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Sunday, June 25, 2017


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."

Friday, June 23, 2017

Sunday, June 18, 2017

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.”

Wednesday, June 14, 2017

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:

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

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

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):

Sunday, June 11, 2017

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.'"

Monday, June 5, 2017

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: 

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:

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."]

Sunday, June 4, 2017

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.”

Thursday, June 1, 2017

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.

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'?"

Tuesday, April 11, 2017

For Your Funnybone…

 At a recent large used book sale I headed to the math section and picked up a few older volumes, including John Allen Paulos’ “Once Upon a Number” from 1998. Of course mathematics is timeless, but I was pleasantly surprised to see how much of the less-mathematical content of this volume is still relevant today (heck, maybe even more so since Nov. 8, 2016); much of it concerning logic, reasoning, meaning, information, clear/critical thinking and the like.
Anyway, I’ll put all that aside to only pass along this non-math joke Paulos tosses in at one point (the book is sprinkled with his typical humor):

A young man is on vacation and calls home to speak to his brother.

 “How’s Oscar the cat?”

 “The cat’s dead, died this morning.”

 “That’s terrible. You know how attached I was to him. Couldn’t you have broken the news more gently?”


“You could’ve said that he’s on the roof. Then the next time I called you could have said that you haven’t been able to get him down, and gradually like this you could’ve broken the news."

‘Okay, I see. Sorry.”

 “Anyway, how’s Mom?”

 “She’s on the roof.”

Sunday, April 9, 2017

Science As Uncertainty Reduction

Sunday reflection:

“It feels like there are two opposite things that the public thinks about science: that it’s a magic wand that turns everything it touches to truth, or that it’s all bullshit because what we used to think has changed… The truth is in between. Science is a process of uncertainty reduction. If you don’t show that uncertainty is part of the process, you allow doubt-makers to take genuine uncertainty and use it to undermine things…
“And it’s absolutely crucial that we continue to call out bad science. If this environment forces scientists to be more rigorous, that’s not a bad thing.”

— Christie Aschwanden (of FiveThirtyEight )

Wednesday, April 5, 2017

Riemann In the News...

Lots of interesting mathy stuff out there this week, but hey, you can’t go wrong with the crown jewel of number theory, so I’ll direct you to two pieces on the Riemann Hypothesis, if you’ve not seen them:
First, a brief interview with Barry Mazur and William Stein, authors of “Prime Numbers and the Riemann Hypothesis” (one of my favorite 2016 books):
…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:
The actual (physics) work was published last year but is just now being widely disseminated on popular media:
There are a great many other introductions to RH on the internet, including some video ones such as these:
From Numberphile:

…and from 3 Blue1Brown:

Sunday, April 2, 2017

Beyond the Boundary of Logic

For a beautiful Sunday reflection, the ending words from Eugenia Cheng in "Beyond Infinity":
"The most beautiful things to me are the things just beyond that boundary of logic. It's the things we can get quite a long way towrd explaining, but then in the end they just elude us. I can get quite a long way toward explaining why a certain piece of music makes me cry, but after a certain point there's something my analysis can't explain. The same goes for why looking at the sea makes me so ecstatic. Or why love is so glorious. Or why infinity is so fascinating. There are things we can't even get close to explaining, in the realm far from the logical center of our universe of ideas. But for me all the beauty is right there on that boundary. As we move more and more things into the realm of logic, the sphere of logic grows, and so its surface grows. That interface between the inside and the outside grows, and so we actually have access to more and more beauty. That, for me, is what this is all about.
"In life and in mathematics there is often a trade-off between beauty and practicality, along with a contrast between dreams and reality, between the explicable and the inexplicable. Infinity is a beautiful dream, inside the beautiful dream that is mathematics."

Thursday, March 30, 2017

For Mathematics Lovers

A great read for anyone who wants to learn what math really is, no prerequisites required. And those of us in the field are reminded of what first drew us to it."    Maria Chudnovsky, Princeton University and 2012 MacArthur Fellow
An elegant sampler of many beautiful and interesting mathematical topics. This could become one of the best books available for a popular audience interested in what mathematics really is.”  —Jayadev Athreya, University of Washington

Apologies for my slightly redundant book blurbs lately, but I again want to re-mention a new volume I already praised. The above publicity blurbs are for “The Mathematics Lover’s Companion” by Ed Scheinerman, and it is so far, my favorite book of this young year. I’m a sucker for volumes that offer up a buffet of interesting math topics without lingering too long on any one. You can view the Table of Contents (and some of the content) here:
I especially like Part 3 of the volume on “Uncertainty” (though it is the shortest section), but the other two parts, covering a nice selection of algebraic and geometry topics, are very good as well.
The reason for even bothering to note this book again, is my disappointment at how little “buzz” I see it getting in cyberspace (I do see it regularly in my local Barnes & Noble outlets, so I know it’s being well-distributed). My only guess is that the publisher, Yale University Press, just doesn’t put as much energy into promotion as some of its counterparts. This is a fabulous book, especially for a lay audience, but also for folks farther along their mathematics journey, so I'd hate to see it ignored... especially if you already love mathematics!

Wednesday, March 29, 2017

Fun From Fibonacci (via Keith Devlin)

Keith Devlin’s new volume, “Finding Fibonacci” is more historiography than math, but there is some simple fun math sprinkled along the way deriving from Fibonacci’s writings. One old problem (translated from Fibonacci’s “Liber abbaci”) that Keith quotes runs as follows [I’m re-wording it in updated English]:
A man buys 30 birds composed of partridges, pigeons, and sparrows, for 30 denari. A partridge costs 3 denari, one pigeon costs 2 denari, and 2 sparrows cost 1 denaro, or 1/2 denaro/each. How many birds of each kind has the man purchased?
Of course this looks like a classic multi-equation problem, except there are only 2 equations, yet 3 unknowns:
1)  x + y + z = 30  (number of birds; x partridges, y pigeons, z sparrows)
2)  3x + 2y + z/2 = 30 (total price)
You can solve it, or look at the answer below…
Keith notes, there is a third hidden piece of knowledge buried in the problem: 
namely, that x, y, and z must be positive integers (because birds don’t come in fractions!)
Thus, the two equations above are easily reduced to:
5x + 3y = 30 (where both x and y must be whole positive numbers)

Keith notes the first and third terms are divisible by 5 and so the second term (3y) must also be divisible by 5. In turn, that means y must equal 5, 10, 15 etc… but 10 or more is too large to work in the equation, so only 5 can be the correct answer (and x = 3, and z = 22).
As I indicated previously, the book may be more appealing to math history buffs than for mathematicians themselves (I found the last few chapters, containing more math bits, the most interesting of the book), though, with warm weather approaching, I'm tempted to call it a good beach read for the nerdier among us. In it, Dr. Devlin makes a case that "Liber abbaci," from the under-appreciated Leonardo Bonacci ("Fibonacci") is "a book that changed the course of Western civilization," and seeing him build that case (including making an analogy between Fibonacci and Steve Jobs) is interesting in its own right.

Monday, March 27, 2017

Re-playing Testosterone

To start the week, a blast from the past….

This weekend’s “This American Life” episode was a replay of a crowd-favorite dating back 15 years to August 2002, on the subject of testosterone. The whole episode is wonderfully entertaining, but I always especially enjoyed “Act 2” with a transgender man, born as a female, but reporting on the results of undergoing years of testosterone treatments:

I’ve written about it here before because of one brief section that is fascinating (in a non-PC sort of way). I’ll simply re-quote from that earlier posting:
….After relating a lot of already interesting stories to host Ira Glass about how the change in gender affected him, the interviewee is asked by Ira if there are any other alterations due to testosterone he thinks worth mentioning. The individual responds that after taking testosterone he "became interested in science; I was never interested in science before." To this Ira can't help but chuckle and respond, "NO WAY!" adding that such a response is "setting us back 100 years." The individual goes on to insist that testosterone resulted in "understanding physics in a way I never did before."  (…the specific exchange occurs around the 22:30 point of the whole episode)
It is concerning, but also funny, to think of our abilities/skills being so subservient to our biochemistry, even if this is just one lone anecdotal case. Anyway, the entire segment is fascinating and worth a listen (assuming you enjoy the style of “This American Life;” however the part I’m noting above is the only bit that actually relates back to math or science; the rest is mostly social/psychological in nature).

In a quite long interview over at the Edge site a few years ago, Simon Baron-Cohen, a major researcher in this area, had this to say following a question from Marcus du Sautoy about the relationship between tendencies toward math, and human biology/testosterone:
“…you mentioned mathematicians, and I think you're right, that there are these areas of human activity, math is one of them, where we do see very disproportionate sex ratios. My understanding is that in mathematics, at university level, it's about 14 males for every one female sitting in the audience in those lectures. That's a very big difference. And there are other sciences, as we know, which used to be like that but which have changed dramatically. Medicine is a very good example. It used to be male dominated and it's now certainly 50-50, or if anything, it's gone beyond and there are now more female applicants and thankfully, successful applicants. If you look at the audience in medical lectures, the sexes are, if not equally represented, maybe even more women than men. But there remains this puzzle why mathematics, physics, computer science, engineering, the so-called STEM subjects, why they still remain very male biased. I’m the first to be open to anything we can do to change the selection processes at university, or change the way we teach science and technology at school level, high school level, to make it more friendly to females, to encourage more women to go into these fields. But there remains a puzzle as to why some sciences are attracting women at very healthy levels, and other sciences, including mathematics, remain much more biased towards males. Whether that's reflecting more than just environmental factors, and something about our biology, is something that I think we need to investigate.”

Controversy continues....

[...much of Baron-Cohen's research, by the way, studies the possible link between high pre-natal testosterone exposure and autism]

Sunday, March 26, 2017

Mathematical Values

This week's Sunday reflection, courtesy of Roger Penrose:
"How, in fact, does one decide which things in mathematics are important and which are not? Ultimately, the criteria have to be aesthetic ones. There are other values in mathematics, such as depth, generality, and utility. But these are not so much ends in themselves. Their significance would seem to rest on the values of the other things to which they relate. The ultimate values seem simply to be aesthetic; that is, artistic values such as one has in music or painting or any other art form."

Thursday, March 23, 2017

About Infinity...

I blurbed a little bit earlier about Eugenia Cheng’s new book “Beyond Infinity.” Very much enjoying it, now that I’m farther in (…but do realize it’s an entire book about infinity — so you need a significant interest in the topic to enjoy it; the typical popular math book might only have a chapter or two on infinity, touching a few highlights; this volume goes deeper).
For now just wanted to mention one small matter that came up:
Quite awhile back on Twitter I asked if there was any sort of “proof” that aleph-null must in fact be the ‘smallest’ infinity; i.e. infinity is full of so many counterintuitive outcomes, and the whole question of whether aleph1 really is the second infinity is so complicated, that I wondered how we could even be sure that the natural numbers, for example, represent the lowest degree of infinity.
The few replies I got implied that the minimalness of aleph-null was axiomatic or established by definition. BUT Dr. Cheng does offer a short form of something like a proof in her volume. Her basic argument is simply to indicate that there is no subset of the natural numbers that can be put into one-to-one correspondence with the natural numbers and have anything leftover (sort of a reversed diagonalization argument). Or as she concludes, “This means that every subset of natural numbers is either finite or has the same cardinality as the natural numbers. There is no infinity in between. So we have found the smallest possible infinity: it’s the size of the natural numbers.”
I don’t know if I’m quite fully convinced (that there is much more than tautology or definition at work here), but I was glad to at least see an argument put forth. Dr. Cheng herself admits “This is not quite a proof, but is the idea of a proof…” It’s at least better than saying that the natural numbers are the lowest infinity by edict ;)
A lot of the difficulty in wrapping one’s brain around infinity lies in our deep-seated entrenchment in one view of what “numbers” are. As Cheng writes at one point, “Infinity isn’t a natural number, an integer, a rational number, or a real number. Infinity is a cardinal number and an ordinal number. Cardinal and ordinal numbers do not have to obey all the rules that earlier types of number obey.” 
I still have several chapters to go, and they look like they will be quite good. As with her earlier work ("How To Bake Pi") Dr. Cheng writes in an off-hand, almost conversational style meant to draw readers in to sometimes difficult or abstract ideas. I don't think she is always successful, but admire her making the effort. And her own passion for her subject-matter is clear.

Sunday, March 19, 2017

The Universe as a Mathematically-designed Machine

"To the divine understanding, all phenomena are coexisting and are comprehended in one mathematical structure. The senses, however, recognize events one by one and regard some as the causes of others. We can understand now, said Descartes, why mathematical prediction of the future is possible; it is because the mathematical relationships are preexisting. The mathematical relationship is the clearest physical explanation of a relationship. In brief, the real world is the totality of mathematically expressible motions of objects in space and time, and the entire universe is a great, harmonious, and mathematically designed machine. Moreover, many philosophers, including Descartes, insisted that these mathematical laws are fixed because God had so designed the universe and God's will is invariable. Whether or not humans could decipher God's will or penetrate God's design, the world functioned according to law, and lawfulness was undeniable, at least until the 1800s."

-- Morris Kline (in "Mathematics and the Search For Knowledge")

Tuesday, March 14, 2017

Just For Fun

Just for fun today, some simple arithmetic….

Math Tricks” blog put up the somewhat classic '30-cows-in-a-field' riddle yesterday, and I’ll post another video of it here today for any not familiar with it:

(what I love about this is that it is simple, appropriate for all ages, and nicely demonstrates language/speech ambiguity)

Sunday, March 12, 2017

Atoms and Primes

 Beginning of Edward Scheinerman's new book, "The Mathematics Lover's Companion":
"The physicist Richard Feynman believed that if humanity were to be faced with the loss of all scientific knowledge but was able to pass on just one sentence about science to this postapocalyptic world, that sentence should describe how matter is composed of atoms. In that spirit, if we could pass on only one bit of mathematics to the next generation, it should be the solution to the problem: How many prime numbers are there?"
[...he goes on to describe some of the proofs for the infinity of primes, before embarking on a wide array of other topics throughout the book.]