Math Fun

[Carleas]

James, I agree with that; I made that same point earlier. But that applies to scientific inductions, i.e. taking a series of data points and extrapolating a general theory that fits them. It doesn't apply to deductive methods, including mathematical induction (so it doesn't apply to the reasoning used in the Blue Eye problem or the MI portion of the Master Logician problem).

And the point supports my SR argument: the fact that there are an infinite number of equations that fit any given set of points means that there is no certain argument that generalizes from a set of data points to find a non-given data point. In other words, if that were what the logician were expected to do, the problem would be impossible. And since it is a given that the problem is not impossible...

[James S Saint]

James, I agree with that; I made that same point earlier. But that applies to scientific inductions, i.e. taking a series of data points and extrapolating a general theory that fits them. It doesn't apply to deductive methods, including mathematical induction (so it doesn't apply to the reasoning used in the Blue Eye problem or the MI portion of the Master Logician problem).

Your still not getting it. You first choose a theory concerning the puzzle. Then you find that the theory fits the puzzle. Then you declare that the theory is the answer merely because it fit the puzzle. What I have been telling you is that many theories might fit the puzzle. You have to prove that yours is the ONLY one, else yours might not be the one that the master is using. The video was expressing that just because something fits within given certain limited knowledge, doesn't mean that it is the true answer. The puzzle requires that you prove your theory to be the only option.

[James S Saint]

The Italian Book.

Last week I bought a book in Italy. The cashier got hundred Euros and gave me twice as much and five cents more back than my entitlement was. Obviously the cashier had confused the amount in euros with the amount in cents.

How expensive was the book?

Another effort to clarify/verify something on that puzzle:

untab

if the proper entitlement is x,
e is the received euros, and
c is the received cents, then

2x + 5 = e+c

And if the received euros and received cents were confused then the proper entitlement is,

x = e/100 + c*100

That seems to be the stated situation. But is that the intent?

[Arminius]

Another effort to clarify/verify something on that puzzle:

untab

if the proper entitlement is x,
e is the received euros, and
c is the received cents, then

2x + 5 = e+c

And if the received euros and received cents were confused then the proper entitlement is,

x = e/100 + c*100

That seems to be the stated situation. But is that the intent?

As I already (indirectly) said:
You need one unknown for the euros and one unknown for the cents, because they are confused by the cashier, and you need them on both sides of the equation, James. The word "confused" is the most important word in that said story ("The Italian Book") when it comes to translate the lingusitic text into a mathematic "text" (equations and so on).

Good luck!

Or should I give the whole solution?

[James S Saint]

Arminius,

untab

Another effort to clarify/verify something on that puzzle:

if the proper entitlement is x,
e is the received euros, and
c is the received cents, then

2x + 5 = e+c

And if the received euros and received cents were confused then the proper entitlement is,

x = e/100 + c*100

That seems to be the stated situation. But is that the intent?

As I already said:

A linguistical hint:
According to the text the cashier did not confuse one thing with itself.
A linguistical-mathematical hint:
If the cashier confuses two things, then they have to be considered as two "things" in a mathematical sense too.
A mathematical hint:
There is merely one unknown in your equations.

So you need one unknown for the euros and one unknown for the cents, because they are confused by the cashier, and you Need them on both sides of the equation, James. The word "confused" is the most important word in that said text when it comes to translate the lingusitic text into a mathematic "text" (equations and so on).
Good luck!

Or should I give the whole solution?

No. Just answer my question.
I was not asking about the solution. I was asking if the equations that I gave (in blue) represent the situation that you are trying to express. If not, why not?

[Carleas]

You have to prove that yours is the ONLY one, else yours might not be the one that the master is using. The video was expressing that just because something fits within given certain limited knowledge, doesn't mean that it is the true answer. The puzzle requires that you prove your theory to be the only option.

We've had this portion of the conversation before. The last time you made this point, I brought up the example of the Pythagorean Theorem, which has many proofs, none of which depend on or are threatened by the existence of any other.

If the solution is by way of a deductive proof, then you don't have to prove that your solution is the only option. If it is deductively provable that, e.g., the blue-eyed islanders will know their eye color on day N and no sooner, then no other solution needs to be considered. Even if other syllogisms exist, they will arrive at the same conclusion.

[James S Saint]

You have to prove that yours is the ONLY one, else yours might not be the one that the master is using. The video was expressing that just because something fits within given certain limited knowledge, doesn't mean that it is the true answer. The puzzle requires that you prove your theory to be the only option.

We've had this portion of the conversation before. The last time you made this point, I brought up the example of the Pythagorean Theorem, which has many proofs, none of which depend on or are threatened by the existence of any other.

I can't believe that you are still saying that. I told you why your example is nonsense. It does NOT APPLY to what we are talking about. We know that the deduction works. That is merely the theory that you picked ("MY theory works!! My theory works!! My theory works!!!"). NOW you have to prove that the master is using THAT SAME ONE!!! But apparently that is beyond your grasp, so I'll once again leave it alone. You are still incorrect for the exact same reason as before, merely completely blind to it, it seems.

[Arminius]

Arminius,

untab

if the proper entitlement is x,
e is the received euros, and
c is the received cents, then

2x + 5 = e+c

And if the received euros and received cents were confused then the proper entitlement is,

x = e/100 + c*100

That seems to be the stated situation. But is that the intent?

As I already said:

A linguistical hint:
According to the text the cashier did not confuse one thing with itself.
A linguistical-mathematical hint:
If the cashier confuses two things, then they have to be considered as two "things" in a mathematical sense too.
A mathematical hint:
There is merely one unknown in your equations.

So you need one unknown for the euros and one unknown for the cents, because they are confused by the cashier, and you Need them on both sides of the equation, James. The word "confused" is the most important word in that said text when it comes to translate the lingusitic text into a mathematic "text" (equations and so on).
Good luck!

Or should I give the whole solution?

No. Just answer my question.
I was not asking about the solution. I was asking if the equations that I gave (in blue) represent the situation that you are trying to express.

Again: There is no text problem.

If not, why not?

Okay, James.

According to the said story it is right that "if the proper entitlement is x, e is the received euros, and c is the received cents, then 2x + 5 = e + c", and it is also right that "if the received euros and received cents were confused, then the proper entitlement is, x = e/100 + c*100".

You wanted me to answer your question, and I have just answered your question. Is that alright for you, James?

I wish you success!

untab

I think your next post will contain the right solution, James.

[Carleas]

We know that the deduction [in the case of the Pythagorean Theorem] works.

Here's my point, James. If the deduction works, then we don't need to prove that everyone's using it. That's how deductions work.

So your argument here is at best question begging: If it's a deduction, if it's sound, then there's nothing else to prove. Your argument here goes something like, "it isn't a sound deduction, so you need to prove that everyone uses the theorem, so the deduction is flawed, so it isn't a sound deduction." That's not a proper argument.

At worst, you're just calling a syllogism a "theory" and then refusing to credit it with everything that follows from a valid logical deduction. That's not a proper argument either.

[James S Saint]

We know that the deduction [in the case of the Pythagorean Theorem] works.

Here's my point, James. If the deduction works, then we don't need to prove that everyone's using it. That's how deductions work.

That was a misquote of me. And you are wrong either way. You have at least two people who have to be "in sync" with the bells. If they are not using the same theory (making a different enabling assumption), they will possibly not be in sync.

The issue, as from the beginning, is that the premise assumption must be the same for everyone, else it is like saying because Pythagorean theorem works, it doesn't matter how many sides the shape has.

Your THEORY is that your syllogism applies (not whether your syllogism is valid in itself). The master might have a better theory using a different enabling assumption and syllogism. You have to prove that the master cannot be doing that, because he controls who leaves on which bell.

untab

I think your next post will contain the right solution, James.

untab

No because those equations do not work out to sensible values for e and c. You end up with fractions of cents and also that;
Received back = e + c = 2.55102040816327

which obviously cannot be right And that is what has been holding us both up.

[Arminius]

untab

I think your next post will contain the right solution, James.

untab

No because those equations do not work out to sensible values for e and c. You end up with fractions of cents and also that;
Received back = e + c = 2.55102040816327

which obviously cannot be right And that is what has been holding us both up.

Those equations you mentioned work, but note: they are merely abstract examples and not the solution for my concrete example. I hope you know that. You asked me to answer your question, then I answered your question. Now I hope you do not confuse my answer with the complete solution of the said task. Note: I merely answered your question.

untab

As i said several times: You need to have both "e" and "c" on both sides of the equation, and then you have to find out which number (amount) the only correct one for the example is. Please note: You have both euros and cents, and your basis should be cents (just for the sake of convenience, because if you used euros as basis, then you would have to change the number "5" in your equation). Your equations work. You do not have the right numbers, James. In my concrete example are merely two whole numbers for "e" and "c" possible.

Good luck!

[James S Saint]

untab

I think your next post will contain the right solution, James.

untab

No because those equations do not work out to sensible values for e and c. You end up with fractions of cents and also that;
Received back = e + c = 2.55102040816327

which obviously cannot be right And that is what has been holding us both up.

Those equations you mentioned work...

No, they don't and that is why I asked if that is what you really meant.

untab

Given those equations:
x = e/100 + 100c
2x + 5 = e + c

2(e/100 + 100c) + 5 = e + c
e/100 + 100c + 2.5 = e/2 + c/2
e(1/100 - 1/2) + 2.5 = c(1/2 - 1/100)
e(-49/50) + 2.5 = c(49/50)
2.5 - e(49/50) = c(49/50)
e(49/50) + c(49/50) = 2.5
e + c = 2.5(50/49) = 2.55102040816327

e is the number of euros in your hand and c is the number of cent coins in your hand, two integers. Yet they do not calculate out to be integers given the original equations.

[phyllo]

x = e/100 + 100c

You used 100 twice and you should only use it once. Which one you use (/100 or 100*) depends on if the answer is in euros or cents.

[Arminius]

untab

Given those equations:
x = e/100 + 100c
2x + 5 = e + c

2(e/100 + 100c) + 5 = e + c
e/100 + 100c + 2.5 = e/2 + c/2
e(1/100 - 1/2) + 2.5 = c(1/2 - 1/100)
e(-49/50) + 2.5 = c(49/50)
2.5 - e(49/50) = c(49/50)
e(49/50) + c(49/50) = 2.5
e + c = 2.5(50/49) = 2.55102040816327

e is the number of euros in your hand and c is the number of cent coins in your hand, two integers. Yet they do not calculate out to be integers given the original equations.

James, please read what I wrote in my last post again:

Those equations you mentioned work, but note: they are merely abstract examples and not the solution for my concrete example. I hope you know that.

untab

What has been holding us both up is more the fact that you did not take my advice. For example this:

You ... have to find out which number (amount) the only correct one for the example is.

That must be a whole number.

In my concrete example are merely two whole numbers for "e" and "c" possible.

[Carleas]

You have at least two people who have to be "in sync" with the bells. If they are not using the same theory (making a different enabling assumption), they will possibly not be in sync.

The issue, as from the beginning, is that the premise assumption must be the same for everyone[...]

This objection is overcome by the stipulations about how good a logician the participants are. To turn again the 2 Blue Eye problem as the simplest case of the logic, if there are 2 blues on the island, it seems clear that two perfect logicians, for whom it is common knowledge that they are perfect logicians, will know their eye color on the second day. There is no reliance on a everyone being "in sync" or using the same "theory", there's is pure deductive logic about what the islanders know and how they must behave and how they must deduce from their knowledge and behavior.

Returning to the case of the Master Logician, I take the same level of logical perfection to be a given, as the Master will enforce accurate deduction on the other logicians. It's not whim or fancy that he's enforcing, but what the logicians must deduce based on what they know and how they must behave and how they must deduce from their knowledge and behavior (or have such deduction forced on them by the master). The effect is the same.

Is that what you're taking issue with here? I admit the premise is implicit; a more-clear statement of the problem would be explicit in the way the Blue Eyes problem is.

[James S Saint]

x = e/100 + 100c

You used 100 twice and you should only use it once. Which one you use (/100 or 100*) depends on if the answer is in euros or cents.

No.. "e" is merely a number of objects received and so is "c", although different objects.

The statement was that those two numbers got reversed: the e number was supposed to be the c number and vsvrsa.

But when translating those numbers into money values, the e number was supposed to be the number of cents, thus

e/100 = amount of cents (proper)

And also the c number was supposed to be the number of euros, thus

c*100 = amount of euros (proper)

So "e/100 + c*100" should be the amount of proper returned change ("entitlement").

But if that was true, it would mean that the received NUMBER of objects, "e + c" had to be 2.55102040816327, which is obviously not possible.

Thus there is a miscommunication going on, as I said in the beginning.

[James S Saint]

You have at least two people who have to be "in sync" with the bells. If they are not using the same theory (making a different enabling assumption), they will possibly not be in sync.

The issue, as from the beginning, is that the premise assumption must be the same for everyone[...]

This objection is overcome by the stipulations about how good a logician the participants are. To turn again the 2 Blue Eye problem as the simplest case of the logic, if there are 2 blues on the island, it seems clear that two perfect logicians, for whom it is common knowledge that they are perfect logicians, will know their eye color on the second day. There is no reliance on a everyone being "in sync" or using the same "theory", there's is pure deductive logic about what the islanders know and how they must behave and how they must deduce from their knowledge and behavior.

The stipulation that the logicians were perfect demanded that they were in sync as a premise. This puzzle doesn't have that stipulation. But even if it did, the puzzle would be unsolvable (or at least by you).

Returning to the case of the Master Logician, I take the same level of logical perfection to be a given, as the Master will enforce accurate deduction on the other logicians.

Well, that was not a given, but like I said, even if it was...

It's not whim or fancy that he's enforcing, but what the logicians must deduce based on what they know and how they must behave and how they must deduce from their knowledge and behavior (or have such deduction forced on them by the master). The effect is the same.

Is that what you're taking issue with here? I admit the premise is implicit; a more-clear statement of the problem would be explicit in the way the Blue Eyes problem is.

You didn't solve the Blued problem either, so referring it doesn't get you anywhere.

The fundamental problem is that you must PROVE that no other possible means for solving the problem can exist. And I have already provided you with several ways equal to yours (you merely deny that yours is just as good). One of those ways directly defeats your assumption by allowing you to make that assumption while also proving a means for everyone to leave by the second bell.

Again, again, and again, you merely keep saying the same thing over and over, that you think your assumed premise is the only possible valid assumption. You haven't proven that. And repeating it over and over does not prove anything other than inference that there actually isn't the actual proof that you need.

[phyllo]

x = e/100 + 100c

You used 100 twice and you should only use it once. Which one you use (/100 or 100*) depends on if the answer is in euros or cents.

No.. "e" is merely a number of objects received and so is "c", although different objects.

The statement was that those two numbers got reversed: the e number was supposed to be the c number and vsvrsa.

But when translating those numbers into money values, the e number was supposed to be the number of cents, thus

e/100 = amount of cents (proper)

And also the c number was supposed to be the number of euros, thus

c*100 = amount of euros (proper)

So "e/100 + c*100" should be the amount of proper returned change ("entitlement").

But if that was true, it would mean that the received NUMBER of objects, "e + c" had to be 2.55102040816327, which is obviously not possible.

Thus there is a miscommunication going on, as I said in the beginning.

Please don't say no when you don't know.

If you have amount represented as e.c ...
then the amount in cent is equal to 100e+c
and the amount in euros is equal to e+c/100

For example :
10.27
e=10, c=27
10.27 (in cents) = 1027 = 100*10 + 27
10.27 (in euros) = 10.27 = 10 + 27/100

[James S Saint]

Please don't say no when you don't know.

I DO know. I am the one who defined it.

If you have amount represented as e.c ...
then the amount in cent is equal to 100e+c
and the amount in euros is equal to e+c/100

That I know too. But that is not what I had asked Arminius.
To which he answered:
"if the proper entitlement is x, e is the received euros, and c is the received cents, then 2x + 5 = e + c" and
"if the received euros and received cents were confused, then the proper entitlement is, x = e/100 + c*100"

[Carleas]

But even if it did, the puzzle would be unsolvable (or at least by you).

That seems like a big proviso. Unsolvable-in-theory is a relatively tiny subset of unsolvable-by-me-in-practice

You didn't solve the Blued problem either, so referring it doesn't get you anywhere.

Happy to keep that conversation going, if you would deign to respond to my syllogisms. Note, off the bat, that they are pure deductive syllogisms relying on the islander's common knowledge and required behavior, and not on any islander being "in sync" with any other. Like with the Pythagorean theorem, the deduction itself is the proof that no other solution is possible.

[James S Saint]

But even if it did, the puzzle would be unsolvable (or at least by you).

That seems like a big proviso. Unsolvable-in-theory is a relatively tiny subset of unsolvable-by-me-in-practice

You didn't solve the Blued problem either, so referring it doesn't get you anywhere.

Happy to keep that conversation going, if you would deign to respond to my syllogisms. Note, off the bat, that they are pure deductive syllogisms relying on the islander's common knowledge and required behavior, and not on any islander being "in sync" with any other. Like with the Pythagorean theorem, the deduction itself is the proof that no other solution is possible.

Nope. You merely jump ahead ignoring that your presumption is merely one possibility that allows you to then rush into deductive reasoning. You do that on both problems, "rush to satisfying judgement". It is a very common mental ailment of the day, especially of attorneys who intentionally try to bias themselves toward their client (prosecution or defendant).

There is no "judge" on this site to call you down to order. So you just keep repeating yourself and being non-responsive. I have explained your errors at every turn. You simply don't care ... not a big issue on a site like this.

[Carleas]

It is a very common mental ailment of the day, especially of attorneys who intentionally try to bias themselves toward their client

Yeah, and of mathematicians and Nobel prize-winning economists and authors of encyclopedias on philosophy, who apparently also foolishly believe that deducing a proposition X is the same thing as deducing a proposition ~(~X). How wacky!

I do care, James. Deeply. I care because failures of reason irk me. Society works best when people are rational, and rationality works best when people are honest with themselves.

And, be honest, you've never pointed to a specific part of the deductive proof of the solution to the Blue Eye problem. You've never grappled with the mathematical induction being used. You've avoided treating in an intellectually honest way, preferring to mock and hand-wave and declare the discussion over and insinuate a failure of reasoning on the part of anyone who disagrees with you. But none of that changes the validity of the solution. None of that even addresses the solution. The solution to the Blue Eye problem can be shown in many ways, has been shown in many ways, has been analyzed by people much more accomplished in math and logic than you or I and has stood the test of honest and rigorous attempts at defeat. If you think you can defeat it, do so honestly, do so rigorously, do so academically. It would honestly be an accomplishment.

[phyllo]

Please don't say no when you don't know.

I DO know. I am the one who defined it.

If you have amount represented as e.c ...
then the amount in cent is equal to 100e+c
and the amount in euros is equal to e+c/100

That I know too. But that is not what I had asked Arminius.
To which he answered:
"if the proper entitlement is x, e is the received euros, and c is the received cents, then 2x + 5 = e + c" and
"if the received euros and received cents were confused, then the proper entitlement is, x = e/100 + c*100"

He gave you a bum steer. All his equations are wrong.

[James S Saint]

It is a very common mental ailment of the day, especially of attorneys who intentionally try to bias themselves toward their client

Yeah, and of mathematicians and Nobel prize-winning economists and authors of encyclopedias on philosophy, who apparently also foolishly believe that deducing a proposition X is the same thing as deducing a proposition ~(~X). How wacky!

Complete strawman.

I do care, James. Deeply. I care because failures of reason irk me.

I don't believe that one for an instant. You never go to the trouble of looking down that other logic road/option. People who care that they are really right, not merely appearing to be right, go to that trouble. You obstinately avoid it. Either you are hard-hypnotized into being hard-blind to specific concerns, or you simply care more about something else enough to not bother checking out the other option.

Society works best when people are rational, and rationality works best when people are honest with themselves.

Well, at least you know the words .. a shame that you don't care.

And, be honest, you've never pointed to a specific part of the deductive proof of the solution to the Blue Eye problem.

Be honest. I pointed to specific reasons why your deduction is irrelevant, not fallacious.
Blued puzzled:
"If everyone is thinking this ..."
But what IF they weren't?

Master Logician Puzzle:
"If we assume that only the colors we see apply, then..."
But what if something else is assumed?

You consistently jump to an easy assumption and go from there, never looking back to verify that your assumption is provable. You just get to an answer and then, apparently in glee, ignore that your starting point was presumptuous. In a courtroom with a jury, that tactic probably would allow an attorney to get away with a great many injustices, as is normal throughout the USA at least. But in a court of logicians, that tactic simply doesn't fly. Logicians don't easily forget that your premise assertion was never proven to be necessarily true.

Logicians are not merely jurors.

You've never grappled with the mathematical induction being used.

Bullshit. I had more math in college than you .. and then proceeded into engineering, using it.

This is just more of your rhetoric. Great for attorneys in a court of law, not a court of justice.

===============================================================================

He gave you a bum steer. All his equations are wrong.

Yeah, that is all that I was trying to say.

[phyllo]

phyllo wrote: He gave you a bum steer. All his equations are wrong.

Yeah, that is all that I was trying to say.

Then why do you keep using those equations???