This is a little test of reasoning ability. I would prefer that for the first few days, only ID advocates post answers. These questions, and the underlying reasoning, are widely discussed on the internet, so you may have encountered them. If you have, I would appreciate knowing that fact. Also, for those who have seen them before, I would like to know how you did the first time you encountered them.
If anyone spots a typo or logical error, I’d appreciate hearing about is so it can be corrected.
The answers I’m looking for are in three parts:
First — yes or no — can the puzzles be solved by reason, assuming ordinary knowledge of the vocabulary. There are no tricks or unusual meanings involved.
Second, provide the answer.
Third, the provide the reasoning or proof.
Uncommon Descent frequently invokes logic and reason. this is a challenge to anyone who posts at UD. Feel free to post your answers on this thread or at UD.
Here are the questions:
1. [The original editor has been sacked. Re-Edited to straighten out the mess: The price of a cheeseburger is $2.20, the price of a plain hamburger plus the price of the added cheese.] A plain hamburger costs two dollars more than the added cheese. How much does a plain hamburger cost?
2. In Elbonia, one person in ten thousand has Ebola. A new test is so good that anyone who is infected will test positive. But three percent of uninfected people will also test positive. John, a citizen of Elbonia tests positive. What is the probability that John has Ebola?
3. I have a deck of picture cards. They have automobiles on one side and living things on the other side. I have looked through them, and I think they follow the following rule: if a card has a GM automobile on one side, it will have an animal on the other side. After shuffling, I deal out four cards.
Cat, Ford, Petunia, Chevy
What cards must I turn over to test my hypothesis?
4. William is tweeting Betty, but Betty is tweeting John. William is in love, but John is not. Is a person in love tweeting a person who is not in love?
5. Elbonia has invented a treatment for Psoriasis. During a recent blind test, of the patients who were given the treatment 197 improved and 95 did not.
Of the patients who were given a placebo, 45 improved and 20 did not.
Is the treatment effective?
Yes, I have been amused by that. Several years ago, I was accused of being a dualist because of my view that mathematics and information are abstract.
From a mathematician’s perspective, there is nothing materialist or physicalist about Cantor’s mathematics.
I also laughed at the expression “Darwinist mathematics” which I assume socle was using tongue in cheek. We see something like natural selection in physics, where a lot of Aristotle’s science has been thrown out and replaced by newer better theories. But mathematics is rich in very old ideas that seem to have staying power.
Petrushka, turning over any card except the Ford is a test of the hypothesis, which is what the question asked.
The rule does not say if there’s an animal on one side there is a GM car on the other side. That is not part of the hypothesis being tested.
Surely whether the card is face down or up is irrelevant. To test the hypothesis you look at both sides. You turn over the cat card and you may now have a card with a GM car on one side and an animal on the other. If a card is cat n GM car, it is also GM car n cat.
It’s not the best test, but it is a test.
You turn over the cat and there’s a GM car. That conforms to the hypothesis, but you can’t prove — or even statistically support — the hypothesis with a small sample.
You turn over the cat and there’s not a GM car. That is irrelevant.
Edit to add:
Let’s start with just four cards examined by an outside observer and known to have cars on one side and living things on the other. Let’s say I claim that by using psychic powers I can determine what is on the other side of the four cards. I look at the four cards and say, “If there is a GM car on one side, there will be an animal on the other side.”
Would you consider turning over a cat to be a test of my psychic powers? Anyone want to chime in on this?
I assumed that we should turn over only the cards that could potentially falsify the hypothesis. Turning over the first two cards would give you some additional information, but wouldn’t put the hypothesis at risk.
The statement of the problem may not be bullet proof, but that is the intended reading. Since the problem states that the hypothesis was formed after sampling the cards, not much is added by one confirming data point in a tiny sample.
A single disconfirming example, however, kills the hypothesis.
petrushka,
” Betty either loves someone or she doesn’t. Since John is not in love, yes indeed, someone is tweeting someone who is not in love That took ten seconds and still doesn’t confirm anything about Cantors ideas.
But that doesn’t answer the question posed. Your proof needs a bit more thought.”
UH, no actually it doesn’t. The question asked if someone in love is tweeting someone who is not in love. WE already know William is in love and John is not. Problem solved.
I am sorry if this was hard for you and your brethren. Cantor was a crank, but lots of people believe things if enough people say its true. One infinity is not bigger than another, Cantor doesn’t own the meaning of words. Materialists are easily fooled by card tricks.
phoodoo
Quick question for you. What is the etymology of “cardinality”, as used in the mathematical sense?
phoodoo,
Actually, William J Murry, one of your cohorts at UD, thinks there is something to Uri Geller, a famed PSI faker.
I’ve not read any one here posting in favour of Uri’s PSI power, so seems no “materialists” (sorry all, keeping it simple) are fooled by that particular card trick.
Unless, of course, you think there’s more to Uri then tricks phoodoo? Dare take a position?
Does Uri have powers beyond the and/or over the material phoodoo?
Another curious question. Referencing the bolded part, does this mean phoodoo is not one of those Cantor denialists who insist that the set of all natural even numbers must be “smaller” than the set of all natural numbers?
I’m pretty sure mathematicians do own the meaning of words as they are used in mathematics.
You haven’t said anything that is untrue, but you haven’t presented your case as a proof. William is not tweeting john. Just present your reasoning as a proof.
OMagain said:
Link?
Steve,
What is the formula and the calculations which have been input into that formula which shows that one set of infinities is bigger than another, ala Cantor?
Can you write the mathematical formula?
OMagain,
Do you have some specific information to show that Geller has no psychic abilities? Or are you just a “skeptic” so you don’t need information, you just believe what you are told to believe?
William,
That would be supporting evidence. I’m glad you can recognize the need on occasion.
Rather then me provide a link to where I believe you’ve said this, why don’t you restate your position on Uri’s ability for clarity?
phoodoo,
Yes, psychic abilities don’t appear to exist despite much research. Therefore he can’t have them if they don’t exist! The likely non-existence of such has been demonstrated by the inability of any person or system to demonstrate such powers in a convincing manner (i.e. under something approaching scientific conditions).
Uri has make a claim. I am skeptical of it. That’s what’s supposed to happen here, right?
You should ask some of the people he convinced he had genuine ability, who then, after spending time with him, realized he was fake. They had “specific information” that Geller was faking it. Do you need names? I’m sure you can email these people.
I can’t prove that Geller has no psychic abilities, all I can say if he has them he’s done a great job of doing things that you could also achieve through deception, trickery and some good sleight of hand.
Much like your purported designer has done it’s work by what looks very much like evolution.
So, anyway, phoodoo, if you have evidence for Uri’s ability, please present it!
Then, if nothing else, I’ll have an idea of the sort of evidence that you find convincing.
William,
Link
Here you are saying that you can bend a spoon via PSI and therefore so can Uri.
Link
Uri has performed this trick. You have performed this trick. You believe there is something “extra” going on here, therefore when Uri performs this trick he is not performing a trick to you – he’s really doing it.
Link
Here you note you’ve experienced all the PSI. So if it’s real, it’s real for Uri, no?
So, however you want to parse it, you believe in PSI despite not because of the evidence.
No need to respond William, unless you want to take an actual position rather then “I just don’t know and it does not matter anyway”.
Omagain,
I did actually state my position, which you seem to have carefully omitted. Here is where it occurred: http://theskepticalzone.fr/?p=4483&cpage=2#comment-47670
And here is the exchange:
Omagain said:
WJM responded:
Please note how you attempt to falsely connect what I have said I have done myself with what I believe others have done when I have expressly stated that I have no idea if others are being truthful or fraudulent in their claims, in an apparent attempt to make it seem as if I “believe” that Uri Gellar is not fraudulent.
My experience of PSI phenomena and knowledge that it exists does not logically necessitate that I believe every such claim is valid. I suspect many if not most PSI claims are mistaken or fraudulent.
Just because I know it exists doesn’t mean I believe Gellar or any other PSI claimant is valid. Your “logic” needs cleaning up.
phoodoo,
Huh? I’m not a mathematician. Why would I have memorized Cantor’s proof? And what does that have to do with my rather banal claim that mathematicians define words in mathematics to suit themselves? You’re not a mathematician, are you?
William,
I heard it said, that in fact Randi probably doesn’t actually even have the million to pay out, even if someone did prove their case-it’s really just a publicity stunt.
Also, I know of some people who have attempted to fulfill his challenge, only to have him say that they were too far down on the list, so he didn’t have time to respond to them, and stopped contact with them.
Steve,
What does being a mathematician mean? That one is a teacher? Does an accountant count as one? How about a nutritionist? A card dealer? An architect?
You made a claim that mathematicians get to define the terms of math, and yet what Cantor did is not even math-its a type of logic (as poor as it is). There is no math involved-there are no equations to prove his idea.
No, it is possible to be a mathematician without being a teacher. A mathematician does mathematics.
No. It is possible to be both an accountant and a mathematician. But that is uncommon. Mathematics is far broader than what an accountant uses. It includes geometry, algebra, topology, set theory, functional analysis.
Of course they do. Steve was right about that.
Nonsense. Cantor’s work is very clearly mathematics.
Neil,
All you have made is blind assertions, with no valid reasons for saying so.
Why isn’t an accountant a mathematician, they don’t do mathematics? An architect doesn’t do mathematics? Says who, you?
If Cantors proof is about mathematics, and not just logic, then tell me the mathematical formula employed. Tell me the calculations? You can’t there aren’t any.
Your whole post was one of, “I say…”
William,
But you do! If you can bend spoons and your friends can bend spoons why can’t Uri also bend spoons?
phoodoo,
Idle speculation on what may or not may be is somewhat pointless.
When somebody proves their case and he refuses to pay up them perhaps you’ll have a point. Until them, irrelevant.
So what? There are many people claiming such abilities and only so much time in the world.
Perhaps you can name the people and we can discuss if their claims stand up to the most basic scrutiny?
Do you believe Uri has PSI powers phoodoo?
William,
I understand that. So I ask explicitly now.
Do you, personally, believe that Gellar’s claims of PSI ability are true based on the sum total knowledge you have about him?
It doesn’t require reasons.
The issue is the appropriate use of words such as “mathematician” and “mathematics”. This is a matter of social convention (linguistic convention), not a matter of logic. And the current social practice is to defer to mathematicians on this. I expressed what I see as the common view among mathematicians.
If you prefer to have your own private meaning for “mathematician”, I guess you are entitled to that. But you won’t be an effective communicator. Instead, you will come across as looking rather foolish and ignorant. But hey, you are entitled to that if you want to be seen that way.
The mathematics that accountants use is only a tiny part of mathematics. And it is only a tiny part of the accountant’s job.
I will take that as your clear assertion that you are clueless about mathematics, and that you are damned proud of your ignorance.
But, hey, you are entitled to that, too.
If you are not interested in what I have to say, then you can stop reading my comments.
William,
So, to be clear, you and a room full of random people can bend spoons but you’ve *no idea* if Uri Geller can do the same?
Your “logic” needs cleaning up.
OMagain said:
There are a lot of people that apparently cannot do it. I don’t know if Uri can or not.
Not all the random people can do it. In the videos I directed you to, not everyone was successful. In my limited reading of the phenomena, again, not everyone is successful, and I don’t know what the overall success rate is.
Your logical error is the assumption that if one person can do it, everyone can. That doesn’t appear to be the case in virtually any of the information on the matter. Just because some humans can do some things does not mean alll humans can do all things any other human can do.
A basic flaw in logical reasoning on your part.
I have no idea if they are true or not. I don’t believe or disbelieve his claims; I’m agnostic about them. It’s possible he can bend spoons; it’s possible he’s a fraud. I have no personal belief either way about it. I hope that’s explicit enough.
phoodoo
The word “mathematician” has a reasonably clear meaning in English. Wikipedia has an article on the concept, as does the Bureau of Labor Statistics; try reading them.
Some mathematicians teach, as do some biologists, some poets and some carpenters. Some do not. Competent speakers of English do not normally refer to accountants, nutritionists, car(?) dealers or architects as mathematicians. That’s how words have meaning: by the way people use them. I use lots of math in my work, but I’m not a mathematician, nor has anyone ever called me a mathematician.
Really, do you always have this much difficulty with frequently used nouns?
Mathematicians disagree with you.
Steve,
Mathematicians disagree with me on whether or not Cantor used math formulas to create his idea about different sets? Well, then this is simple, the mathematicians are wrong! There is no math formula!
Is it the mathematicians who get to decide who mathematicians are? Now we really have a problem, that would be logically impossible!
Are you a scientist? A biologist? A statistician?
Neil,
Well, clearly you are NOT a mathematician. You don’t even know what a math formula is.
phoodoo,
We are talking about a purported proof of the theorem stating that the set of real numbers is uncountable.
In this diagram illustrating the key step of the proof, you will see an equals sign at the bottom, hence there’s an equation.
If you compare the numbers along the diagonal with the digits of x at the bottom, you will see that there’s a simple rule or function that determines the nth digit of x: Take the nth diagonal entry, add 1, and take the remainder mod 10.
In equation form, x_n = (d_n + 1) % 10.
Is that mathy enough for you?
Socle,
x-5(d) + nd2^2 >={2-x^} n+xy^2 When used on a linear scale of approaching densities of cardinality.
So I guess Cantor must be wrong. Thanks for that.
Socle,
Does -%10 x = (d + 1) ?
William,
I claim I can walk on water. Are you agnostic about that claim also?
phoodoo,
As someone once said:
phoodoo, you just don’t know when to quit.
You ‘say’:
“I heard it said…”
Well La Dee Freakin’ Da.
“…, that in fact Randi probably doesn’t actually even have the million to pay out, even if someone did prove their case-it’s really just a publicity stunt.”
In “fact”? Really? So you ‘say’, because you claim to have “heard it said”. Again, well La Dee Freakin’ Da.
“Also, I know of some people who have attempted to fulfill his challenge, only to have him say that they were too far down on the list, so he didn’t have time to respond to them, and stopped contact with them.”
So you ‘say’. Again, well La Dee Freakin’ Da.
“All you have made is blind assertions, with no valid reasons for saying so.”
ROFLMAO! Look who’s SAYing “blind assertions, with no valid reasons for saying so.”
“Why isn’t an accountant a mathematician, they don’t do mathematics? An architect doesn’t do mathematics? Says who, you?”
Who SAYs they ARE mathematicians, you? Who are YOU to ‘say’?
“If Cantors proof is about mathematics, and not just logic, then tell me the mathematical formula employed. Tell me the calculations? You can’t there aren’t any.”
You’re in no credible position to lecture on the subjects of logic and mathematics.
“Your whole post was one of, “I say…””
Wow. As is typical of IDiots, self-awareness obviously isn’t your strong suit.
Hey phoodoo, show your mathematical skills by calculating/measuring the ‘bits’ of so-called CSI-FSCI-dFSCI-FSCO/I in a banana, a rock, an atom, a human, a gorilla, and a spoon.
Socle,
You didn’t answer the question?
Omagain:
Yes, I am.
phoodoo said:
“Socle,
You didn’t answer the question?”
phoodoo, you wanted an equation and socle showed you one, so what is the point of your question?
Phoodoo has the same attitude toward math as he does toward logic. Make a claim that it was obvious after ten seconds, then fail to demonstrate any reasoning or proof.
phoodoo,
Shall I add that to the paper I’m having to write for you?
Oh, remind me again about the origin of the word cardinality, as used by Cantor?
Or are you afraid, for some reason, to address that?
William,
Bought any bridges lately?
Omagain said”
Why would I buy a bridge the existence of which I’m agnostic about?
Do you know what “agnostic” means?
If you guys want to have a Cantor thread, please start one.
This one is asking whether people who can’t solve logical and simple mathematical puzzles should be taken seriously when they challenge Cantor or Einstein or Darwin.
Since the zero thread at UD has been brought up, I read some of the comments in it and there are some things that I want to say.
Mapou said:
“Saying that you can imagine it in your head is not proof of existence.”
Apply that to ‘God(s)’.
“Nothing exists unless it does. It’s that simple.”
Apply that to ‘God(s)’.
“Physicists can claim anything they want. In the end they die like everybody else but the truth remains.”
Well, ‘God’ pushers (like you) can claim anything they want. In the end they die like everybody else but the truth (reality) remains. Lots of so-called ‘Gods’ and associated fairy tales have been invented, believed in, worshiped, and promoted, and most have been discarded and forgotten. Which ‘God(s)’ and associated fairy tales were/are “the truth”, and which of the many millions of ‘believers’ who are now dead believed in true or false ‘God(s)’ and associated fairy tales?
“Nothing ever reaches infinity.”
So much for an infinite ‘God’ then. Since ‘God’ therefor cannot be infinite, ‘God’ must have had a beginning and will have an end, so who or what created ‘God’ and who or what will eliminate ‘God’?
phoodoo said:
“When you start taking mental concepts such as a “beginning”, or an “end”, or “unlimited”, or “all” or “large” , and then you pretend that you can make this into a definitive math problem, you are already just playing a game. There is no number for large, or a number for middle, or a number for vast or a number for time…these are things we imagine in our brains.”
Is ‘God’ unlimited, all, vast, timeless, and large? Does ‘God’ have a beginning and an end? Isn’t ‘God’ just a mental concept and imagined in some peoples’ brains? And when you pretend that you can make that into a definitive existence of ‘God’, aren’t you just playing a game? Is gordo (kairosfocus) right when he calls ‘God’ a mathematician?
“The mathematicians are not using logic, or reason, or facts, they are just saying it is so, and hoping no one calls them on it. NO realities in life are solved by this math problem. NO new information about the world is gained by this absurd use of definitions. It makes a complete mockery of the concept of knowledge.”
‘God’ pushers are not using logic, or reason, or facts, they are just saying it is so, and hoping no one calls them on it. NO realities in life are solved by this ‘God’ and associated fairy tale pushing. NO new information about the world is gained by this absurd use of ‘God’ and associated fairy tales. It makes a complete mockery of the concept of knowledge.
William,
Do you neither believe nor disbelieve in Santa Claus?