Math Fun

Let me explain to you how this works, since you seem to be confused.

If someone says “answer this question for me” and you don’t question the question, you are accepting the assumptions behind it.
What I have done is look at your question and say “hang on, this question doesn’t fly”. I proceed to unravel the question and clearly show why it is a duff question, and to go along with it would lead to a duff answer. I don’t like duff answers. I did, however, do you the courtesy of giving you an answer that was related to your question, but fundamentally based on my criticism of it, that in a way actually answers your question just fine!

Thus, can I pick a number that would fail to work in the same way as you were demonstrating in your example of picking 97?
Yes, I can pick any number that would fail to work in that way, as shown by my criticism of your whole process in dealing with any given number.
Question answered.
Is there any number between 0-99 that would cause them to deduce improperly? Yes, all of them ^ question answered. Zero fearing on my part.

What more do you want?

Yes, a PhD is completely irrelevant to either assuming there is only one solution in this case, or understanding that there is only one. That was my point: your hyperbole about PhDs was irrelevant.
I already did prove there were no alternatives (without a math or logic PhD) - I suspected that you hadn’t been reading anything but your own posts for many pages now. Or maybe you just didn’t understand the concept of dealing with all and only the certainties from any given islander’s own perceptions, and only when that fails to lead to a solution on its own, adding in the Guru’s communication and seeing if that led to a solution… which it just so happens to do - and considering ALL the certainties that arise from a combination of both of those factors, we get one single solution. This, by default, eliminates any other possibilities, because by process of elimination, they would necessarily rely on something other than certainty/definite knowledge and flawless deduction.

Case closed. Get over it.

More rhetoric.
Can’t answer a simple question.

Show me one number that doesn’t work and show it not working.

I showed you one that did work. It is your turn.

…

what?

I… just answered your question. Right… there…

Q: IF everyone chooses the exact same number to use in counting days before deducing that they are the remaining blue (or not), is there any number between 0-99 that would cause them to deduce improperly?
A: Yes. All of them.

How is that… “not answering a simple question”?

Okaaay…
Here:

This is an example of one number that doesn’t work.

(D) is the best example of why it doesn’t work. (D) doesn’t work because if there were only 97, (A) would not be true. Yet (D) claims both to be true.
In (D), “if there were only 97”, each one would see 96 other blues. They would not know that they all know that there were at least 97 blues because they can only see 96 (which is less than 97, not “at least 97”) blues because each do not know their own eye colour and they are not seeing blues that are not there. Not even seeing 97 blues means they do not “all know [that they all know] [that there are at least 97 blues]”, which is (A). (D) says that they do.

I… cannot be any more clear here. This doesn’t work. It is “one number that doesn’t work”. I just showed it not working. Just like you asked.

You’re telling me a clear answer to your question is “just rhetoric”? Really?

Sure looks like it works to me. They all correctly discovered their own eye color.
So no. You didn’t show anything not working.
“Not working” means they make a mistake in their color.

Was there ever only one person on the island when the guru said there was one?
Noooo.
It is irrelevant.

The issue is the ability to know your color. They each learned of their color.
And they will every time… IF they ALL choose the SAME number.

Again, James, what you’re saying suggests that you don’t understand the logic behind the canonical solution.

In the cases where 1, 2, or 3 islanders learn their eye color, they all know that there are at least two islanders on the island with blue eyes. But the counterfactual, “if there were only 1 he would leave on day 1”, is still necessary to reach the conclusion. You seem to understand and accept this.

For some reason, though, when we get to 4, you and Fixed Cross discount the value of the same information, and suppose that the 1 would just spontaneously know that he has blue eyes. What is the difference between the case of 3 blue islanders and the case of 4 blue islanders such that the guru is necessary in the former but not in the latter?

If we use your example for the case of 4 blue eyed islanders, I think what Silhouette is saying becomes even clearer:

It should be obvious that while they all know that they all know that there is at least 1 blue, there is no 1 who knows that he has blue eyes. We are given that no one knows their eye color, and D requires that there is at least 1 who does. The contradiction, as Silhouette has repeatedly explained, is that you’re taking the knowledge out of context. 4 know there is at least 1. But if there were only 1, he would not know that there is at least 1 - until the guru speaks.

The Guru never said that there was only one person on the island.

She said she can see someone who has blue eyes. Not only one someone.

Sounds to me like you’d be happy with anything that looks like “a working out” of one’s eye colour just so long as the end step “says” that they definitely worked out their own eye colour. I’m not disputing that your solution looks like it works to you.
At the end of your process, it DOES say they all leave (and therefore know their own eye colour, as this is the sole stated condition on which their leaving depends).
It’s just that the process doesn’t deductively lead to that end through only logic, observation and certainty.

This issue raises an epistemological point about what constitutes knowledge.
Is it knowledge when someone uses uncertain assumptions in order to determine a fact?
How about if someone uses uncertain assumptions in order to come to the same conclusion as someone who has used certain deductions?

I do not class these two things as the same, though perhaps you do, and perhaps you are vouching for a utilitarian “ends justify the means” approach to the puzzle. You do say that the ability to know their own eye colour is all that matters, and you say nothing about how they arrive at this information. Perhaps your point, all along, has been that as long as you think you know your own eye colour, even if through flawed logic, information counter to observation, and assumption, that’s good enough?

You have only the choices of logical deduction, spontaneity, and pot luck.
The only one of those that always works is logical deduction.

You can’t think of a single number that they could all choose that would not work because any number they all choose will work every time. As logicians, they know what always works because they can deduce what always works. I didn’t try every number, I deduced that any number would work. So if I can deduce what always works, why can’t perfect logicians?

Who said anything about the guru saying that there was ONLY one??

What it suggests to me is that You do not understand logic.

You should explain that to Sil.

What FC and I know is that it is not spontaneous, else it could not work every time they do it. Why don’t you know that?

You have trouble with patterns, I take it…?

(D) does NOT require that anyone know their own eye color. It makes the exact same hypothesis as the canonical version, “IF there were only one and he knew there was one.”

I haven’t taken any more out of context than you;
“If there was only one when the guru said that was one” (which never took place).

But the question, still incorrectly answered was simply;
“IF everyone chose the SAME number to start counting the days before deducing their color, would it always lead to an accurate deduction of their color?”

So far, you have each said “no”, yet you cannot come up with a single number that doesn’t lead to an accurate deduction of the color, every time they do it.

It is a yes or no question with a request for what number that would be if your answer is “no”.
Let me guess, you guys took an oath to ban yourselves if you turn out to be wrong…??
Seems it has to be something like that for you to go to such extreme lengths to avoid such a simple question.

Then why didn’t you choose it?

^

^ Clearly

Oh, I can think of a reason.

What’s the point in talking to you?

I seem to remember telling You that there isn’t much point in You talking to me, because you merely attempt to be contrary to anything I say… and that usually leaves you being seriously wrong, and in this case, looking a bit foolish.

I am blue out of 4 blues; A) They all know [that they all know] [that there are at least 1 blues]. B) They all know [that they all know] [that they don't know their own color]. C) They all know [that they all know] [that they will leave if they could deduce their color]. D) They all know [that they all know] [that if there were only 1 and (A) were true, they would leave the 1st day]. E) They all know [that they all know] [that if there were only 2 and (A) were true, they would leave the 2nd day]. F) They all know [that they all know] [that if there were only 3 and (A) were true, they would leave the 3rd day]. G) They all know [that they all know] [that if there were only 4 and (A) were true, we leave the 4th day].
Compare with

I am blue out of 3 blues; A) They all know [that they all know] [that there are at least 1 blues]. B) They all know [that they all know] [that they don't know their own color]. C) They all know [that they all know] [that they will leave if they could deduce their color]. D) They all know [that they all know] [that if there were only 1 and (A) were true, they would leave the 1st day]. E) They all know [that they all know] [that if there were only 2 and (A) were true, they would leave the 2nd day]. F) They all know [that they all know] [that if there were only 3 and (A) were true, we leave the 3rd day].
What is the difference here? Every blue knows that every other blue knows that there is at least 1 blue in both the three and the 4 example. Why don’t the three leave the island on the first day?
The canonical answer is that they can’t leave because they need to know [that they know [that they know [that there are at least 1 blue]]], and that the 4 islanders need to know [that they know [that they know [that they know [that there is at least 1 blue]]]]. Your answer seems to reject the nested knowledge requirement, so I’m curious why you see these cases as requiring a different logical process.

What is the difference between an island with 3 blues and an island with 4 such that the four can divine their own eye color just by counting the days?

That seems like a silly question.
They don’t leave the 1st day because they see another uncounted blue.
That is the same reason they don’t leave in the canonical version.

You are STILL trying to argue about HOW they know what number to choose.
THAT is NOT my question.
The question is a mere hypothetical, “IF they ALL choose the SAME number.…”

IF they ALL choose the SAME number to count with/from (within each group), there would be no difference in the result.
They would both accurately deduce their own eye color.
What makes you think that it wouldn’t work for 3 or 2?

Just to be clear, are you saying that your method would work for an island on which there are only two people, both of whom have blue eyes?

“Method”??

I have asked you a question. Why is that so hard to comprehend???
If you had only 2 on the island and you used that same “counting and deducing” procedure, given that everyone on the island is starting with the same count, would they still deduce their proper color? The obvious answer is “yes”, but it seems you are hell bound to never say it.
Hell, any 3rd grader could figure that much out.

Without the guru saying anything? Really?
Every blue sees 1 blue, and has no idea what color his own eyes are. He also knows that that blue has no idea what color his own eyes are. Why would counting the days change the this sum total of knowledge on the island? Neither of them should expect anything to happen on the first day: they don’t know that the person they’re looking at knows anything about blue eyes. What have they gained on the second day? Even if they start with the same number, they number isn’t related in a meaningful way to the color of their eyes, or to the color of anyone’s eyes.

Can you make a syllogism for this case?

Geeezzz…

Carleas, if someone asked you “If the Earth was flat, would the water spill off the edge?”, how many pages would you continue to argue and demand to know how the Earth got flat before you actually addressed the question?

If everyone knew there was at least one blue on the island, even if they didn’t know that everyone else knew it, and there really was only one, he would deduce that it was him immediately.

Agreed. But, if everyone on the island knew that there was at least 1 blue when there are 2 blues, they don’t deduce anything unless they know that the other person knows it.

For 3 blues, the change is similar: they must know that each other islander knows that each other islander knows.

For 4, they must know that the others know that the others know that the others know that the others know.

With each additional blue eyed islander, they must nest the knowledge an additional degree.

Still just can’t bring yourself to address a hypothetical question, huh.
I think you have lived in DC too long.

Your proposed solution doesn’t work because you are taking knowledge out of context, and missing the distinction between someone knowing something, and someone knowing that someone knows something.

On an island with 100 blues, everyone knows that 99 have blue eyes, but no one know how many blues the 99 blues they see see. And no one knows how many blues the 98 blues that the 99 blues see see see. And no one knows how many blues the 97 blues the 98 blues the 99 blues see see see see.

Let me repeat that, with parentheses, because it is both grammatically and logically complex: No one knows how many blues (the 97 blues (the 98 blues (the 99 blues see) see) see) see.

I am not “proposing a solution”. I am asking a question which seems to be akin to having a discussion that starts with “IF men really were superior to women,…”… with a woman.

I have asked a hypothetical question (about 50 times now it seems). Your only answer has been “no”, yet you cannot demonstrate your answer’s correctness. Instead you repeat (probably close to 50 times now) the same argument that everyone involved in this thread knew very long ago and hasn’t argued against… with the exception of another possibility, which you totally avoid discussing.

I am not interested in the canonical proposed solution. That one has been beat to death and is “old news”. I am proposing a hypothetical “new angle” from which to discuss a possible new solution. How often you repeat the old solution isn’t going to change anything nor address the actual question being asked.

How about actually answer the question with a demonstration of any single value (between 0-99) that didn’t produce the correct color deduction.

As I did pages ago, I will take 98. 98 is a number between 0-99, and even if all islanders start counting from 98, they will not deduce the color of their eyes.

Let’s start in the position of an islander. He doesn’t know how many blue eyes there are. He doesn’t know his eye color. He sees 99 blue eyes and he knows that “there are at least 98 blue eyes, and all of the people with blue eyes that I can see know that there are at least 98 blue eyes.”
He knows that if there were 98 blue eyed islanders, and they know “there are at least 98 blue eyes”, they would leave the island.
But his knowledge that there are at least 98 blue eyes doesn’t transfer to this hypothetical group: he knows there are at least 98 blues because he can see at least 98 blues. But if there were only 98, they could not see at least 98 blues. By the very assumptions made to construct the hypothetical, they would not have access to the information that one of the 100 blues on the island has access to. They thus would not leave on day 1, so nothing would be learned when they haven’t left on day 2, and on day 3 he would still not know what color his eyes are.

Contrast this case with a case in which the guru says “I see at least 98 blues”. There, he knows that if there were only 98 blues, they would leave, because they derive their knowledge from what the guru says, and not from what they see.

In short, the reason that people can’t deduce their eye color even if they all start with 98 is that if there were 98 blues, they could not see 98 blues. The fact that every blue islander can see 98 blues does not change this.