Revving Up The Brain on Monday

 A few logic problems (via Alex Bellos) to start your week:

https://www.theguardian.com/science/2021/aug/23/can-you-solve-it-logical-philosophers

Reasoning, Induction, Math Stories...

 The latest essay from Jim Propp:

https://mathenchant.wordpress.com/2021/08/16/reckoning-and-reasoning/

Continuum Hypothesis via Scott Aaronson

 

Bored with the headlines, folks?... Is election news getting you down?... Or the blitz of political ads scraping on your last nerve?... Is that what's buggin' you, kiddies??? 

Well, OK, this won’t be everyone’s cup-a-tea, or fun distraction, but in a longish post (first of multiple) Scott Aaronson tackles the independence of the Continuum Hypothesis:

https://www.scottaaronson.com/blog/?p=4974

I knew it would be a great read when he started off with this quote from Bertrand Russell:

 ;)

“in adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics.”

He Said She Said

Yo, logic enthusiasts, when I saw a post entitled, “Smullyan and the President’s Sanity” listed on the mathbogging.org feed this morning it caught my attention. Have fun:

http://www.neverendingbooks.org/smullyan-on-the-presidents-sanity

"Given These Premises"....Inference

Am copying this verbatim from a recent Futility Closet posting about a hand of cards:

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Given these premises, what can you infer?

1. If there is a king in the hand then there is an ace, or if there isn’t a king in the hand then there is an ace, but not both.
2. There is a king in the hand.

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What is your answer???

—————————————

Now, go read the Futility Closet post:

https://www.futilitycloset.com/2016/09/07/a-cognitive-illusion/

As you will see, the post claims that “almost no one sees” the correct answer, and "practically everyone" infers (wrongly) instead that “there is an ace in the hand.”  The correct answer seems fairly obvious to me, but the post implies that most all fall for this “cognitive illusion.” Unfortunately there’s no way for me to know how many readers here immediately see the proper answer, but I’m wondering if math fans, perhaps more grounded in logic than the general populace, don’t answer this correctly at a much higher rate than other groups of people... IF that were indeed the case, it would be another indication of why training in mathematical thinking ought be encouraged.

The article says it is "unclear" why people mess up on this particular problem, though I think it's just one more example of how verbal cues are often very ambiguous or misleading for people... language is rarely as precise as individuals tend to assume. It all even reminds me a bit of a very old classic math conundrum that throws most people off (most of you will be familiar with it), which in one version (from Wikipedia) runs like this:

"Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop $5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $1 and keep $2 as a tip for himself. Each guest got $1 back, so now each guest only paid $9, bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?"

OR, alternatively, here's a more recent example from the Web that many of you will recall, where the answer is actually fairly simple, yet many people, once again, are misdirected by the wording**:

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Jack is looking at Anne, but Anne is looking at George.
Jack is married, but George is not.
Is a married person looking at an unmarried person?

A) Yes       B)  No       C)  Cannot be determined

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...I s'pose the ability of language to hinder or interfere with rational thought has never been better demonstrated than by the current American presidential election :-( 

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**  the correct answer is "A"

Put On Your Thinking Caps

Wonderful new Brian Gallagher article in Nautilus yesterday covers some classic Ray Smullyan/George Boolos logic conundrums:

http://nautil.us/issue/30/identity/how-to-solve-the-hardest-logic-puzzle-ever?

Gives an overview of what is famously-designated "the hardest logic puzzle ever" (created by Smullyan and solved by Boolos).

Gallagher ends the piece noting the puzzle demonstrates "how essential one of the supposed fundamental laws of logic -- the law of excluded middle -- seems to be" (which assumes that "every statement is either true or false -- there is no middle ground"), or in Boolos' words, “Our ability to reason about alternative possibilities, even in everyday life, would be almost completely paralyzed were we to be denied the use of the law of excluded middle.”

A practical problem of course is that the law of the excluded middle only operates within narrow, well-defined contexts, and NOT in most of day-to-day life... language and life are far more characterized by ambiguity, continuity, and gray areas, than the discrete black-and-whiteness implied by a simplistic excluded-middle law. Thus, my own increased recent interest in so-called "fuzzy logic" (mentioned awhile back) over classic Aristotelian logic... but still, for puzzle and logic purposes, a great article.

Still Fuzzy on Fuzzy Logic

"Classical logic is like a person who comes to a play dressed in a black suit, a white, starched shirt, a black tie, shiny shoes, and so forth. And fuzzy logic is a little bit like a person dressed informally, in jeans, tee shirt, and sneakers. In the past, this informal dress wouldn't have been acceptable. Today, it's the other way around."
-- Lofti Zadeh (1984)

Though it's been around for a good while I only recently began dabbling in "fuzzy logic," and now enjoying it as an approach that makes a lot of sense (reminds me also of the non-Aristotelian approach of General Semantics, and getting rid of the "law of the excluded middle"). I've enjoyed various essays by Bart Kosko in the past, but only recently learned of his connection to fuzzy logic (which drew me to the subject). Kosko's 1993 read, "Fuzzy Thinking" is a great introductory volume.
Another popular old-read (also 1993) on the topic is "Fuzzy Logic" by McNeill and Freiberger, but I didn't find it nearly as satisfying as Kosko's volume.

There are also many web videos available on fuzzy logic, but the few I've looked at didn't seem all that helpful or effective. I'd still like to find a good visual presentation. So if someone cares to recommend a good video, feel free to (and save me some time ;-) Or feel free to recommend other books and websites for the interested layperson.

Simple Logic...

List:

   1.  This sentence contains eight words.

   2.  This sentence contains five words.

   3.  Exactly one sentence in this list is true.

Euler... Thinker Extraordinaire

1) Probably any mathematician would rank Leonhard Euler as one of the greatest mathematicians of all time (he pretty much shows up on anyone's top five list). And Professor William Dunham reveres Euler as the very top of the heap. In this video he entertainingly makes the convincing case for Euler -- Dunham talks for about an hour, followed by 30 mins. of question-answers -- for such a long presentation it maintains interest well:

You'll learn what i^i equals (and how Euler computed it), as well as Euler's better-known contributions, all accomplished in an era long before computers and calculators.
One odd, quirky side-note: it turns out that of the top 5 mathematical formulas/results (as voted on somewhere) 3 come from Euler and the other 2 from Euclid... just an interesting twist that mathematicians with 2-syllable names beginning with "Eu" are responsible for so much powerful math! (...uhh, in the future, please just call me Eushek).




2) No promises that it'll get you thinking as sharply as Euler, but the below post offers a nice primer on mathematical thinking/proof (based on logic and abstract reasoning), which I think may be similar to the approach Keith Devlin takes in his Coursera MOOC course on the same subject (but someone feel free to correct me if the comparison is off-base).

http://amininima.wordpress.com/2013/07/25/how-to-prove-somethin/

Mind-wrenching….

Call me a masochist, but I do love these self-referential or recursive logic puzzles (that shred my brain to itty-bitty, miniscule, pulsating pieces!). The one below I've encountered multiple times over the last year-or-so, and seen several different answers offered as solutions. Most of them don't check out when applied carefully, but one does seem to work well, and because there is at least some wiggle room in interpretation of the statements there may be one or more other successful answers.

So I'm curious if anyone knows the precise origin of this logic puzzle, and definitively what the proper answer(s) are supposed to be?

And, if you've never seen it before, well, knock yourself out…:

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Given the following 12 statements which of the statements below are true?

1.  This is a numbered list of twelve statements.
2.  Exactly 3 of the last 6 statements are true.
3.  Exactly 2 of the even-numbered statements are true.
4.  If statement 5 is true, then statements 6 and 7 are both true.
5.  The 3 preceding statements are all false.
6.  Exactly 4 of the odd-numbered statements are true.
7.  Either statement 2 or 3 is true, but not both.
8.  If statement 7 is true, then 5 and 6 are both true.
9.  Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.

***********************************

...I've already used several of these type puzzles on the site before, but if you have a favorite you think I may like send it along to me via email and I'll consider posting it and acknowledging the sender.

[...In a few days I'll state one of the answers that works on the above in the comments section, for those interested.]

Questioning Peano

For those inclined toward epistemology and formal logic, another wonderful post from RJ Lipton below, this time on the possible inconsistency of Peano Arithmetic:

http://rjlipton.wordpress.com/2011/10/01/what-if-peano-is-inconsistent/

This stuff makes my head hurt... but I always enjoy watching others pursue it!

Diagnosing Logicians

Fun post from Bill Gasarch on the sometimes popular (and paradoxical) notion that professional logicians tend to be CRAZY ("a few axioms short of a complete set")... or, NOT! :

http://blog.computationalcomplexity.org/2011/07/disproofing-myth-that-many-early.html