Dimensions... and Illusion

Today's Sunday reflection:

"What about four-dimensional space? Is there such a thing? If we're asking whether four-dimensional space is real we might as well ask about three-dimensional space: Is there such a thing? I suppose it appears that there is. We're walking around (apparently), and things certainly look and feel as though they are part of a three-dimensional universe, but when you come right down to it, three-dimensional space is really an abstract mathematical object -- inspired by our perception of reality, to be sure, but imaginary nonetheless. So I don't think we should put four-dimensional space in any special mystical category. Spaces come in all sorts of dimensions, and none are any more real than any other. There are no one-dimensional or two-dimensional spaces in real life, and the only thing that gives the number 3 any special status is that our senses appear to offer us that particular illusion."

-- Paul Lockhart, from "Measurement"

Tangential to Math

A couple of not-very-mathy postings that I think worth passing along, both dealing in some sense with the future, near-term or not-so-near-term:

a)  The first is an interesting and timely philosophical/psychological piece, that I'll (tenuously) justify as math-related because it involves the perception of probabilities: applying the "trolley car problem" (a famous ethics-thought-experiment) to self-driving cars and real life...
https://medium.com/@tanayj/self-driving-cars-and-the-trolley-problem-5363b86cb82d

b)  And this long, fascinating, articulate discussion from physicist Sean Carroll for Edge.org that I definitely recommend to anyone with an interest in the current state of physics (or, just an interest in Sean's viewpoints):
http://edge.org/conversation/sean_carroll-layers-of-reality

The Power of Biography

courtesy of pixabay.com

 I always enjoy learning which popular math books have inspired other mathematicians along their way. One of the volumes that most frequently comes up, to my surprise, is E.T. Bell's "Men of Mathematics." In fact, it arose again in an obituary for John Nash over the weekend, mentioning it as one of his favorites. (Of course both the Sylvia Nasar bio and the movie about Nash's own life were huge popular hits as well.)

I do like biographies, but Bell's 80-year-old volume never much grabbed me, probably because it runs from Zeno to Cantor, missing all those who would interest me since Cantor's time. Paul Erdös, Ramanujan, Benoit Mandelbroit, H.M.S. Coxeter, John von Neumann, and of course Gödel, are some other 20th century mathematicians with bios or autobios available. Stanislaw Ulam and Norbert Wiener also wrote autobiographical accounts, and there have been some more recent compendiums, like Bell's original book, as well. Anyway, it's interesting how easily moved we are by the lives of others, and I wonder which of our modern day mathematicians will one day have important bios written about them, inspiring future generations of mathematicians.

Surely, there will be a biography of Terence Tao at some distant point in time. Andrew Wiles seems another likely candidate; perhaps Donald Knuth or Bill Thurston.  Of course, Martin Gardner (more strictly a philosopher and writer than a mathematician), did his own autobiography and also has an official biography in the works. I suspect Raymond Smullyan, again more strictly a logician than a mathematician, will eventually have a biography of his incredible life as well. And John Nash's too-sudden death may spur an additional biography or two of him. Who else? There are a lot of possibilities... one for certain, and highly anticipated (due out in just a couple of months), is Siobhan Roberts' bio of John Conway ("Genius At Play"), who is still with us, and no doubt a fantastic character for a book-length profile!

And speaking of biographical stuff, any of you 60s/70s fans out there may enjoy this recent profile of a one-time math teacher by the name of... Art Garfunkel:

http://tinyurl.com/mzfw8hu

A Happy Birthday and a Little Logic

via Ripounet/WikimediaCommons

For starters, I'll just note that this is a verrrry Memorial Day indeed (apart from the American Holiday)... it is Raymond Smullyan's 96th birthday... and THAT is cause for celebration (as well as a day off), be you a knave or a knight!
In Dr. Smullyan's honor I'll offer this thought-provoking question:  IF Ray confided in you that he was a pathological liar... would you believe him?...

Moving on, as a professional logician, I'm sure Ray appreciates the "Wason Selection Task," a simple-seeming reasoning test that most of you have likely seen in one form or another, and that most people err on the first time they attempt it (picking 2 cards out of 4 to confirm an initial supposition). But do people mess up because of the logic involved or simply because of the wording? A wonderful piece on the Wason test from Nautilus tries to address that question:

http://nautil.us/blog/the-simple-logical-puzzle-that-shows-how-illogical-people-are

Parochial Math

Today's 'Sunday reflection' from Ian Stewart, in "Letters to a Young Mathematician":

"I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry's points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, but not shape, to be fundamental to its perception of the world. And while discrete numbers like 1, 2, 3, seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of my sheep been stolen?...

"What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit."


[p.s... also, another new interview up at MathTango this morning!]

Set Theory, Type Theory, HoTT, Univalent Foundations...

Quanta Magazine consistently offers some of the best STEM writing around for lay readers.  Remarkably, over half of their contributing writers are individuals I'd never even heard of prior to reading them in Quanta. This recent piece from Kevin Hartnett is perhaps the best quick introduction to "type theory" I've ever seen for a general audience:
https://www.quantamagazine.org/20150519-will-computers-redefine-the-roots-of-math/

(The above covered, briefly, several of the foundational topics I was hoping Eugenia Cheng's "How To Bake Pi" book might cover, but didn't -- I reviewed it yesterday HERE).

Should anyone care to see a bit more advanced discussion of some of these topics, check out the comments/discussion on a recent Michael Harris posting pertaining to such:
https://mathematicswithoutapologies.wordpress.com/2015/05/13/univalent-foundations-no-comment/#comments

Anyway, the Quanta piece made me want to know more about Kevin Hartnett, so I checked up a bit. His only prior piece for Quanta is here (also mathematically-inclined):
https://www.quantamagazine.org/20150113-a-proof-that-some-spaces-cant-be-cut/

But he's been writing a diverse column for the Boston Globe (entitled "Brainiac") for awhile:
http://www.bostonglobe.com/ideas/brainiac

He also writes a far more personal blog about fatherhood here:
http://growingsideways.net/

A Nigerian Prince Needs My Bank Account Number...

"Hello, my name is Prince Kwami Sovay Kysong and I seek your asistance and discretion, as a known trustworthy, dependible individual for a transaction of utmost importance and profit for yourself and my own family sufering the ravages of our war-torn village...."

A couple years back I noticed that the Nigerian scams I was abundantly receiving in email were, contrary to what one might expect,  getting stupider and stoopider, and full of spelling/grammar mistakes. It seemed baffling to me that these missives were still being sent out in such numbers, until a couple months passed and somewhere online I read the explanation.

Today, Presh Talwalkar repeated the explanation (if you've never heard it before) of why Nigerian scammers are so insistent on letting you know that the bad-grammar, misspelled, insipid emails you receive, originate... from Nigeria -- because they want to filter out all those folks with a modicum of intelligence(!), and Presh uses an analogy to medical test results to help explain it:

http://tinyurl.com/lwoot39

Postal Pigeonholing

To start the week, another thought problem I've adapted again from Posamentier & Krulik's "Problem-solving Strategies in Mathematics." It's actually quite simple, though the language can throw people off:

A certain neighborhood includes 50 condominiums (1 mailbox/household per condominium). One afternoon, the postman has 151 pieces of mail to deliver to those 50 households. What is the largest number of letters that some household is guaranteed to receive?
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Answer:   4
...of course there are many possible (but UNguaranteed) scenarios in which one (or more) of the households could get far more than 4 items, but if we look at the 'most evenly spread out case,' 50 recipients will get 3 items, and there is still one more remaining that must go to someone, who thus ends up with 4; i.e., SOMEone is guaranteed to get at least 4 items.

Paradise...?

In deference to my newest interviewee over at MathTango today, Sunday reflections from David Hilbert:

"The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite."

"No one shall expel us from the Paradise that Cantor has created for us."

There's Something About Sitan...

A VERY interesting/enjoyable (transcribed) interview with 20-year-old Harvard student Sitan Chen, a young STEM whiz who combines prodigious talents in math, music, and computer science:
http://tinyurl.com/lhzg7wv

He lists his current interests as  "pseudorandomness, algebraic topology, and algebraic geometry." And when he's not working on complex math and computer science problems, he's playing classical piano at Carnegie Hall.

He has his own blog here (but not for the average math doodler):
https://formalreasons.wordpress.com/

And a previous short Khan Academy interview with Chen here:
http://life.khanacademy.org/post/55172352451/a-khanversation-with-super-humble-super

It's good to know that someone like Sitan is coming along, because, approaching his 40th birthday, Terry Tao is of course almost over-the-math-hill. ;-)

Below, Chen is in action a couple years back in a 19-minute video from the first article:

Legends of Math and Movies

Interesting little article about the quirkiness of mathematicians and certain mathematical problems:

https://www.bbvaopenmind.com/en/what-purpose-do-the-great-mathematical-problems-serve/

It starts off by telling the legend of George Dantzig, who arrived late to a UC Berkeley math class one day in 1939, and copied a couple of problems off the blackboard, thinking they were homework assignments. He went home, struggled to figure out their solutions which he turned in, only to discover 6 weeks later that they were famous unsolved problems in statistics... now resolved by a student assuming them "homework."

In the below Numberphile clip, James Grime talks about the Dantzig legend in relation to the popular movie Good Will Hunting; and also talks about American genius William Sidis and Indian prodigy Ramanujan, all of whom were thought to be possible models for the Hunting main character.

Listening To Feynman...

Tomorrow is Richard Feynman's birthday... in his honor, for this week's 'Sunday Reflection,' a blurb from Freeman Dyson's "The Scientist as Rebel":

"Before I met Feynman, I had published a number of mathematical papers, full of clever tricks but totally lacking in importance. When I met Feynman, I knew at once that I had entered another world. He was not interested in publishing pretty papers. He was struggling, more intensely than I had ever seen anyone struggle, to understand the workings of nature by rebuilding physics from the bottom up. I was lucky to meet him near the end of his eight-year struggle. The new physics that he had imagined as a student of John Wheeler seven years earlier was finally coalescing into a coherent vision of nature, the vision that he called 'the space-time approach.' The vision was in 1947 still unfinished, full of loose ends and inconsistencies, but I saw at once that it had to be right. I seized every opportunity to listen to Feynman talk, to learn to swim in the deluge of his ideas. He loved to talk, and he welcomed me as a listener. So we became friends for life.
"For a year I watched as Feynman perfected his way of describing nature with pictures and diagrams, until he had tied down the loose ends and removed the inconsistencies. Then he began to calculate the numbers, using his diagrams as a guide. With astonishing speed he was able to calculate physical quantities that could be compared directly with experiment. The experiments agreed with his numbers."

...and from Feynman himself:

"We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degrees of certainty — some most unsure, some nearly sure, but none absolutely certain."....

"It is in the admission of ignorance and the admission of uncertainty that there is a hope for the continuous motion of human beings in some direction that doesn't get confined, permanently blocked, as it has so many times before in various periods in the history of man."....

"I can live with doubt, and uncertainty, and not knowing. I think it's much more interesting to live not knowing than to have answers which might be wrong. I have approximate answers, and possible beliefs, and different degrees of certainty about different things, but I'm not absolutely sure of anything. There are many things I don't know anything about, such as whether it means anything to ask 'Why are we here?' I might think about it a little bit, and if I can't figure it out then I go on to something else. But I don't have to know an answer. I don't feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose — which is the way it really is, as far as I can tell. Possibly. It doesn't frighten me."

An "Exemplary" Solution

Another quick problem today adapted from Alfred Posamentier's simple volume, "Problem-solving Strategies in Mathematics," that I'm enjoying (...the volume is very similar to an older Posamentier/Krulik work you might already have). This is actually a classic puzzle most readers are likely familiar with. The key is not so much getting the answer... but HOW you get the answer:

25 basketball teams compete in a single-elimination tournament (as soon as you lose a game you're out of the tournament, and with 25 teams one team will get a bye in the first round).
How many games total will be played by the end in order to crown the champion?
Answer down below... but first a li'l more about the book.

This is a great volume for middle and high school teachers to have on-hand to draw problems from, and many of the problems could be suitably presented to younger math-inclined pupils as well (I'm talking to you Mike Lawler! ;-))

The approach of the book is to offer a problem, and then give a somewhat brute-force (or what they call "common") strategy for solving it, followed by a simpler, quicker, more elegant (or what the authors call "exemplary") solution. Many of the problems are classics, or offshoots of classics, generally requiring nothing more advanced than logical, arithmetic, algebraic, or geometric solutions.

The authors illustrate a total of 10 "strategies" for solving math problems, starting off with "logical reasoning," "pattern recognition," and "working backwards." Students will often experience facepalms when they see the "exemplary" solutions given. The book is supposed to be the first of a series (edited by Posamentier) on mathematical problem-solving from World Scientific Publishers.
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Answer:   24  ...the answer can be tediously found using arithmetic, but all you need realize for the "exemplary" solution is that to arrive at 1 champion in a single-elimination tournament of 25 teams means there MUST be 24 losers... and at one loser per game, there must be 24 games played by the end; no pencil and paper required.

Get Your Pencil and Paper...

Currently reading a couple of books simultaneously, including Alfred Posamentier's and Stephen Krulik's new "Problem Solving Strategies in Mathematics" (so far very good... but then that hardly needs saying if it's from Posamentier). Anyway, it looks to include several nice puzzles/problems along the way, one of which I'll adapt to my own wording today:

Donald Chump rolls an ordinary (6-sided) die repeatedly, and records the result of each outcome. He is asked to stop as soon as one number has come up 3 separate times. He thus ends up stopping after the 12th roll, at which point the sum of ALL the rolls is 47.
Which number appeared three times?
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Answer:  6
(I've given explanation in the comments section)

(image via David Vignoni/WikimediaCommons)

"...positively spooky..."

From a few scientists (Sunday reflection):

“It is positively spooky how the physicist finds the mathematician has been there before him or her.” — Steven Weinberg

“I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.” — Richard Feynman

“One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.” — Heinrich Hertz

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner

A Couple From Today

Couple of posts that appeared too late to make it into my MathTango weekly wrap-up:

a)  If The Cantor Set doesn't already blow your mind enough, don't despair, 'cuz Evelyn Lamb is here to blow your mind with the "fat Cantor Set":
http://blogs.scientificamerican.com/roots-of-unity/2015/04/30/a-few-of-my-favorite-spaces-fat-cantor-sets/

b)  and in some more deep stuff, Michael Harris has an ongoing discussion of homotopy theory:
https://mathematicswithoutapologies.wordpress.com/2015/05/01/pantheism-and-homotopy-theory-part-3/