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....

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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!]

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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:

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"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)
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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.

Noson Yanofsky... The Limits of Reason

Math-Frolic Interview #18

 "Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In "The Outer Limits of Reason" Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes."  
                            
Dr. Noson Yanofsky is a professor, computer scientist, and author of the book from which the above book-flap quotation is lifted, "The Outer Limits of Reason." It is quite simply one of the BEST books I've ever read, cutting across so many important categories (logic, math, computer science, physics, philosophy). I'm thrilled to be able to present him here to give a hint of what his work is about -- you'll still need to read the book to fully appreciate the range of material he has brought together in one volume, and how well it is presented.

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1) Can you tell readers a little about your background and how you came to be interested in the subjects that are the focus of your current book (paradox, self-reference, uncertainty, logic, infinity, limits of knowledge...)?

 

I grew up in Brooklyn, New York which remains my home. I always loved popular science books so it was a big thrill to actually write one. As a teenager I was a computer geek and my undergraduate degree is in computers. My father was a mathematics professor so, to some extent, math is in the genes. (My daughter is also very good at math!) My PhD was in pure mathematics. I worked in category theory and algebraic topology. Presently I teach computers in Brooklyn College and in the Graduate Center of the City University of New York.
My interest in the area goes back to 2003 when I published a paper titled “A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points” The paper basically contains a single theorem that is the core of all self-referential phenomena. After stating and proving the simple theorem, I give 19 different instances of the theorem from different aspects of language, computer science, and mathematics. The purpose of the paper was to show that all these different self-referential phenomena have the same scheme. The paper was very well received. The book shows -- in a non-technical way -- what this scheme is all about.  

Another inspiration of the book is the type of classes that I teach in college. I teach computability theory which is a field in computer science that deals with what tasks computers can -- and more importantly -- cannot perform. I also teach complexity theory. This field discusses what tasks computers can and cannot perform efficiently. These two topics form two chapters of the book.
At some point I realized that all these different things are discussing limitations to reason. The book shows how all these different limitations are related.


2) The fundamental take-home message of your book seems to be encapsulated in the book's subtitle, which states that there are things that "Science, Mathematics, and Logic CANNOT Tell Us" [added emphasis]. How widespread do you feel that notion is among your colleagues?
 And relatedly, some in the artificial intelligence (AI) community believe that it is only a matter of time before computer scientists will be able to duplicate the workings of the human brain, while others believe that the full capabilities and consciousness of the brain will NEVER be duplicated nor fully understood. Care to comment on that schism?


There is, in fact, much that science, mathematics and logic can tell us. Our lives and our civilization are, for the most part, so much more advanced and better because of all the things that science, math and logic tell us. The point of my book is that there is a lot of information that is beyond the ability of science, math and logic to tell us. It is only over the past few decades that various fields are recognizing its limitations. This is not to take away from what these fields do tell us. Nor am I saying that somehow these fields or the things they tell us are faulty. I also do not want to give the impression that science, math and logic might come to an end. Rather I am saying there are things that science, math, and logic tell us is beyond their ability.

As to how widespread this inability to know is, it is hard to measure. There are reasons to believe that there is a lot more “out there” that we cannot know than what we can know. I give a few of these reasons in the book. Nevertheless, it is hard to speculate. Isaac Newton says “What we know is a drop, what we don’t know is an ocean.” Similarly John Archibald Wheeler is quoted as saying “As the island of our knowledge grows, so does the shore of our ignorance.” Newton and Wheeler are talking about what we do not know. What about what we cannot know?

The vast majority of scientists, mathematicians and logicians are working in areas where they are telling us new things. There are, however, some who work in foundational issues and looking at what is beyond the ability. I think researchers are becoming more familiar with such limitations of reason.
Artificial intelligence is mentioned in the book. The discussion is from the point of view that it is a provable fact that there are some tasks that cannot be performed by computers. The obvious question arises as whether human beings can perform these tasks. For the most part, it looks as though these tasks are also beyond the ability of human beings. To that extent, I tend to lean towards the point-of-view that human beings are just as limited as any computer and that the human brain is simply a very very complicated machine. From this point of view, AI should be possible. We have to get our machines to be as complicated as our mind.

Although computers will never be able to do everything, I think to some extent we already have AI. Computers answer phones, Siri answers your questions, robots sweep your floor. These are amazing things that computers of even ten years ago could not do. Now someone might respond and say “This is not really intelligence! This is just following rules. Humans have intelligence.” Perhaps they are right. But perhaps they are wrong. Maybe humans are also just following rules. I think people who think that we do not already have AI just want magic. As long as they understand what a computer is doing, they do not consider it magic. As Arthur C. Clarke says, “Any sufficiently advanced technology is indistinguishable from magic.” We might already have magic.


3) Early in the book you label yourself an "extreme nominalist," meaning that you fall in a camp not only rejecting mathematical Platonism, but opting for a view that NEITHER abstract nor physical objects exist as we perceive them to. Traditionally, most working mathematicians have probably been Platonists, but I see more-and-more mathematicians today unabashedly adopting the NON-Platonist viewpoint. Again, how much resistance do you find among your peers to your viewpoint?

 
I am not sure I would label myself an “extreme nominalist.” I think I lean that way. But to firmly put me in one camp over another is troublesome. The question of nominalism vs. Platonism/Realism is essentially an unanswerable metaphysical question. There is no way that we can tell which position is correct. For all I know I could be wrong and there is a Platonic universe (I dismissively call it “Plato’s attic”) that does have abstract ideas neatly categorized and ordered. Extreme Platonism demands that even physical objects have little Platonist tags that tell what it is. The point in the book is that there is really no reason to make that assumption. People do not have clear definitions in their head of what things are.

Recently, a friend, Mark Zelcer, and I wrote a paper titled “Mathematics via Symmetry” (available on the arxiv and my web page). One of the main ideas in the paper is that the seeming objectivity and universality of mathematics that justifies most of the Platonist ideology can easily be explained in another way. Anyone interested in the nominalism vs Platonism battle would gain from looking at that paper. 

As for what most mathematicians think, I suspect most do not worry about the nominalism vs. Platonism battle. Those who are Platonists are usually very accepting of my heterodoxy. I think we all realize it is an unanswerable metaphysical issue that has no relevance to anything important.
Whenever I feel absolutely firm in my nominalism stance, I like to think of Martin Gardner’s rock-solid defense of Platonism: “. . . if two dinosaurs met two other dinosaurs in a clearing there would have been four there even if no humans were around to observe them. The equation 2 + 2 = 4 is a timeless truth.” Does he have a point?


4) Another writer I very much enjoy, and who I believe is on the same page as you, is retired mathematician William Byers. I was quite surprised not to see him referenced in your book nor bibliography. Just curious if you are familiar with his work ("How Mathematicians Think" and "The Blind Spot") which seems to me quite similar to yours?


I am embarrassed to say, that I am unfamiliar with his works. I just looked up the two books you mentioned and they seem very interesting. I ordered them. Thank you for the recommendations. 


5) Doug Hofstadter is another well-known computer scientist/writer who has dealt a lot with self-reference and human cognition (you note him in your book a few times). I'm curious if you have anything to say about his ideas, or if the two of you have had occasion to discuss your interests together?


His books are a tremendous source of enlightenment and enjoyment. I think I read every book he published. I really cannot judge his ideas about self-reference, consciousness and AI. All I can say is that I hope he is right. I never had the good fortune of actually meeting him or communicating with him.



6) What are some of your own favorite popular math (or related philosophical) books to read, and to recommend to other readers?


Here is a short list of my favorites in alphabetical order.

Barrow, John D. Impossibility: The Limits of Science and the Science of Limits.
Barrow, John D., and Frank J. Tipler. The Anthropic Cosmological Principle.
Bell, E. T. Men of Mathematics.
Burtt, E. A. The Metaphysical Foundations of Modern Science: The Scientific Thinking of Copernicus, Galileo, Newton, and Their Contemporaries.
Davies, Paul. The Goldilocks Enigma: Why Is the Universe Just Right for Life?
Absolutely anything by Paul Davies
Fogelin, Robert. Walking the Tightrope of Reason.
Anything by Brian Greene.
Herbert, Nick. Quantum Reality: Beyond the New Physics.
Anything by Douglas R. Hofstadter.
Kline, Morris. Mathematics: The Loss of Certainty.
Poundstone, William. Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge.
Anything by Rudy Rucker.
Weinberg, Steven. Dreams of a Final Theory.
Anything by Hao Wang.
All these are great and worth reading and rereading. But I left out a lot of other great books.

[That's quite a mix! ...of these, I'll just mention that I'm especially fond of the Poundstone volume, which probably isn't as well-known as several of the others.]


7) Your volume is the best compendium I've ever seen of the kind of information/topics you've pulled together. It can certainly be used for a classroom course, but is also written in a style that lay readers may enjoy and gain much from. Unfortunately, I haven't seen it receive the publicity it deserves, compared to other 'popular' math-science publications. Is that because you intend it mostly for an academic audience, or does MIT Press (the publisher) just not generally engage in a lot of public outreach/publicity, or some other reason?


The book started as a textbook for a Core course in Brooklyn College. The course is essentially a science class for non-science majors. The title of the course is "Paradoxes and the Limits of Knowledge." It is an immensely successful course that is always fully enrolled. The students were very helpful in editing the book and making sure it is easy to read. Any part that seemed complicated to my students, I had to rewrite until it met their approval. It was a lot of fun writing the book with the students looking over my shoulder.
The book is only two months old. But it has gotten much publicity. MIT Press has advertised it in many different venues and it has been getting much press. This week there was a very positive review in the New Scientist. Judging from emails and peoples responses, the book is selling and is well received by both academic audiences and popular lay readers.


8) To round yourself out a bit, when you're not engaged in mathematical or computer matters, what are some of your main interests/hobbies/activities?

Outside of work and research I mostly help raise three kids. That is a full time job in itself. Other than that I do like to chill out by watching an old movie or taking a walk with my wife.

9) Personally, I wish every high-schooler in the country could be exposed to the sorts of ideas covered in your volume. What might you say about the pertinence of your material to developing an educated, thinking citizenry?

I truly believe that this book could be read by any intelligent high school student. I tried to make it as easy to read as possible. I agree with you that as many people as possible should be exposed to these ideas. The preface starts off with “With understanding comes ambivalence. Once we know something, we often find it boring and trite. On the other hand, the mysterious and unknown fascinates us and holds our attention. That which we do not know or understand is what interests us, and what we cannot know intrigues us even more.” I am not sure about developing good citizens. But it is interesting stuff!

 [...I highly recommend this thought-provoking book to everyone from bright high-schoolers to graduate-level mathematicians/scientists!]
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I am so glad Dr. Yanofsky took the time to take part here -- and I truly LOVE his book! -- it may not entertain you as much as Simon Singh's new book on The Simpsons, nor take you to the cutting edge of mathematics as Ed Frenkel's recent volume does, nor thrill Martin Gardner fans as much as his recent autobiography has (...the 3 other volumes I've touted a lot lately), but this is a fabulous, rich read that cuts across so many important boundaries, it ought NOT be missed!

ADDENDUM:  My review of the book is now posted here:

http://mathtango.blogspot.com/2013/11/phenomenal-book.html

In Love... With Math (Frenkel's new book)

Mini-review of Edward Frenkel's "Love and Math":

   It took me two-and-a-half days to read Martin Gardner's autobiography and jot down enough notes to write a lengthy review of it. I purchased Edward Frenkel's new, rich volume, "Love and Math: The Heart of Hidden Reality," a few days later and took over three weeks to finish it. I won't do a detailed review simply because there's so much of it I don't grasp well enough! That might sound like a negative…but in this case it ISN'T; in fact, it's a glowing positive… I love the book, from its Vincent van Gogh cover to its endnotes and "glossary of terms" (even though there is much in-between I don't yet comprehend)!

Let me explain: I once commented to a friend that most people seem to enjoy attending talks/lectures where they understand (or agree with) everything that is being said… I find those b-b-boring! Rather, I enjoy going to talks/lectures that are way over my head (or offer viewpoints I'd never considered), and being challenged to pull out of the air whatever bits of new learning I can draw from them… a talk that is 80% incomprehensible to me, but that I learn new ideas from, and stretch my mind from the other 10-20%, is a great, almost exhilarating, talk to me. THAT'S the sort of feeling I get reading Frenkel's new volume.

I've seen references on math and physics blogs for awhile now to the "Langlands Program," but without getting much sense of what it is. Frenkel's book covers a lot of ground, but with a primary purpose of elucidating to a general audience the Langlands Program (his specialty), and why it is so important.
The book is an odd mix of personal history and introduction to real and advanced mathematical ideas. The early chapters are interesting and important foundationally, but the volume really takes off on its mathematical excursion with chapter 9 on Andre' Weil's "Rosetta Stone" of the connection between number theory and geometry through the three "parallel tracks" of "number theory," "curves over finite fields," and "Riemann surfaces." From there on, lots of discussion of manifolds, sheaves, Kac-Moody algebra, Lie groups, gauge theory, SO(3) groups.... latter chapters becoming increasingly difficult if you haven't fully fathomed earlier chapters…. heavy-duty stuff for the average person who finds this book lying next to Tom Clancy or Barbara Kingsolver on a table at their local bookstore!!

Here's an example of the sort of content you can expect along the way:

"As we established in the previous chapter (see the diagram on p. 161), in the version of the Langlands relation that plays out in this column, the cast of characters has 'automorphic sheaves' in the role of automorphic functions (or automorphic representations) associated to a Lie group G. It turns out that these automorphic sheaves 'live' in a certain space attached to a Riemann surface X and the group G, called the moduli space of G-bundles on X. It's not important to us at the moment what it is."

Got that!? Me neither, but I have fun trying. Frenkel ardently tries to walk the reader through many ideas and how they tie together, and to do so at a level that a lay person can follow. His own fervent passion for his subject exudes off almost every page.

Frenkel's life-story, which is embedded in the volume, is itself fascinating, from growing up and being educated in the Soviet Union to his current position at UC Berkeley, but the mathematical portions are clearly not a fast-read… much of the book is a slow-savor-contemplate-re-read endeavor… BUT, worth it! I plan to re-read parts of this volume several times. The final chapter covers Frenkel's award-winning short film "Rites of Love and Math" quite a different topic, but at least a more reader-comprehensible narrative to end the book with. Prior to that he summarizes the volume this way:

"The Langlands Program has been the focus of this book. I think it provides a good panoramic view of modern mathematics: its deep conceptual structure, groundbreaking insights, tantalizing conjectures, profound theorems, and unexpected connections between different fields. It also illustrates the intricate links between math and physics and the mutually enriching dialogue between these two subjects. Thus, the Langlands Program exemplifies the four qualities of mathematical theories that we discussed in Chapter 2: universality, objectivity, endurance, and relevance to the physical world."

Last year I commented that one of the many things I liked about Paul Lockhart's book Measurement was that the author made no pretense that math is easy… he warned readers at the start to be prepared to really slow down and think while proceeding through the book… that parts would be tough-going. Too many popular math books hype themselves as volumes that will finally make you enjoy or connect with math… when it just ain't so. I still believe that Steven Strogatz's book, "The Joy of X" is one of the ONLY books that actually achieves such a goal of wide accessibility to non-mathy readers.
Frenkel likewise falls short of this noble goal, BUT what I love about it is his utter sincerity and hard work in wishing/striving to reach the masses. One can tell by the way the sentences and paragraphs are crafted that he (or a good editor ;-) is truly attempting to make plain to a general audience very, verrrry deep, rich, often labyrinthian mathematical ideas. The old Dr. Seuss adage says, "Don't be sad that it's over, be happy that it happened!" Well, I would say of Frenkel's volume, 'Don't be unhappy that he fails to make everything crystal clear, be thrilled that he's made such an arduous effort!'

And here are some other Web reviews of the book:

http://tinyurl.com/muungoj

http://www.math.columbia.edu/~woit/wordpress/?p=6266

Undiluted Martin Gardner…

Part of Douglas Hofstadter's tribute to Martin Gardner upon learning of his death back in 2010 (this doesn't come from Gardner's autobiography, but from online sources):

"This is really a sad day. Not so much sad that Martin died, since we all knew it had to come pretty soon, but sad because his spirit was so important to so many of us, and because he had such a profound influence on so many of us. He is totally unreproducible -- he was sui generis -- and what's so strange is that so few people today are really aware of what a giant he was in so many fields -- to name some of them, the propagation of truly deep and beautiful mathematical ideas (not just "mathematical games", far from it!), the intense battling of pseudoscience and related ideas, the invention of superb magic tricks, the love for beautiful poetry, the fascination with profound philosophical ideas (Newcomb's paradox, free will, etc. etc.), the elusive border between nonsense and sense, the idea of intellectual hoaxes done in order to make serious points (for example, one time, at my instigation, he wrote a scathing review of his own book "The Whys of a Philosophical Scrivener" in "The New York Review of Books", and the idea was to talk about the ideas seriously even though he was attacking the ideas that he himself believed in), and on and on and on and on. Martin Gardner was so profoundly influential on so many top-notch thinkers in so many disciplines -- just a remarkable human being -- and at the same time he was so unbelievably modest and unassuming. Totally. So it is a very sad day to think that such a person is gone, and that so many of us owe him so much, and that so few people -- even extremely intelligent, well-informed people -- realize who he was or have even ever heard of him. Very strange. But I guess that when you are a total non-self-trumpeter like Martin, that's what you want and that's what you get." 

If you missed it, you can read my initial 'broad-brush' take on Martin Gardner's autobiography HERE; but ahead (as promised), a much l-l-longer read and heart-felt tribute to Gardner and his biography: 

That Martin Gardner was a "fideist" and a "mysterian," that his literary tastes were far-flung (including G.K. Chesterton, H.G. Wells, Frank Baum, Lord Dunsany, Miguel de Unamuno, Lewis Carroll), that he had little formal mathematical training yet inspired a slew of others to pursue such a course, that he was opinionated and outspoken while also being shy, humble, and unassuming, and that he was one of the finest thought-provoking writers I've ever encountered... made him, for me, one of the most remarkable individuals in all of Americana. I only wish his autobiography was twice as long, for I never tire of reading him and feeling enriched.

 This book is both simple and complex, befitting the simple and complex person it is about. I can just imagine Gardner begrudgingly laboring on this volume at the behest of others, and wondering why anyone would find his vanilla life interesting. The writing is terse, succinct, matter-of-fact, conversational, even rambly at times, neither flowery nor scintillating, yet still fluent and interesting; sprinkled throughout with Gardner's subtle humor, illustrative anecdotes and encounters with other interesting people. These words and stories coming from some ordinary individual might not even be worth relating, but coming from Gardner they rise to another level.

In a very brief single-page Preface to the volume, Gardner ends with these telling lines:

"The best known remark of stand-up comedian Lenny Bruce was that people are leaving their churches and going back to God. What follows here is a rambling autobiography of one such person -- me."

One thing I was elatedly surprised to read in Gardner's book was that he regarded "The Whys of a Philosophical Scrivener" as his "most important book" and "The Night Is Large" as his "second best book." I have always regarded these as my two favorite Gardner reads, but was flabbergasted that he would pick them out as well from all his prolific writings (…apparently great minds do think alike ;-)
"The Night Is Large" is a superb anthology of many of his best and most varied essays stretching from 1938 to 1995. Whenever I meet people who know of Gardner only as a recreational mathematician, this is the first book I recommend that they additionally read. "The Whys of a Philosophical Scrivener" is Gardner's 1983 treatise on his many underlying philosophical notions; never one of his most popular or well-known books, it is must-reading if you wish to understand the man behind the math and the skepticism. It is a book that surprised many when, despite that outspoken skepticism, he came out as a theist or what he characterized as a "fideist" -- believing in an indefinable God despite having no rational reason or argument for doing so (but simply out of emotional comfort). I was emerging from my own longstanding agnostic/atheist phase when I read this volume and discovered that "fideism" was about as close to any term I could find for my own newly-evolving belief (and apparently I had some good company).
Gardner's near-infatuation with Spanish philosopher Miguel de Unamuno was another surprise from "The Whys..." book. Prankishly, and no doubt realizing how many of his acquaintances would be shocked by the contents of the volume, Gardner wrote a scathing tour de force critique of the book, under an alias ("George Groth"), for the NY Times (only at the very end of the biting piece is the shenanigan divulged) -- one of his all-time best stunts (and there were many), apparently at the behest of Doug Hofstadter.
If you're interested in Gardner and haven't read these two volumes, they go on your to-do list.

I'll briefly overview the current autobiography's content, touching upon a few high points:

Chapter 1 begins with some almost random reminiscences from childhood, and establishes the author's penchant for dropping in little sidebar musings along the way. In telling of his mother's love of rainbows, Gardner reflects:

"Now that I am an old man, my heart still leaps up when I, too, see a rainbow. It made a high leap one morning when I saw a secondary bow. The wonderful thing about a rainbow is that it is not something 'out there' in the sky. It exists only on the retinas of eyes or on photographic film. Your image in a mirror is similar. It's not a thing behind the looking glass."

The next 3 chapters take us through Gardner's high school years (where he didn't get great grades, by the way), before chapter 5 has him heading off to the University of Chicago, where he would major in philosophy. Chapters 5-9 cover his life in Chicago (academic and beyond). Robert Hutchins was the then-famous President of U. of Chicago (creator of the "Great Books" movement), who developed many critics along the way… including Gardner. Gardner also reflects on Mortimer Adler, Richard McKeon, and Charles Hartshorne, three more philosophers at Chicago toward whom he is not particularly favorable. He was an acolyte though of Rudolf Carnap who visited Chicago. (...Ray Smullyan, Doug Hofstadter, John Conway, James Randi, Ron Graham, Persi Diaconis, are among the many others who do fare well in this chronicle.)
Gardner entered Chicago "in the grip of a crude Protestant fundamentalism" as he puts it, but there (as often happens in college) he 'lost his faith.' 
One particular story in these chapters helped explain an oddity to me:  One of the most unusual finds I ever made in a thrift store book section was a 470-page volume Gardner authored on the Urantia Group, "Urantia: The Great Cult Mystery" -- one of the oddest cults of all time; one based upon the bizarre 2000+ page "The Urantia Book." Why Gardner bothered to plow through 2000 pages of craziness and devote time writing an entire book to debunking the cult (and by his own admission he knew there would be limited audience for such a tome) I could never fathom; but in chapter 7 he explains (almost embarrassingly) that the Seventh-day Adventist origins of the cult is what drew his attention because he'd once been attracted to the religion himself, even though he admits the book project was probably a waste-of-time.

 For all my admiration of Gardner I sometimes found the severity of his skepticism objectionable. Several topics that he viewed as nonsense or rubbish I'd be less completely dismissive of (always leaving a crack open for more future information). My biggest disappointment with Martin was his complete rejection of "General Semantics." In chapter 8 he briefly discusses the subject (which he's covered more extensively elsewhere). His blunt criticism of its founder, Alfred Korzybski, may be partially on target, but I believe his negativity toward the more general movement was greatly off-base.
Where I attended high school a "course" in general semantics, based on S.I. Hayakawa's popularization of it, "Language In Thought and Action," was a one-semester requirement (embedded into an English course) -- to this day I consider it the single MOST IMPORTANT course I took in my entire academic life!; it perturbs me that Gardner (and so many others for that matter) failed to see the societal significance of it.  So while Gardner sang the praises of many literary figures for reasons I could barely fathom, he dissed one of the most important schools of thought I'd ever encountered (although he wasn't particularly critical of Hayakawa as one of its proponents)! Moreover, General Semantics passes along some of the critical thinking skills that makes one less susceptible to the very sort of 'pseudoscience' Gardner spent his life combating (but admittedly, Gardner, not I, was in the majority, as GS's strongest proponents over the years have been few-and-far-between).

Chapter 10 covers Gardner's turn to journalism and writing (having decided he didn't want to teach philosophy) and taking his first job in Tulsa, Oklahoma. Chapter 11 veers off to tell us about his parents (mostly his dad) and mention of brother Jim and sister Judith is also made.
Chapters 12 and 13 cover Gardner's time (4 years) in the U.S. Navy, which, to my surprise, he seemed to enjoy quite satisfactorily, and where he learned the nature of, and some control over, migraine headaches that had plagued him. He also tells amusingly about learning to 'toss in four-letter words' in conversation while in the Navy.

From the Navy, Gardner returned to Chicago where he wrote fiction for Esquire and then, believe-it-or-not, wrote for the children's magazine "Humpty Dumpty." He mentions that the last sentence of one rejected Esquire story, about a man who commits suicide, ran something like this: "The cut was a perfect geodesic, the shortest and simplest curve joining two points on his neck" -- ah hahh, hints of the Scientific American columnist yet to come!! The big break to SA came in 1956 when they published his first famous piece on "flexagons." ...And the rest, as they say, is history! Gardner explains, touchingly, that his own mathematical ignorance helped him, as he had "to struggle to understand what I wrote, and this helped me write in ways that others could understand."
Chapter 15, recounting so many of the highlights of his time with Scientific American is a wonderful read, and a great walk down memory lane for all who followed Gardner during those years. (By the way, I noticed that an early Rouse Ball puzzle book that Gardner references as one of his own early math recreation inspirations is freely available on the Web here: http://www.gutenberg.org/files/26839/26839-pdf.pdf ).

Chapter 16 delves into Gardner's "skepticism" and his unrelenting censure of "pseudoscience."  He was an early member of what today goes under the name "Committee For Skeptical Inquiry," a major "skeptics" group that he played a leading, vocal role in. I joined that group early as a dues-paying member, but dropped out some years later when it seemed clear to me that the Committee was not so much interested in "skepticism" as in "debunking" -- there's a difference -- true skeptics will still investigate a phenomena with open-minded objectivity in search of the truth, whereas debunkers go in with their minds pretty much already made-up and a specific purpose to debunk (one of the founding members quit the group around the same time and for the same reason I let my membership lapse -- I should add that I still have admiration overall for the work the Committee does). Moreover, the group aimed most of its fire at fringe or pseudo science where the pickings were easy, instead of academic science where it takes more bravery (and work) to call out critical shots.  True skeptics have to also be skeptical of science, of evidence, of data, of human endeavors, and even skeptical of skeptics! A lot of very weak/poor science creeps into so-called 'evidence-based science' and the line between the latter and 'pseudoscience' is simply blurrier than most pretend. In short, I often see 49 shades of gray ;-) where Gardner and others regularly painted things in black-and-white. It was, by the way, the sharpness and strict rationalism of Gardner's skeptical stances that caused many to be stunned by his religious disclosures in the "Whys … Scrivener" volume.

Chapters 15 - 21 are the volume's best parts (or perhaps just seem so because they cover the years when most of us were aware of Gardner). In addition to his SA experience, he covers his wife, family and friends a bit, and then in the very last (21st) chapter ("My Philosophy") recapitulates some of his deepest philosophical views, including mysterianism and theism -- I consider it the best chapter of the book and would almost recommend reading it first, except that it's likely better, as icing on the cake, after having read from start to finish. He classifies himself as a "Democratic Socialist" and names Norman Thomas as one of his heroes. I also enjoy his little rant against "modern art" in the final chapter. (An afterword to the book, by the way, is written by Gardner's long-time friend and fellow magician/skeptic James Randi.)

A couple of miscellaneous final remarks:

 Colm Mulcahy noted in a podcast interview that the headshot photos used for Gardner books always displayed a serious, dour-looking, even sour, face (and the same is true of the photos in this volume), even though in-person Gardner was warm, friendly, and puckishly humorous -- I've always presumed that Martin picked out those pics himself, and it says something about how he saw himself (but just my guess).

That Gardner lacked formal education in mathematics (taking no math courses beyond high school) has always been an inspiration to me, as a math blogger lacking the pertinent academic background for my own online endeavor. There are of course other math and science popularizers who cover their subject despite a lack of specific academic credentials, but Gardner was in a league of his own.

My only two disappointments with this volume are that 1) Gardner doesn't discuss math Platonism to any significant extent; a subject I love to hear him expound on; and 2) the book is too short! I could've easily read another 100 stories/anecdotes/memories from his past.

This particular volume, variously described by Gardner himself as "slovenly," "rambling," and "disheveled," probably won't win any national book awards for writing, but to his many fans it is most assuredly a prize!
Gardner softened his skeptical rationalism enough to carry a belief that there might be some sort of afterlife... a view many of his readers likely doubt (but perhaps he and Harry Houdini are playing cards somewhere as I write this ;-)). He certainly remains alive via the extensive legacy of books he left providing reading/thinking/learning material to last our own lifetimes. And thank you Martin for this last, final, further peek into your brilliant, fertile, curious, nimble, incisive, probing, captivating life and mind...

Finally, in case you wonder where the title of the autobiography stems from, it is from a stanza of poetry (actually known as a "grook") of scientist/writer Piet Hein that Gardner opens the book with:

"We glibly talk of nature's laws
but do things have a natural cause?
Black earth turned into yellow crocus
is undiluted hocus-pocus."

[On or around October 21 (his birthday), in various cities worldwide, annual "Celebration Of Mind" events take place in honor of Martin Gardner.]

Gardner Sans Math

There's so much I want to say about Martin Gardner's new autobiography ("Undiluted Hocus-Pocus"); it will take much more application of pen to paper (or pixel to screen) to cover much of the ground, but a few broad-brush remarks today.

First, if you're a Gardner 'groupie' (and by that I mean someone who has enjoyed some of the full breadth of Gardner's prolific writings), of course, OF COURSE!!, read this book. It will be like having Martin over to sit in your living room in a big ol' lounge chair by the fireplace and casually pass along stories of his past to you. BUT (and this is where it gets tricky), if you ONLY know of Gardner through his recreational math writings, or if you're too young to even know those and only know him as a famous name, well, I'm less certain just how much joy this volume will bring. If you like biographies, you may still enjoy it, though it is not the rollicking tale some may want in a good biography. Gardner very largely led a 'life of the mind,' not a life of high, flashy adventure; almost a sedate life in many ways. And there is paltry little of mathematics in these pages (much more on philosophy, religion, literature), though he does relate stories about some of his most popular, well-known columns for Scientific American (his entire time spent with SA though, is but a fraction of the book). So if you're in search of mathematics or rousing life stories, you just might be disappointed. This is a quiet, mostly soft-spoken (even understated) chronicle from a man in his 90's looking back and re-telling scattered memories (no doubt because some people advised him there would be an audience for such a recounting). In his humility, he calls the volume at one point "this slovenly autobiography" and then later his "disheveled memoirs." His fans will love it; it's just harder for me to predict the reaction of those less admiring or knowledgeable of him in advance.

The book reminds me just a tad of Michelle Feynman's compendium of her famous father's correspondences, "Perfectly Reasonable Deviations from the Beaten Track: The Letters of Richard P. Feynman," another volume that contained many mundane, innocuous elements, but also so many bits of pure lovable Richard Feynman -- probably not very meaningful or engaging for a non-Feynman fan, but a treasure-trove to relish for anyone who is one.  Gardner's volume also has a fair amount of run-of-the-mill material, but always peppered with his thoughtfulness, intellect, and humor (no knee-slapping guffaws here, just anecdotes that put a twinkle in your eyes and a curl at the corners of your lips).

This book also reminded me of all the things I disagreed with Gardner on, yet how much I enjoyed reading his viewpoint even when I bristled at it (and there was of course far more to agree with than disagree with; moreover, we both regard ourselves as "Mysterians"). Those are some of the things I might touch on in a broader, more detailed review later. And then too I always found his eclectic taste in literature almost bizarre, but that too just made him all-the-more interesting a character to me. This book pulls back the curtain and reveals a little more of the almost inexplicable, unpredictable wizard that was Martin Gardner. He was not merely a man of numbers, but a brilliant man of letters and thought. And 3+ years after his death this book arrives as truly a gift.
Thank you, thank you, to Princeton University Press for bringing it to us, and at a future point I'll say more about its contents.


ADDENDUM: my much longer, more detailed review of Gardner's book is now posted here:

http://mathtango.blogspot.com/2013/10/undiluted-martin-gardner.html

Shecky Riemann... NOT Bernhard's Great Grandkid

 Math-Frolic Interview #17 

 'I'm not sure I'd want to belong to any club that would have me as a member.'
-- Groucho Marx

Several of the question-sets I've sent out to potential interviewees of late haven't been returned, so I'm turning my attention to someone I knew I could count on; someone dependable, reliable, patient, and wise beyond his years ;-) …Shecky Riemann of course!!

Yes, I'll be interviewing my alter-ego today for the satisfaction of all those who occasionally send inquiries via email. ...Without foither adieu:

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Well, Shecky you astute, dashing gent you, how ya doin' these days?

Pretty good, let me just get my teeth back in and I'll be right with ya... ;-)

1) So, where did the name "Shecky Riemann" come from anyway?

When I started "Math-Frolic," inspired at the time by the recent death of Martin Gardner, I knew I wanted "Math" in the name of the blog, yet didn't want that word scaring people off (especially the sorts of lay people I sought as readers). So I picked a title ("Math-Frolic") that I thought sounded light-hearted and non-threatening, and then wanted to reinforce it with a humorous-sounding name for myself.
Bernhard Riemann was one of my favorite mathematicians, and I'm of an age where I associate the name "Shecky" with a number of stand-up and Borscht-belt comedians from my youth. For me, "Shecky" is a comic's name, and I immediately thought "Shecky Riemann" had a humorous ring to it, so it was born.
After the blog went up, I was surprised to find that many didn't catch the intended humor, and thought I might really be related to the famous Riemann (I WISH!), so I had to be more clear that it was strictly a pseudonym for entertainment purposes (the name still gives me a chuckle!).

2) What in your background brought you to math-blogging? And what is your 'day job'?

Well, not much… that is to say I only got as far as a year of college calculus before being diverted to other things academically (B.A. psychology Pomona College; M.A. Communications plus some doctoral work).  Nonetheless, since childhood I've been fascinated by both mathematics and mathematicians. As indicated above, that latent fascination was re-kindled when Martin Gardner died, and I recalled all the joy his writings brought me in years prior. I'd also been reading/enjoying Sol Lederman's "Wild About Math" blog for quite awhile which convinced me that a math blog which was "fun and accessible" and directed at non-professionals, could attract an audience. Still, I didn't expect "Math-Frolic" to last even a year (I've done LOTS of short-lived blogs in the past), but lo-and-behold, 3+ years later I continue having a blast with it, and loving the contact it provides me with 'real' mathematicians.
One of the great things too about math-blogging is that so much of the material is timeless... where the posts of say a political blog may already be obsolete in a month's time, math bloggers can write about the Pythagorean Theorem, 2000 years after the fact, and still be saying something that's both true and interesting!
As far as a day job, I've spent most of adulthood as a lab technologist in different areas (primarily clinical genetics), oddly unrelated to my prior academic work, but for some time now have done odd jobs. If the right opportunity arose I'd probably return to a more sciencey/math environment.

3) How did your interest in mathematics originally come about?

I really have no idea, except that when I was very young my mother would read bedtime books to me and, unlike other kids, I gravitated to the books that were most-filled with numbers or even arithmetic. She kidded me in later years that the books I asked her to read would put HER to sleep, instead of putting me to sleep, as intended! :-) Math was easily my best subject from early on through high-school, before it fell by the wayside.

4) What are your favorite aspects of mathematics to study or read about?

When I was younger it was certainly geometry… the beauty and logic of geometry I think initially  attracts a great many people. As I grew older, and especially as I got acquainted with the ideas of Cantor, Gödel, and others, I've become a lot more interested in the foundations of math and in mathematical paradoxes and uncertainty. I think this is sort of a natural progression, from the seeming precision and certainty of something like geometry to the abstraction of what math (and even "knowledge" more generally) is at a deeper cognitive level; it even ties into other interests I have in psycholinguistics and semantics.

5) Is your blog principally "a labor of love" or is it more than that for you?

For the surprising number of hours that it takes up, accompanied by very little compensation gotten out, yes I think of it as a 'labor of love.' It also provides me, in an odd way, with a feeling of returning to childhood, and what interested me so much way back when. I've said before that I think mathematicians, more than most folks, are eternal children-at-heart (who see the whole world, indeed the Universe, as a playground for exploration)!

6) How do you select the topics you post about?

no real rhyme or reason, except of course picking things that I personally find interesting (and not too technical) and wish to pass along. There are large swaths of math that don't interest me, which means a lot of topics, which are perfectly valid blogging fare, don't get covered by my blog.
When I began blogging, my plan was to avoid delving much into math education, since I'm not an educator myself, and there are SO MANY other bloggers doing a good job covering that controversial area… still, that arena is so fascinating and important, I often do find myself doing education-related posts; it's always a hot-topic.

7) You actually run two math blogs now, "Math-Frolic" and "MathTango;" can you explain the differences between them?

Yes, it's a little crazy that someone with my background is actually doing two math blogs! (Go figure!) Math-Frolic started as what people often call a "link-blog" -- a blog primarily linking readers to other posts/pages of interest… with all the right-hand column side-links I include, I also wanted lay readers to use my blog as a useful/informative math "portal."
I average over 5 posts/week at Math-Frolic which is a LOT. MathTango was started specifically so I'd have a repository to put up some much longer posts that could sit there fer a spell and 'percolate;' it only averages about 3 posts per month. I like the combination and flexibility that affords me for now, but can imagine that someday, I may post fewer but longer posts at Math-Frolic, and the two blogs might be re-merged into one.

8) Are there certain blog posts you've done that stand out for you as personal favorites or ones that were the most fun to work on? And from the other side, which posts seem to have been most popular for your readers?

I have to admit my readers and I have DIFFERENT tastes! :-( My own favorite posts at Math-Frolic tend to deal with paradoxes and philosophical conundrums or puzzles, but often the greatest traffic, I s'pose understandably, comes from coverage of specific timely math matters that are in the news, or alternatively, humorous posts frequently draw a lot of hits (even though I often think of them as trivial space-fillers!).
At MathTango I've had special fun writing occasional off-the-cuff commentaries about some matter weighing on my mind, but those don't usually get nearly the traffic (one recent exception though was this one on 'skepticism') as interviews with 'name' mathematicians or certain of the book reviews … I very much enjoy doing the interviews and book reviews too, just not sure that my favorite ones coincide with reader choices.

9) You bring up a lot of popular math books on your sites, but when you're not reading math what do you like to peruse, and what are your other interests/activities/hobbies?

I'm strictly a non-fiction reader, and in last few years pretty much restricted to math and science! (I wish I could enjoy fiction more, but really I blame mandatory high school English lit for turning me off to fiction! -- in my mind, I think I wondered, 'when will I ever use this?' ;-) I don't even like science fiction which so many of my science friends savor). As far as other hobbies, I'm a birdwatcher (and like other animals as well), enjoy some hiking, tennis, flea markets, and generally pretty simple things.

10) Any parting words, not covered above, you'd want to pass along to a math-oriented audience?

just that cyberspace has opened a world of mathematics that really wasn't accessible when I was growing up… it's a WONDERFUL thing to witness and to have at one's fingertips... I hope anyone with even an inkling of interest in math takes advantage of it. I just wish I was now 9-years-old, instead of, uhhh, well, 39+.

Well, thank you Shecky, for doing what you do so well... talking to yourself! 

....and with that, Shecky sauntered off, mumbling something about turning coffee into theorems.

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[....Let that be fair warning folks -- this is what can happen if people DON'T return their interview questionnaires!]