Wikipedia:Reference desk/Archives/Mathematics/2015 August 16

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August 16

Need some help at ITNC

There's an article up for nomination at WP:ITNC with a mathematical focus, pentagon tiling. Before it can be posted, however, it needs some work with referencing and fleshing out the latest developments. Is there anyone who frequents here that could maybe to a little work on it? Thanks in advance! --Jayron32 01:50, 16 August 2015 (UTC)

Logical fallacy identification

What is the name for the logical fallacy of confusing the intended use(s) of something and the acceptable use(s) thereof?

Examples:

1) "Sex was invented for the purpose of procreation, therefore homosexuality and all other forms of sex for pleasure are wrong."

2) "God made Adam and Eve, not Adam and Steve."

3) A cop stops a drug dealer: "Laws were not made to be broken." The drug dealer's response: "Heroin was not made to be condiscated."

4) On page 149 of the rock musical The Bittersweet Generation, Asst. Principal Pittman says, when defending school uniforms: "I’d like to state that, contrary to what young Bryce said, we needn’t worry about the message we send with regards to creativity. You go to school to learn, not for self-expression." Bryce's reductio ad absurdum is: "So? You could also say, oh, you go to school to learn, not to eat, so let’s abolish school lunches!

5) "Nuclear weapons were invented for cowing other countries into submission, so it's OK to use nukes as long as you're using them to win wars." Anthony the Thinker (talk) 06:33, 16 August 2015 (UTC)

Genetic fallacy, perhaps. Sławomir
Biały 11:10, 16 August 2015 (UTC)

(I numbered your items.) In every case above the premise seems highly suspect:

1) Sex wasn't "invented" at all. As for why it evolved, reproduction is certainly the primary reason, but it also serves a social purpose in many species, especially in primates like bonobos and humans.

2) You can't even prove that God exists, much less made anyone.

3) Many laws can't be 100% enforced, or society would break down. Drug laws are a prime example, where if everyone who ever used a drug illegally was imprisoned, society would cease to function. Then there are laws that, in addition to being unwise and unenforceable, are also unjust. There we have a duty to engage in passive resistance, according to Martin Luther King and Gandhi. There are also laws that arguably have been written intentionally so people will break them. US immigration laws may be an example, where business interests want there to be immigration, to provide the glut of labor that drives wages down, but they want those immigrants to be illegal, so they can't claim benefits (paid for by the employer) which would be guaranteed by law.

3a) Here we are looking at the motivations of drug cartels, which shouldn't be considered as very important, compared to the motivations of the rest of society. As for why heroine was made, it was originally produced as a medical pain killer. It is now made for profit (or perhaps for survival in the case of the peasants who grow the poppy), not to benefit society.

4) The premise seems to imply that learning can be done without any self expression. Seems highly suspect, to me, at least in classes like creative writing.

4a) The reducto ad absurdum seems quite valid to me. That is, just because something has a primary purpose does not eliminate any other side purpose.

5) Nuclear weapons were invented to win one specific war (WW2), not for any other reason. See Manhattan Project. StuRat (talk) 18:12, 16 August 2015 (UTC)

SR - I agree with your comments, but that isn't what the OP was asking - they were just examples, and as such a bit irrelevant. However I'm not sure what the OP is seeking. If we take "I like chocolate cake" as the statement (whether true or not, I'll not disclose) "Therefore I like carrot cake" may be the kind of thing - that is a "non-sequitur". "Hammers are heavy" and "My spanner is heavy so it is a good hammer" may be closer to what the OP seeks - I don't know what this is called. -- SGBailey (talk) 13:52, 17 August 2015 (UTC)

My point is, you can't assume that just because the conclusion is obviously false, that there must have been a logic error. It's also possible that the logic is valid, but that the initial premise is false. StuRat (talk) 15:44, 17 August 2015 (UTC)

Logic is actually a hard and very misunderstood topic, I've been studying it for years and still get confused. What StuRat is talking about refers to Logical Soundness. Most logically fallacious arguments do indeed follow valid logic, that's why they are frequently deceptively convincing, but I think they are still classed as a "logic error". Even in the case of a pure non sequitur where the conclusion is true AND the premises are all true, but the conclusion does not follow, that's still a "logic error". I think it's more correct to say you can't assume that just because the conclusion is obviously false, that the argument was invalid. Vespine (talk) 02:03, 21 August 2015 (UTC)

Logic puzzle

This logic puzzle was posted to WP:RSN a few days ago, and appears on page 24 of this document.

I have three friends, Alan, Bert and Curt. I write an integer greater than zero on the forehead of each of them and I tell them that one of the numbers is the sum of the other two. They take it in turns in alphabetical order to attempt to deduce their own number. The conversation goes as follows:

Alan: "I cannot deduce my number."

Bert: "I cannot deduce my number."

Curt: "I cannot deduce my number."

Alan: "My number is 50."

What are Bert’s and Curt’s numbers?

I've been thinking about the solution. I can easily eliminate (50,25,25), (50,100,50) and (50,50,100), as A, B and C respectively could have called them on Round 1 without having to do any deduction. I can eliminate (50,75,25), (50,25,75), (50,100,150) and (50,150,100) with a little more work:

For example, for 50, 75, 25:

B knows that it's either 50, 75, 25 or 50, 25, 25

But if it were 50, 25, 25, then A would have called it, and he didn't.

So B would know it was 50, 75, 25 on Round 1 and call it, but he didn't.

So it's not 50, 75, 25.

For 50, 150, 100:

A knows that it's either 50, 150, 100 or 250, 150, 100

B knows that it's either 50, 150, 100 or 50, 50, 100

If it's 50, 50, 100, then C will call it on Round 1.

As C didn't call it on Round 1, B (but not A) would know that it was 50, 150, 100 on Round 2.

But A (not B) calls it on Round 2. So it's not 50, 150, 100.

However, that still leaves several hundred possible answers that I can't see how to eliminate (I can't imagine the largest number appearing being more than about 500). Is the problem actually solvable as stated? If so, what's the answer? (A hint would be OK, or just the numbers without the deduction...) Tevildo (talk) 17:04, 16 August 2015 (UTC)

Spoiler alert: The solution is here, page 10. Surprisingly complex. --Wrongfilter (talk) 17:26, 16 August 2015 (UTC)

Ahh, thanks. I got to step 6, but not to step 7 - expressing it algebraically does make it easier to follow. Tevildo (talk) 17:58, 16 August 2015 (UTC)

To correct my previous reply, I got to step 7, but not step 6. And the answer is, if anyone can't follow the link:

Answer
50, 20, 30 A knows it's 50, 20, 30 or 10, 20, 30 If it's 10, 20, 30: C would know it's 10, 20, 30 or 10, 20, 10 If it were 10, 20, 10, B would have called it on Round 1. But B didn't call it. So C would have known it was 10, 20, 30 and called it on Round 1. But C didn't call it. So A knows on Round 2 that it's 50, 20, 30.

Tevildo (talk) 23:32, 16 August 2015 (UTC)

Consistent equations

I have a mutually inconsistent set of N linear equations. I want to know the cardinality of the largest subset having the property that all its members are consistent--i.e., that there exists a joint solution for them. Is there any efficient way to find this out? 208.49.194.161 (talk) 20:34, 16 August 2015 (UTC)

Start with Rank_(linear_algebra)#Applications. --JBL (talk) 20:58, 16 August 2015 (UTC)

That doesn't address this. It just gives the conditions for an entire system to have any solutions. But I already know that mine has none. 208.49.194.161 (talk) 00:04, 17 August 2015 (UTC)

I think that your problem is NP-complete because if you could solve it then you could also solve the Max-Cut Problem as follows: Given a graph $G=(V,E)$, create one variable $x_{i}$ for each vertex. For each edge $(v_{i},v_{j})$, create the equation $x_{i}+x_{j}=1$. To force each variable $x_{i}$ to assume either the value 0 or 1 (and not something else like 0.5), add the equations $x_{i}=0$ and $x_{i}=1$ (yes, these are inconsistent). Replicate these equations $|E|+1$ times to make sure that enforcing 0-1 has priority over maximizing the cut. The largest subset of consistent equations should correspond to a maximum cut. I kept this brief for now, tell me if you want more details. Egnau (talk) 04:58, 17 August 2015 (UTC)

I get the idea--thanks very much! 208.49.194.161 (talk) 13:28, 17 August 2015 (UTC)