Stimulus-Response-Reality

An old bit today from Paul Watzlawick's 1976 volume, "How Real Is Real?":

"There is a joke, known to most psychology students, in which a laboratory rat says of its experimenter, 'I have trained that man so that every time I press this lever, he gives me food.' Obviously the rat sees the S-R (stimulus-response) sequence quite differently than the experimenter does. To the experimenter, the rat's pressing the lever is a conditioned reaction to a preceding stimulus administered by him, while to the rat, the pressing of the lever is its stimulus administered to the experimenter. To the human, the food is a reward; to the rat, a reaction. In other words, the two punctuate the communicational sequence differently. Ordering sequences in one way or another creates what, without undue exaggeration, may be called different realities."

An Anniversary

24 years ago today Andrew Wiles announced “I think I’ll stop here” and the rest, as they say, dear mathematicians, is history:

http://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html?pagewanted=1

Blank Crossword Puzzle, No Clues

From the fertile mind of Lee Sallows this "self-enumerating crossword puzzle" (via Futility Closet):
https://www.futilitycloset.com/2015/11/27/a-self-enumerating-crossword/

Prove or Disprove

John Allen Paulos, from his older volume, “Once Upon a Number”:

“The joke about the mathematics professor who gave a test consisting of four problems is apt. The first three problems required proofs of theorems and the last one was a statement prefaced with the directions ‘prove or disprove.’ One student toiled for a while and then came up to the professor’s desk and asked, ‘On that last problem, do you want me to prove it or disprove it?’ The professor responded, ‘Whichever is the right thing to do.’ ‘Oh,’ replied the student, ‘I can do either one. I was just asking which one you preferred.’ The interchange, of course, would not be a joke if the subject were history or literature.”

Puzzles To While Away Time

In the event you need something to distract you from the news-of-the-day, a few puzzles from the week…

1)  First some puzzle site suggestions via AMS:

http://blogs.ams.org/blogonmathblogs/2017/06/13/math-puzzles-to-pass-the-time/

2)  Jim Propp begins his own series of video math puzzles here (you know it’ll be good):

https://www.youtube.com/watch?v=epbmEXr1o6s

3)  And from Futility Closet this 1976 prime number puzzle:

https://www.futilitycloset.com/2017/06/14/prime-magic/

4)  And lastly, not exactly a puzzle, but a game using a variation on Tic-Tac-Toe to make it more interesting (or maybe not):

https://mindyourdecisions.com/blog/2017/06/13/who-wins-tic-tac-toe-if-the-second-player-wins-only-for-ties-ft-scamschool-game-theory-tuesdays/

The Infinite and Perfect, Studied by the Finite and Flawed

Today's Sunday reflection comes from Bruce Schechter's biography of Paul Erdös, "My Brain Is Open":

"Mathematicians are finite, flawed beings who spend their lives trying to understand the infinite and perfect. That kind of thing is bound to result in problems and misunderstandings. Trends and fashion, politics and pig-headedness all affect the lurching progress of  mathematical knowledge. None of them, however, affect the validity of mathematical knowledge. 'There are many ways,' [Edward] Rothstein writes, 'to show that the ratio between the circumference of any any circle and its diameter is always the same, a number known as pi. The priests, farmers, and builders who first used that ratio may have had various intentions and goals. And the ratio may be given names like pi or zed or Milwaukee, for that matter. But the number and its meaning are unaffected by cultural apparatus and influence.'"

Be On the Watch!

I hardly have time anymore for all the excellent, polished math videos that are showing up these days. Hopefully, anyone reading this blog is already well-aware of these wonderful presenters: 

3Blue1Brown

Numberphile

Mathologer

Infinite Series (from PBS)

A few others I’ll mention are:

Singing Banana from James Grime, well-known from Numberphile, but still also going strong on his own site (similarly, another Numberphile contributor, Matt Parker, has his own site for fun math at StandUpMaths).

Mind Your Decisions, Presh Talwalkar’s less fancy and more recreational site.

PatrickJMT and ProfRobBob, teachers with plenty of basic instructional videos.

Finally, this Pinterest site has links to tons more math-related videos of varying quality/interest:

https://www.pinterest.com/mathematicsprof/

With the rapidly-rising quality of such video presentations it makes one wonder exactly what the future holds for the role of live human teachers in the classroom! Like brick-and-mortar shopping, brick-and-mortar education likely has major changes coming.

We’ve advanced a long way since Khan Academy, which continues with its own evolving site (...and give Salmon Khan credit for early on recognizing/promoting the value of free, widely-distributed learning videos). Amazing to think of the youngsters (and adults) worldwide who weren't previously exposed to good schools, teachers, or textbooks, but now do potentially have 24/7 access to entertaining and instructional resources. Sometimes I think/fear we're in a race between fascism sweeping across the globe or good education (perhaps an antidote) sweeping across the globe!

[...As if math videos weren't time-consuming enough, there are also math audio podcasts, a handful of which I list in the right-hand column to this blog, but many more available. These are usually less instructional, but still covering topics or people of interest.
One for book-lovers, that I only recently discovered, though it's been around for quite awhile, is "New Books In Mathematics."]

The Day the Light Dawned

For Sunday reflection, this from Paul Halmos (and h/t to Jim Propp for bringing this quote to my attention):

“The day when the light dawned… I suddenly understood epsilons and limits, it was all clear, it was all beautiful, it was all exciting… It all clicked and fell into place. I still had everything in the world to learn, but nothing was going to stop me from learning it. I just knew I could. I had become a mathematician.”

A Few Book Notes

Princeton University Press is working hard to keep all of us math fans happy!:

Two of my favorite PUP books from 2015 are newly-out in paperback:

Marc Chamberlain’s “Single Digits,” a fun read covering lots of examples/ideas, and Michael Harris’s thought-provoking, quirky, even unique, “Mathematics Without Apologies.” If somehow you’ve missed these, no better time to catch up then when the paperback arrives.

A book I’m not familiar with is Oystein Linnebo’s “Philosophy of Mathematics,” but among many choices in this genre this looks like it ought be a good introduction.

I’ve already mentioned “Power Up” by Matthew Lane, a lively read on math and video games, a topic not geared to my interests, but which is getting good buzz from the many who do hold such an interest.

I also previously mentioned “The Probability Lifesaver” by Steven Miller, a massive (700+ pgs. volume I’m just dabbling in as time permits), specifically for those with a penchant obviously for probability; loads of problems/examples/explanations. Miller spends an entire introduction basically trying to make the book seem user-friendly and less imposing/intimidating than it appears. Likely a must-have for the stats-crowd.

The last 3 volumes above I would say are more suitable for niche audiences (that will love them), while the first two books (from Chamberlain and Harris) are more appropriate for a wider, lay and professional crowd of math fans.

Finally, and also from PUP, is the new 500+ page “Unsolved” by Craig Bauer, on unsolved cryptographic messages; some famous, others lesser known — little direct mathematics in it, but of course the actual methodologies for solving cryptograms involve very-largely mathematical thinking, and who among us didn't enjoy cryptograms sometime in our youth. 

I’m close to finished with it and have to admit much of it was more spellbinding than I’d expected. For lovers of cryptography certainly another must-have. Since the majority of examples in the book are unsolved messages (from various times/places) you have plenty of work to attempt if you so choose, or just enjoy reading the mysteries. There's also a website that ties into the book with additional material.  I'll say more about the volume in the near future.

As usual, thanks to Princeton U. Press for such a wonderful, ongoing and varied array of mathy offerings.

My impression, thus far, of this year in popular math books, is that there are more 'specialty' books aimed at specific interests, and fewer general interest math offerings showing up than usual, but the year isn't even half-over so we'll see what happens.

Life Lessons From One Who Succeeded

From Edward O. Thorp's “A Man For All Markets”:

“Education has made all the difference for me. Mathematics taught me to reason logically and to understand numbers, tables, charts, and calculations as second nature. Physics, chemistry, astronomy, and biology revealed the wonders of the world, and showed me how to build models and theories to describe and to predict. This paid off for me in both gambling and investing.

 “Education builds software for your brain. When you’re born, think of yourself as a computer with a basic operating system and not much else. Learning is like adding programs, big and small, to this computer, from drawing a face to riding a bicycle to reading to mastering calculus. You will use these programs to make your way in the world. Much of what I’ve learned came from schools and teachers. Even more valuable, I learned at an early age to teach myself. This paid off later on because there weren’t any courses in how to beat blackjack, build a computer for roulette. or launch a market-neutral hedge fund.”

[...and over at MathTango this morning I have a further look at Thorp's recent volume.]

Ben and Jim Deliver (good Wednesday stuff!)

These are both toooooooo good to hold them until the Friday potpourri, so passing them along now:

1) Ben Orlin, a bit more serious than some Wednesdays, on “the three phases of the mathematical life” (competition, mentorship, and collaboration):

https://mathwithbaddrawings.com/2017/05/24/the-three-phases-of-the-mathematical-life/ …

2)  Jim Propp on “Math, magic, and mystery” in this 36-min. video describing math being “liberated from (physical) reality”:

A couple of pieces I think might also make good adjunct readings to Jim’s talk are this Evelyn Lamb piece from a couple years back on epsilons and deltas:

https://blogs.scientificamerican.com/roots-of-unity/the-subterfuge-of-epsilon-and-delta/

…and this old Terry Tao piece on rigor and intuition in math:

https://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/

Cantor Weirdness

Fantastic treatment of the fractal Cantor Set and the “Devil’s Staircase” (Cantor function) from PBS’s “Infinite Series." Is it any wonder Cantor was driven to a sanatorium!:

Math Melancholy

For Sunday reflection, this from Marcus du Sautoy’s “The Great Unknown”:

“The importance of the unattained destination is illustrated by the strange reaction many mathematicians have when a great theorem is finally proved. Just as there is a sense of sadness when you finish a great novel, the closure of a mathematical quest can have its own sense of melancholy. I think we were enjoying the challenge of Fermat’s equations so much that there was a sense of depression mixed with the elation that greeted Andrew Wiles’s solution of this 350-year-old enigma.”

Happy 15th Anniversary for Stephen Wolfram

When Stephen Wolfram’s tome, “A New Kind of Science,” came out 15 years ago, I saw more critical reviews of it than positive ones, but its sheer size (~1200 pgs.) and technicality made it a very difficult volume to review adequately at all.

Now on the 15th anniversary of his opus, polymath Wolfram, who’s accomplishments are multi-fold, is out with a long post reviewing matters. PLENTY to consider and chew on here, including “computational equivalence” and the “computational universe,” machine learning, neural networks, artificial intelligence, language design, and the nature of mathematics and physics.  You’ll need to set aside some significant time to read and digest it all:

http://blog.stephenwolfram.com/2017/05/a-new-kind-of-science-a-15-year-view/

Courtesy of Car Talk

A recent Car Talk re-run on NPR had a nice, simple-to-state mathematical puzzler I’ll pass along if you missed it:

I hand you one thousand $1 bills and 10 separate envelopes.

Your chore is to put some number of those single bills into each envelope such that if someone asks you for any whole amount of money between $1 and a $1000 you are able to hand them a set of envelopes that, added together, constitute that exact sum of money! 

How is the $1000 divided up among the 10 envelopes?

And for the answer, go to their site:

http://www.cartalk.com/puzzler/1000-1-bills-and-10-envelopes

The Saintly Erdös...

For Sunday reflection, commentary on Paul Erdös from Bruce Schechter in “My Brain Is Open”:

“To [Joel] Spencer and many other mathematicians, Erdös was a modern version of a medieval mendicant monk. Erdös is frequently called without a trace of irony, a saint. Indeed, there was something saintly in Erdös’s generosity, in his honesty and his support of the rights of the individual. But the essence of the saintliness his friends speak of was his total devotion to the mathematical pursuit of pure beauty. Erdös often said that ‘property is a nuisance.’ In fact, to Erdös all aspects of life — jobs, money, property, and intimate personal attachments — that interfered with his devotion to mathematics were a nuisance to be avoided. While few people would choose to emulate him, Erdös’s life was an example cherished by many.”

A Couple of Books I Won’t Be Reading and One I Will

Well, that’s an exaggeration, but 2 books I recently received from Princeton University Press I at least won’t be reading from cover-to-cover for different reasons:

1)  “Power Up” from Matthew Lane (subtitled, “Unlocking the mathematics in video games”), is, obviously, focused on video games; a topic that simply has never held much interest for me (beyond Pong and Space Invaders... seriously, by Pac-Man I was already bored with them) — have never quite understood their attraction! Having said that, I’ve been leafing randomly through this volume, reading miscellaneous paragraphs, and the writing is lively, engaging, and interesting -- I can see how the book will hold the attention of all the folks who are drawn to video games.  As the publicity sheet for the book says the game world is “steeped in mathematics” and I’m sure plenty will find this volume to offer a whole new level of appreciation for the gaming experience. As best I can tell, the approach is not so much to use math to explain or describe video games, as to use video games as a stepping stone to discuss interesting mathematics.

With all that said I do have a big beef with the book... Princeton U. Press's overall presentation is as usual, beautiful with a major exception: the book is entirely in an oddball (“Archer Book”) font that I find aesthetically very annoying and unappealing! (and I'm not very picky about fonts) — I suspect there is some reason, I’m unaware of, related to video games, that this font was used (feel free to explain it in the comments if you know), but I found the font very off-putting.

2)  Many are likely familiar with Adrian Banner's somewhat classic “The Calculus Lifesaver” and now Princeton is out with a similar tome, “The Probability Lifesaver” by Steven Williams of Williams College — certainly more of a textbook or adjunct text in 700+ pages than a “popular” math read. But of course probability is a very hot and fascinating topic these days, and this comprehensive treatment seems to cover plenty of topics — again, I won’t be reading it cover-to-cover, but picking out sections to read as interest directs over time.

To my eyes it looks like an excellent addition to the math shelf, but I’m no expert on the tricky area of probability (and statistics, in general, is controversial these days), so one small concern I have is that of the many publicity blurbs out for for this text, none seem to be from the many prominent recognizable names in statistics; not sure why there is a lack of endorsements from “big” names (it may mean nothing, but I have seen cases where that’s not a good sign). If statistics IS your field and you've seen this volume, feel free to weigh in on it below. Looks marvelous to my naive eyes, but what do I know!

3)  Finally, the book I am looking forward to reading, but don’t know how much mathematical content is included, is “A Man For All Markets” by and about Edward Thorp, a famous (and self-made rich) Wall Street trader AND mathematics professor. This book should be interesting as a bio, and I’m presuming there will also be interesting financial math and probability along the way as well.

If anyone is familiar with any of these 3 books feel free to comment in greater detail than I have done, below, with your own plusses or minuses.

"We Didn't Start the Fire" (as the Emperor fiddles)

Too worn down by current events for mathematics, so just more music today....
(hope to have some math book blurbs up by end of week)

“We didn’t start the fire

It was always burning

Since the world’s been turning

We didn’t start the fire

No we didn’t light it

But we tried to fight it”

— Billy Joel