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Sunday, May 28, 2017

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.]

Wednesday, May 24, 2017

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

Tuesday, May 23, 2017

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!:

Sunday, May 21, 2017

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.”

Wednesday, May 17, 2017

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:

Tuesday, May 16, 2017

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:

Sunday, May 14, 2017

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.”

Friday, May 12, 2017

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.

Wednesday, May 10, 2017

"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

Sunday, May 7, 2017

Wisdom Versus Quantification

This week’s Sunday reflection from Philip Tetlock’s and Dan Gardner’s “Superforecasting”:
“Numbers are fine and useful things, I would say in that alternate universe, but we must be careful not to be smitten with them. ‘Not everything that counts can be counted,’ goes a famous saying, ‘and not everything that can be counted counts.’ In this era of computers and algorithms, some social scientists have forgotten that. As the cultural critic Leon Wieseltier put it in the New York Times, ‘There are ‘metrics’ for phenomena that cannot be metrically measured. Numerical values are assigned to things that cannot be captured by numbers.’ This naive positivism is running rampant, taking over domains it has no business being in. As Wieseltier poetically put it, ‘Where wisdom once was, quantification will now be.’”

Wednesday, May 3, 2017

America’s Future….

"...this is for the ones who stand their ground
When the lines in the sand get deeper
When the whole world seems to be upside down
And the shots being taken get cheaper, cheaper..."

-- Mary Chapin Carpenter

“I’m not ready to make nice
I'm not ready to back down
I'm still mad as hell, and I don't have time
To go 'round and 'round and 'round.”
— The Dixie Chicks

Sans math today…. Am in a mood :( …increasingly pessimistic at the simplistic lure fascism holds for this country/world... with disastrous consequences for our children.
…sooooo, just taking a moment to thank a few of those maintaining the good fight and vigilance on Twitter:

(one could expect writers/journalists/politicians to stand up to Trumpism, but I highlighted three of the mathematicians also consistently doing such, because they don't have to do so... EXCEPT for feeling compelled as American citizens -- and interestingly, two of them, Keith and Ed, are naturalized, not native-born, citizens).
(…plenty of others, of course, carry on the resistance in venues other than Twitter).

Have posted the below musical bits previously, so apologies for the redundancy, but suspect I may be listening to these a lot over the next 4 years.

....and thank you, Mary, Martie, Emily, Natalie!

Monday, May 1, 2017

Two From The Weekend

Passing along a couple of things from the weekend…

1)  It’s been around for awhile, but I only learned of the UK’s Chalkface Blog this weekend (through Twitter); worth checking out:

2)  Occasionally over the years, someone asks me to recommend a video series for learning calculus. Since I’m out of the teaching loop I feel hesitant or unqualified to make recommendations of some sites over others… BUT now, without hesitation, I feel free to recommend Grant Sanderson’s (3Blue1Brown) new beautifully-done series on calculus that begins here (and is still in production):

On Twitter, Mike Lawler writes, "The new calculus series from Grant Sanderson is so good that I basically have no words to describe it. Never seen anything comparable."

And if you wish to support Grant's brilliant work, he has a Patreon account here:

As I blurbed elsewhere this week, I can’t help but wonder if this sort of graphic presentation doesn’t represent what the eventual future of primary/secondary math education may look like in this country.

Sunday, April 30, 2017

Some Classic Thoughts

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that 'laws of nature' exist, much less that man is able to discover them. 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 P. Wigner

"I believe that scientific knowledge has fractal properties, that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think, is the secret of the Universe."   -- Isaac Asimov

"I think it's important to regard science not as an enterprise for the purpose of making predictions but as an enterprise for the purpose of discovering what the world is really like, what is really there, how it behaves and why. Which is tested by observation. But it's absolutely amazing that the tiny little parochial and weak and error-prone access that we have to observations is capable of testing theories and knowledge of the whole of reality, which has tremendous reach far beyond our experience. And yet we know about it. That's the amazing thing about science. That's the aspect of science that I want to pursue."   
-- David Deutsch

Thursday, April 27, 2017

No Largest Prime Gap

I've reported on this in the distant past, but since Mike Lawler recently asked bloggers to post some entries that might be of interest to both mathematicians and students, I’ll re-run this simple, old demonstration that you can have ANY size gap between two prime numbers that you want. I’ve always liked it, for its simplicity, since first seeing it in a popular 1984 volume from Laurie Buxton called “Mathematics For Everyone.” It runs like this (using Buxton’s example):

Hopefully you know what 600! means, i.e. the product of 600 x 599 x 598 x ….. x 2 x 1.
A pretty large number, but we need not actually multiply it out. Now consider the following string of consecutive numbers:

600! + 2
600! + 3
600! + 4
600! + 5
600! + 600

The above produces a list of 599 consecutive integers, NONE of which can be prime. Every number here will be divisible by at least the number on the right (because 600! is divisible, without remainder, by every number UP TO 600, and adding anything between 1 and 600 simply includes one of those divisors). Thus, in this example we have a gap of at least 599 integers without a prime appearing. BUT clearly one need not start with 600. One can start with a number as large as one likes in order to generate a prime gap as large as one wants. There will never be a largest gap. Simple and convincing!

Multiplying It Out

I’m currently in rerun mode, just replaying some posts from the past. Here’s a previous puzzle from an old “Scam School” episode. You may get it right away (it’s simple), or if you don’t, you’ll facepalm yourself when you see the answer (below).
Here goes:

You are to multiply together a long sequence as follows:

(a-x) X (b-x) X (c-x) X (d-x)...... (y-x) X (z-x)  i.e., utilizing ALL the letters of the alphabet once.

What will be the end product of this sequence when multiplied out???

.Answer below

Answer = 0 ...just before the final two sequence entries listed, would be (x-x)

Tuesday, April 25, 2017

"Gauss can recite all of pi -- backwards"

Blogging may continue to be a bit slow while I'm catching up on a number of things (and waiting very patiently for impeachment hearings ;), so may just re-run some old posts in the meantime. Anyway, will start by referencing this favorite old "Gauss Facts" site that's always good for a chuckle:


Sunday, April 23, 2017

Truth, Certainty, Explanation... and Mathematics

Physicist David Deutsch reflecting on mathematics (from his "The Fabric of Reality"):
[There is] "...an ancient and widespread confusion between the methods of mathematics and its subject-matter. Let me explain. Unlike the relationships between physical entities, relationships between abstract entities are independent of any contingent facts and any laws of physics. They are determined absolutely and objectively by the autonomous properties of the abstract entities themselves. Mathematics, the study of these relationships and properties, is therefore the study of absolutely necessary truths. In other words, the truths that mathematics studies are absolutely certain. But that does not mean that our knowledge of those necessary truths is itself certain, nor does it mean that the methods of mathematics confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and paradoxes. And that does not mean that the  conclusions of such study are necessarily false or paradoxical.
"Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing  mathematics. The objective of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be mathematical explanation."

Sunday, April 16, 2017

"Whence Certainty?"

Sunday reflection... from Rebecca Goldstein in "Incompleteness: the proof and paradox of Kurt Gödel":
"So the question is: Whence certainty? What is our source for mathematical certainty? The bedrock of empirical knowledge consists of sense perceptions: what I am directly given to know -- or at least to think -- of the external world through my senses of sight and hearing and touch and smell. Sense perception allows us to make contact with what's out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock? Is our faculty for intuition the means for making contact with what's out there in mathematical reality? Or is there just no 'there'?"