Intro To Set Theory

An introduction to set theory from plus.maths.org here:

http://plus.maths.org/content/searching-missing-truth&src=fpii

In Memoriam...

Image via Wikipedia

No math today....


This day marks the 25th anniversary of the Space Shuttle Challenger disaster. The shuttle broke apart 73 seconds after lift-off, killing all 7 crewmen on board, including America's first civilian/teacher astronaut...
In their memory, from John Denver:





If you're too young to remember the tragedy, learn about it here:

http://tinyurl.com/97zst

A Seemingly Impossible Task, That Isn't

 Raymond Smullyan must be a popular fellow. A months-old prior post I did on him continues to be one of the most visited entries on this blog daily. So I won't argue with success. Here's another post related to him, recounting one of the multitude of paradoxical logic puzzles he has played with....

I've re-written this, from Martin Gardner's version in his "The Colossal Book of Mathematics":

Imagine you have access to an infinite supply of ping pong balls, each of which bears a positive integer label on it, which is its 'rank.' And for every integer there are an INFINITE number of such balls available; i.e. an infinite no. of "#1" balls, an infinite no. of "#523" balls, an infinite no. of "#1,356,729" balls, etc. etc. You also have a box that contains some FINITE no. of these very same-type balls. You have as a goal to empty out that box, given the following procedure:

You get to remove one ball at a time, but once you remove it, you must replace it with any finite no. of your choice of balls of 'lesser' rank. Thus you can take out a ball labelled (or ranked) #768, and you could replace it with 27 million balls labelled, say #563, just as one of a multitude of examples. The sole exceptions are the #1 balls, because obviously there are no 'ranks' below one, so there are NO replacements for a #1 ball.

Is it possible to empty out the box in a finite no. of steps??? Or posing the question in reverse, as Gardner asks: "Can you not prolong the emptying of the box forever?" And then his answer: "Incredible as it seems at first, there is NO WAY to avoid completing the task." [bold added]
Although completion of the task is "unbounded" (there is no way to predict the number of steps needed to complete it, and indeed it could be a VERY large number), the box MUST empty out within a finite number of steps!

There are various proofs of this amazing result (which Raymond Smullyan originally published in the "Annals of the New York Academy of Sciences" in 1979, Vol. 321), but it only requires logical induction to see the general reasoning involved:

Once there are only #1 balls left in the box you simply discard them one by one (no replacement allowed) until the box is empty --- that's a given. In the simplest case we can start with only #2 and #1 balls in the box. Every time you remove a #2 ball, you can ONLY replace it with a #1, thus at some point (it could take a long time, but it must come) ONLY #1 balls will remain, and then essentially the task is over. S'pose we start with just #1, #2, and #3 balls in the box... Every time a #3 ball is tossed, it can only be replaced with  #1 or #2 balls. Eventually, inevitably, we will be back to the #1 and #2 only scenario (all #3 balls removed), and we already know that situation must then terminate. The same logic applies no matter how high up you go (you will always at some point run out of the very 'highest-ranked' balls and then be working on the next rank until they run out, and then the next...); eventually you will of necessity work your way back to the state of just #1 and #2 balls, which then convert to just #1 balls and game over (even if you remove ALL the #1 and #2 balls first, you will eventually work back and be using them as replacements). Of course no human being could live long enough to actually carry out such a procedure, but the process must nonetheless amazingly conclude after some mathematically finite no. of steps. Incredible! (too bad Cantor isn't around to appreciate this intuition-defying problem).

If you wish to read about the problem in Gardner's volume (which is available for free on the Web, BTW) it is near the beginning of his Chapter 34. But again, we have logician Raymond Smullyan to thank for this wonderful thought paradox. I'm just using Gardner as the great explicator that he is.

Partitions

Nice, brief article on "partitions" and the recent innovative work of Ken Ono, especially as it pertains to number theory, here:

http://tinyurl.com/4lhr4ng

more here:

http://tinyurl.com/4gcwhdz

the Wikipedia entry for partitions here:

http://en.wikipedia.org/wiki/Partition_%28number_theory%29

and, finally, Wolfram MathWorld here:

http://mathworld.wolfram.com/Partition.html

Geomagic Squares

Lee Sallows has written his own brief, self-deprecating bio on the Web as follows:

"Born in 1944 and raised in post-war London, Lee Sallows has lived in Nijmegen in The Netherlands for the past 40 years. Until recently he worked as an electronics engineer for the Radboud University. A handful of published articles on computational wordplay and recreational mathematics are the only fruits of an idle, if occasionally inventive, life."

I'm familiar with Lee from his fascinating, creative work with "self-enumerating" sentences and pangrams covered in some of Douglas Hofstadter's books (but also available online). Here's one sentence he invented that became famous in certain circles:

"Only the fool would take trouble to verify that his sentence was composed of ten a's, three b's, four c's, four d's, forty-six e's, sixteen f's, four g's, thirteen h's, fifteen i's, two k's, nine l's, four m's, twenty-five n's, twenty-four o's, five p's, sixteen r's, forty-one s's, thirty-seven t's, ten u's, eight v's, eight w's, four x's, eleven y's, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !"

Recently, Alex Bellos posted about Lee's latest creative venture into "geomagic squares":

http://alexbellos.com/?p=1495

For some reason that post didn't initially grab me much, 'til I noticed the concept getting more and more play around the Web. Peter Cameron's post drove home for me that once again Sallows is on to something deeper than first meets the eye:

http://cameroncounts.wordpress.com/2011/01/21/geomagic-squares/

Sallows' own intro to the subject is here:

http://www.geomagicsquares.com/index.php

Geomagic squares have to do, in a sense, with patterns... of patterns; and 'magic number squares' (with which most of you are familiar) are simply a special case of Sallows' new broader category. I think Martin Gardner would've gotten quite a kick out of this new li'l creation, and hopefully many of you will as well!

New Math Book For Kids

I haven't seen it, but if Keith Devlin calls it "...one of the most amazing math books for kids I have ever seen," (and he does!) that's good enough for me. A new math book for children, entitled "You Can Count On Monsters" by Richard Schwartz reported on here (including a link to Devlin's rousing NPR audio piece on it):

http://tinyurl.com/4czfws3

Amazon lists it as a "hybrid math/art" book for grades 4-8, also using the words "delightful," "innovative," "ambitious," and "imaginative," in describing it. Got young kids? Sounds like a book to get.

ADDENDUM: another glowing review of the book, from a homeschooler, now appears here:

http://tinyurl.com/6afqdfr

Peer Review... Reviewed

This is a tad askew from the usual subject matter of this blog, but yet related... "Nature" has a recent online article about peer review and the trial-by-fire that scientific papers sometimes now go through via blogger and social media commentary that can be rapid, highly critical, and not necessarily polite. The article mentions a couple of the more famous recent examples of this, and there will undoubtedly be far more cases down the road.
The traditional print-form peer review and discussion (that some favor) can be painfully slow and wearisome. As someone who thinks the new sort of 'open science' internet debate is therefore mostly to the good, and certainly the wave of the future, I find this 'fast feedback' over the Web a positive and irreversible development, even though it can strain nerves/feelings.

What I found especially interesting in the "Nature" piece was the following admission that math and physics have already long dealt with (if not even relished) rapid (and early) feedback, and it is primarily the biological sciences where most resistance lies:

"In some fields, notably mathematics and physics, this sort of public discourse on a paper has long been the norm, both before and after publication. Most researchers in those fields have been depositing their draft papers in the preprint server arXiv.org for two decades. And when blogging became popular around the turn of the millennium, they were quick to start debating their research in that form.

"Scientists in other fields seem less willing to get involved in pre-publication discussion. Biologists, in particular, are notoriously reluctant to publicly discuss their own work or comment on the work of others for fear of being scooped by competitors or of offending future reviewers of their own work. Adding to the disincentive is the knowledge that tenure committees and funding agencies do not explicitly reward online activity."

In general, biologists ('notorious' indeed! ;-)) often appear both more critical of one another, and more sensitive to criticism, than physicists and math-types tend to be, and thus are naturally more fearful of the kind of wrangling free-for-all the open Web offers. Anyway, food for thought....

The Pareto Principle

In answer to the Edge question for 2011 ("What Scientific Concept Would Improve Everybody's Cognitive Toolkit?) Clay Shirky responds with the non-Gaussian "Pareto Principle" (the general notion that ~80% of results or effects in some circumstances originate from ~20% of causes):

http://www.edge.org/q2011/q11_6.html#shirky

...and the Wikipedia entry for "Pareto Principle" here:

http://en.wikipedia.org/wiki/Pareto_principle

The Endlessly Fascinating Zeta Function

Matt Springer elucidates Riemann's zeta function and why prime numbers are built into it, in this post:

http://scienceblogs.com/builtonfacts/2010/11/sunday_function_79.php

Gödel

Nice video intro to Gödel from the BBC here:

http://beauty.yamnul.com/bbc-dangerous-knowledge-part-7-10

(This entire BBC series from 2007, "Dangerous Knowledge," looks to be quite good.)

Fun With Numbers

A couple of recent entries over at "Futility Closet" that I found entertaining...

First, a sort of prime-number magic square of sorts:

http://www.futilitycloset.com/2011/01/14/through-and-through/

Who figures this stuff out... and are they still sane by the time they finish!??? ;-))
...actually, the person who concocted this square has a homepage here:

http://users.cybercity.dk/~dsl522332/

And then there's this deceptively-simple li'l story problem where all is not quite as it seems:

http://www.futilitycloset.com/2011/01/13/petty-cash/

Couple a Books...

Always on the lookout for good popular math books, and recently noticed "Math For Love" blog touting one I've not seen, nor even heard of. Still, feel I can't help but pass along their recommendation for what they term "the all-time best math book ever" (entitled, "Mathematics, A Human Endeavor," by Harold Jacobs):

http://mathforlove.com/2010/12/the-all-time-best-math-book-ever/

Speaking of books, I'm about 2/3 of the way through "The Best Writing on Mathematics 2010" and won't do a full review, but will give the following blurb and a general thumbs-up for it!

Anthologies covering broad areas, like this volume, can be difficult to pull off. I most like anthologies of a single individual's writing (think of any essay compendium from Stephen Jay Gould) or anthologies on very focused topics (within math, an anthology on infinity or prime numbers or paradox, would interest me). Too often in wider-reaching volumes, and true for me here, several pieces are very interesting, several not-so-much, and of course the rest lying somewhere in-between --- but of course that experience differs for each reader depending on their own subjective interests or background (so I won't even mention which essays most appealed to me here).

The book contains around 35 varied pieces separated into 6 chapters or categories. Most of the authors here were new to me and their writing styles can differ quite a bit from one to another. I'd call the writing more cerebral than scintillating. Topic-areas addressed include computational theory, information theory, foundational mathematics, proofs, math teaching, math and the internet, intuition, aesthetics, and many more diverse areas. The variety of material is both the volume's strongpoint (something here for everyone) and possibly its weakpoint (everyone will likely find something of little interest).  I recommend the book though, in part, for that very breadth of what it attempts to sample and introduce the reader to. And I simply like the idea of a "Best of ..." volume for mathematics each year. This is a fine first go at it... one can easily imagine it getting better and better each year, so hoping sales are good enough to promote a series.

Lastly though, I'll caution that the book is not particularly for those with only a casual interest in math. You don't need to be a professional nor have a strong academic math background (though there are some technical passages), but you probably do need a fairly strong interest in mathematics to appreciate this volume.

Statistics Scrutinized

Longish, worthwhile piece on the hazards of statistics, pointing to some of the inherent problems with much research:

http://tinyurl.com/yz22ldq
(lot of differing opinions expressed in comments section)


(H/T to John McGowan for pointing me to it)

Math Specialization et. al...

Cliff Pickover recently tweeted a link to this interesting older post (2009) by Doron Zeilberger; timely, with the current joint math meetings just ended in New Orleans (it deals with the degree to which math is now composed of highly-specialized areas, and what this means for large conferences):

http://www.math.rutgers.edu/~zeilberg/Opinion104.html

(Zeilberger urges more "generalists" in math "who can see the big picture;" of course, extreme specialization is a problem faced today by most of the sciences)

And a few months after the above posting, Zeilberger wrote a followup with suggestions for improving professional math meetings:

http://www.math.rutgers.edu/~zeilberg/Opinion106.html

(...I just wish he'd tell us how he REALLY feels. ;-))


Meanwhile, "polygeek" has been playing around with prime numbers and asks for comments to his recent  findings/patterns here:

http://polygeek.com/3327_flex_prime-patterns-2


Lastly, on a side-note, since most math-folks are also interested in physics, I'll pass along this person's top 10 list of popular physics books from 2010:

http://physicsworld.com/cws/article/news/44623

Wherefore Art Thou America?

Sorry, but not in a very frolicky mood this morning....



...with thoughts and prayers for the most recent victims of American political/gun violence.

Of 'Stomach Ulcers' and 'Prions'

RJ Lipton discusses mathematical proof and efforts by 'amateurs,' especially with pertinence to the P vs. NP problem, and proof by contradiction, here:

http://tinyurl.com/22mnsnr

Math Miscellany

The annual joint mathematics meeting of the Mathematical Association of America, American Mathematical Society, and four other groups has begun in New Orleans. Even if you're not there you may find something of interest at the website for the meeting:

http://www.ams.org/meetings/national/jmm/2125_intro.html

And at the opposite end of the spectrum, I recently came across this page where examples of mathematical 'crackpottery' are listed (looks like there might be some "fun" reading here, though I haven't had time to really look it over enough yet):

http://www.crank.net/maths.html

Finally (and swinging the pendulum back again), for many years now, Harper-Collins has put out a series of end-of-year anthologies on a variety of subjects: "Best Science Writing of 2010," "Best Spiritual Writing of ....," "Best Essay Writing of ....," et. al. Though I like many anthologies, I often find these particular volumes a bit disjointed and only partially engaging --- worthy of a single reading and then off to a used bookstore or thrift shop (as opposed to going up on my shelf for re-relishing multiple times).
Still, I'm happy to see that Princeton University Press has filled a void left by Harper-Collins, by now putting out "The Best Writing On Mathematics 2010." I received a review copy, and scanning the chapters and authors, look forward to reading it; just don't know how long that will take, or if I'll review it here. At first glance it looks to have something for everyone (who's interested in math).

Liking... "Loving and Hating Mathematics"

 "Loving and Hating Mathematics" by Reuben Hersh & Vera John-Steiner -- a review


I received a review copy of "Loving and Hating Mathematics" from Princeton University Press, a new volume from Reuben Hersh and Vera John-Steiner. Any new volume from Hersh is anxiously anticipated, and for the sake of simplicity I'll simply refer to his name throughout this review (it is also my suspicion, though I don't know for certain, that the parts of the book I liked most were primarily penned by him, and the parts I liked less came from the secondary author, who is listed as a linguist and educator, not a mathematician).
I have liked Hersh's writing in the past, although never quite finding it as engaging as I'd expect for the material he covers. The same is true in this volume which, for me, has a sort of Bell Curve of appeal. It starts off so-so, but quickly builds through some interesting, well-done middle chapters, only to then trail off into some shallowness in the last few chapters. For the right audience I do still recommend the volume, as a likeable, if not necessarily lovable volume.

The book is especially for those interested in mathematical culture, persons, and modern math history. It is about the milieu and nature, the sociology and personalities, of mathematics, more than it is about mathematics itself. Mathematical professionals, or those otherwise focused on doing math, may find less of interest here. The lay person with a side interest in mathematics will find over 300 pages about math in a loose sense, but with no significant amount of applied mathematics to get in the way of the narration.
The writing style is a tad dry, although much of the subject matter is so inherently interesting as to carry the material along without flashy writing. For those well-versed in mathematical history (or Hersh's past writings), the book may not add greatly to your knowledge, but it is nice to have so much of this material brought together in an orderly manner in a single volume.

The subtitle of the book, "challenging the myths of the mathematical life," implies that one purpose of the offering is to counter many of the images/stereotypes that people carry around of mathematicians and math study. While the book certainly contains a lot of diverse examples I'm not so sure but that a surprising number of them don't do more to reinforce, rather than counter, the generalizations people often make of the introspective, eccentric, oddball, isolated, anti-social or loner math-type. There are a lot of quirky stories told herein. One entire chapter on "Mathematics As An Addiction" almost makes the problematic lives led by so many math prodigies and logicians seem to be the norm. But other chapters do communicate the aesthetics, feeling, creativity, and sheer joy, that are (to the surprise of some I s'pose) an underlying aspect of doing math. Still, the subject of recreational mathematics, one of the most 'normal,' social, fun areas in all of math is oddly entirely absent.

You get some sense of the overall subject matter of the book from the middle chapter titles which I most enjoyed, as Hersh builds from more singular aspects of mathematics to its more social context:

Ch.2  -- Mathematical Culture
Ch. 3 -- Mathematics as Solace
Ch. 4 -- Mathematics as an Addiction
Ch. 5 -- Friendships and Partnerships
Ch. 6 -- Mathematical Communities

(The latter two chapters above, dealing with mathematicians in association with others, were for me, the best chapters of the book, but all these chapters were quite good.)

Chapter 7 on "Gender and Age in Mathematics" seemed less engaging, but might perhaps mean more to those most directly affected by these components (of math study when being female or older). I felt the last two chapters ("The Teaching of Mathematics" and "Loving and Hating School Mathematics") were a bit more shallow in their material, causing the volume to end somewhat weakly (though on some very important topics). Had the book's final few chapters been as strong as the middle chapters it would have been a great read; as it is, it is still a very good read. The brief biographic section of mathematicians at the end is nice as well, except that it is hard to decipher just why certain of the individuals were included, while a number of modern popularizers are omitted.

I have just two minor beefs with the volume:

1) I'm not sure the title is all that pertinent to the contents. The title is probably intended to be 'catchy,' but almost seems a bit glib. And just by including the phrase "hating mathematics" it could even turn off a segment of its prospective audience, who may indeed harbor long-ago school memories of hating mathematics. In short, I'm not so sure the title won't chase away more readers than it attracts.

2) A second minor concern is that Hersh puts out almost 400 pages on "math culture," including extensive bibliographic listings and an end-compendium of famous mathematicians, and yet never mentions Martin Gardner at all. Gardner and Hersh feuded across the years over the underlying nature of mathematics (with Gardner harshly reviewing Hersh on occasion), so it may be no surprise that Hersh affords him no recognition here. Moreover, Gardner was never a 'professional' mathematician, and, as mentioned above, Hersh essentially doesn't touch on recreational mathematics in this volume at all.
There are many other modern "popularizers" of math who also are not included here besides Gardner, however none have had the impact of Gardner on the current math community. Gardner's fame, contributions, and influence on many mathematicians, is undeniable, and his absence here almost seems a petty oversight, even detracting from the volume's credibility --- to the degree that this book addresses educating, motivating, and encouraging future mathematicians, Gardner and recreational math ought be mentioned.

With those two criticisms out of the way (and they are minor overall), I do recommend the work to the interested mathematics observer... especially if you already love mathematics; this volume will let you soak up more math-thought without having to work at it. If you 'hate' math, well, I don't think this volume will transform you, though even then, some parts may intrigue.

The book, BTW, has a Facebook page devoted to it here:

http://tinyurl.com/2b348e5

Finally, on a side-note, in the course of reading Hersh's book, I discovered that William Byers, author of one of my all-time favorite math books, "How Mathematicians Think," has a new book coming out around May 2011, entitled "The Blind Spot: Science and the Crisis of Uncertainty" --- something definitely to look forward to, especially if you are sympathetic to Hersh's view of mathematics.

Happy Prime Year!!

As a prime number that is the sum of 11 consecutive prime numbers, 2011 looks to be an amazing year (...one can hope):

http://tinyurl.com/39hfbm2