Life In 19x19

Reference: http://www.lifein19x19.com/forum/viewto ... 05#p179105

vier wrote:How can a definition be false?

Instead of "false", one speaks of "not well-defined". The ko definition http://home.snafu.de/jasiek/ko.pdf would be not well-defined if at least one example in line with the used methodology exists that a) is a ko while the definition says it is not a ko or b) is not a ko while the definition says it is a ko.

The paper shows samples of all known repetitive shape classes, with a few known exceptions, for which verification that the definition works well should be straightforward if somebody spends the time to check. For final knowledge, one must 1) discover new repetitive shape classes and find some such class where the definition does not fit or 2) classify all possible shapes into shape classes to identify all possible forced-repetitive classes or detect that no further exist.

However, look at the last condition of global-ko-intersection and perceive how close to the final solution my definition must be! If anything, I would search for a possibly wrongly calibrated minor condition. Happy hunting, but, for each changed definition, do not forget to test and positively check at least all the examples and negatively check at least all the counter-examples I have tested!

RobertJasiek wrote:Reference: http://www.lifein19x19.com/forum/viewto ... 05#p179105

vier wrote:How can a definition be false?

Instead of "false", one speaks of "not well-defined". The ko definition http://home.snafu.de/jasiek/ko.pdf would be not well-defined if at least one example in line with the used methodology exists that a) is a ko while the definition says it is not a ko or b) is not a ko while the definition says it is a ko.

The paper shows samples of all known repetitive shape classes, with a few known exceptions, for which verification that the definition works well should be straightforward if somebody spends the time to check. For final knowledge, one must 1) discover new repetitive shape classes and find some such class where the definition does not fit or 2) classify all possible shapes into shape classes to identify all possible forced-repetitive classes or detect that no further exist.

However, look at the last condition of global-ko-intersection and perceive how close to the final solution my definition must be! If anything, I would search for a possibly wrongly calibrated minor condition. Happy hunting, but, for each changed definition, do not forget to test and positively check at least all the examples and negatively check at least all the counter-examples I have tested!

IIRC we've had this discussion on terminology in the past, but just for completeness I'll state it here again. A definition is always well-defined. It's a definition. If a definition is false, then it is not a definition. If I formally define a chair to be a frog, the chair is a frog. and it is true in the scope of my work. My work may bear no connection with the frogness of the chair, but it doesn't mean the definition is false, because it is not.

From your description (I read your ko definition a long time ago but don't remember it right now and don't want to read it again... it wasn't specially appealing,) you don't have a definition.

Definition: We'll call a situation satisfying

• Hypothesis 1: blabla
• Hypothesis 2: blablabla
• ... Hypothesis N: blabla...bla

a ko.

This is a definition, and it is true, in any instance we'll call *that* a ko. Just like the chair will be called a frog even if it doesn't jump!

It seems like you have not studied maths at the FU Berlin:) A definition itself was not considered well-defined because of being a definition. It was considered well-defined if it fulfilled its intention in the given framework.

The given framework for a definition of ko is the set of all known relevant examples and counter-examples and the related understanding of what ko is about.

If you think that any definition of ko could be stated, state a second definition, prove that it is unequal to my definition and prove by application that it distinguishes examples from counter-examples as well as my definition. You cannot define "a ko is a chair but not a frog" and claim to have a well-defined definition, because you cannot successfully apply it to the known examples.

RBerenguel wrote:A definition is always well-defined.

Exhibit A:

The smallest positive integer not definable in under eleven words.

The set of all sets that are not members of themselves.

Not to mention pink invisible unicorns!

RobertJasiek wrote:It seems like you have not studied maths at the FU Berlin:) A definition itself was not considered well-defined because of being a definition. It was considered well-defined if it fulfilled its intention in the given framework.

The given framework for a definition of ko is the set of all known relevant examples and counter-examples and the related understanding of what ko is about.

If you think that any definition of ko could be stated, state a second definition, prove that it is unequal to my definition and prove by application that it distinguishes examples from counter-examples as well as my definition. You cannot define "a ko is a chair but not a frog" and claim to have a well-defined definition, because you cannot successfully apply it to the known examples.

Nope, I didn't, my German is not good enough for it In my university, most (I can't put my hand on all I have read, since my memory is not *that* good) math and papers I have read, a definition is essentially a labelling of something with a name. It can be something with a set of hypothesis on how it behaves, or the properties it has.

The only instance when I call something well-defined or ill-defined is with mappings/applications/functions, where you have to actually check the definition of a function (from one space to another) satisfies the constraints of the source and target spaces.

Some reference is always useful.

Russell & Whitehead, Principia Mathematica wrote: Definitions. A definition is a declaration that a certain newly-introduced
symbol or combination of symbols is to mean the same as a certain other
combination of symbols of which the meaning is already known. Or, if the
defining combination of symbols is one which only acquires meaning when
combined in a suitable manner with other symbols*, what is meant is that
any combination of symbols in which the newly-defined symbol or combination
of symbols occurs is to have that meaning (if any) which results from substi-
tuting the defining combination of symbols for the newly-defined symbol or
combination of symbols wherever the latter occurs.

The smallest positive integer not definable in under eleven words.

This is an object (an integer). Can or cannot exist, but we can define it.

The set of all sets that are not members of themselves.

This is a set which is not a set... and the contradiction can be shaved with class theory instead. It's not a "definition" in the mathematically assumed sense (see my quote above.)

http://de.wikipedia.org/wiki/Wohldefiniertheit
http://en.wikipedia.org/wiki/Well-defined

Not surprisingly, the German page has the kind of well-definedness I need:

"Eine Menge ist wohldefiniert, wenn das Definiens für jedes beliebige Objekt eindeutig festlegt, dass es entweder Element der Menge ist oder nicht Element der Menge ist."

Translation:

"A set is well-defined if the definition specifies unequivocally for each arbitrary object that it is either element of the set or not element of the set."

Here, the considered implied set is the set of all objects that are 'ko', where objects can be parts of go positions.

A set is well-defined if the definition specifies unequivocally for each arbitrary object that it is either element of the set or not element of the set.

Thus the definition

A "ko" shall be defined to be any kind of frog.

is perfectly well-defined (assuming biology is clear enough on what constitutes and doesn't constitute a frog).

I agree that you're somewhat misusing the word "definition". Whether or not your definition captures colloquial usage of the term you're (re)defining is quite irrelevant to the definition itself.

RobertJasiek wrote:http://de.wikipedia.org/wiki/Wohldefiniertheit
http://en.wikipedia.org/wiki/Well-defined

Not surprisingly, the German page has the kind of well-definedness I need:

"Eine Menge ist wohldefiniert, wenn das Definiens für jedes beliebige Objekt eindeutig festlegt, dass es entweder Element der Menge ist oder nicht Element der Menge ist."

Translation:

"A set is well-defined if the definition specifies unequivocally for each arbitrary object that it is either element of the set or not element of the set."

Here, the considered implied set is the set of all objects that are 'ko', where objects can be parts of go positions.

Robert, this is exactly like the example of well-definedness I explained above. This does not apply to a "definition," as in the proper usage of \begin{definition} you'd use in a paper or book. A definition is a enumeration of properties something satisfies (or not.) and which may exist, or not. What you are quoting is well-definedness, and is about *sets* (and thus extends to functions, or a specific function and a wide array of things.)

This is mainly (in my field) used in "let f:some_set to another_set" (this "let" is an informal definition since it's usually "local" and thus doesn't need a proper name but just a dummy name.) Then, "lemma: f is well-defined" when you check it actually satisfies the set constraints. Nothing prevents it from being *defined* since defining something doesn't mean it exists or makes sense (or is *well-defined*).

leichtloeslich wrote:

A set is well-defined if the definition specifies unequivocally for each arbitrary object that it is either element of the set or not element of the set.

Thus the definition

A "ko" shall be defined to be any kind of frog.

is perfectly well-defined (assuming biology is clear enough on what constitutes and doesn't constitute a frog).

I agree that you're somewhat misusing the word "definition". Whether or not your definition captures colloquial usage of the term you're (re)defining is quite irrelevant to the definition itself.

Exactly! The problem is connecting real-world usage with the proper sense of definition. Also, we can perfectly define pink flying unicorns. Actually:

Definition: We'll call P the set of pink flying unicorns.

Perfectly well defined, as in "this is a definition, since it uses known properties and assigns them a name for later".

Lemma: The set P is empty.

I'll leave the proof to the reader.

RBerenguel wrote:This does not apply to a "definition,"

Wait, so you're actually nitpicking that the object of a definition may or may not be well-defined, but never the definition itself?

Seems like an irrelevant language-usage point, but nevertheless.. challenge accepted!

Let me just do some horrible things to Berry's paradox.. and voila:

The shortest definition of the empty set that is not definable in under sixteen words.

So here the object of our definition is itself a definition, but this definition isn't well-defined because the defining meta-definition definitely defines this very definition, which was supposed to be un-definable in under 16 words, in just 15 words!

edit: miscounted the number of words by 1 :[

How can anyone write 46 pages on the definition of a ko and expect to be taken seriously? I know what a ko is and so does every other player with more than minimal experience. It does not take 46 pages. In my opinion, that fact alone means your arguments lose all credibility.

DrStrawman wrote: In my opinion, that fact alone means your arguments lose all credibility.

... said the gent who bases vital parts of his beliefs about reality on anecdotes involving his cat!

leichtloeslich, "A "ko" shall be defined to be any kind of frog." is well-defined in a weak sense of defining a term by using (presumably) defined other terms, but not well-defined in a stricter sense because the word "ko" is abused. Call it "all_kinds_of_frog" and there is no ambiguity WRT the intention of "ko".

RBerenguel, given my definition of "ko" and given the known examples informally called 'ko'. Proposition of well-definedness of my definition: "a) Each example has a "ko" according to my definition on exactly the intersections informally perceived to belong to the ko. b) None of the intersections in each counter-example has a "ko" according to my definition." Currently the proposition is proven empirically on the high definition level (not strictly on the low level of left-parts of strategies).

I agree that the problem is connecting real-world usage with the proper sense of definition. It is like doing maths as a physicist. Until we have checked each example in the universe (e.g., each point of the time-space of the universe), we cannot be absolutely sure that the general theory of relativity always applies. We can only say that it always applies on the non-quantum scale in each example experiment ever done. Proving the ko proposition is easier because the space of all subsets of all positions is much smaller. Currently the proof is only partial for all known (classes of) examples.

In the strongest sense of well-definedness, the proposition will be proven in general or rejected by at least one counter-example to the proposition.

DrStraw, do not forget the 13.5 years needed to write those 46 pages; how can anybody be taken seriously needing so long to define ko when it is something that can be understood within seconds?;) What can be understood within seconds is basic-ko. Long cycle kos are much more difficult to understand. Anyway, I do not expect to be taken seriously by people resorting to meta-discussion instead of discussion of what is written in the paper.

You claim to know what a ko is and that every player would know it. You (or anybody), since you know it, write it down, apply your writing and check it for the examples in my paper. Does your definition identify exactly those examples and same-shape examples in arbitrary other positions? We will see. Credibility is measured by the success of identifying such examples in a re-producable manner. This requires sharing knowledge explicitly (such as in writing); a mere claim (by you) to just know what ko is lacks the possibility for re-production by others. Credibility is also measured by this possibility. (BTW, of course, you are not even impressed that finding some of those examples took many years of thinking. If you really did know what ko is, you would have told us about those ko shapes long before we researchers found them.)

RobertJasiek wrote:It seems like you have not studied maths at the FU Berlin:) A definition itself was not considered well-defined because of being a definition. It was considered well-defined if it fulfilled its intention in the given framework.

The given framework for a definition of ko is the set of all known relevant examples and counter-examples and the related understanding of what ko is about.

If you think that any definition of ko could be stated, state a second definition, prove that it is unequal to my definition and prove by application that it distinguishes examples from counter-examples as well as my definition. You cannot define "a ko is a chair but not a frog" and claim to have a well-defined definition, because you cannot successfully apply it to the known examples.

The problem with "definition by known examples" is that, like "known examples" themselves, it is more of a hypothesis. We can never be sure that "all possible examples" and "known examples" are the same set.

And anyways, how do you define which examples are KO and which are not, if you don't have a definition? How do you determine the set of "known examples"? It seems to me that you must already have a definition, explicit or implicit, or you would not be able to differentiate between KO and non-KO examples. So all you are trying to do is to find another wording for it, but still make it equivalent to the other definition.

So in general, I agree with RBerenguel that a definition is what defines a set or condition.
If the set or condition is not what you had in mind, this makes the definition possibly inappropriate, but not necessarily wrong. It just defines a different set or condition that you wanted. Maybe its your "want" that is wrong?