Note that although Elroch depicted it as a full mathematical representation of draughts, it was not that at all.
It was in the correct sense. I can understand why you are confused about this.
So we can understand that at that time, weakly solving it in this way was at the limit of practical possibilities.
Yes, solving checkers took 18 years and over a thousand CPU years. Hundreds of thousands of dollars worth of computing.
Zermelo represented his Theorem as a proof that chess may be solved similarly.
No. He simply presented a mathematical proof that serves for all combinatorial games. [A technicality meant it only applied to chess with a drawing rule which forces games to be finite. Later this was extended to basic chess, where games can go on forever]
Firstly, Elroch has knowingly misrepresented the question, since he had claimed that Zermelo proved that chess could be mathematically represented, which involves representing chess as a series of equations.
This is a major misunderstanding of what mathematics is.
In general mathematics is about abstract truth, In the main this is revealed by the deduction of propositions from sets of axioms that define the properties of an object or a class of objects. For example, you can write a set of axioms that defines a vector space, then derive an infinite number of theorems that apply to all vector spaces. But it also incorporates more specific results, such as the result of an arbitrary calculation like 134798174 * 1382382. Solving a specific game is a bit like the latter.
The representations in Schaeffer's proof are part of the working of the proof that there is a drawing strategy for white and a drawing strategy for black. If you think of it as being like the working of a big arithmetic calculation, you won't go wrong.
This is a rather petty result to mathematics, which is interested in generalities rather than arbitrary examples. But to humans, the solution of a single classic game is of interest. By contrast, Zermelo's result is general. But it does not tell us the result of any game, nor provide any strategies - it just proves they exist! Mathematics is full of existence proofs, as well as more explicit results.
Schaeffer's work is in truth a huge proof most of which is done by a computer. This is perfectly normal - we can easily write a program to check things and be confident of the result even though the working is too big to check. Say a huge arithmetic calculation. Or, for example, the mathematical result that 2^82,589,933 − 1 is prime requires a large amount of computer checking to verify. It is certainly important that programs used to derive mathematical results are checked thoroughly. Ideally redundancy should be used, but the computational cost of results like solving checkers is too big for this to be fully done until the cost falls a lot.
Zermelo's theorem doesn't rely on any representation - it relies on the axioms defining a class of games.
I am quite sure that it is impossible to expect that a simple proof by mathematical induction demonstrates that a simple, linear game such as noughts and crosses may be mapped to an extremely complex, non-linear (no, that's not a valid use of the term 'non-linear') game such as chess. Zermelo's claim was definitely bogus in this respect.
Let me be quite blunt - that is ignorant narcissism. Zermelo was a mathematician who developed the set theoretic foundation of all of mathematics, and you are a guy who boasts about IQ tests you took when you were young. More importantly, Zermelo's theorem stands today, tested by several generations of mathematicians, all more capable than you. If you disagree, show me your best work.
This is backed up by my son's judgement that representing chess mathematically is completely impossible.
I recall this from a few years back and understand what he meant - that you cannot simplify chess in a way which would permit a compact proof - it is too arbitrary. I think your son would understand that generating the 32-piece tablebase is conceptually possible (just impractical) and you can inform him that in the relevant sense this is "representing chess mathematically". This is the right sense, because it is the sense that determines if a proof is possible. Do discuss tablebases with him.
Elroch is a statistician,
I am not. My original specialisation was in mathematical analysis, and my MMath is in this area. I use probability theory in my work, and am a proponent of Bayesian analysis.
as against a very gifted mathematical analyst
he's a physicist, but I am sure his mathematical skills are good. Here the relevant field is game theory and I am not aware of his level of knowledge.
who has performed groundbreaking mathematical procedures including representing magnetism mathematically as a product of fermionic spin.
Good for him. And irrelevant to this, to be frank.
I want to say a word about Elroch's behaviour. Elroch constantly switches stories.
I warrant that you are unable to exhibit a single example of two posts of mine that support this.
[deleted drivel]
"Looked at it. It's an hypothesis. It may be considered by Zermelo to be an axiom and there's no syllogistic proof to support it."
If it wasnt proved it wouldnt be called "zermelo's theorem"
"It was an inductive proof, you fool. Can't you even read? It wasn't deductive. Means it's an assumption."
in mathematics inductive proofs are literally logically equivalent to deductive proofs. "Induction" is just referring to the techniques used.
for example, one of the most basic inductive proofs is to prove that the sum of the first N integers is equal to N(N+1)/2.
let f(N) = N(N+1)/2. Basic arithmetic shows that f(N+1) - f(N) = N+1. therefore, if f(K) = the sum of the first K integers, then f(K+1) = sum of first K+1 integers (where K is a known constant).
then, we start by verifying that f(1)=1.
finally, mathematical induction refers to the step where N can be extended from 1 to all natural numbers. this too is mathematically rigorous, for any M that we claim is the lowest integer for which a statement is false, since M-1 must be true, M must also be true.
All in all optimissed i think your struggles come from imprinting different definitions to mathematical terminology and methods.
On reflection, I was completely right. The mathematically inductive proof that Zermelo used for his simplistic ideas can only be extended to solving chess via a process of philosophically inductive reasoning, which happens to be false since like isn't being mapped to like. It's as though a crumpet is being mapped to a falcon.
I was pretty sure I was right all along but when challenged by both you and Elroch I thought I better look into it and have a think about it. It turns out that Zermelo was not using the mathematically inductive process that Elroch represented him to be using. Noughts and crosses and chess are not mutually commensurable and so one cannot be mapped to another in that way. Noughts and crosses can be solved by linear arithmetic and chess is immensely complex and certainly can't. Zermelo was using philosophical induction, as I stated. so at that point I had won the argument with Elroch completely.
I have shown that he was inconsistent all the way through, constantly shifting his position so much that it seems deliberate. He claimed that chess can be represented mathematically by means of equations and that Zermelo's Theorem supports that. He is completely incorrect. His entire argument depended on the necessity that chess can be represented mathematically. Otherwise he has to fall back on heuristics, which he's condemned many times in ty. Hence his entire argument has fallen apart and he's lost.
Most people here know I'm cleverer than Elroch so it isn't a big deal. Only Dio and Player stick up for him and one wonders if they count. Well, I stopped wondering that years ago.