A Helping of Friday Potpourri

A compendium of mathy links you might want to check out over the weekend, if you've missed….

1) A "fiendish" set of brain teasers here:

http://www.independent.co.uk/arts-entertainment/tv/features/dara-obriains-brain-teasers-try-the-comedians-fiendishly-tough-maths-quiz-9144496.html

2) If you missed this CBS News clip on math teacher Jim O'Connor, well, stop whatever you're doing and watch it (…and have a kleenex ready):
http://www.upinspire.com/inspire/1184/this-math-teacher-kept-a-big-s

3) A creative approach to trigonometry here (from the BetterExplained site):

http://betterexplained.com/articles/intuitive-trigonometry/

4) Interestingly, just weeks after a Kazakh mathematician claimed a proof (still being studied) of Navier-Stokes equations, none-other than Terry Tao has offered a new approach to the problem:

http://tinyurl.com/ljf37kn

5) I won't pretend to understand all this digital currency stuff… but that doesn't stop me from finding it fascinating... enter "Riecoin": http://tinyurl.com/kgjg7q4

6) just some fractal fun, Sierpinski style: http://beautyandthemaths.tumblr.com/post/77265096812

...as always, a lot of math education ideas bouncing around the mathosphere, including:

7) another Top 10 issues in math education, this time from Brie Finegold:
http://blogs.ams.org/blogonmathblogs/2014/02/25/my-top-ten-issues-in-mathematics-education/

8) more on the troubles in math education here, via Slate:  http://tinyurl.com/kfbkedf

9) Meanwhile, Arizona is dumping Common Core; read all about it in these 2 places:

http://www.goodmath.org/blog/?p=2882  (from Mark Chu-Carroll)

http://tinyurl.com/kky7z46  (AZ. newspaper)

10) And Alex Bellos gets schooled on current math education in Britain in this 1/2 hr. podcast:
http://www.bbc.co.uk/programmes/b03szv89 , and I already linked (over at Math-Frolic) to this Conrad Wolfram piece on reforming British math education: http://tinyurl.com/kyp8yto

11) Lastly, not math, but still an interesting post, on who gets hired by Google... where "expertise," even a college degree, is not so important, but here's what is:  http://tinyurl.com/lxny6g9


-- Sunday, or Monday at latest, I should have another interview up here with another cyber math enthusiast, meanwhile enjoy some of the above links… and, your weekend! (and let me know ASAP of any broken links)

Cathy O'Neil (mathbabe)... Math Quant Turned Blogger/Activist

 Math-Frolic Interview #20

 "Exploring and venting about quantitative issues." 
   -- "Mathbabe" blog

I've been reading Cathy O'Neil's "Mathbabe" blog off-and-on pretty much since its inception, but either I've changed or her blog has, because for the last several months almost every entry seems like a gem to me.  Cathy is somewhat outside-the-box of the typical math bloggers I follow...  a blogger with a tad more 'attitude' and range of issues.  She is a Harvard (PhD) graduate (also Berkeley and MIT) and a data scientist, who left the finance industry when disillusioned.
Political candidates often talk of having a "fire in the belly," and that's also the sense I've had of Cathy's blog for awhile now. So I was very happy to learn more about the life of the blogosphere's mathbabe, and think you will as well:

 ************************************

1) To start, could you tell readers a little about your diverse background and how you came to be a sort of math "freelancer" and blogger… including when did your interest in mathematics originally arise, and when did you know you wished to pursue it professionally?

I started liking math when I was 4 or 5. I remember thinking about which numbers could be divided into two equal parts and which couldn't, and I also remember understanding about primes versus composites, and for that matter g.c.d., when I played with spirographs and taking note of different kinds of periodicities and when things overlap. Of course I didn't have words for any of this at that point. 

Later on in elementary school I got really into base 2 arithmetics in 3rd grade, and I was fascinated by the representation of the number 1 by 0.9999... in 7th grade. I was actually planning on becoming a pianist until I went to a math camp after 9th grade (HCSSiM), and ever since then I've known. In fact it was in that summer, when I turned 15, that I decided to become a math professor.

Long story short I spent the next 20 years achieving that goal, and then when I got there I realized it wasn't the right speed for me. I went into finance in the Spring of 2007 and was there throughout the crisis. It opened my eyes to a lot of things that I'd been ignoring about the real world, and when I left finance in 2011 I decided to start a blog to expose some of the stuff I'd seen, and to explain it as well. I joined Occupy when it started and I've been an activist since then.

[Because so many carry the stereotyped image of a mathematician as someone standing at a blackboard writing inscrutable, abstract symbols, I think Cathy's "activism" has been one of the most appealing aspects of her blog!]

2) You're involved in quite a number of important activities/issues… what would you list as your most ardent (math-related) goals, for say the next year, and then also longer-term?

My short- or medium- term goal is to write a book called "Weapons of Math Destruction" which I recently sold to Random House. It's for a general audience but I've been giving a kind of mathematical version of it to various math departments. The idea is that the modeling we're seeing proliferate in all kinds of industries has a dark side and could be quite destructive. We need to stop blindly assuming that because it has a mathematical aspect to it that it should be considered objective or benign.

[...Love the title of the book.]

Longer term I want to promote the concept of open models, where the public has meaningful access to any models that are being used on them that are high impact and high stakes. So credit scoring models or Value-Added Teacher models are good examples of that kind of thing. I think it's a crime that these models are opaque and yet have so much power over people's lives. It's like having secret laws.

3) Related to the above, you've been especially outspoken about various financial/banking issues and the "Occupy Wall Street" movement… I have to believe that there are both very rewarding and very frustrating/exasperating aspects to tackling those issues… care to comment?

I'd definitely say more rewarding than frustrating. Of course things don't change overnight, especially when it comes to the public's perception and understanding of complex issues. But I've seen a lot of change in the past 7 years around finance, and I expect to see more skepticism around the kind of modeling I worry about, especially in light of the NSA surveillance programs that people are up in arms over.

4) Your blog covers a wider diversity of topics than most "math" blogs. Sometimes your blogposts seem to be a combination of educating the public while also simultaneously, venting! (indeed your subheading hints at such)… how might you describe your feelings/attitude/mood when writing typical posts? And what are your favorite (math-related) subjects to write about or study?

Honestly blogging has crept into my daily schedule like a cup of coffee in the morning. It would be really hard for me to stop doing it. One way of thinking about it is that I'm naturally a person who gets kind of worked up about how people just don't think about a subject X the right way, and if I don't blog about those vents then they get stuck in my system and I can't move past them. So maybe a better way of saying it is that getting my daily blog on is kind of like having an awesome poop. But then again maybe that's too gross. Sorry if that's too gross.

[Let's just say that I may never think about composing blog posts in quite the same way again! ;-) ]

5) Is "Mathbabe" blog principally "a labor of love" or is it more than that for you (some sort of means to an end)? i.e., You're writing a book and you do speaking engagements, along with other activities… is the blog a mechanism to help promote/sustain those other endeavors, or do you view it as just a recreational side activity?

I've been really happy with a decision to never let mathbabe be anything except fun for me. There's no money involved at all, ever, and there never will be. Nobody pays me for anything, nobody gets paid for anything. I do it because I learn more quickly that way, and it forces me to organize my half-thoughts in a way that people can understand. And although the thinking and learning and discussions have made a bunch of things possible, I never had those goals until they just came to me. 

At the same time I wouldn't call it a side activity either. It's more of a central activity in my life that has no other purpose than being itself.

6)  Go ahead and tell us about the book you have in the works and its timetable...

It's fun to write! I can't believe people are willing to let me interview them! It won't be out for a couple of years. At first I thought that was way too long but now I'm glad I have the time to do the research.

7) How do you select the topic you post about on any given day? And are there certain blogposts you've done that stand out as personal favorites or ones that were the most fun to work on? From the other side, which posts seem to have been most popular or attention-getting with readers?

I send myself emails with ideas. Then I wake up in the morning and look at my notes and decide which issue is exciting me or infuriating me the most. 

I have different audiences that get excited about different things. The math education community is fun, they have a LOT to say on comments. People seem to like Aunt Pythia but nobody comments -- I think it's a guilty pleasure.

[Yes, I was skeptical of Aunt Pythia when you announced it (seemed a bit of a stretch), but it too is a fun read... though I most enjoy the passionate posts about issues tangential to mathematics.]

I guess it's fair to say that people like it when I combine venting with strong political views and argumentation. My most-viewed post ever was when I complained about Nate Silver's book.

8) What are some of the math-related books you've most enjoyed reading and/or ones you would particularly recommend to lay folks?

I don't read very many math books to be honest. I've always enjoyed talking math with people more than reading about it. 

But I have been reading a lot of mathish books in preparation for my writing. For example, I really enjoyed "How to Lie with Statistics" which I read recently and blogged about. 

Most of the time I kind of hate books written about modeling, to be honest, because usually they are written by people who are big data cheerleaders. I guess the best counterexamples of that would be "The Filter Bubble," by Eli Pariser which is great and is a kind of prequel to my book, and "Super Sad True Love Story" by Gary Shteyngart which is a dystopian sci-fi novel that isn't actually technical but has amazing prescience with respect to the kind of modeling and surveillance -- and for that matter political unrest -- that I think about all the time.

9) Anything else you'd want to say to a captive audience of math-lovers, that you haven't covered above?

Math is awesome!

[INDEED!]

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Thanks so much, Cathy, for filling in a bit about yourself here. Good luck in all your endeavors!

Cathy tweets, BTW, at @mathbabedotorg 

And she did this fascinating interview for PBS's "Frontline" in 2012 (largely on the financial crisis):

http://www.pbs.org/wgbh/pages/frontline/oral-history/financial-crisis/cathy-oneil/ 
(I highly recommend this!)

Still More Beauty.... and Appreciation

The previous MathTango posting touched upon the frequent topic of "beauty" in mathematics, and then lo-and-behold just yesterday Nobel physicist Frank Wilczek posted his own brief take on that very topic:

http://frankwilczek.com/2014/mathematicalBeauty02.pdf

Wilczek links "beauty, prediction and reward" and also "novelty" in a simple and interesting way (though I'm not sure others haven't written similarly before, with different words). Wilczek speculates on why so many people "don't find mathematics beautiful at all, and who in fact fear and hate it," based on his prediction/reward emphasis, and believes his ideas will have "important practical implications for teaching and learning." His piece is very brief but he also mentions having a forthcoming book in-the-works fleshing out the subject more, so something to look forward to!

Slightly relatedly, this weekend I was reading a 1981 essay by R.P. Boas, which included this passage:

"…it was not until I became editor of the [American Mathematical] Monthly that I quite realized how hard it is for mathematicians to write so as to be understood even by other mathematicians (outside of fellow specialists). The number of manuscripts rejected, not for mathematical deficiencies but for general lack of intelligibility, has been shocking. One of my predecessors had much the same experience 35 years earlier.
"To put it another way, why do we speak and write about mathematics in ways that interfere so dramatically with what we ostensibly want to accomplish? I wish I knew."

Knowing and understanding math, has never been a guarantee of ability to effectively communicate mathematical content or beauty to others -- and if it is hard to communicate to one's mathematical peers, how much harder is it to communicate to the wider public. Thus, I think it worthwhile on occasion to salute those who do accomplish that goal, and also acknowledge those who toil away less prominently striving toward such a goal.

So to the Strogatz's and Stewarts, the Devlins and du Sautoys, the Pickovers and Posamentiers, the Hershes and Hofstadters, (and of course to Martin Gardner, who may stand in a class all by himself), and so very many others who do a great job of communicating mathematical material/thinking for a wider audience... and also to those in the blogosphere, and in their personal classrooms, who similarly endeavor to do the same, THANK YOU! Even the titan-likes of Tim Gowers and Terry Tao, when they put their mind to communicating to a broader audience, do so splendidly! It remains astonishing to me, in this culture of such widespread math dread and discomfort, how many wonderful books for a general audience yet appear every year! (thank you also to editors at Princeton University Press and Basic Books, et.al.)  I don't know that there's ever been a better, richer time in history to be a student (or just an interested follower) of mathematics. Enjoy.

Should Mathematical Explanation Entail Mathematical Beauty?

                                          
                       Math
                         Logic
                          Beauty?

Is there any math writer/blogger left who hasn't written about the beauty of mathematics… on multiple occasions? It is an almost overwrought topic that audiences either 'get' by now… or likely will never comprehend. I have an acquaintance who, upon once being told that I was reading a book entitled "Love and Math," responded, "well, THAT'S an oxymoron; those two words should NEVER go together"… I presume she would say the same thing about the phrase 'math and beauty.' :-(

Nonetheless, "Mathbabe" blog has a wonderful new post up (from a guest-poster, not from mathbabe Cathy O'Neil herself) which is a great take on this matter; I recommend it to all even if you've read a plethora of these 'math and beauty' type posts before, not because it necessarily offers something new or profound, but just because it is so well-composed:

http://mathbabe.org/2014/02/04/guest-post-beauty-even-in-the-teaching-of-mathematics/
 
Further, the post notes that there will be a conference in Sweden, March 10-12, on this very connection of math and beauty, or as the author puts it, "Specifically, we will look at the question of whether mathematical beauty has anything to do with mathematical explanation. And if so, whether the two might have anything to do with visualization".

By coincidence, this post came along just as I was also reading some old 1960/70's essays on the history and nature of formal/symbolic logic, somewhat dividing that history into the pre- and post-Gödel periods. Anyway, this all got me to thinking how differently (I believe) logic is viewed from mathematics in this "beauty" regard... even though logic is very much at the foundation of mathematics.

Math enthusiasts easily perceive the beauty of their field (and wish everyone beheld it). But I suspect logicians don't perceive their subject in the same way (...but professional logicians please let me know if you do see beauty as an integral, prevalent feature of your field as well).
Formal logic seems to much more aptly fit the cold, dry, analytical (dare I say, boring) stereotype people so often apply to mathematics (not that there isn't any beauty to logic, but that you have to more deliberately look for it to see it, than in math).
The difference between logic and math seems somewhat akin to the difference between machine code (all 1s and 0s) versus the richer, higher level programming languages most people learn, even though the latter are ultimately based, in some sense, on the former. 
I guess I'm wondering if others (especially those more regularly working with academic logic) agree with my impression that logic is less pervaded by "beauty," and more befitting of the common stereotypical view many hold toward mathematics… or, alternatively, if taught and approached correctly, is logic also a matter of under-appreciated beauty?

ADDENDUM: I'll close this out with a concluding quotation from one of the essays I was reading (by Leon Henkin, 1962), which may bear some pertinence: 

"…perhaps of greater significance is the consensus of mathematicians that there is much more to their field than is indicated by such a reduction of mathematics to logic and set theory. The fact that certain concepts are selected for investigation, from among all logically possible notions definable in set theory, is of the essence. A true understanding of mathematics must involve an explanation of which set-theory notions have 'mathematical content,' and this question is manifestly not reducible to a problem of logic, however broadly conceived.
"Logic, rather than being all of mathematics, seems to be but one branch. But it is a vigorous and growing branch, and there is reason to hope that it may in time provide an element of unity to oppose the fragmentation which seems to beset contemporary mathematics -- and indeed every branch of scholarship."

 

Of Education and Game-playing

 I often try to steer clear of these education debates, but did enjoy Cathy O'Neil's recent quick take on Diane Ravitch and Common Core standards:

http://mathbabe.org/2014/01/29/diane-ravitch-speaks-in-westchester/

A stupendous amount has been written (pro and con) about Common Core in the last year-or-two, so don't mean to single out Cathy's piece above other views, except I like the approach she's taken.

I grew up at a time when several of my peers were graduating high school without basic reading, writing, and math skills… how these particular students were even being passed along from grade-to-grade, let alone graduating high school, was hard to comprehend. It was, frankly, an embarrassing, deplorable (even fraudulent) situation. Universities found, to their surprise, that entering freshman sometimes lacked the necessary skills for college work… significant remedial programs had to be instituted.

So when, understandably, standardized proficiency-testing programs began implementation state-by-state, I eagerly supported it. Since then, I've often participated in the 'grading' of these standardized tests, and what became apparent within a few years was that teachers were 'teaching to the test.' It had all become, almost inevitably, a sort of game for teachers, whose own evaluation was often based in part on how well their own students did on such tests. So they gamed the system, likely skirting some teaching responsibilities, creativity, and effectiveness in the process.

I've written earlier that I believe the use of 'flipped classrooms,' MOOCs, and digital resources in general is one of the most fascinating, even 'paradigm-shifting' changes in education coming along, yet grappling with "standardized testing" remains a hugely difficult nut to solve. Not only must there be some sort of standardized requirements (especially for mathematics) that all students should meet before graduating high school (…and really, LONG before graduating high school), but I believe they should indeed be national, and not variable state-by-state standards. BUT these should be truly bare, minimalist levels of literacy and numeracy for our adult population (or each grade level), not "high-bar" or accomplished standards.
In some ways Common Core seems like one of those near-comical results portrayed when 'designing things by committee': http://tinyurl.com/mcay7uq

I once attended a college that required a "swim test" (long since dropped) for graduation. The idea was not that everyone should be a good or fast or very capable swimmer, but simply that everyone ought have some ability to float and dog paddle and move through the water, in the event of an emergency -- that this was simply a life skill (even if not an academic skill) one ought have as a college graduate.
The whole nature of "literacy" is rapidly changing… in the future, basic societal "literacy" won't so heavily entail reading-and-writing skills, but rather computer, coding, and office-suite sorts of skills -- THESE will be basic needs to successfully 'swim' in society (having read and discussed "Hamlet" in high school, or knowing how to diagram a sentence, will be of virtually no use!) So I still believe some form of standardized proficiency testing is necessary, but I too have reservations about rigid Common Core standards, as configured -- almost inevitably, they will set in motion another round of diversion and educational game-playing, stifling creativity... while raking in big bucks for the private enterprises developing/administering them.

Perhaps we have become such a society of manipulators, shirkers, and system-gamers that
there is simply no good solution to such a double-edged problem (both no-uniform-testing and mandatory standardized testing are problematic), but only least-bad solutions…. but as Cathy concludes, the first, necessary steps really involve, not reforming our educational system, but alleviating poverty and related underlying conditions that undermine learning. And on THAT score, our nation seems to be moving entirely in the wrong direction. (...Long, one of the most lunatic things to me, has been the way our public schools are funded largely through local property taxes -- a system insuring wide disparity in quality between schools... but that's a topic for a whole different discussion.)

Of course, a LOT more thoughts/news on math Common Core available through Google:
http://tinyurl.com/l547xsb

Mathematics In 2013... Read All About It

Another edition of "The Best Writing On Mathematics" (for 2013) is now out, from Mircea Pitici (4th in the series since 2010). It is slimmed down from the last issue, which was slimmed down from the prior one etc. -- don't know if Pitici is becoming more selective in what he chooses, or just doesn't have the time to read as many potential entries as in the beginning (or perhaps it's just an economic publishing decision?). In any event, it is another eclectic collection, that, as I often warn about anthologies, will include some chapters to please most any math-lover... and, some that won't; those two categories will simply vary per person by what interests you bring to the table to begin with.

There are some very big names in this compilation: Roger Penrose writes a wonderful Foreward (a quotation from which I used a few days ago on Math-Frolic), and I much enjoyed the overview of Pitici's own Introduction, which includes this ebullient passage:

"Mathematicians are mavericks -- inventors and explorers of sorts; they create new things and discover novel ways of looking at old things; they believe things hard to believe, and question what seems to be obvious. Mathematicians also disrupt patterns of entrenched thinking; their work concerns vast streams of physical and mental phenomena from which they pick the proportions that make up a customized blend of abstractions, glued by tight reasoning and augmented with clues glanced from the natural universe."

Philip Davis follows with one of the longest pieces of the anthology reviewing the place of mathematics across many cultural aspects of the modern world. Ian Stewart covers one of his favorite topics, symmetry, in chapter 2, and after that, the always-interesting Terence Tao overviews the dual tug of complexity and universality in mathematics. In one fine example he tells the "legendary" story from a 1972 gathering, of mathematician Hugh Montgomery meeting physicist Freeman Dyson for the first time and discovering, in that "chance meeting" (where Montgomery feared they'd have nothing in common to talk about), that remarkably, they had both deduced the same formula, but Montgomery working from the arena of number theory and the zeta function, while Dyson, contrarily, had approached it from "the study of energy levels in the mathematics of matrices."

I thought these first three pieces were a strong beginning to the volume, after which followed a real mixed bag of selections of varying interest (and including a few I wouldn't have included). Charles Seife, Donald Knuth, and John Pavlus are some of the other author-names that might be most familiar to readers (about half the writers included were folks I was unfamiliar with).

 The hot topic of education merited only one entry (from Frank Quinn), which seemed a bit unfair -- that subject almost requires a much wider sampling of opinion if you're going to touch upon it at all (but then an entire anthology on math education perspectives could easily be put together separately if one so chose!). On-the-other-hand there are several entries touching on probability, perhaps a bit overweighting that always-interesting topic.

Appropriately, the volume ends with one of the brief but fascinating updates to one of 2013's math highlights: Shinichi Mochizuki's "proof" of the abc conjecture (which other mathematicians are only very, very slowly coming to dissect).
Anyway, I enjoyed the first and last few chapters of the volume most, with less enthusiasm for some of the middle pieces (…but your mileage may vary ;-)
Worth noting that a few entries involve some significant math background, but most are fairly accessible for a lay reader.
If you've enjoyed the Pitici anthologies from prior years you'll certainly want to add this one to your collection, or if unfamiliar, by all means begin with this one.

In closing, I applaud Pitici for taking on the difficult (even unenviable) task each year of distilling from 12-month's worth of math writings a collection that can appeal to the broad spectrum of varied math enthusiasts that are out there.

(I might just note that Pitici's series is published by Princeton University Press, and is NOT part of the "Best Writing In ______ " series that comes out yearly on several other subjects, and which most readers are likely familiar with.)

Artful Geometry…

I'll confess to not being a big fan of coffee-table books; pictures of animals and exotic places are okay and appreciated in certain contexts, but outside of those two subjects, I'm usually not much enamored of large picture books collecting dust on a tabletop as a snooty bit of decor. And that goes double for math and science books pretending to be table-toppers… I want to read and ponder math and science, more than I want to stare at pretty pictures, so I'm always a bit skeptical of coffee-table-like books that approach these subjects... and appropriately such books are a bit rare.

Having said all that, I have another confession to make: I'm a sucker for geometry!! ...My initial thought when I first glanced at the new volume "Beautiful Geometry" was, "oh good a new Alfred Posamentier book!"… but it isn't from Dr. Posamentier; it is from Eli Maor (and artwork by Eugen Jost), another fabulous math-popularist. And if I owned a coffee table (I don't) this is a volume I'd be very pleased to plop upon it! Princeton University Press has done its usual splendid job of presentation with this large striking offering of indeed beautiful geometry (beginning with the gorgeously-rendered cover of a Sierpinski triangle -- which, though you might not recognize it from that cover, is characterized by the delightful fractal paradox of having an infinite perimeter bounding an area of zero!).
 
I could be mistaken, but it seems as if the wide popularity of Clifford Pickover's "The Math Book" has spawned a mini-proliferation of these sorts of artsy-math compilations, that combine math text with eye-catching art and graphics. The artwork in this volume is consistently exquisite -- sometimes its meaning being clear and obvious, and other times requiring closer examination or pondering. Maor's text is excellent (and very accessible) as well, and does raise the volume above the level of a mere look-see coffee table book -- you can definitely learn some interesting and real math along the way, which is beautiful in its own way, apart from the artwork.

Although the topics covered (over 50 gems) run mostly in chronological order, you can easily hopscotch around the volume jumping to topics which interest you most, or which are new to you. Of course there are many classic bits of plane geometry included here, but also some subjects that might be less familiar to many readers, like "Ceva's theorem," "Steiner's prism," "Lissajous figures," and the "Reuleaux triangle."
It's not often that math books can be described as "delicious," "scrumptious," or "delectable," but such words fit in this case, or, as Ian Stewart calls it, "a feast for the eyes."
Mind you, you're average neighbor who stops by for a beer on Sunday afternoon may not be much moved by these pages, but for math enthusiasts and especially young people delving into geometry this is A BUY! Geometry has, perhaps, never been rendered more beautifully!

From indications I've seen, it looks like another banner year ahead for popular math books (this particular volume is scheduled to hit bookstores around end-of-January)… I can't imagine it can equal 2013, but there's some great stuff coming down the pike (the one I'm currently most anticipating is Max Tegmark's "Our Mathematical Universe" due out in a couple weeks). Yeah, I'm already likin' the looks of 2014!
So a Happy New Year to all!!, as MathTango, which started as an experimental offshoot of Math-Frolic, essentially completes it's first year of postings.

James Tanton... Making Math Accessible

Math-Frolic Interview #19

"The high-school English curriculum teaches both the grammar and the poetry. Why can’t a high-school mathematics curriculum teach the poetry and artistry of its discipline as well?
"The goal of this site is to demonstrate the beauty of mathematics, its wonder and its intellectual playfulness, and to work towards bringing true joy into mathematics learning and mathematics doing for one and all."
-- from James Tanton's "Thinking Mathematics" website


 Dr. James Tanton might be deemed one of the 'rock stars' of math cyberspace. I first learned of him (as I suspect many did) when Sol Lederman of Wild About Math began drawing attention to his work (p.s.: Sol podcast-interviewed Dr. Tanton over a year ago). He is one of the most creative, thoughtful educators on the Web, and you can sense the joy he draws from math whenever he presents the subject to others.
Besides, anyone who likes border collies, has an Aussie accent, and enjoys teaching math, is a pretty cool dude in my book! Read on to learn more about him....

****************************************

1) When did you know that you wanted to do mathematics professionally, and can you give a brief synopsis of your life's road to your current math duties?

I didn't know I wanted to be a mathematician until university when it dawned on me during a course on abstract algebra that I actually was a mathematician, had been one all my life, and just didn't know it! (I wrote a piece about this at http://www.jamestanton.com/?attachment_id=917.)

I found mathematics in school dreary beyond belief, disconnected, and joyless. I knew I could do the work asked of me, and I did it well, but it was primarily procedure and rote doing in an unenlightened curriculum experience. I was never really interested in "what" questions ("What is the height of a tree viewed at an angle of elevation on 34 degrees from 60 meters away?" "What is 76622 divided by 13?" "What is my monthly interest payment if I take out a loan of ....?"). I was much more fascinated by "why" questions,and really loved to ponder the "what if" questions too. But I was well-trained not to ask these sorts of  questions, and thus understood mathematics to be about technique and algorithm and computation. Abstract algebra was all about the "whys" and invited many a "what if." I was dumbfounded and delighted by this, that this is the sort of thing mathematicians really think about. I felt I was "home" with this material.

SO ... I finished up my theoretical physics undergraduate degree, switched over and did a pure mathematics "honours degree" (Australia offered three-year undergraduate degrees, with an option to extend to a fourth honours year), and then headed over to the U.S. for a PhD in mathematics at Princeton.

But I decided not to pursue the pure high-powered research route in mathematics. I've always felt a call for teaching, which, I understand now, is actually a call for sharing the "joy" -- to do whatever I can to share with the world the joyful, and mind-blowing, experience thinking about, pondering on, and doing mathematics can bring. I taught at liberal arts colleges for phase one of my working career, somehow fell into doing a great deal of consulting work for high-school and middle-school educators, and then realized that I didn't really know what I was talking about when advising on matters of K-12 mathematics curriculum and education.

I felt the overwhelming pull to be honest in this work. I left the college world and became a full-time high-school teacher myself. I wanted to truly understand the demands and frustrations of working as a high-school educator, to truly understand the nuances of the curriculum and of teaching that curriculum to young students, to find the wiggle-room within the rigidity of the curriculum and the frenetic, all emotionally consuming school culture, and to truly follow my call for the "sharing of joy" at what I think is an absolutely vitally important level of mathematics education. I joined St. Mark's School in Southborough, MA. While there, I founded the St. Mark's Institute of Mathematics, which I linked with Northeastern University's School of Continuing Education. Through the Institute, I offered extracurricular research classes for middle-school and high-school students across the Boston area, gave graduate course for educators on the content of K-12 mathematics (and beyond), offered general lectures and workshops, wrote math essays, published books, and did whatever I could to share the joy of mathematics at all levels of thinking and doing.

And then life recently brought me to Washington D.C.

2) You currently work with the Mathematical Association of America… what is your role and goals with that organization, and is that position open-ended or are you on a certain time-frame?

I was absolutely delighted and honored to have been offered a visiting position at the Mathematical Association of America. Word seemed to be out on the street that I was moving to the city (my fabulous wife is a top-notch planetary geologist and was offered a plum job here) and this position was basically waiting for me upon my arrival. It has since turned into a long-term open-ended position and comes with the stunning title of "Mathematician in Residence."

The MAA has long been interested in matters of good quality, joyful, mathematics at all levels, and joyful education, at all levels. Of course, its primary focus is on supporting work and education at the undergraduate level, but that is not its only focus. For over 60 years, for example, the MAA has been supporting the American Mathematics Competitions for middle- and high-schoolers.

My position at the MAA is only half-time. I do a lot of travel and consulting work now, workshops, talks, lectures, short courses, etc for educators and students across the nation, and I am happy to promote the work of the MAA as I do so. I also have started the "Curriculum Inspirations" project with the MAA to show how the content developed by the AMC is relevant for the classroom, connects beautifully with the Common Core State Standards, and, most important to me, how each "what" question asked in those competitions is actually an invitation to deeper thinking into the "whys" and "what ifs." I make videos and write essays about how to see each concrete piece as a portal to joyful wonder and further exploration. The AMC content is not really about competition at all. (Any sense of competition turns me off, and would have done even more so as a student). The AMC is really about innovative thought and play of ideas. "Curriculum Inspirations" is about that.  

I should also mention that the doors of Math for America, DC, were laid open for me too upon my arrival here at DC. (Amazing!) I do the professional development work for all their fellows. They have such a wonderful program and are truly influencing the local departmental cultures of mathematics teaching - making joy, personal understanding, personal confidence, the willingness to rely on one's wits to "nut things out," the top goal for students in their teaching.

3) You have a PhD. in mathematics (from Princeton, no less), but your interest seems to have always been focused on secondary education (not college instruction)… can you say a little about the dichotomy between college and secondary level math and what draws you to the latter? And do you foresee ever teaching again at the college level?

Having spent basically ten years as a college professor and ten years as a high-school teacher, I feel I have my "creds" in both worlds now. These two cultures, in the past, really haven't talked to each other in meaningful ways. (I remember many a college department meeting complaining about how high-school teachers don't teach this or that, of if they do, they don't teach it in the way that is true to mathematics.)  Part of my mission of "mathematics joy" is about connecting these two worlds. I feel I truly sit between the two now.

I know how hard it is being a high-school teacher (it is way harder than being a college professor re matters of time, control of one's schedule, freedom of mental space, time to reflect and ponder, time to enjoy your own subject!) and I truly admire those folk that work in more demanding situations than I did. (I am not actually qualified to teach in the public schools -- even though I train people to do so.) I like to think that my conversations with educators across nation are seen as genuine and real. When I talk about "the curriculum," I really am talking about what I actually do in the classroom! And my mathematician-self is still doing it in a way that is true to the mathematics. (My prime example of what I mean by this is how I taught quadratics in algebra II. I still did all that I was told to do, but I did it in a way that follows the sensibilities of a mathematician. See the online course www.gdaymath.com on quadratics.)  

I don't know where I'll be in ten years' time, but this in-between spot is a powerful and helpful place to be. College departments talk to me, high-school departments talk to me, and I feel I can speak to both with understanding and, more important, offer actual concrete ideas that might be of help.

And all through this, I am drawn to the high-school level education as the place that really can't afford to forget about the idea of "joy" in mathematics.   

4) I think you are likely regarded as one of the most astute instructors of math on the Web today… Is that just a 'natural' talent that you've always had, or have you had to work hard at it and change your approaches over time to find what works best with students and on the internet?

Wow! Am I?

The honest truth is that I don't think too hard on what I do. Well, that's not quite true. I believe in "mulling" -- so I mull a lot.

When I think of a topic in the curriculum, my guiding principles are "get rid of the clutter" and to ask "What is really going on?"

My ultimate goal as an educator is to teach kids the confidence to rely on their wits, to be confident enough to try something., to get it wrong, to flail, to turn flailing into "successful flailing," make educated guesses, to find successes, and enjoy the success (and then wonder about more!)  Life comes with no answers in the back of the book -- I have no idea how to do most things I encounter in life. Success in business and in research is about flailing, asking new questions, and getting things wrong most of the time.

So when I look at a curriculum topic and ask "What is really going on?" I am really asking: "What are the key one or two ideas that make this topic click - so that the rest just follows as common sense?" I ask this because it is the reliance on common sense I want to teach.

Quadratics (again - sorry) is really about symmetry. Once I realize that all quadratic graphs are symmetrical, then all that horrible algebra II memorized formula stuff can go out the window! (e.g. y = (x-3)(x-9) + 15. Hmm. x=3 and x=9 look interesting in this formula. They both give y=15. Oh, two symmetrical points on a symmetrical graph -- the vertex must be halfway between at x=6. I can sketch the thing easily now. Oh ... Can I do the same sort of clever thinking for y = x^2 - 4x + 15? Are there any interesting x-values staring me in the face? etc.)

Of course, when I do teach a topic, I do get a sense of what works well for students and what not so well and I do make adjustments and tweaks.

5) You're a native of Australia… do you see any significant differences (pluses or minuses) between secondary math education in your home country and here in the U.S.?

Oh I really can't comment on this one. I haven't set foot in a school in Australia for over 30 years now and my visits to Australia since coming here in 1988 have been short, and family focused. I am not up on how the mathematics curriculum has changed in Australia since my personal unenlightened days. (Of course, I do have the sense that much has changed, and much of what I am hearing and seeing I like. But I don't have enough of a sense of things to make any comments. Maybe I should spend a semester teaching back home in a school?)

6) Can you summarize where you think the future of secondary math education is headed in relation to digital resources, especially in regards to 'Khan Academy' type sites and the interest in "flipped classrooms?" And might something along the lines of "MOOCs" ever apply at the secondary education level?

I have no idea.

I would love students to enjoy a sense of control of their own leaning in the classroom (and, therefore, classroom cultures with the flexibility for that), so that if a student really is struggling with a topic, or wants to fly with a topic, he or she has the time and mental space to make use of the online resources out there to pursue it.

I would like to see homework change from "Do these 40 problems on logarithms" to perhaps something like: "Do enough of these problems until you feel you really get it. And then do just one more, the one in the rest of the list that looks hardest to you. When done,  if there is still time, look at some of these videos and tell me something interesting about logarithms you saw."

I've always thought it would be interesting to hand out to students the final exam to the course on the first day of class, and say: "The first week's assignment is to make as much sense of this as you can, using whatever resources you want. We'll start discussing all these ideas next week." Actually, that's a good way to prepare for SATs. Just start doing past exams, and figure stuff out as you go along. (Ahh. The culture of assessment!)

Even though I am failing to give a meaningful response to this question, I obviously have some thoughts on the matter re the videos I make and the short courses I have started posting on line. I've always said to myself that these are for educators, for their own personal enrichment and rekindling with a love for mathematics, and that they are for students too. But have I ever thought through how and when students might use these resources? Hmm. I do know that a number of educators have shown my videos in their classes, or have assigned them as viewing for homework. And I like this idea in general, this flipped classroom idea. I will predict that that, at least, will happen more and more often in the years to come.

7) What are your own favorite aspects of mathematics to study or read about? And are there any particular books you'd especially recommend to the lay person with an interest in math?

My degree is in algebraic topology, the study of shapes and surfaces and how you might be able to use mathematics to detect what shape you are living on. (Columbus, in sailing West to return from the East, would have verified that the Earth is round. But would that have proved it is a sphere? Could the Earth still be a donut?)

But I really do have a love for number theory. As Mr. Honner says, I am overly obsessed with triangular numbers.

Readings: Well, of course, read anything and everything by Martin Gardner. (I remember the delight of discovering his columns in back copies of old Scientific Americans stored in the basement of my university library.)

Read the works of Bill Dunham. He's a super guy who writes with great clarity on matters of history of mathematics AND the mathematics itself.

Speaking of mathematics history: "Math through the Ages: A Gentle History for Teachers and Others" by Berlinghoff and Gouvea is a gem.

"Mathematical Circles: Russian Experience" by D.Fomin, et al, is a brilliant guide of fascinating and beautiful mathematical ponderings.

"The Queen of Mathematics: A Historically Motivated Guide to Number Theory" by J. Goldman gets a tad advanced, but it is brilliant.

Is this enough for a start on material you might not have encountered? (I know you recommend some winners of books too!)

[Yes, interesting list of books that I'm largely unfamiliar with!]

 8) You've written a number of excellent books which are available through Web sources but not otherwise widely distributed. I'm curious if that is by your choice, or some other reason a traditional publisher has not taken them up and given them a wider audience?

I am hopeless at self advertising!

I have really enjoyed the flexibility of self-published books. They are all "charmingly human" (meaning, that they have never been copy-edited), but if ever I find one chapter is a bit too "charming" I have the means to quickly re-upload a corrected chapter and the issue is dealt with.

And I love the complete control I have on what I write and how I present it in my self-published pieces. 

Plus, despite not advertising these things, my self-published pieces do sell at a fairly constant rate - and that's lovely and helps support the work I do.

I have published three books the traditional way and have never really felt they reached a good market. Somehow I have a sense that my my self-published books are "out there" more so than I can detect with my company-published books. I find that curious.

However, if a company were to approach me, I am up for a discussion. (My "Math Without Words" book, for example, is under contract with Tarquin Press in the UK right now.)

9) When you're not bringing the joy and wonder of math to young people, what are some of your other main interests/hobbies/activities?

I would like to say border collies. My family and I have always had border collies in our lives (and my wife is a trained sheep herder too!), but it has been a sad year this year as our three dogs all passed on. With the amount of travel my wife and I each do, it is actually easiest being dogless right now. But it feels oh so wrong!

I have an overly sweet sweet tooth and I bake desserts. My other mission is to bring the pavlova to the American consciousness. (I am somewhat obsessed  with meringue, that and triangular numbers.)

Apart from that, family and math really is about it for me. And that's pretty darn good!

[Well, we share sweet-tooths and a love of border collies... now if I could just acquire your talent for math!
Thanks for taking part here... your resounding passion for mathematics and teaching shines through!]

****************************************

Hope everyone enjoyed learning more about Dr. Tanton and his approach to math and instruction, as much as I did!
If you're not already familiar with it, be sure to play around on James' main site (it's chockfull of good stuff!): http://www.jamestanton.com
Also, check out his YouTube channel here: http://www.youtube.com/user/DrJamesTanton/videos
Learn about his books here: http://www.jamestanton.com/?page_id=15 
...and he's on Twitter: @jamestanton

Of Dinosaurs and Mathematics... and Pathos

Anyone up for some philosophical reflection on a lazy Sunday morning...?

(image via Danny Cicchetti/WikimediaCommons)

Vocal math Platonist Martin Gardner famously wrote that if 2 dinosaurs met 2 other dinosaurs in a clearing, there would then be 4 dinosaurs present… whether or not there was any human mind around to appreciate the fact, and whether or not there existed any such words as "two" or "four," the relationship would still exist completely apart from human recognition of it. In fact, Gardner (who was actually more a philosopher than a mathematician) largely scoffed at the few professional mathematicians in his day who claimed that mathematics was merely some sort of human/cognitive creation. The case against Platonism has grown since then, with fewer, though still many adherents.

Non-Platonism comes in several different varieties and degrees, but Platonists tend to more uniformly feel that mathematics is a real, ubiquitous component of the Universe (or, in the case of Max Tegmark, they believe math is ALL there is -- it is ultimately the only component or structure of the Universe; p.s., Tegmark's new book "Our Mathematical Universe" will soon be out).

Much of this debate continues to seem semantic, hinging on what one means by words like "real," "existence," "component," and other words that simply can't be defined in terribly rigorous/consistent ways (even "mathematics" is not that easy to define). Is mathematics 'out there' in the Universe, apart from us and our existence, or is it only inside our heads, generated from neurochemical processes?

I bring all this up because recently Jason Rosenhouse broached the Platonist/Non-Platonist divide in this post where he addresses interview comments from Platonist/recent-author Ed Frenkel:

http://scienceblogs.com/evolutionblog/2013/12/13/in-what-sense-do-mathematical-objects-exist/

I tweeted a link to the above post when it first came out, but deliberately held off posting about it here right away, because I suspected it might lead to some engaging discussion in Jason's comment section… and, it has. Eventually, Ed Frenkel himself responded (twice), and all the discussion has been quite interesting... unless that is, you're in the camp that finds such philosophical debate sleep-inducing! ;-)

It certainly seems to me that if "science" has any reality in the Universe (and we're not just living in a simulation imposed by highly-advanced aliens) then there are elements of mathematics that must be real and integral to the Universe's operation -- if math ISN'T "real" in some sense, then "science" (which is based upon it) must also not be real, which implies that the Universe, rather than having 'order,' 'laws,' and causation as we perceive, is a rather hopelessly random/chaotic place (...which in turn assumes that "randomness" can truly exist!???)... but whether all of mathematics exists in some Platonic sense or only elements of it is more of a leap, and perhaps, as I say, more an argument over words and meaning, than mathematics itself... In any event, read Jason's take and the conversation that follows it.

....I wrote the above words yesterday for posting today, then this morning woke up to a tweet from Alexander Bogomolny linking to this quote on a discussion forum, from "Thelonious Mac," which somehow seems worth closing with:

"I just saw a television program in which a mathematician lost his daughter. Unable to express himself in the language of platitudes that most people use at such a time, he created a series of equations to represent her life, a work of art, expression, in math. Yes math is beautiful. There is absolutely no aspect of our lives for which you will not find math at its foundation. If I have a glass of clean water today, it is because of the math behind the engineering that brought it to me.
Math is the mother of all science. Without it, our lives would be incomprehensibly pathetic."

Puzzles, Puzzles!

I usually do puzzles over at the Math-Frolic site, but will switch it around this time....

First, I'll just link to a couple of recent puzzle offerings I liked from Richard Wiseman and Presh Talwalkar to warm you up, in the event you missed them:

1)  http://richardwiseman.wordpress.com/2013/12/02/answer-to-the-friday-puzzle-234/

2)  http://tinyurl.com/o8rufko

By the way, if you're into game theory, I notice that Presh has a new game theory eBook out, "The Joy of Game Theory" -- (Presh is great at finding interesting problems and explaining them well): http://ow.ly/rCtJ6  

3) As we approach the year-end, I realized I haven't re-run one of my all-time favorite Raymond Smullyan brain twisters lately (originally published by Smullyan in the "Annals of the New York Academy of Sciences" in 1979, Vol. 321). Apologies to long-time readers here, who didn't even like this puzzle the first time around! ;-) But what I love about it, is that it is rather involved, and requires some fairly heavy-duty math to prove the very counter-intuitive outcome, yet can be verbally explained so as to be comprehended logically without employing any real mathematics whatsoever. I've re-written it, from Martin Gardner's excellent treatment of it in his "The Colossal Book of Mathematics" (chapter 34)... without further adieu:

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. etc. You also have a box that contains some FINITE number 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 or #767 or #5 if you so desired, just as a few 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 does: "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!
This amazing result 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 having been 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, 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).

Mind… blown….

Caption Contest!!!

Sorting through some bookshelves last week found a few (paperback) math books I have duplicate copies of and don't need, so figured I'd give away to some lucky reader! Not terribly recent, but hey, mathematics is timeless (and in this instance, free)!! The four books are:

*********************
"The Mathematics Of Life" by Ian Stewart (2011)
"Mathematics: The New Golden Age" by Keith Devlin (1988)
"Beyond Numeracy" by John Allen Paulos (1991)
"The Kingdom of Infinite Number" by Bryan Bunch (2000)
*********************
     
I've seen "Caption Contests" work well at other blogs, so will give it a try here.
Send in, via the comments, your caption(s) (limit 3 per person) for the below image, and I'll award the books to whoever's wit most moves me.
Depending how many good punchlines I get, not sure if all books will go to one person or be divided among top couple of entries (will let contest run around 3 weeks). Start your keyboards:

                         (the image has traveled around the Web for some years; I'd give credit, but not sure what it's origin is...?)

[and sorry, but probably only willing to ship books within US or at least N. America; though others can enter for sheer glory of possible win! ;-)]

[Sunday, Dec. 15 will be the last day for entries.]

Phenomenal Book...

"Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself."  -- MIT Press

In 1979, like a lot of people, I picked up a book entitled "Gödel, Escher, Bach" by an unknown author named Douglas Hofstadter... and was blown away. The book was easily one of the most creative, thought-provoking I'd ever encountered; over the years it became internationally famous (often simply referred to as "GEB") as did its author who won a Pulitzer and National Book Award his first time out-of-the-gate. (Interestingly, years later, Hofstadter would write that almost all reviewers and discussers of GEB miconstrued his own goals with the book -- people often read into it whatever they wanted -- but nonetheless it remains a keen treatise on human cognition, and Hofstadter has written several volumes since. [p.s. -- I recently discovered a great 1982 read (pdf) from Hofstadter on self-reference and Gödel theory HERE.]

34 years later I've finally been bowled over by another book… not as creative, ground-breaking, or Rorschach-like as Hofstadter's effort, but still a vitally important, rich read, and the author, not surprisingly, is also a Hofstadter fan.
A brief look at:

"The Outer Limits of Reason"  by Noson Yanofsky

Before picking up this book I'd not heard of "Noson Yanofsky," so I was astounded that this is the best, most lucidly-written volume for lay readers I've ever encountered on the underlying or foundational topics I most enjoy, related to mathematics; including issues that cross the boundaries of math, logic, philosophy, physics, and computer science.

In terse summary:
After an introductory chapter, Chapter 2 delves into "language paradoxes" and self-reference (a topic that runs throughout the volume), including the Berry Paradox, Richards Paradox, and the 'interesting-number paradox,' in addition to even more common ones. Chapter 3 moves on to "philosophical conundrums," followed by "infinity" in Chapter 4. Chapters 5 and 6 delve into a range of computer science issues. Chapter 7, the longest and perhaps most difficult one (50+ pages), covers "scientific limitations," including quantum mechanics and multiverse ideas. This is followed, in turn, by chapters on "metascientific perplexities," "mathematical obstructions," and a final wrap-up chapter on "reason" and its limits.

More specifically, all the following topics (and more) are brought into focus in this volume:

paradoxes
self-reference
infinity
epistemology
logic
sets
axioms
algorithms
uncertainty
P vs. NP
quantum mechanics
relativity
entanglement
Halting problem
Galois theory
Mandelbrot set
chaos
Anthropic principle
Platonism
scientific induction
Thomas Kuhn/paradigm shifts
Hume/Hempel/Karl Popper/ falsifiability
Gödel incompleteness

The writing is clear, interesting, and comprehensible, covering a lot of ground, without proceeding to such advanced elements as to throttle the reader along the way (the editor has done a fantastic job!). The book ends with 15 pages of excellent "notes" to the individual chapters, and a dozen pages of good bibliographical references (each chapter ends with suggestions for further reading as well).

One Amazon reviewer wrote "Reading this book could be a mini education!" and that captures my own feeling as well.
Having said all this I should note that the typical mathematician won't learn any new math here; a typical physicist won't learn any new physics, and a philosopher won't find new philosophy here... Rather, what is wonderful and well-crafted (and rare) is the weaving together of all these (and more) areas into a single tapestry on the nature of human rationality across such fields -- something I believe all students should have exposure too. To some degree most of the chapters are self-contained units that can almost be read in any order and be enjoyed, but reading from beginning to end is likely the best way to appreciate Yanofsky's progression of thought and complexity, as he puts it "from concrete to abstract."

The book's subtitle is, "What Science, Mathematics, and Logic Cannot Tell Us" and that is the central, important theme of this offering: that despite the success that science, math, and logic meet in providing us with information, there exist "truths" or information which are not only very difficult to gather, but which are inherently beyond our capacity to attain.

"Certainty," and the hubris that often follows it, is one of the most perilous dispositions humans can have… especially so in politics, religion, and other arenas of culture… but even within science, where empirical evidence reigns supreme, there are real limits to certainty and knowledge that need to be recognized -- I know of no more important science lesson a book can pass along, and I know of no book that does it as well as this one!

New Scientist reviewed Yanofsky's book here:

http://tinyurl.com/n5ld6yv

And in the blog-post just prior to this one I interviewed Dr. Yanofsky.