True but not yet proven.
Will the Sun rise tomorrow? We do not know, it might be swallowed by a wormhole tonight.
True or False Chess is a Draw with Best Play from Both Sides
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Feb 9, 2022
0
#8283
@tygxc is basically correct.
Let me say it again, @Optimissed:
Certainty is always inappropriate from inductive reasoning
All knowledge of the real world is inductive. The same is true of artificial domains where rigorous proof has not been achieved.
It would be helpful to recognise when beliefs are based on inductive reasoning.
"Chess is a draw", the Riemann Hypothesis, the Goldbach Conjecture:
all known to be true, but not yet proven.
"Checkers is a draw", Fermat's Last Theorem, the Four Color Theorem:
all proven, but known to be true before the proof.
It’s relative, but there is a simple answer. No, it does not mean each player made perfect moves if you draw.
I can try to explain why.
Could a draw mean each player made perfect moves, absolutely. However, when playing chess you don’t start off by thinking, “ I’m going to draw my opponent.” This would mean that you are trying to win first and foremost and if you don’t succeed and end in a draw then I guess someone made a mistake. Maybe it’s not a mistake at all and it’s only because someone missed a good opportunity to take the advantage. At the end of the day you have all played a game of chess when you were down (because of mistakes) but you’re able to hold your composure and draw at times. Consider this for the same scenario, if you were still able to draw then your opponent most likely made a mistake as well. Unless your mistake was so minor it didn’t give the opponent and advantage which means it didn’t really effect the outcome of the game. I’ll give you this, if both players make mistakes and it still ends up in a draw, in some way it does mean that you or both you and the opponent made the perfect moves to get to a the draw if you are in a situation where you are trying to draw. I will say in most cases whether mid game or end game, one player is trying for a draw and the other is trying to mate.
In conclusion, it is to broad of a question and it’s relative to ever say a draw is ended on and because of perfect play on either account alone.
The real problem with the "chess is a draw with best play theory" is that in all the myriad possible chess games there only has to be one combination where white (or black) has a forced win for the whole theory to fall flat. The question will not be solved until chess is solved, maybe never? Chess is a draw with best play is just an opinion not a fact yet, even though most of us, including me, "believe" that chess is a draw with best play.
Logically, if there were one such winning line, chess would need to be solved to find it and if there were no such winning line then solving chess entirely wouldn't prove anything. It's also probably impossible to solve chess and to be absolutely sure that the solution one has is correct, since there may have been an error somewhere and if so, it couldn't be found until chess is solved "even more" and so on .... so it's actually impossible to know that chess has been solved.
All of that isn't relevant if chess is a draw. It looks like a draw and every indication is for there to exist a drawing margin for both sides, sufficient that the question of perfect play becomes irrelevant. As BlueEmu says, the onus is on those who claim chess is a win to prove it, because it's a far-fetched idea; similar to the idea that God may in fact really exist or that although it seems that every human is mortal, I might just live forever. How do you KNOW I will die?
lfPatriotGames
0
#9445
blueemu wrote:hoodoothere wrote:The real problem with the "chess is a draw with best play theory" is that in all the myriad possible chess games there only has to be one combination where white (or black) has a forced win for the whole theory to fall flat. The question will not be solved until chess is solved, maybe never? Chess is a draw with best play is just an opinion not a fact yet, even though most of us, including me, "believe" that chess is a draw with best play.
The "chess is a draw with best play" theory is pretty well supported by indirect evidence. As we move to higher and higher rating categories... as we approach closer and closer to "best play"... the percentage of draws increases steadily. At the world championship level, it is common for EVERY game in the match to be drawn. In matches between top-level engines, draws account for more than 95% of the games. Not proof positive, of course, but good circumstancial evidence. Anyone familiar with calculus will recognize the approach to a drawn limit.
The theory that "a winning line exists for one player or the other" is supported by nothing whatever.
Given the preponderance of indirect evidence, the burden of proof lies with the "not a draw" faction. It is up to them to offer evidence, or at least a convincing rationale.
i agree with everyone but i dont know why we are boing truth or false
Apr 12, 2022
0
#8287
tygxc wrote:
"Chess is a draw", the Riemann Hypothesis, the Goldbach Conjecture:
all known to be true, but not yet proven.
Laughable.
As a mathematician, I can say that is not the status of belief about the Riemann Hypothesis or the Goldbach Conjecture. The status of these conjectures (there is a big hint in their names) is that their truth is unknown. They are unproven and their negations are unproven. Many believe it is probable that each is true. "Probable" is not "certain".
One thing they have in common is that they are statements that have infinite scope - a neighbourhood of the entire negative real line and the range of all natural numbers. Another is that in both cases we have proven results for finite portions of those domains. Extrapolation of those results is obviously uncertain, rather than 100% reliable!
This is similar to the past case of Fermat's Last Theorem (ironically named for 400 years, entirely because of a falsehood written by Fermat!) where there were proofs of the conjecture for increasingly large finite exponents long before there was a proof for all exponents.
The unreliability of intuitive extrapolation bolstered by finite evidence is shown very clearly by a conjecture of Euler that is like a stronger version of Fermat's Last Theorem, thought up a couple of centuries later. This conjecture says that for any number n > 2 and 1 < k < n, there is no solution in integers of the equation:
a^n = b_0^n + b_1^n + ... + b_k^n
In English - which may be clearer - no power of n is a sum of k powers of n when k < n and n > 2.
This conjecture was stated by Euler - viewed by many as the greatest mathematician in history - in 1769. It was proven for n=3 in 1802 (Euler himself thought he had proven it earlier, but unusually there was a flaw in his proof).
Yet the conjecture was not resolved until 1966 when a counterexample was found for n=5.
Apr 12, 2022
0
#8288
The intuitive reason is simple: it is always possible for inductive reasoning to be wrong. It doesn't matter how many white swans you see, you can't be sure there is no black swan. This is obvious abstractly, because we can see it is feasible that there is a very large number of white swans and one black swan.
The quantitative reason arises thus:
First you show that Bayesian probability is the only consistent way of quantifying belief and induction with evidence (see Jaynes - Probability, the logic of science)
Secondly you observe that it would be ridiculous to have a prior that was certain about a fact about which you lack knowledge or evidence
Thirdly you see that Bayes rule applied to an uncertain prior and a piece of evidence always leaves uncertainty.
Fourthly repeating this process with additional evidence continues to leave uncertainty.
Apr 12, 2022
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#8289
Apr 12, 2022
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#8290
@Optimissed, I suggest reading #9594.
But if you want the full story, read Jaynes.
It is copyright free, a gift to the world.
Every time I read Elroch's posts (and God knows I try to keep up) , I hear this:
Apr 13, 2022
0
#8292
Optimissed wrote:
Elroch wrote:
The intuitive reason is simple: it is always possible for inductive reasoning to be wrong. It doesn't matter how many white swans you see, you can't be sure there is no black swan. This is obvious abstractly, because we can see it is feasible that there is a very large number of white swans and one black swan.
The quantitative reason arises thus:
First you show that Bayesian probability is the only consistent way of quantifying belief and induction with evidence (see Jaynes - Probability, the logic of science)
Secondly you observe that it would be ridiculous to have a prior that was certain about a fact about which you lack knowledge or evidence
Thirdly you see that Bayes rule applied to an uncertain prior and a piece of evidence always leaves uncertainty.
Fourthly repeating this process with additional evidence continues to leave uncertainty.
It's possible for deductive reasoning to be wrong too, of course. A premise, thought correct, might turn out not so.
The right way to express it is at all deduced conclusions are CONDITIONAL on those things that are ASSUMED true ("axioms" in mathematics).
The theory of deduction and induction you describe is correct in an ideal world. I mean by that a World composed of ideals or a World that is ideal in such a way that things like Boyle's Law are exact and perfect.
No. Rather it is correct in the worlds of abstraction - mathematics, computer science, information theory (and any others that slip my mind!)
In reality, they aren't: and we use observation to draw conclusions that are always, to some extent, theoretically unsure.
Try to find a valid example and you will fail. 
You, as a mathematician, place your faith in ideals: ideal laws, premises, logic etc.
You need to find a single example where the actual things I put faith in mislead me.
You will fail!
So you're reiterating your own opinion but it may differ from that of others. An engineer isn't really interested in ideals, though. They're interested in what works. Why it works can be left until later and so it is with the assumption that chess is a draw.
I too am interested in what works. For example, I am interested in wind turbines. I am interested in space exploration. All these and many, many other examples of engineering start with understanding that leads to design and successful implementation.
Apr 13, 2022
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#8293
That's EVERY proof, back to the time of the genius Euclid who may have been the first to understand the power of deduction in a formal system.
Apr 13, 2022
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#8294
Optimissed wrote:
You need to find a single example where the actual things I put faith in mislead me.
You will fail!>>
A year or so ago, you tried to teach me my subject ... Theory of Knowledge. Your grasp of epistemology would have fitted in nicely with the energetically naive view of a philosophy student, just beginning his second year.
I hope that gives you great pleasure, but I don't acknowledge it to be of any substance.
Anyhow, we don't want to start going round in more circles, so let's just be happy about axioms and how they facilitate proofs. I don't think that the handbook of official axioms is closed.
LOL. It is charming that you think it might be.
More to the point, there isn't a group of people who dictate what they are, because within the field of philosophy, fundamentally diverse ways of analysing mankind's relationship with reality are possible.
Axiom schemes are selected for a specific purpose. Others can accept them and use them for the same purpose or come up with a different scheme for an alternative.
For example, the Zermelo-Fraenkel axiom scheme is widely used to formalise set theory, but needs enhancing with the axiom of choice for many purposes and with the Continuum hypothesis for others. By contrast the axiom scheme called Constructive Set Theory serves a similar purpose but does not.
Apr 13, 2022
0
#8295
It seems that modern computers might be indicating, not proving, that chess could be a win for white. Most people have assumed that it is a draw, but co.punters show that it is possible that best play produces a win by slowly improving on the initial advantage of moving first until you have a won position.
#9613
"It seems that modern computers might be indicating, not proving, that chess could be a win for white."
++ No, on the contrary: the stronger engines become, and the more time they get, the more they draw and the less they win. 60 times longer time yields 5.6 times less wins.
Most engines start at an evaluation around +0.33 for white just reflecting that white is one tempo up. 3 tempi = 1 pawn. After a pair of trades this advantage dies out to +0.00.
Apr 14, 2022
0
#8297
Risen from the dead! It will again turn out that most people think chess is a draw with best play and posters here will argue whether or not our present state of knowledge is sufficient to assert tis for CERTAIN. Soon there will be insults flying back and forth. Tres amusant.
Apr 14, 2022
0
#8298
Optimissed wrote:
Elroch wrote:
Optimissed wrote:
You need to find a single example where the actual things I put faith in mislead me.
You will fail!>>
A year or so ago, you tried to teach me my subject ... Theory of Knowledge. Your grasp of epistemology would have fitted in nicely with the energetically naive view of a philosophy student, just beginning his second year.
I hope that gives you great pleasure, but I don't acknowledge it to be of any substance.
We can all play that game. I wonder if you've thought it through? Whether other people acknowledge what you decree to be the case, to "be of any substance"? I think you have a lot of very bad ideas and I've noticed that others think that too. They aren't all stupids, either. Try again?
Anyhow, we don't want to start going round in more circles, so let's just be happy about axioms and how they facilitate proofs. I don't think that the handbook of official axioms is closed.
LOL. It is charming that you think it might be.
More to the point, there isn't a group of people who dictate what they are, because within the field of philosophy, fundamentally diverse ways of analysing mankind's relationship with reality are possible.
Axiom schemes are selected for a specific purpose. Others can accept them and use them for the same purpose or come up with a different scheme for an alternative.
For example, the Zermelo-Fraenkel axiom scheme is widely used to formalise set theory, but needs enhancing with the axiom of choice for many purposes and with the Continuum hypothesis for others. By contrast the axiom scheme called Constructive Set Theory serves a similar purpose but does not.
The more axiom sets you describe, the more you make it obvious that we can invent them either for pleasure or for profit. And, presumably, that people do just that.
No, my examples certainly do not do that - you seem unfamiliar with their importance.
I described important axiom schemes that are the foundation of large amounts of mathematical work. Most branches of mathematics are built on some version of set theory.
It is certainly true that you could invent an axiom scheme "for pleasure". You would be ignored unless it does something important. As for doing it for profit, there really is no market. Even research funding is only available for work recognised as being significant.
In practice, mathematicians choose axioms to serve specific purposes. For example, each class of algebraic object has a set of axioms that defines it. All mathematicians would understand that the axioms are chosen to represent important aspects of abstract truth. Physics has an influence on which objects are interesting: sometimes the physical need motivates the development of the mathematics, such as in the case of "quantum groups".
Apr 14, 2022
0
#8299
A lot of lecture courses on mathematics start with a few axioms.
For example, group theory.
A group G is a set of objects with a product
* : G x G -> G
where the product of elements h and g (the order matters) is written
h * g
The product satisfies a few axioms:
1. Associativity:
For all g, h, k in G:
(h * g) * k = h * (g * k)
2. identity:
There's an element e in G such that for all g in G:
e * g = g * e = g
3. Inverse
Every member of g has an inverse written g^-1 such that:
g * (g^-1) = (g^-1) * g = e
And that's all. (Set theory is also assumed so that notions like membership, the product set G x G and what a function is are already known. This could be made explicit with more axioms).
There is an important point that is worth making. There are many different objects that satisfy the above axioms and are thus groups. For example, the signed integers with addition as the product is a group. So is the set of invertible 7 x 7 matrices of real numbers with the usual matrix product.
Any theorem derived from the axioms will apply to all such groups, even though they are very different.
A little of this sort of abstract mathematics has crept into high school curricula, as "modern mathematics", but it makes up much more of university maths.
Isn't this just a tautology? Like, of course with perfect play on both sides it's a draw, but rather that in a lot chess there isn't perfect play but rather degrees of inaccuracies and the difference is whether you can exploit those inaccuracies for a win or not.
It seems obvious to us that with best play chess is a draw, but several of the posters in this thread dispute that idea.
They tell us that we can't possibly know it's a draw. That's one result of equating knowledge with a deductive proof, which must always contain the knowledge, of course, in the form of premises which need to be rearranged in a way that is in agreement with logical necessity. Obviously, since it was known that draughts (checkers) is a draw before the deductive proof was achieved, that attitude is incorrect and not in accord with reality.
It was definitely known, without doubt, that draughts was a draw. It wasn't just a good guess. Another example is evolution, which was known, without doubt, by a large group of scientists a century before Darwin proved it. There are many such examples and it's incorrect to assert that we cannot be sure that chess is a draw.
I could say with certainty that if I buy a lottery ticket, I won't win the big prize. Then I could go and test this belief thousands of times and be right every time. Even so, I would have been WRONG to be certain.
Certainty is always inappropriate from inductive reasoning. That doesn't stop it being right a lot of the time, but the cost of being wrong very occasionally when you are "certain" is infinite, by an appropriate measure.
What is the correct alternative? To acknowledge there is a small probability of being wrong. While it may often be useful to quantify it, merely acknowledging it is greater than zero is a big step in the right direction.