Mathematical vs Metaphorical Understanding


This is a page to discuss two different ways of approaching and understanding the game of Go.



Naustin: I want to make clear that I think that neither is inferior though I have a bias. I also think it is interesting to recognize that they are different. These two approaches are often necessarily complementary but I do not think that they always are. There seem to be some concepts that are more aptly or easily conveyed in one way than the other. In order to express more clearly the division I see and to provoke discussion if possible I have provided links to illustrate the two sides of the coin and have begun the discussion below.

Naustin: Of course one of the first things that leaps to mind is the old left versus right brain discussion. Historically, there seems to be a deep tradition of metaphorical description of the game. In Kawabata's Master of Go the traditional Japanese view of Go emphasizing aesthetics is seen as passing with the passing of the Master. At the very least its ascendancy.

As far as I am aware, the mathematical approach has been a result of two factors: first, the intersection of the game with Western culture, and secondly, the advent of computers and the ability to carry out really large scale pattern searches and statistical analyses. The question would then become are these really two separate ways of looking at the game, or are they really just different names for the same thing? That is, do these two different viewpoints really imply a different way of "thinking" about the game or are they simply different name sets? Kami No Itte seems like a good example in a way because it is something that is expressed in a metaphorical or poetic manner but is being studied from the mathematical viewpoint now.

Another question of interest is, do different ways of thinking about the game actually translate into different styles of play? Are there certain styles of play that tend to be adopted by the more mathematically (metaphorically) minded Go enthusiasts?

Analogies are another relationship associated with the metaphor idea. It seems to me that there are lots of people out there who can find in Go analogies for all sorts of things in intellectual and general culture and life. A question that seems to spring out of this is: Does this proneness to analogy spring more from the nature of the human mind as it views any complex system or is there a particular character to this game which rewards such thinking?

starline: Maybe Go is an analogy for the working of the human mind itself!

ilan: I don't think there is a mathematical way to Go, except for small board sizes up to 7x7. My observation of other mathematicians who play Go is that they usually ignore the mathematical aspects. In fact, in almost all cases, Go players are much too caught up in improving their play to make the necessary abstraction and simplification for mathematics to work. It is not surprising then, that the mathematical theory was developed by a non-Go player (J.H. Conway).

JG My understanding is that Conway can play go, but not well. He certainly had access to Go equipment in 1970.

This directly addresses the mathematical approach to Go, but I believe that you have confused "mathematical thinking" with "analytical thinking." Mathematics is about finding elegant universal results in very simple settings, e.g., a triangle. Playing Go well is about dealing with a mess of technical details and is intractable mathematically.

On the other hand, playing Go well definitely requires analytical thinking, in particular reading and counting, and playing the game without this is pretty much garbage. The "metaphorical" aspect you refer to is meaningful only when these analytical aspects are present.

Alex Weldon: Math underlies Go the way physics underlies, say, soccer. In both cases, the system is so complex and chaotic that the large-scale behaviour that results has little resemblance to the equations that are actually responsible for it, especially because of human involvement. Math might tell you which of two endgame plays is bigger, as physics can tell you the trajectory of a soccer ball... but approach Go from a purely mathematical point of view and you'll end up playing about as well as Stephen Hawking plays soccer. :-)

ilan: You can also consider billiards. There is a mathematical theory of billiards which has absolutely no application to the actual game of billiards, since friction is not considered in the mathematical theory. Of the thousands of mathematical articles written about billiards, there are only two which apply to the game: the book of Coriolis 1840, and a theorem of Euler (son), 18th century. Interestingly, most mathematicians and billiard players are unaware of this dichotomy.

Naustin-- On the other hand I think it is important to notice that as a complete information game no idealisation (i.e. departure from the true nature of the object of study) is required. Go is an ideal system in that sense.

It makes sense to me that a mathematician might not try to solve go in a mathematical way when he plays particularly as there is no theory for go as of yet at the 19x19 size. However I think that though not completely separate it does make sense to talk about how someone understands the game in a way that is not completely equivalent to talking about how they decide to make specific moves in games. One of the places I see this kind of thing is here at senseis in how different people study the game. Notice above how I stated I don't think these approaches are probably even possibly completely separable.

This is one of the very enjoyable things about go to me. I can play games with other people which is great but studying go is also very enjoyable and in a way is a sort of solitaire.

I guess basically what I am trying to do is say that I think there is a more interesting definition of understanding than informs the replies above but that could still yield a meaningful discussion.

Naustin-- [ext] this is a link to a series of articles, these aren't very deep but several of them do talk about the idea of metaphor using different examples.

Naustin-- I have been thinking about this page some recently. I am dissatisfied with it because I think it could be better. I would like to redo it to capitalize on the parts I like and to strengthen my own arguments and idea at least. I guess what is called for is a master edit, but as I have never done this before and have seen some of the worst of Sensei's library coming out of such master edits, so I have a little trepidation. I feel I am a logical person to do a master edit as I am both the page creator and someone who would like to see more out of the page. I guess I have read somewhere it is nice to post that a master edit will happen so I am hereby doing so. This page may not be controversial enough or visited often enough for this to be a problem but there it is anyway. Any advice would also be appreciated, or input either from people who have already posted here or others. Thanks.

ilan: I have been thinking about these issues recently. My observation about most Go players is a total inability to be imprecise. The ability to be imprecise is the basis of big conceptual advances in mathematics, in particular, differential calculus.

Agilis: to add my small piece, I'd like to make a controversial claim that all learning comes from extending from what one already understands, and one of the easiest (though perhaps not the only) way of extending understanding is through metaphor. Because learning happens this way, humans are very adept at spotting similarities and differences. With a metaphor, we get 90% of the idea into our heads quickly ("it is like..."), before understanding the last 10% where the metaphor breaks down through experience ("it is not like ... when...").
For example, the word "dragon" easily paints a picture of something big, scary, and extremely hard to kill, along with "common sense" like not placing small edible things (groups) near one. Instead of explaining why people should not attack large strings of stones, we simply say "this is a dragon, beware of them" and the metaphor if thought about just slightly, conveys most of our meaning concisely. But most importantly, talk of everyday things sticks in our minds with far less effort than abstract talk.

Naustin-- I think ilan's point is pretty interesting. I'm not sure I agree, I will have to think about it, but it is an neat reversal of the usual stereotype of mathematical advance. I think it is an idea with some application here. I would like to ask ilan this question though: part of what I am trying to talk about here is the idea that in fact one way that go differs from chess is a much more developed way of being imprecise. The metaphorical ways of discussing strategic qualities for example seem to be a clear cut case of very valuable vagueness. They are vague in just the right way to make them interesting concepts rather than rules for playing in a certain situation. This is part of the initial problem I thought we were discussing. If these thoughts are true however it would contradict your impression of go players. What do you have to say to that?

On the other hand I would like to be more precise on this page. As long as your comment doesn't constitute an objection to the master edit I proposed I will keep thinking about what I would like to do to improve it.

ilan: Well, there is not as much room for imprecision in Go as there is in Math, but the general principle is still the same, that too much focus on details prevents you from understand what is really going on. Usually the reason for this focus is that you don't understand the details well enough. So, what I believe I am observing among Go players is the phenomenon in mathematics, that until you reach a high enough level, you need to see all the details to understand an argument. In mathematics, the ability to follow what is going on despite incomplete details is usually referred to as mathematical maturity. It is not surprising to see this in Go players, since in math, you will rarely get a simple explanation from a recent Ph.D. of what his subject is about (too involved in the technicalities), not to mention established mathematicians. Go ask a dozen mathematicians what the Riemann Hypothesis is about, and I bet you that only a couple can answer in a very simple yet meaningful way. So it is normal to see similar behaviour from strong amateur go players. So far, the only person I have noted who seems to be able to explain Go in simple, accurate but imprecise terms, is Michael Redmond.

JG Achim Flammenkamp has published a mathematical definition of the rules of go on the following web page. [ext] (There is no period after www in the URL, and you will need a DVI viewer.)

Flammenkamp explains his definition in this way: ''The simplest (and already totally sufficient) aim in playing Go is to color as many points of the board as possible with the color of your own stones. As a mathematician I like to strip rules down to their basics; thereby freeing them from their historical, metaphysical, religious, cultural and fashionable influences.''

Flammenkamp's mathematical rules correspond to the following informal description.

  • Positional super-ko.
  • Suicide is allowed.
  • Traditional stone counting, with a two-point group tax.
  • Points in a seki are counted.
  • Play ends after two consecutive passes or by resignation.

Under these rules, counting a game requires all the dame to be filled, followed by all but two of the empty points inside each group (apart from seki.) It would be tedious to do this correctly over the board with clocks running, but the procedure allows a simpler mathematical formulation.

Flammenkamp's definition seems to be a purely intellectual exercise, but it would be interesting to know Robert Jasiek's opinion.

Mathematical vs Metaphorical Understanding last edited by on January 23, 2015 - 09:04
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