"David Richerby" wrote in message
...
Chess One wrote:
"David Richerby" wrote:
I don't know if any mathematician has ever seriously proposed which
of these two states, potential :: action, is superior from any
theoretical model.

Again, what's this double-colon you're using?

Its an advanced colon, as used by ... well, no jokes occur to me. It
is very like a single colon, though more signalled.

A colon isn't grammatical at that point in that sentence, either!
Does it, in fact, mean `and'?

David, my real comments on the two proposition are at the bottom, but in
between we encounter the dangers of transference.

The term " :: " indicates among philosophic types that it is not a colon
used as in writing, but used in writing in the same way as a colon is used
in mathematics, that is, one suggests a ratio, or proportional relationship,
rather than alternative listings of items with or without stated
relationship, as normally used in writing.

Chess is not even conveniently describable /as a process/ as a
Finite or Infinite Game.

Chess is trivially describable as an infinite game by omitting the
three drawing rules (dead position, three-fold repetition, 50-move
rule). It is then easy to prove that it suffices to consider only
finite games because of the drawing rules.

But finite is itself an arguable determination.

No it isn't! A collection of objects is finite if, and only if, you

a metaphysic! is the you any you, or a hypothetical you? an 'if-you' which
may not actually exist! or could not exist. [pain in the ass, no? these
logical formations in writing, but their advantage of symbol displacements
is that impossible 'you's' can be discerned. In math the positied term could
go unchallenged, and it doesn't concern anyone if it is a hypothetical value
of 'you'. But in writing of actual 'you' values it must relate to what is
possible, not metaphysically supposed.

Otherwise the sort of person referenced as 'you' may or may not exist, and
such statements do not even imply they /could/ exist. But this renders the
formulation undifferentiated from either the impossible or the imaginary.

can (in principle)

is 'if you can (in principle)' a conditional statement where the 'if' means
'when' rather than 'if you could', and the '(in principle)' is a known
principle than than a supposition?

otherwise in formal logic you posit an hypothesis which, true, /would/ be
finite, but is not known if it actually exists. this is why finite is
arguable - to wit: to whom?

example: red and blue are colours. The 'and' does not tell us if they are
the same colour, as in the case if we had written 'light blue and azure are
colours'

if c is the category, 'a colour', then the statements:-
a=c
b=c
does not mean a=b except in relationship to c, which is to say, they are
equal only in set-theory, and the term "=" does not mean 'same' as in
identical, but 'equivalent quality' related to their set, which is not the
only set they can belong to...

count them (1, 2, 3, 4...) and stop at some number,
having counted all the objects. So, the number of pieces on a
chessboard is finite, as is the number of grains of sand in the Sahara
Desert.

but time is not finite. and if time taken exceeds time permitted, eg, the
entirety of an individual's life, or beyond the extent of the life of the
entire universe for all people, then the term finite runs into trouble -
since know other time is available or knowable to cite as a possibility

Is it used synonymnously for 'theoretical',

No.

even though there are more moves than atoms in the universe

Even though there are more chess games than atoms in the universe.
This means that the finiteness of chess is of largely theoretical
interest, since knowing that it's finite doesn't help much in
practice. But `finite' does not mean `theoretical'.

Except of course in the terms I pose above - especially if 'you' were to
attemtp to solve anything, then it can be that 'you' lack sufficient finite
time to solve it. This is also true of collective efforts, where finite runs
into time available for all people

So if you were to answer that chess would be finite if there were infinite
time to solve it, do we not encounter a paradox common in Large Number
Theory? There are conditions where only from infinity we can define finite.
But if we acknowledge we have no infinity available to us, is what would be
finite, actually so?

Do the parallel lines oscillate, [what does not?] is their amplitude such
that they cross over?

but there is no /fixed/ prescription for when pawn promotions become
a consideration to actual play - and 'continuing the play' is what
happens in balanced or even dynamically unbalanced positions -

You've misunderstood the phrase `continuing the play' and are trying
to apply it outside its context. That context is a particular class
of infinite games. Infinite games are often defined as follows.
The two players

Or two or more, but yes...

For the purposes of the theory of games of perfect information, any
finite number of players is equivalent to two. Perfect information
means that all the players have full information about the position
(i.e., know where all of everyone's pieces are and any other relevant
facts).

Okay

Why does it suffice to consider two-player games? Suppose we have n2
players. We can analyze the game from the point of view of player 1
by assuming that some super-human plays the moves for players 2, ...,
n himself. If player 1 can beat the super-human, he can also beat any
collection of ordinary people acting individually. Likewise for each
of the other players.

Okay, given our suppositional super-human, but still if all other players do
not act 'individually, but gang up...

(This doesn't apply for games of imperfect information because, in
such a game, player 2 and player 3 cannot effectively co-operate
because they don't know where each other's pieces are or there is some
other key fact they don't know. But the super-human knows where all
the pieces are for players 2, ..., n, so he should be more powerful.)

But positing super-human ability must mean something to do with process, and
again this assumes that other players are now relegated to less process or
ability. And this is to subvert the equation of 'best play' into a
relativity, a socio-drama of participants.

The tricky part of inifinite games is that the rules themselves
evolve, so that some moves may condition new rules. This may except
alternating moves. This is a quibble, but still...

No, no, no! NO! There is nothing in the concept of `infinite game'
that means the rules change.

I am sorry, but this was a factor Carse provided for Infinite Games - of
course their can be others, but I am citing Carse who launched the IG ship.

Further, every game where the rules are
allowed to change is equivalent to one where the rules are static:
suppose we have rule sets 1, 2, ... (this may be finite or infinite).
Just make the number of the rule set currently in operation part of
the position and produce a master rule set that says, in effect, `If
the current rule set is number 1, you're allowed to do this; if it's
number 2, you can do this; ...' So we never need to even consider
games with changing rules in order to develop a theory of how games
work.

The very rules of nature seem to contradict this assertion. Is any new thing
possible by anticipating it from the sum of its parts? I think so, it is our
awareness of the interrrelationship of the parts which is new, and also an
awareness of the dynamic of the whole shebang [this is almost a statement
about science this past quarter century]compared with the previous rules we
had related to our, then, understanding. Now we attain to seeing the
relationship of the part to the whole, we also have a holistic paradign to
include in our awareness. The original 'rules' of behavior may not be
discarded, but included in a greater dynamic.

Psychology eg, has much to do with what goes on in people, whereas
anthropolgy with what goes on between people.


(Not all infinite games work like this. Consider the following
game, the name of which I'm afraid I don't recall. The game is
defined by a set S of real numbers between, but not including, 0
and 1. We start with `0.' written on a piece of paper and take
turns to add one digit to the end of the number written so far. I
win if the infinitely long decimal we produce is in S; you win if
it isn't.)

This again is an aside, but is this infinite, or simply an
undetermined linear extension of a theme?

This is genuinely infinite. The technical term for for your
`undetermined linear extension' is `unbounded'. An unbounded finite
game is one where there's no limit to how long a game can last but
every game must eventually finish.

Or evolve, said Darwin.

Eventually, there is some move
that is played that is the last move of the game. An infinite game is
one where there are necessarily an infinite number of moves:

yes in a way, but not necessarily because the moves are infinite in number,
but the rules governing moving can change. you probably know about
probability theory, the more complex the equation, the less forseeable or
predictable the result? This is straight Godel

there is also the implicit factor in your statement that although there is
linear play the game will continue linear, and since this tends to
contradict nature, by which i also mean physics! then such assertions are
questionable these days

it is not as much as if what you say is not true, but not possible in this
world, and therefore some care might be taken with these terms finite and
infinite, since again we encounter the paradox of even deciding what is
finite without referring to what is infinite

this great interested my author, Dr. Carse, who incidentally is a professor
of religion who would, I should imagine, get on ratehr well with Bohm.

the game
will never finish and there is no last move. After every move,
another must be made.

[Technical note to any mathematicians reading: yes, I'm aware that not
all infinite ordinals are limit ordinals but introducing the idea of
games of length omega+1 in the current discussion will just confuse
everyone.]

An example of an unbounded game is as follows. I think of a number
(an ordinary, positive integer: 1, 2, 3, ...) and you try to guess it.
After each guess, I tell you whether my number is smaller than or
greater than your guess and the game continues until you get the
correct number. On the assumption that your guesses are consistent
with the information I've given you (so you never guess 10 at any
point after I've told you the number is less than three, for example),
the game is unbounded but not infinite.

yes I understand what is considered unbounded in mathematics, yet that is
not to address chess, but hypothetical possibilities which may or may not
apply to chess, or indeed to any application! so while you may raise
possibilities of unbounded finites, these are not abstracted - that is, not
ab stracto, not 'taken form' any in vivo or real circumstance necessarily,
but from a hypothetical universe of relationships

It is not infinite because there are only finitely many numbers
smaller than the one I chose. If I choose the number `n', there are
only n-1 numbers smaller than it so, if you don't repeat a guess,
after n guesses, you must have either found out my number or guessed
something bigger than it. If you guessed something bigger (say, some
number `m'), there are only finitely many (m-1, in fact) numbers
smaller than it so, after at most m-1 more guesses, you must have hit
on the right number. There's no way that the game can go on
forever but, on the other hand there's no bound on how long the game
can be: I can choose an arbitrarily large number and your guesses can
be 1, 2, 3, ... .

(The condition I stated is critical: without it, the game becomes
potentially infinite as you could just guess `1' every time, even
though I tell you that the number is bigger.)

Sorry - I think analogous suppositions are abridged too far :0
They are rather indulgences which may or may not have any application to the
nature of chess. To say they could is not to say they do.

and there are no fixed number of moves to any game of chess,
except as metaphysically decided by others.

What on earth is the word `metaphysically' doing in the middle of
that sentence? This has nothing to do with metaphysics.

I use the word in its technical sense - 'without person'

Oh. `Non-corporeal' might have made your meaning clearer.

No. The statement, 'you are wrong' is metaphysical, since the missing words
are 'In my opinion. There is no subject to the sentence, no person. Nothing
to do with corporeal factors, all to do with specific ownership. It is a
term used in logic. It does not imply that the person speaking has any
experience whatever, and especvially no bio-organic sense of what they
address. Heiddegger called such statements 'pathic' ones [and added, as if
conducted by idiots].

There is nothing 'wrong' about supposing things weith no knowledge, except
when this is confused with the knowledge that is based on experience.
Denying any personal involvement has the implicit criticism of it that what
is spoken abnout metaphysically may not be spoken about in any other way!
That is not a paradox, merely funny!

Anyway,
what does `there are no fixed number of moves to any game of chess,
except as decided by others without person' mean? The number of moves
in a game of chess is decided (antagonistically) by the players
themselves.

But not by prescription. Players don't collude that one or the other is
mated at move 17. The number of moves is decided by the previous moves and
is indeterminate.

In this sense the game of chess is decided not by the players [the
persons] but by non-players implementing their determination of
rules.

I disagree. Ultimately, it's the players who implement the rules. An
arbiter can say, `You're not using the rules that we agreed so I do
not consider what you are doing to be part of the tournament,' but he
can't implement the rules for them.

Sorry - I attemtped to say too briefly something about the influence of ex
cathedra decisions. I'll let it pass.


small snip on similar theme

Can the statement, "black always wins" be refuted

Yes. Consider the game 1.e4 e5 2.Bc4 Bc5 3.Qh5 a6 4.Qxf7#. The
statement `Black always wins with perfect play' can be refuted, in
principle, by case analysis.

Though its also true that black can mate quicker than white - so
perhaps the explicit statement 'with best play' needs to be added,
and if so, is there any answer?

Yes, `with best play' needs to be added to make the question
non-trivial. The answer is that, in principle, the statement `with
best play, Black always wins' can be refuted if, and only if, it is
false. If it is true, clearly, it cannot be refuted; if it is false,
it can be refuted by demonstrating that, with best play, the game is
drawn or won by White.

Okay - there are the means. What is our answer?

Whether there is a practical refutation, I do not know. But, again,
as soon as you start asking for practicalities (rather than mere
existence), you're straying into the world of computer science.

Gawd! They don't even know an emulation from the real. That's not even
science, and resembles cargo-cultism.

or can it be shown to be unprovable?

No because it is not unprovable.

shrug So can you prove with best play - instead of one-sided
manipulation to invoke worst play - that your assertion is true?

`Unprovable' has a technical meaning.

Yes.

A statement is unprovable if
there exists no proof of its truth or falsity. (Goedel's famous
incompleteness theorem says that any system powerful enough to
formalize a system called `first-order arithmetic' can either prove
something that is false or cannot prove something that is true.

BTW - you read Oxford Murders? All about Goedel and complexity, and even
IQ][

The
effect is that any such system that cannot prove any falsehood must be
unable to prove some truths: these are said to be `unprovable'.)

And...

I have shown how one could, in principle, prove or disprove the
assertion (depending on whether it is true or false) so it is not
unprovable.

You have introduced a suppositional condition to answer the question, but
ended with a determinate. If I said one could answer the question, but did
not answer the question, then this would be a self-refuting tautology, as if
my statement was to deny itself by its own terms.

As for whether there exists a practical (dis)proof, I don't know.

Yeah.

Phil


Dave.

--
David Richerby Mexi-Ghost (TM): it's like a
haunting
www.chiark.greenend.org.uk/~davidr/ spirit that comes from Mexico!