Theorem vs. Paradox

 In case you're in the mood to twist your mind a bit into knots, there's this:  ;)

https://inference-review.com/letter/a-theorem-and-a-paradox

Lob's Theorem and Curry's Paradox

 Fitting coincidence that after yesterday's post this piece also on self-reference should come up:

https://inference-review.com/article/loebs-theorem-and-currys-paradox

Philosophers At (Word)Play...

 From Futility Closet today, the paradox of a paradoxical paradox (if you catch my drift):

https://www.futilitycloset.com/2021/05/26/the-paradox-of-the-question/

Making Connections Between Paradoxes

Connecting Berry's Paradox to Gödel's Incompleteness Theorem:

https://thatsmaths.com/2020/07/30/berrys-paradox-and-gdels-incompleteness-theorem/

Rambling About Sorites

[Any resemblance between this graphic & the author of this post is purely coincidental]

Suppose a man starts off with100,000 hairs on his head, but on Wednesday he loses 50 hairs. Is he now “bald.” Of course not. But say you don’t see him for a decade and when you next run into him he has only 50 hairs left on his head. Is he now effectively “bald.” Yes. But when, in time, did he “become” bald. This is just one of many ways of stating the ancient “Sorites” paradox, also known as the “heap paradox” or trying to describe how many grains of sand constitute a “heap” of sand.

This all came to my mind a bit ago when Mike Lawler tweeted out, “It is almost impossible to imagine -> 4:40 per mile pace for the entire marathon,” while referencing a newly-set record for a marathon race. Indeed, I do find it impossible to imagine… yet, it was accomplished:

https://www.theatlantic.com/health/archive/2018/09/eliud-kipchoges-world-marathon-record/570400/

I’m always amazed at how, over time, so many track-and-field records keep falling. There are of course improvements made in nutrition, training, equipment, etc. but still limits to human performance must, in the end, rule — there is never going to be a 5-second 100-yard dash, nor a 1-hour marathon (at least not as humans are currently constituted). Yet finding that ‘boundary’ which can be asymptotically-approached but not crossed seems a difficult task.

Famously, Roger Bannister barely broke the 4-minute barrier for the mile-run in 1954, following a century of efforts by others. In the 60+ years since, the record has gradually dropped to almost 3:43. How much lower can it go (can it break the 3:40 level)? Read the history/progression for this track event here:

https://en.wikipedia.org/wiki/Mile_run_world_record_progression

Of course the longer the race, the more likely there is room for improvement: easier to imagine shaving a second off a marathon or 10K run (or even a mile) than the 100-yard dash. How about the pole vault, the long jump, the hammer throw, the shot put?... how easy to keep setting records there?
[One philosophical approach to Sorites is to argue that a definite boundary exists, but that it is unknowable. Perhaps a similar take exists for athletic activities: there are human (physical/physiological) limits, but it's unknowable exactly what they are...?]

...In logic, the law of the excluded middle, is both a staple, but also controversial. Claiming that a statement can’t be both true and false, the law seems simple and innocent… except that in normal discourse meaning and language are rarely so binary. Is it true that John is tall, or smart, or fast, or…. Obviously, it depends on how you define “tall…etc.”, but moreover no definition will likely neatly fit precisely all cases (especially since you also enter into issues over measurement, precision, and context). As applied in math and logic the law is somewhat more clean, but still controversial, and Sorites, with its boundary-ambiguity, gives some indication of its ongoing murkiness.

https://en.wikipedia.org/wiki/Law_of_excluded_middle

Loosely, this all also reminds me a bit of mathematical “surreal numbers” and “Dedekind cuts” where it is the ‘boundary' or middle ground that again becomes all-important. Like many ancient paradoxes, the Sorites paradox has a lot of depth.
"Fuzzy logic" and other multi-valued logics (which include 3 or more truth-values) are one alternative to the classical logic of two truth-values.

...In the meantime, there are enough variables at work in running a marathon that record-breaking can probably go on for quite awhile!

To Toss or Not to Toss

The paradox known as Buridan’s Bridge:

Bridge Gatekeeper to the approaching passerby: “If the next sentence you utter is true I will permit you to cross. But if you speak falsely, I will throw you into the water."

Passerby: “You shall throw me into the water.”

Two For One...

Nice explanation today of Banach-Tarski here:
https://infinityplusonemath.wordpress.com/2017/11/09/double-for-nothing-the-banach-tarski-paradox/

The Universe... Even or Odd

I should be writing a blurb about the various 2017 mathy books that have passed my way the last few months, but instead the volume I just finished reading is an older classic, Roy Sorensen’s 2003 A Brief History of the Paradox. Toward the end comes a ‘paradox’ (perhaps known by some as 'the odd universe' paradox) I was unfamiliar with and frankly don’t quite understand, though it doesn't appear too difficult. Am passing it along because some of you may find it interesting (…or be able to explain it better to me!).

Verbatim from the book (I’ve bolded a few bits that I especially have difficulty following):

“Meanwhile, Nelson Goodman kept sharpening the knife of nominalism. In 1951 he published The Structure of Appearances. This book contains a logic of parts and wholes. Goodman denies that there are sets. Instead, there are fusions built up from smaller things. Unlike a set, a fusion has a position in space and time. You can touch a fusion. I’m a fusion. So are you. Goodman’s ‘calculus of individuals’ says that there are only finitely many atomic individuals and that any combination of atoms is an individual. Objects do not need to have all their parts connected, for instance, Alaska and Hawaii are parts of the United States of America. Goodman does not let human intuition dictate what counts as an object; he also thinks that there is the fusion of his ear and the moon. In a seminar Goodman taught at the University of Pennsylvania around 1965, John Robison pointed out that The Structure of Appearances implies an answer to ‘Is the number of individuals in the universe odd or even?’ Since there are only finitely many atoms and each individual is identical to a combination of atoms, there are exactly as many individuals as there are combinations of atoms. If there are n atoms, there are 2n - 1 combinations of individuals. No matter which number we choose for n, 2n - 1 is an odd number. Therefore, the number of individuals in the universe is odd! The exclamation point is not for the oddness per se. Aside from those who think the universe is infinite, people agree that the universe contains either an odd number of individuals or an even number of individuals. What they find absurd is that there could be a proof that the number of individuals is odd. ‘Is the number of individuals in the universe odd or even?’ illustrates the possibility of one good answer being too many. Our expectation is that this question is unanswerable. The lone good answer confounds beliefs about what arguments can accomplish.”

Anyway, seems like an interesting thought exercise to play with.

(If you can explain it any more lucidly in the comments feel free to give it a go. The primary part I'm unclear about is, in the 2nd part that I've bolded, why does the 2nd sentence necessarily follow from the prior sentence?)

Beware of Dating Logicians

            
                       YES    NO   ????


Got a hot date this weekend?
For the men out there (...though I s'pose women could turn the tables and use this as well), two questions to pose to your next date after that romantic dinner:

1)  Will you answer THIS question the very SAME way you answer the NEXT question?

and

2)  Will you come back to my place and make mad, passionate love to me tonight?


(I adapted this specifically from a fine little volume, "Paradoxes: Adventures In the Impossible" by Gary Hayden and Michael Picard, but have run across versions of it in other places.)

Be Afraid, Be Very Very Afraid

Two for the price of one today:

1)  First, in time for Halloween, DO NOT miss the frightful tale of Differentiation... as only Ben Orlin can tell it (bwaaahaaaahaaaaa):
http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/


2)  Less scary, but more mind-racking perhaps than differentiation, is the 'Sleeping Beauty Problem/Paradox,' which I haven't mentioned for awhile, but do now (...at least one version of it):



The correct answer is: 1/2, 1/3?; 1/2, 1/3?; 1/2 or 1/3???.... two different logical answers, splitting the mind in two, with no final resolution. Spine-tingling stuff! ;-)
My original post on it was back in 2012 with a number of additional links:
http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

Also, Tanya Khovanova had lengthy previous discussion of it on her blog here:
http://blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/

And even physicist Sean Carroll covered it a year ago, drawing 240+ comments:
http://www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/

Pick your side... you'll find some good arguments (and thinkers) backing you up either way.  Spooky indeed!

Gladly Paying $1.10 For a Dollar Bill...

....or why rational choices ain't always so rational:

Another lovely puzzle/paradox today from Greg Ross's "Futility Closet" volume. It's known as the "dollar auction" paradox created by economist Martin Schubik. The setup (I've adapted from Wikipedia):

An auctioneer is to auction off a single dollar bill with the following rule: the bill goes to the highest bidder, AND the second-highest bidder LOSES the amount that they bid (to the auctioneer). The winner could gain a dollar for say 20 cents, for example, but only if no one else bids higher. The second-highest bidder is the biggest loser since they pay out their bid and get nothing in return.
The opening, minimum bid is 5 cents (with 5-cent increments thereafter) from one player, who would make a 95-cent profit if no one else bid. But it's sensible for another player to bid, say 10 cents, and still make a 90-cent profit. Then similarly, another bidder may now bid 15 cents, making 85-cents profit.
Whoever is the second-highest bidder at any point in time will wish to convert his potential loss to a gain by bidding higher than the highest-bidder, and so on. Obviously, if this keeps up, at some point, the dollar will COST someone a dollar to purchase -- but at least they will suffer no loss, while the 2nd highest bidder will lose 95 cents, giving them an incentive to bid $1.05 and thus decrease their loss to a nickel... at which point, the other bidder loses a whole dollar... and on and on. Bids beyond $1.00 mean that both top bidders lose money, thus minimizing the amount of loss then becomes the focus. A series of rational bids will reach and ultimately surpass the one dollar point, as the bidders seek to minimize their losses. Thus, "rational" bidding leads inevitably to both the two highest bidders losing money (while the auctioneer makes out well).
No wonder some call economics "the dismal science." ;-)

More of Life's Paradoxes

Long-time readers here know that I'm especially fond of paradoxes, so was naturally drawn to a post from blogger Tony Mann where he employs some logic from Martin Gardner and Curry's paradox to demonstrate his knack for predicting sport outcomes, and beating "the pundits" :-):
http://tonysmaths.blogspot.com/2015/01/logical-paradoxes.html

Meanwhile, NPR has another new hour-long science-oriented show called "Invisibilia" -- the first two episodes that I've heard have been fantastic. Check your local station to see if it's available in your area, or you can pick it up off the Web here:
http://www.npr.org/programs/invisibilia/

Anyway, the latest episode on "fear" included a segment that ended with an odd little paradox of its own.  Turns out there is a very rare amygdala-destroying genetic disorder known as "Urbach-Wiethe disease" which eliminates the human experience of fear -- literally, individuals feel NO FEAR because they lack the biological requisites for its sensation. The end result of this bizarre condition is fascinating: it means that such a sufferer has many MORE "bad" experiences in their life, because they lack the necessary fear to avoid such experiences. On-the-other-hand, a normal person, with proper fear response, avoids a lot more of life's dangers, BUT experiences MUCH MORE fear/stress, and essentially unhappiness, via the fewer instances that they do experience... i.e., the individual who suffers more often (or has more bad things happen to them) is happier than the individual who suffers less often, because of how the suffering is experienced. Anyway, no math, just interesting, and counterintuitive. And paradoxes are part of life, not just logic textbooks.

Moravec's Paradox

This isn't exactly math, but it's artificial intelligence (AI), and that's close enough... especially since a few posts back I wrote about IBM's "Deep Blue" and its 1997 defeat of chess grandmaster Gary Kasparov (at the time, a long-held goal of AI). Well, Moravec's paradox is the interesting idea that advanced or high-level reasoning and logic is much more easily mimicked by a computer system than are low-level sensori-motor skills that have evolved over millions of years... it's easier for a computer to learn to play chess, than to recognize human faces. This is one of those things that is fairly obvious when you stop to think about it... but, we often don't stop to think about it!
Here's what Steven Pinker wrote in "The Language Instinct":

“The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face, lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest engineering problems ever conceived…. As the new generation of intelligent devices appears, it will be the stock analysts and petrochemical engineers and parole board members who are in danger of being replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to come.”   

A more recent blog piece applies the paradox to Google's self-driving cars, a creation I've certainly had trouble comprehending, given the countless issues/variables involved:

http://www.eugenewei.com/blog/2014/10/13/moravecs-paradox-and-self-driving-cars

[p.s. -- actually, where are the dang flying jetpacks I grew up believing we would all have by now... forget the cars Google, I want my personal commuting jetpack!]

anyway, below, another somewhat provocative post applying Moravec's paradox to brain processing:

http://blog.jim.com/science/moravecs-paradox-rna-and-uploads/

"The Upside-Down World Paradox"

Given my fondness for paradox, just linking today to this quirky, fun little (non-mathematical, but logical) post about the 'upside-down world game' (an offshoot of 'the liar paradox'):

http://m-phi.blogspot.com/2014/10/the-upside-down-world-paradox.html

The blogger's seven-year-old daughter enjoys the game in the post, so if you have young kids maybe they will as well.

"Why Study Paradoxes?"

Given my fondness for paradoxes… and, for Raymond Smullyan... I couldn't help but love this weekend post from Ray T. Cook on paradoxes (and why to study them) at Oxford University Press blog:

http://tinyurl.com/ppvg4cp

After offering his own example that he once posed to the master logician Smullyan, Cook goes on to talk about the mathematical complexity of paradoxes before ending thusly:

"...that's why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered."

Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

And that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

that’s why I work on paradoxes: their surprising mathematical complexity and mathematical beauty. Fortunately for me, there is still a lot of work that remains to be done, and a lot of complexity and beauty remaining to be discovered. - See more at: http://blog.oup.com/2014/09/why-study-paradoxes/?utm_source=feedblitz&utm_medium=FeedBlitzRss&utm_campaign=oupblogmathematics#sthash.VxyOQRJg.dpuf

Also, Ray has apparently written a short book entirely on Yablo's Paradox which I've mentioned here before (and which the above post is related to):
http://ukcatalogue.oup.com/product/9780199669608.do

Through the Looking Glass

"Why, sometimes I've believed as many as six impossible things before breakfast."
-- Lewis Carroll (Alice In Wonderland)

I love it when disparate mathy things I stumble across over a couple of days start bouncing off one-another in my head....

First, let me point readers to this per-usual great response today from Keith Devlin to a NY Times piece I had referenced here a couple days back (about Common Core):
http://devlinsangle.blogspot.com/2014/07/the-power-of-dots.html
(DO NOT miss this post... whatever side of the issues you're on!)

...Next, a h/t to Vi Hart for introducing me to the YouTube channel of "Looking Glass Universe":

https://www.youtube.com/channel/UCFk__1iexL3T5gvGcMpeHNA

Vi mentioned it for its excellent short vids on physics topics, but there are also some more strictly math videos as well, including the below, wonderful treatment of one of my favorite, mind-boggling, math puzzles/conundrums (...Keith Devlin writes above that "understanding structure... requires human insight. It is not trivial and it is difficult" -- I think this video possibly captures that in 3 minutes):



And finally, the above video, in turn, led me to recall a brief posting from a couple days back on the exquisite "Painter's Paradox" of Gabriel's Horn (another lovely mind-bender):

http://tinyurl.com/qhqphpa

The sudden inter-meshing (cross-fertilization, if-you-will, in my mind) of Gabriel's Horn, the diagonal paradox, and Dr. Devlin's emphasis on structure, pattern, and insight/creativity (...which can make the seemingly impossible, possible), was for me, a beautiful thing!...

Truthiness From Yablo

This isn't everyone's cup-o-tea, but I've mentioned Yablo's Paradox before, and, since I love it, will do so again! As Sam Alexander states, Yablo's Paradox is "a cute version of the Liar’s Paradox" that manages "to achieve paradox without any direct self-reference." The simply-stated paradox involves a countably infinite number of sentences, each of which refer only to sentences that come after it:

• Sentence 1:  Sentence n is false for every n > 1
• Sentence 2:  Sentence n is false for every n > 2
• Sentence 3:  Sentence n is false for every n > 3
• Etc....

Read Alexander's post here:  http://www.xamuel.com/dangerous-graphs/  (he discusses it in terms of "graphs").
And here is (Stephen) Yablo's amazingly short, original (1993) piece introducing the paradox:

http://www.mit.edu/~yablo/pwsr.pdf

Worth noting, that while Yablo claims his paradox involves neither self-reference nor circularity (because all steps along the way reference sentences that are yet to come), others disagree with this contention. If you're logically-inclined, see Graham Priest here:

http://www.accionfilosofica.com/misc/1183297103crs.pdf

and JC Beall here: http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_circular.pdf

(This is another good example of how things become muddled when infinity is involved; in the case of Yablo's Paradox, an infinite number of sentences.)

Re-visiting Sleeping Beauty

(via Rachel CALMUSA/WikimediaCommons)

In chapter 11 ("Is Time an Illusion") of Max Tegmark's new book "The Mathematical Universe" the author, while discussing the nature of time and human consciousness, touches upon the "Sleeping Beauty" puzzle/paradox, which I mentioned here almost two years ago:

http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

This is one of the most interesting and delicious (perhaps even complicated, in some ways) puzzles around, as people argue vociferously for either of two different answers (1/2 or 1/3), because of the conditional probabilities involved.

[Here is one statement of the puzzle: Sleeping Beauty undergoes the following experiment, being told all these details ahead of time. On Sunday she will be put to sleep. A fair coin will then be tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday AND Tuesday. But when she is put to sleep again on Monday, she is given an amnesia-causing drug which ensures she cannot remember the prior awakening. In this case, the experiment ends after she is interviewed on Tuesday.  Whenever Sleeping Beauty is awakened and interviewed, she is asked, "What do you believe is the probability that the tossed coin landed on  heads?" -- What is her answer?]

In re-visiting the links I provided in my original blog post I discovered that the "Tanya Khovanova" link has since added further lo-o-ong discussion of the issues by two commenters back-and-forth, which is probably worth checking out if you are especially interested in probability in general, or this problem in particular (if these areas don't interest you, don't visit it, lest you fall into a deep, deep coma, or alternatively, your head explode ;-)

Speaking of Leakers...

Nice explication of the Ellsberg paradox with a simple card gambit from Presh Talwalkar. I like the way this little problem interweaves risk (or ambiguity) aversion, uncertainty, and probability all together in one presentation, and doesn't require any deep math to comprehend:

http://mindyourdecisions.com/blog/2013/07/31/ellsberg-paradox-facing-the-uncertain/#.UfjmHFOhRo4

(p.s., this paradox indeed comes from Daniel Ellsberg, who before his fame as the 'Pentagon Papers' leaker, was an economic theorist specializing in decision theory)

Your Lovers Have More Lovers Than You Do

"You spend your time tweeting, friending, liking, poking, and in the few minutes left, cultivating friends in the flesh. Yet sadly, despite all your efforts, you probably have fewer friends than most of your friends have. But don’t despair — the same is true for almost all of us. Our friends are typically more popular than we are."                                                                     -- Steven Strogatz

The above was Steven Strogatz's lead-in to his piece on the "friendship paradox" for the NY Times back in 2012:

http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can-count-on/?_r=0

Presh Talwalkar did a nice treatment of the paradox as well at his blog awhile back:

http://tinyurl.com/lba7qu7

The friendship paradox was first recorded in 1991 by sociologist Scott L. Feld, demonstrating that most people, on average, have fewer friends than their friends have!  It is essentially a result of sampling bias in social networks.

The basic notion can be used to show that, on average, most of your social media contacts as well (Facebook, Twitter, Google+, etc.) have more contacts/followers than you do (assuming your name isn't "Justin Bieber"); or even perhaps (if you care to think about it) that most of your sex partners have had more sex partners than you. In a more serious vein, the paradox has been used to study the course of epidemics, spread through human contact:

“We think this may have significant implications for public health,” said Christakis [Harvard researcher]. “Public health officials often track epidemics by following random samples of people or monitoring people after they get sick. But that approach only provides a snapshot of what’s currently happening. By simply asking members of the random group to name friends, and then tracking and comparing both groups, we can predict epidemics before they strike the population at large. This would allow an earlier, more vigorous, and more effective response.”

From HERE.