So, connected passed pawns, rook behind them...
Chess will never be solved, here's why
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Elroch
Sep 15, 2024
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#12841
Sep 15, 2024
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#12842
Just to spoil the fun, https://syzygy-tables.info/?fen=8/8/5r2/2K2Pk1/6P1/8/8/6R1_b_-_-_0_1.
It's already been solved. All you have to do is remember 17½ thousand million winning positions, which side wins, and the shortest number of moves to mate for each of them and Bob's your uncle.
(Why Black to move?)
Sep 15, 2024
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#12843
Just an accident. Not crucial. It was the final position in a game and l was surprised at the theoretical result.
white not to checkmate in one lol (white on bottom).
better yet...how did white get to this position and it be their move ?
Sep 16, 2024
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#12845
Got the move. The other question I will pass on. You've set the board up upside down!
Sep 16, 2024
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#12846
Regarding my last diagram, interesting that this one is not drawn. It's not just about the blockade.
Sep 16, 2024
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#12847
Rook and two passed connected pawns against rook is an interesting endgame.
I looked at it many years ago and forgot about how it works.
Am guessing that there are many that are drawn and many won.
Something like rook and bishop versus rook but maybe easier.
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that last one maybe has something to do with the fact that the black rook has less room for lateral checking.
Sep 17, 2024
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#12849
i’d say that white def wins because no matter what you’ll end up at least a rook up if you’re white
Sep 17, 2024
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#12850
Thee_Ghostess_Lola wrote:
white not to checkmate in one lol (white on bottom).
better yet...how did white get to this position and it be their move ?
also just for my peace of mind was your first thought Rb2+ too
Sep 17, 2024
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#12851
"Infinite" means not finite. This encompasses two related but distinct concepts.
A set of objects is finite if it has n objects for some natural number n. An infinite set is one where this is not so.
A magnitude - like the length of a line or the area of some region in the plane - is finite if it corresponds to a real number r. An infinite magnitude (eg the area of the upper half plane) is one that is greater than any real number r.
These are, of course, precisely defined mathematical concepts. Mathematical models are used to model the real world and where ever the notion of infinity applies to the real world it is really about some model of the real world.
Sep 17, 2024
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#12852
They are mathematical entities. Some of them happen to be very useful in models of reality.
Complex numbers have a very intuitive role as transformations of the plane (if you have a complex plane and multiply all the numbers by some complex number z, this amounts to enlarging it by a factor |z| and then rotating it by a factor arg(z). So everything about complex numbers is really about that set of transformations of the plane. Transformations are very important in fundamental physics.
Sep 17, 2024
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#12853
Optimissed wrote:
Personally, I tend to think about it slightly differently from that although it comes out similarly.
But if infinity is uncountable, it's incommensurate with or seperate from real numbers.
With all due respect, this seems confused. "Infinity" is not a single entitity. There are infinite cardinal numbers. An infinite cardinal number is countable if it is the same size as the natural numbers. If it is not (i.e. it is strictly bigger) it is called uncountable. When a term is entirely standard, it is important to use it correctly.
Therefore we wouldn't say that infinity is greater than any real number.
Me neither, unless I was very clear that I was talking about the magnitude +infinity that can be added to the upper end of the real numbers by definition, and defined in a consistent way as the limit of all sequences that increase without bound. There is also usually a -infinity added to the other end for some purposes (not needed for things like areas and lengths), and this is smaller than any real number (by consistent definition).
I don't think it's a grievous error to do so,
It's not an error at all. The extended real numbers are a very useful consistently defined algebraic object with many applications.
since these ideas are notional or abstract in any case and all we do is to choose a form of words which gets the idea across. But I think of infinity not as greater than any real number but simply as uncountable.
And there you are getting confused again. It is infinite cardinals (abstract objects used for counting) that can be uncountable. The term has no meaning for infinite magnitudes (abstract objects used for measuring, as in the examples given).
Optimissed wrote:
If you can't make sense of what I'm saying. I think that you should ponder on the relationship between magnitudes and symbols used to denote them, without being so quick to imagine that I and not you am confused. Perhaps it would be better not to talk in such terms (as "being confused") but to try to reach a mutual understanding which may lead to agreement.
Obviously, all those who believe everything you utter will think you're right. But the opinions of people who can't think are unimportant, as you know very well. Your mind is better than theirs.
Worry less about the ranking and hierarchies of others' intelligence and more about your own competence and you'd be far better off than you are now.
Sep 17, 2024
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#12855
First you need to remember that if you want to use the word "infinity", you need to attach it to a well-defined (mathematical) entity. This word does not have a unique meaning, and statements using a word whose meaning has not been made clear are not meaningful.
And the same for the word "infinite" (except this needs to be associated with a well defined property - the precise mathematical analog of an adjective in English. A property has a scope - one meaning has the scope of cardinal numbers, another has the scope of some selected class of measurable sets).
Sep 17, 2024
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#12857
Optimissed wrote:
Elroch wrote:
First you need to remember that if you want to use the word "infinity", you need to attach it to a well-defined (mathematical) entity. This word does not have a unique meaning, and statements using a word whose meaning has not been made clear are not meaningful.
Yes, I agree. However, this is not a doctoral thesis. It's real life and in real life it takes two people to communicate. Each has to be willing to make at least some effort to understand the other. I certaily don't intend to define every word I use but you are capable of asking me to enlarge on something, I'm sure. At least, talking to you, even if we don't agree, I usually get the feeling I'm talking to someone intelligent. I can't think who I would apply that to among your supporters. Llama maybe, to some extent but he's more of an independent spirit.
And the same for the word "infinite" (except this needs to be associated with a well defined property - the precise mathematical analog of an adjective in English. A property has a scope - one meaning has the scope of cardinal numbers, another has the scope of some selected class of measurable sets).
Since infinite is an adjective, I broadly agree. I'm also not going to be dragged into pretending to be a mathematician when I'm not. My subject is philosophy, as you very well know. Philosophy and mathematics are different from one-another. If both people are careful, they might meet in the middle.
philosophy and math have different definitions for infinity
Sep 17, 2024
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#12858
A lot of arguments could be had over the meaning of the word infinity.
But the word is meant to serve us - not us the word.
As to whether infinity refers to the real world and/or possibilities of the real world - it will never be proven that various quantities of the real world are finite or infinite.
As to the second part - possibilities of real infinity - they definitely exist.
As to 'philosphy' trying to set itself up as an authority that could rule on such things - such authority is false and pretensious.
Sep 17, 2024
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#12859
Again - regarding rook and two connected pawns versus rook -
an issue not discussed so far although mentioned ...
when such positions are drawn it appears that happens when the defending rook has has enough room to keep laterally checking the attacking King and that King cannot escape the checks without taking itself too far from supporting its pawns.
That might be only one element of determining whatever position is a draw but it appears to factor.
