"What is going on here?"

Today's re-run is a passage about the role of intuition in mathematics (from "How Mathematicians Think" by William Byers):

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"For the mathematician, the idea is everything.  Profound ideas are hard to come by, and when they surface they are milked for every possible consequence that one can squeeze out of them. Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary.

"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…

"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."

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In essence, this seems very much what Eugene Wigner's famous notion of "the unreasonable effectiveness of mathematics" revolves around… how is it that the human brain is capable of such seemingly successful intuitions about the world around us….

Another Classic Puzzle

Re-running a fun, simple old golden-oldie from Alfred Posamentier's "Mathematical Amazements and Surprises":

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"You are seated at a table in a dark room. On the table there are twelve pennies, five of which are heads up and seven of which are  tails up. (You know where the coins are, so you can move or flip any coin, but because it is dark you will not know if the coin you are touching was originally heads up or tails up.) You are to separate the coins into two piles (possibly flipping some of them) so that when the lights are turned on there will be an equal number of heads in each pile."

"Your first reaction is 'you must be kidding!' How can anyone do this task without seeing which coins are heads or tails up? This is where a most clever (yet incredibly simple) use of algebra will be the key to the solution."

Posamentier Continues:

"Let's 'cut to the chase' (You might actually want to try it with 12 coins.) Separate the coins into two piles, of 5 and 7 coins each. Then flip over the coins in the smaller pile. Now both piles will have the same number of heads! That's all! You will think this is magic. How did this happen. Well, this is where algebra helps us understand what was actually done."

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Explanation: ...At the start, there are 5 heads showing among 12 coins. After separating into piles, let's say the 7-pile now has "h" heads. The 5-pile then has "5 - h" heads, and "5 - (5 - h)" tails (or, just "h" tails). Once you flip the entire smaller pile, all the tails ("h" of them) become heads, and all the heads become tails. Thus you are left with "h" heads in the "5-pile," the same as the number in the "7-pile." Whaaa-laaahhhh! Math is a beautiful thang!!

Meta-thoughts....

I've used this quote from the irascible David Berlinski several times over the years... I'm sure it's not everyone's cup-o-tea, but it remains a personal favorite:

"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.

"Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.

"These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."

-- David Berlinski (from "The King of Infinite Space")

ASMR Sunday

For April, and a diversion from everything going on around us, time to insert another example of "fast and aggressive" ASMR from one of its practitioners:

The Dismal Science...

Today's golden-oldie re-run I first saw on Greg Ross's delightful "Futility Closet" site:

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....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." ;-)

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Hahhh! I've pre-timed these re-run posts to run on forthcoming days, and on Monday (4/13) noticed that Kyle Evans entered The Aperiodical's "Big Math-Off" contest with a British version of this very same puzzle:
https://aperiodical.com/2020/04/the-big-lockdown-math-off-match-2/

Contemplation versus Dynamite

Today, just re-posting this old math meditation from Jordan Ellenberg:

"Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that's one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960s and '70s, had a different view: 'The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.'

"The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone.

"Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite."

-- Jordan Ellenberg in "How Not To Be Wrong"

Old Richard Wiseman Classic

With so many current distractions, am lacking the energy for posting new material, but will fill time re-posting some old Math-Frolic posts for any who missed them the first go-around, starting with this old Richard Wiseman classic puzzler:

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 Verbatim from Richard Wiseman's blog: 

"Imagine there is a country with a lot of people. These people do not die, the people consists of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance its a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?"

Wiseman's Friday puzzles are frequently devious… but, often once the answer is given and explained, one feels impelled to slap one's forehead and exclaim "DOH!, well, of course!." So perhaps his best offerings are those that, even once explained, are still not totally clear, and generate a lot of ongoing discussion/debate...

The one from last Friday (above) is such an effort, once again proving how tricky and misleading, probabilities can be. I confess to requiring extra time to convince myself that 50% was the correct answer, and it stiiiill rankles my intuition (…reminds me of Cantor's "I see it but I don't believe it" reaction! ;-)). This seems to be one of those quirky puzzles that is patently obvious to many, yet hugely thorny for others (one of the keys, I think, is to remain tightly focused on strict statistical probability, and not let your brain get distracted by what could theoretically happen). Read all the discord for yourself:

http://richardwiseman.wordpress.com/2012/01/16/answer-to-the-friday-puzzle-139/

(...peruse as many of the 270+ comments as you care to -- in fact you have to read some to get the answer, since Richard neglected to post an answer or explanation in his own post).

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Classic

Haven't had much time for math or blogging lately so will just re-post a link to an old fun favorite, lest any have never seen it... George Boolos' "Gödel's Second Incompleteness Theorem Explained in Words of One Syllable":

https://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf