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?
phoodoo,
I know what I am, but what are you?
There’s a whole cottage industry of people who object to Cantor. Some of that is probably religious. Often, the objection seems to be that they have difficulty with the concept of infinity. One wonders how they can talk about their God being omnipotent, omniscient and eternal, yet be unable to grasp the underlying concept.
Alain Badiou, an interesting contemporary French philosopher, makes the rather dubious claim that Dedekind and Cantor undermine a key intuition underlying rational theology: that infinity cannot be comprehended.
The idea that infinity cannot be comprehended plays a significant role in Descartes’s argument for the existence of God, in the Third Meditation. I have no idea if this is a widely-held intuition in rational theology or if (as I in fact suspect) Badiou is over-emphasizing the importance of Descartes.
But it is surely right that Dedekind, Cantor, and others have given us a rigorous definition of an infinite set, which is to say, they have allowed us to comprehend infinity. And Cantor’s discovery of different “sizes” (i.e. cardinalities) of infinity is surely significant!
Why anyone would find Cantor’s work objectionable is entirely beyond me. It seems unobjectionable that if there are infinitely many rational numbers, and yet infinitely many irrational numbers between any two rational numbers, then the cardinality of the real numbers (the union of the rationals and the irrationals) must be greater than the cardinality of the rational numbers.
Cantor denial seems to be one element of generalized denial disorder. Add evolution, AGW, relativity, and vaccination, and you approach Time Cube territory.
It is interesting that denial cranks can’t be bothered with demonstrating their brilliance on simple puzzles.
As for Phoodoo, he seems egar to gloat over a presumed error by me. He would like to think he is smarter than Keith’s, but he can’t really be sure that Keith’s is right about the hamburger problem. Which is to say, he can’t say why Keith’s is right. He can’t be sure.
KN,
On the other hand, there are infinitely many integers, and between any two integers there are infinitely many rational numbers, yet the union of the two sets still has cardinality aleph-0.
That’s true, socle. Let this be a warning: don’t trust your “intuitions” over what can be proven (in a priori domains) or explained (in a posteriori domains).
Petrushka,
What makes you think Cantor could do simple logic? Since you admit you are not capable of assessing such a situation, your just going along with those who you believe are authorities, is simply demonstrating that you are an unquestioning sheep. So what, it doesn’t make Cantor right.
Furthermore your test is totally ingenuous. Were I or anyone else you disagree with to provide you with answers, you would simply device a new test, or claim that it proves nothing, because so and so doesn’t know about this, or that, or well, it still doesn’t mean you are right about Cantor, and off you would go with your ball, and say the game is over.
That none of us are foolish enough to take your little test, shows just how much smarter we are than you. Materialists never, ever want to face being wrong.
Kantian Naturalist: “Why anyone would find Cantor’s work objectionable is entirely beyond me.”
Yes, Kantian, you are absolutely right about that!
I accepted two criticisms of the hamburger puzzle.
Disingenuous. I don’t think that word means what you think it means.
phoohoo,
Yeah, I guess that’s why when I ask questions at UD about evolution that they should be able to answer given the claims they are making about it, they are simply showing me how smart they are by not answering!
Make them face it! Publish your dis-proof of Cantor’s proof in the academic literature and rub their faces in it!
Make them face it! Publish your dis-proof of Cantor’s proof in the academic literature and rub their faces in it!
If any one has a proof that the real numbers and the rational numbers have the same cardinality — which is what I understand Phoodoo to be saying here — then he or she should be published immediately. That overturns one of the most important results of 20th-century mathematics!
Ah, but there is the little problem that all the mathematics journals are organized by the Materialist Cabal, right? Such a shame!
I’m waiting to see what answer is offered for question 5. Better and worse answers are possible (besides the ones that are just wrong, that is).
I’m waiting to see what answer is offered for question 5. Better and worse answers are possible (besides the ones that are just wrong, that is).
Presumably a correct answer involves a way of determining statistical significance, right? 67% of those given the treatment improved, and 69% of those given the placebo improved. Is that 2% difference significant? I don’t see how, but then again my knowledge of statistics is rubbish.
(One of the reasons I left the sciences for the humanities is that my mathematics is extremely weak. Meanings I can grasp, but when it looks like a purely syntactical operation to me, my brain can’t make sense of it. I know, intellectually, that mathematics is not just syntax, but I can’t help it — it feels like syntax to me!)
Not only are all the journals dealing with math and science controlled by the materialist cabal, it appears that UD also has rules against posting anything that contradicts materialist math.
Kariosfocus, of all people, has waded into the fray with what looks like a defence of Cantor. Next thing you know, he’ll be defending relativity.
ETA:
If the placebo has equal or better results than the medicine, do we need a statistical test? Just asking.
The UD thread that inspired this post has been closed to new comments, so this one is now open to posting solutions and other comments.
Two of the problems have been solved, and the hamburger problem has been analyzed into simple algebra. (I think it is now correctly stated.)
It depends on what you are trying to prove.
If your aim is to prove that the medication is actually harmful, then you need a significance test. If you are trying to prove that the medication works, then you have not succeeded.
Mark Chu-Carroll at Good Math/Bad Math has posted several critiques of Cantor-cranks. He hasn’t posted anything in a while bu if you’ve got an interesting disproof of Cantor he’ll probably point out your errors for you.
The only question asked is, “Is the treatment effective?”
An incorrect answer is that the psoriasis treatment doesn’t work. An correct but weak answer is that your data do not support the hypothesis that the treatment works, and in fact suggest that it doesn’t work. A stronger answer would be to give the odds ratio or risk ratio and the confidence interval on it. The point estimate for the odds ratio, for example, is 0.92 (< 1 means you're less likely to respond if you're given the treatment), but the 95% confidence interval covers the range from 0.52 to 1.65. Since the CI extends well above 1.0, it's still possible that your treatment actually works, but that you missed it because of the small sample size.
Question 2: The probability that John has Ebola is 1/301. Doctors frequently get this kind of question wrong, especially if you word it a little differently. If you say a positive result is 98% accurate (and define what that means), a disturbingly large number of doctors think that means you have a 98% probability of having the condition.
Many scientists are similarly clueless about what a significance of p < 0.05 means. Depending on the power of your study and the prevalence of what you're testing for, it's quite easy for most results with p < 0.05 to be false positives.
(Tangential note: For Ebola tests, both false positive and false negatives results are really bad. False negatives mean that the patient is not isolated, and can spread the virus, while false positives mean a patient who does not have Ebola will be confined to an isolation unit with desperately sick, contagious people. The latter happened to one of our Sierra Leonean colleagues.)
I carried this from my source. I suspected I would encounter criticism for incautious wording. But the challenge does ask if the puzzle can be solved, as the first step in the answer.
The problem I found most difficult was the love/tweeting problem. I should stipulate for the purposes of the puzzle, that one is either in love or not in love. Not both or in-between.
Mr. Joey “I have a 150 IQ” G claims that Cantor is wrong. That is proof enough for me.
Steve says that the probability of John having Ebola is 1/301. Why is it that and not 1/300?
Did you use Bayes’ Theorem?
In 10,000 tests, there will be 301 positives, one true and 300 false. Actually there should be a smidgen fewer than 300. I think.
Yeah, a smidgen less, since there are 9999 uninfected for every infected person.
Ok, now I have a question. If I assume the population of Elbonia is 1,000,000, then 100 people have Ebola.
That means there are 999,900 without Ebola, so if everyone gets tested, there will be 0.03*999,900 + 100 = 30097 positive results.
The probability of a person having Ebola given that they test positive should therefore be 100/30097. But that can’t be right, because this number is not 1/301.
What’s my mistake?
You have not made a mistake socle: you have discovered the “smigden less” mentioned above. One in 300.970.
What’s bothering me is that earlier I got 1/301 exactly like Steve by using Bayes’ Theorem:
P(Ebola | Positive) = P(Positive | Ebola) * P(Ebola) / P(Positive)
where the numerator is 1*1/10000 = 1/10000 and the denominator is 1/10000 + 3/100 = 301/10000.
“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?”
What a poorly worded puzzle. Am I limited to how many I can turnover? If I turn over all the cards with non-animals and all the cards with a GM car on them, it may contradict your hypothesis and it may not, you still haven’t learned much, since you have already stated that you have already looked through the deck. You may turn over the Chevy card and find that it actually has a rock on the other side. The petunia may have a GM on the other side. Well, congratulations, your theory was wrong. Or maybe it doesn’t have a rock, and maybe the petunia doesn’t have a corvette, your theory might still be wrong. Poor puzzle.
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.
If a hamburger without cheese costs
2.10. Add the price of 10 cents for cheese and you got a $2.20 cheeseburger. Cantor was a crank.
The placebo effect for psoriasis has too few of a sample to make any good predictions. Many tests in science fail upon second trials or longer testing. Its one reason why so many science tests are bullshit. You often get results that can’t be repeated.
I neither looked up any of these problems, nor did I check for very long the accuracy of them, nor do I know or am I concerned with how long it would have taken Einstein or Cantor to figure them out- because a set that can never be defined as a set is neither larger or smaller than another set that can be defined yet never finalized.
Words have meanings that can’t be hijacked by a select few to make their own little club.
Your denominator is wrong in your application of Bayes’ theorem. P(Positive) = P(Positive|Ebola) * P(Ebola) + P(Positive|not-Ebola) * P(not-Ebola) = 1 * 1/10,000 + .03 * 9999/10,000.
Ah, thanks for clearing that up.
Phoodoo,
“What a poorly worded puzzle. Am I limited to how many I can turnover? If I turn over all the cards with non-animals and all the cards with a GM car on them, it may contradict your hypothesis and it may not, you still haven’t learned much, since you have already stated that you have already looked through the deck. You may turn over the Chevy card and find that it actually has a rock on the other side. The petunia may have a GM on the other side. Well, congratulations, your theory was wrong. Or maybe it doesn’t have a rock, and maybe the petunia doesn’t have a corvette, your theory might still be wrong. Poor puzzle.”
This reply does show a thought process. I like it.
I cannot say the puzzle is poorly worded since there is a correct answer to it as stated and I do not know what question was intended. Turning over the Ford card will not test the hypothesis. So, to test the hypothesis, any one of the other three cards must be turned over. “Must” implying a minimum requirement.
Turning over 1,2, or all 3 of the non-Ford cards tests the hypothesis, but the question asks “must” not “can”.
You do bring up a good point about evidence. There is always room for more! Assuming the hypothesis hasn’t been falsified, more evidence always helps. In this example, there is actually a statistical “confidence level” in the hypothesis that could be calculated if we knew how many cards were in the deck. If we dealt all the cards and the hypothesis was not falsified, the confidence would be 1, or 100%. For all in-tent sand porpoises, the hypothesis would be confirmed as true.
Dave,
Turning over the cat doesn’t really tells you anything-although it might solve some curiosity. “Hm, do other cards besides GM cards also have animals? I ….No, on second thought , I wouldn’t be that curious, I already saw the whole deck of cards!”
One other thing, turning over the cat card is not as good a test as trying to falsify the hypothesis.
Cross posted. Anyhow, a positive result is still a valid result. If there is a GM car on the other side of the cat card, we have evidence in support of the hypothesis.
As for Q4, the question does not state that the love is requited.
More importantly, this video explains Cantor’s proof in ten minutes: Proof some infinities are bigger than others.
Dave,
I don’t know what you mean, I never asserted that the love must be requited. Either William is tweeting to someone who doesn’t love, Or Betty is. Either way, someone is.
Dave,
There are thousands of people who claim to verify Cantor’s little card game. And no matter how many try, its still the same trick. They can’t identify any of the elements in the set of real numbers to list, but insist it is a complete set, which can’t be compared to another. Its either a completed set or its not-but they want it both ways.
Its just silly word games, and the fact that there are so many gullible people in the world doesn’t change anything.
Which set is bigger, the set of all sets, or the set of all sets combined with the empty set? Nonsense definitions give nonsense results.
phoodoo,
I’ll be happy to write a paper on your behalf and submit it to a relevant journal. Is that your entire argument, or is there more?
Q4, your answer is right, Phoodoo. I simply misunderstood your initial explanation.
Phoodoo, re Cantor’s diagonal proof, you could imagine it with cards with the numbers written on I guess. Pretty pointless though. Cantor uses numbers, he isn’t playing cards. The words “card game” are not essential to any critique of the diagonal proof. What card game are you thinking of, out of interest?
Anyone can identify an element in the set of real numbers. Do you like 8? Prefer 49.27? Pi? Square root of 7 perhaps.
When you talk about the ‘completeness’ of the set of real numbers I can’t see that there is any meaning of complete that would be relevant. Cantor’s diagonal proof doesn’t rely on enumerating every element in the set.
As for “can’t be compared”, comparison is the very purpose of mapping elements from each set.
“It’s just silly word games ” is rhetoric not refutation.
As for the last paragraph, you appear to be expressing a dislike for one or both of the axioms of empty set and power set. Dislike is not a disproof. Given the axioms, the diagonal proof follows. It doesn’t mean that there are different sized infinities of anything physical.
phoodoo,
This construction is banned in Darwinist mathematics.
But that doesn’t answer the question posed. Your proof needs a bit more thought.
Hotshoe gave a complete solution to the card puzzle. The question asks which cards “must” be turned over to test the hypothesis. Test, not prove. Two of the cards cannot confirm or disconfirm.
This construction is banned in Darwinist mathematics.
Indeed. The set of all sets necessarily includes the empty set; if it didn’t already include the empty set, then it would not be the set of all sets.
I think it quite amusing that we seem to have arrived at a stage in the dialectic where it is the so-called “materialists” [though no one here is, so far as I can tell] who are insisting on the validity of Cantor’s proof that the cardinality of the real numbers is greater than the cardinality of the natural and rational numbers. Historically speaking, “materialism” has long been associated with empiricism (in epistemology) and nominalism (in metaphysics). [Of course there is also long-standing controversy about whether materialism and empiricism are consistent, and there is a long-standing tradition of non-materialist empiricism, including Berkeley, Hume, and (sometimes) Russell.] And there’s also the long-standing problem of how empiricists can accept mathematics, with its apparent commitment to abstract objects.
Be that as it may, I just thought it funny that in the current squabbles between TSZ and UD, it’s the “materialists” who are on the side of Cantor, and the “anti-materialists” (or however they may characterize themselves) who reject it.
Phoodoo:
It is stated in the problem that Betty is tweeting someone who is not in love.
The question to be deduced is whether someone who is in love is tweeting someone who is not in love. Not just yes or no.
Show the proof.
Why do you say that? Show your work.
ETA:
You may or may not be correct, but you haven’t satisfied the requirement to prove it.
I have accepted criticism for lack of clarity. Now it’s your turn. Show your work.