenge for teaching is probably to arrange the conventional wisdom in a teachable order, that also implies the least unlearning. Can this be done? I'm not sure - perhaps one does have to start young and shift one's ground, to become really strong.

Mathematics is the science of patterns,
Music is the art of patterns,
Go is the game of patterns.

John F. There was a big fad a few decades ago in some parts of the academic world for splitting learning into inductive and deductive methods, with the fashionable ones going inductive. That petered out but was resurrected in a different form a few years back with a fad for practising everything with the right side of the brain. It seems to have evolved now into methods stressing pattern recognition. Go seems to be a useful paradigm. As far as I can see, mathematicians so far have only been able to aspire to master go. Maybe there is a wall they meet if they stick to a rigid logical approach. In any event I don't think I've come across many pros who've studied maths. Literature and foreign languages often feature for those who spend time at university, with a fair number doing economics or information sciences.

Charles I was a professional mathematician for 11 years after a Ph.D. in it - I might still know more about mathematics than go. I'm clear enough on the mathematician's way of thinking about things. The purely-deductive tends to be called axiom-bashing (not in itself a compliment). CGT lends itself to that, for example.[4] I learned more about context when I switched into theoretical computer science, than I ever could have within maths - this is perhaps the key issue in applying patterns in go. Using the left-side/right-side of the brain argument as if it was a knock-down has always annoyed me. I'd see it as simply hiding the problem elsewhere; in fact the Western form of citing Zen as if that explained everything.

John F. I agree entirely with the last point. As it happens there is a relevant and interesting news item today on [ext] http://news.bbc.co.uk/1/hi/health/3025796.stm which shows that Mandarin speakers use both sides of the brain, whereas western language speakers use only the left.[5] There is no claim of either method being superior, but (my interpretation) use of both sides of the brain from childhood for such a task may predispose people towards go. Note that the findings of the news items would not, as reported, seem to apply to (or exclude application to) Japanese or Korean.

[1] Charles: I see this is quite actually an interesting example for Remove double threats before you first capture the ko. See ko threat in seki example.

[2] Charles: Coming back to this, I see reification again. And another intellectual gap: you may have to play inside honest territory, to deny your opponent ko threats.

[3] Charles: Cf. the examples from Guo Juan's teaching that have recently been posted here. They go against some basic planks in the conventional wisdom platform.

[4] Bill: Charles says that CGT lends itself to axiom-bashing in a purely deductive manner. Frankly, I do not see that. Perhaps our difference in viewpoint explains our different treatments of CGT in relation to go on SL. Or maybe it's just a difference in rigor.

Charles I think the surreal numbers page here is precisely axiom-bashing, in the usual sense that this term is applied. Which isn't by itself derogatory: the implication is more of footslogging through mathematical foothills, when you might as well take a helicopter to Everest base-camp if you are serious about getting to the top.

[5] Bill: As reporters are wont to do, this one has oversimplified. The study showed that Mandarin speakers use both temporal lobes to process heard language, while speakers of Western languages use only the left temporal lobe. But Westerners also use the right brain for language processing, even though the principal language centers are in the left brain. It may be that for speakers of Mandarin (and perhaps other tonal languages) there is a third language center in the right temporal lobe. The claim is also made that Mandarin is a difficult language to learn. (Without qualification.) That's questionable.

John F. I have no brief for the reporter but I can't see why you think he has "over" simplified. The main point was that the scientists found a difference between Mandarin and other languages, which you seem to accept. The fact that we westerners also use the right brain doesn't seem to affect that point at all, unless someone comes along and shows we use it in the same way. As to the claim of Mandarin being hard being given "without qualification", I think the report said this was for adult westerners, which seems both sufficient qualification and an observable fact from the amount of time allowed for the so-called "hard" languages in university syllabuses, foreign ministry training, etc all over the west.
Bill: Relevant quotes:

Researchers in Britain have found that people who speak Mandarin Chinese use both sides of their brain to understand the language.

This compares to English-language speakers who only need to use one side of their brain.


Mandarin is a notoriously difficult language to learn.

John F. I don't think selective quotations are relevant. Here is another quotation from the piece: "Native English speakers, for example, find it extraordinarily difficult to learn Mandarin." All the journalist did was to give a headline summary as above, then went on to repeat the points and explain in more detail. Relevance can't be divorced from context. Which was, incidentally, not so much to give an abstract but to flag up a forthcoming scientific conference - this was an example of some interesting work that will be featured there. I still can't see that the charge of "over" simplification sticks, especially given the general audience. I'm sure meteorologists find the weather forecasts oversimplified but that doesn't stop them being useful for the layman.

Bill: How about something like this?

Researchers in Britain have found that people who speak Mandarin Chinese use both sides of their brain to understand the language.

This compares to English-language speakers who mainly use the left side of their brain. Mandarin speakers make heavy use of a part of the right brain that English speakers do not: the right frontal lobe.


Mandarin is a notoriously difficult language for Westerners to learn.

I expect that this can be tightened up a bit. But the point is that the qualifications are easy to make: mainly, heavy use, for Westerners.

John F. I agree your version is more succinct and therefore better stylistically, but I don't really see that you are adding any new information of use to the layman that wasn't buried there in the original. My point was that it was not "over" simplification, at least to the extent it remains of interest to go players. Let us agree to differ on this one - I suspect you have a rather different, professional, perspective on this anyway.

Dieter: It is true that Go, in countries where it is lived professionally, is not commonly taught or discussed in an axiomatic way, at least not as far as we know. It is also true that my ideas on Go theory try to take such an axiom based approach to Go. I think that the relative failure of my project is more due to it being a one-man-effort of a relatively weak player, than its being fallible by nature.

It is not because knowledge is not usually transferred in a certain way, that it cannot or should not be done in that way. The scientific approach does not work with Chinese medecine for example. Yet, when Vesalius began decomposing the body for inspection, it marked the start of Western medecine, compatible with the scientific method. There are surely drawbacks to leaving the holistic approach in medecine, something which many patients suffer from and is making them turn to alternative medicine, but the scientific approach has enabled doctors in the West to establish a firm fundament for the activity. There is no reason to believe that unlike for medecine, a knowledge space like the game of Go is fundamentally incapable of being ordered in the scientific (hierarchichal) way.

Maybe it is inefficient to do so, given the heights reached by professionals without applying a scientific method but a more convoluted kind and the fact that there are no Western professionals who overcame their Eastern counterparts by applying it. Maybe Go is more like riding a bike than like solving math problems. Maybe. But I'm still curious to try it (but have no time these days), preferably with a group of stronger players (than me).

Besides criticizing such attempts, I don't know what this page tries to do. It seems very hard to me to be descriptive in a deepening way, while trying to avoid being derivative. Also, the point of what one should know at a given rank is not at all the point I am trying to make and not one that really interests me, because rank is rather circumstantial. The fact that most 1d know the L-group to be dead, but fail to understand Takemyia's opening, and on the other hand one can perfectly understand the carpenter's square and still be 15k, does not prevent at all from attempting to build Go Theory in a hierarchical way.

Go as hierarchically ordered last edited by Dieter on June 19, 2008 - 10:30
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