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.

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

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

Colm Mulcahy... Let the Mathemagic Begin

Math-Frolic Interview #16 

"Follow your dreams while doing something worthwhile.  Share your toys. Cherish your family and friends..." -- Colm Mulcahy

For any who don't know of Colm Mulcahy I'll let him introduce himself to readers through the synopsis he sent along as part of this interview:

"Colm Mulcahy is a professor of mathematics at Spelman College, in Atlanta. Over the last decade, he has been at the forefront of publishing new 'mathemagical' principles and effects for cards, particularly in his long-running bi-monthly Card Colm for the Mathematical Association of America (MAA). He also blogs at the Aperiodical and the Huffington Post. Dr. Mulcahy has been a recipient of the MAA’s Allendoerfer Award for excellence in expository writing. His interests are broad, ranging from algebra and number theory to geometry. He earned a B.Sc. and M.Sc. in mathematical science from University College Dublin and a PhD from Cornell University for research in the algebraic theory of quadratic forms."

Here's a summary of places on the Web you can find Colm:

Twitter: @CardColm  https://twitter.com/cardcolm
author of "Mathematical Card Magic: Fifty-Two New Effects"
http://www.cardcolm.org
http://www.spelman.edu/academics/faculty/colm-mulcahy
http://www.maa.org/columns/colm/cardcolm.html ("Card Colm")
http://www.huffingtonpost.com/colm-mulcahy/
http://aperiodical.com/category/columns/maths-colm

One of the delights for me in interviewing Dr. Mulcahy, was that he was friends with Martin Gardner during the last decade of Martin's life, and so I took the liberty of inquiring a lot about Martin (almost like getting two interviews for the price of one! ;-) -- especially timely, since the annual "Celebration of Mind" (in tribute to Gardner) comes up next month. Hope you all enjoy these great responses as much as I did:

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

1. You're both a magician and a mathematician… which interest came first, and have the two interests intertwined most of your life? Also, at what point did you know you wished to pursue math professionally?

I came to magic very late in life, in my early 40s.  Hence, combining that with my natural laziness, I’m a terrible magician.  Had I started at 14 like the rest of them, today I’d be good at double lifts, false counts, Charlier passes and the like.  So I have to resort to---gasp!--- mathematics, to entertain with a deck of cards.  What I do (and what’s in my book) is largely original creations. Forget clichés like dealing into three piles, basic addition or subtraction disguised, or casting out nines.  Those are hackeyed and pretty boring in my view!

As it happens, the only sleight-of-hand I can do is a perfect faro shuffle.  That took me a month to learn to get it right, after an intensive one-on-one tutorial from the ever-patient Mark Setteducati a decade back.  The only reason I stuck at that was because the mathematical possibilities fascinated me.

But what really fascinates me are the possibilities with a genuinely shuffled deck, or a slightly rigged deck, even after a riffle shuffle done by a spectator.  That’s what I’ve come to specialize in.  And none of my creations take advantage of perfect faro shuffles; that would put too much pressure on myself. Of course I take a dim view of crimps or other card marking.  I will include a little false shuffling, or tell a few white lies such as “I couldn’t possibly know what these cards are” for entertainment purposes, but the underlying principle of the magic I do has to be mathematical.

As regards my real career---the day job!--- when I started university as a teenager, I didn’t know that it was possible to focus just on mathematics, which was the only subject that made total sense to me then.  I had assumed I’d end up in science, like my brother before me, probably physics in my case.  But after six months I discovered that I could soon drop all yucky labs and do nothing but pure and applied mathematics coursework—what a revelation!   (This was in Ireland, where it was assumed that you already had picked up a decent liberal arts education in secondary school.) I went for it, hook, line and sinker, and never looked back.  I believe I’m still a student at heart, and a life-long learner.  The only real job I’ve ever had was the three years I served as department chair. For the past few decades I’ve actually been getting paid to study---I hope nobody finds out---and also to share the fun with others, they call that teaching, writing, publishing and presenting.

2. You're originally from Ireland… can you say what brought you to the States to begin with and do you get back to Ireland often?

I came over after doing my masters, back then it was impossible to get funded to do a PhD in Ireland.  I've been lucky to be able to get back often over the decades, at least twice a year these days.  Maths Week Ireland (www.mathsweek.ie) sometimes have me over, it's the largest mathematics outreach programme in the world (http://www.huffingtonpost.com/colm-mulcahy/math-weak-try-math-week_b_2008517.html), with over 130,000 kids getting involved last year. I spoke to 1000 of them myself, up and down the length of the country, in the course of my gruelling but fun 9 day "fall break".  The Maths in the Street activities are a hoot, because you get to engage curious people aged 8 to 80, who just go out to do their shopping and they get home with groceries AND a knowledge of the Towers of Hanoi, how to solve a maze, or a card trick they can do for the family.
[--We need more of that in the U.S.!]

3. You were a friend of Martin Gardner through the last decade of his life… are there any behind-the-scenes stories about Martin you can share that might interest my readers?

Martin was amazing. Humble, incredibly focussed and hardworking and productive.  So kind, and generous with his time and wisdom. We chatted on the phone and corresponded on and off from 2000 to 2006, when I finally visited him for the first time. By then he was in his 90s, and probably not as sharp as he’d been a decade or two earlier.  He was certainly a little forgetful: sometimes he didn’t remember that he’d already mastered something and written about it, but he could look it up quickly in his amazingly organized files, and it would come back to him right away.  And his less robust state didn’t stop him writing a half dozen books in his last five years, including the ace up his sleeve, his autobiography.

Back in 2000, I’d sent him about 50 Latexed pages of notes I’d made on mathematical card tricks---Latex is what most mathematicians use to typeset their work---largely stolen from assorted publications of his.  He responded with enthusiasm, and suggested I write a book on the topic.  I was flattered of course, but knew I’d have to come up with a lot more original material for that to be a reasonable proposition.  I set myself a personal goal of getting it published by 2006, the 50th anniversary of his landmark "Mathematics, Magic and Mystery," and found a willing (university) publisher.  I’m deadline driven: without one, a lot of things don’t get done.  With a deadline, they eventually get finished, although seven years late in this case, and with another publisher.  Along the way I decided to try to switch to the premier outlet for recreational mathematics, AK Peters (now part of CRC Press).  I was very happy that Klaus Peters offered me a contract and that I finally delivered the goods this year, before books become obsolete.

4. Gardner was surprisingly unpredictable in some of his viewpoints and interests… in your time with him was there any particular view he held or interest he had that most surprised you, or that the two of you debated over?

He tried to engage me in a debate once about whether I thought mathematics was discovered or invented, which is of course an old conundrum, and one he had definite views on.  Being the shallow person I am, I refused to be drawn in.  It never really interested me, though of course it should have. In more recent years, I have asked myself the same question in relation to mathemagical principles which I have stumbled on.  Did I create them out of nothing, or did I just get lucky and find them because I was kicking around in the leaves where nobody else before me had?  I still don’t know.  I probably should have paid more attention to Martin on this topic.

Following his death, many tributes assumed, as I myself had in earlier years, that he was either agnostic or atheist.  Not so.  I wouldn’t presume to speak on his behalf---we never discussed it and his views here are well documented in his own writings---but I believe he was a theist who did not believe in any organized religion.  That surprised me when I first realized it.
[-- Gardner covered this, and 'outed' himself as a "fideist," in one of my favorite volumes of his, "The Whys of a Philosophical Scrivener" -- a bit of an oddity among his works, and in some ways a dry read -- yet, made fascinating, by the chance to see how his mind worked on a very philosophical (less-empirical) level -- also, Gardner's high regard for Spanish philosopher/writer Miguel de Unamuno is made clear in this volume.]

One sweltering March day when I was visiting him, I spotted a curious object hanging up over the sliding door to the outside, and I asked him what it was.  It was a small musical saw. He confessed to playing it on rare occasion for relaxation.  Despite my gentle encouragement he made it clear that there would be no public or private performance, and I felt a little embarrassed, as if I had intruded on his privacy.

I never wanted to bother him unless I had something to share which I thought he’d really appreciate knowing about, but I did have a habit of phoning him now and then while waiting to board flights. The very last time I did this—about six months before his death---I ended the call as I often did by asking if he’d had any interesting visitors, and he replied in the negative, as usual.  After some prodding he perked up, and said he’d had one surprising visitor, which baffled him: Richard Dawkins.  He added something like, “I don’t know why somebody famous like that came to see me.”  I retorted, “Because YOU’re famous!” but he wasn’t having any of that.  I went on, pointing out that Dawkins had written much about theology and philosophy, just like he had.  He revealed that all they’d done was chat about life, and grandkids and baseball---Dawkins had just given a talk at the university nearby---until  his visitor stood up to leave because he had a plane to catch.  Then they both realized that there was a big topic they were supposed to discuss, and they sat down again and chatted intensely for 15 or 20 more minutes.  At this point I had a plane to catch myself---I was about to fly from Boston to Dublin---and I never got to hear the rest of the story.  Or, sadly, Martin’s voice again.  I'd grown very fond of him.  Such a sweet unassuming man, for such an intellectual giant.

[-- Wow, fascinating Dawkins story… easy to speculate that they could've had an 'intense' chat over the nature of religion or religious belief, but purely a guess; no doubt many possibilities. If any readers, perchance, know Dawkins well enough to ask him about it, I'd LOVE to hear what the "the rest of the story" is!
One of Gardner's most remarkable qualities (in my view) was his balancing act between being a strong skeptic and harsh critic on-the-one-hand, while also being humble/"sweet unassuming" on the other... opinionated, but not in a know-it-all kind of way; curious and uncertain of many things to the end.]

5. What are your own favorite aspects of mathematics to study or read about, and if you could only take say 3 or 4 math books with you to a desert island what might they be?

 I like surprises and counter-intuitive things which every graduate in all disciplines should know, such as not-so-obvious averages (http://www.huffingtonpost.com/colm-mulcahy/mean-questions-with-harmonious-answers_b_2469351.html) and non-Euclidean geometry. Most of us are smart enough to know that the earth isn’t flat, but is space flat?? I especially like things that teach us to distrust simple “numerical evidence”:  The false positive paradox. Simpson’s paradox. The Birthday Paradox.  Benford’s Law.  The Gilbreath Principle. There are probably five more of them out there that we all need to know about.  If anyone knows what they are, please tell me.

I really should have a good grasp of the basic history of mathematics. "Math Through the Ages" by Berlinghoff and Gouvêa is probably a great place to start.  I’m particularly interested in knowing about the contributions of all cultures around the globe, even if much of that was hidden until recently, or doesn’t “fit the mold.”

I’m woefully ignorant of topology, but if Pontryagin and Morin (both blind) could contribute so much to the subject, it wouldn’t kill me to learn a little.  I never really got quantum mechanics as a student all those years ago, it was the one topic I faked understanding, perhaps taking false comfort from the fact that Einstein didn’t buy it either.  I’m not that attracted to it, but if it was an undergraduate topic in the 70s, I should be able to handle it.

It’s time I understood the whole mathematics/music connection.  It might help if I actually knew something about music, but I’m strictly an avid consumer. 
[-- yes, I was surprised back in college days by how many music majors I ran into with math minors, or math majors with music minors! -- definitely an interesting connection]

Of course we should all study Diaconis and Graham’s "Magical Mathematics" daily.  If I could master half of the tricks in it, I’d die happy.

6. Do you have one mathematical-related achievement from your life that you're most proud of above others?

I suppose getting my first book out, twenty-five years after I first considered writing one, and after starting and abandoning two others!  The one just published bears no resemblance to anything I would have considered writing even fifteen years ago, and the people who trained me probably wouldn't recognize it as having much to do with mathematics, but it contains a lot of fresh creations. I hope it turns some youngsters on to the joys and possibilities of mathematics.

Starting in 2012, I've been blogging for the "general public", and not just for card magic aficionados as I do in Card Colm at maa.org.  I've written some things for the Huffington Post http://www.huffingtonpost.com/colm-mulcahy/ and also for The Aperiodical in the UK. http://aperiodical.com/category/columns/maths-colm  In recent times, both have taken a back seat due to the focus I needed to bring to getting my 380-page book manuscript to the printers, set up appearances and book signings and so on, but I plan to get back to that kind of blogging soon.

There's a big world out there that needs to have a better attitude about mathematics.  We've never had the best PR, and I aim to play a role in improving the image of the subject.  I've always had a big issue with the claim that people are either numbers or words people.  Many of us are both.  Some mathematics teachers are reluctant to correct basic spelling and grammar in their students' work, for instance, either on the basis that it's not their job, or it doesn't matter. Wrong!  It's all about communication, about being able to justify your position with a cogent argument using words AND numbers correctly. A string of numbers presented without accompanying text is hardly convincing, and may in fact mask the presenter's confusion or ignorance.  We all need to have a healthy skepticism of numbers and "experts" and learn to make informed decisions ourselves.  And be willing to change our minds in the light of compelling fresh evidence. Ironically, education is no inoculation against frauds and charlatans, whether they're peddling hokey stats or snake oil.  James Randi can confirm!

7. When you're not reading math or magic, what other types of reading do you like? And what other hobbies or activities do you enjoy?

Irish fiction. World history. Cookery books. Music, music, music.  I cook a lot and indulge in craft beers and chocolate more than is good for me.  I jog and am trying to bike more often, to atone for my sins.  Actually, I’ve always found that such outdoors exercise stimulates the mind.  Over the past ten years or so, some of my best ideas for new Card Colms have come to me while puffing and panting.  I am not alone in that regard (see http://www.huffingtonpost.com/colm-mulcahy/sports-and-mathematics_b_2475226.html ).

8. And back to Martin… You've been heavily involved with the "Gathering For Gardner" and "Celebration of Mind" (coming up soon in October) events which honor Martin Gardner's legacy.  Most Gardner fans are probably familiar with these events, but can you give a quick synopsis of them for anyone unfamiliar? And do you have a particularly memorable experience from one of these events that stands out for you over the years?

G4G is an insider and by-invitation-only conference held in Atlanta in March of even-numbered years.  They’ve being going on for 20 years.  They  attract very creative and interdisciplinary mathematicians, puzzlers and magicians as well as skeptics, illusionists, Alice afficionados, and so on.  Four or five days of extreme intellectual stimulation and networking.  You might meet some of your heroes, from Randi to Shortz, Conway to Penrose, Teller to Smullyan.   A few years back there, Erik Demaine announced that any two polygonal regions of the same area are hinge-dissection equivalent, meaning that the first can be cut into pieces and these swung around on hinges to form the second.  He got a standing ovation.  I was present when history was announced.  It was awesome.   I was able to tell my students at Spelman soon thereafter when I proved for them the much easier version ignoring hinges.  Who’d have thought it, a new math result proved in their college days that they could at least understand the statement of?

The 'Celebration of Mind' events were started in 2010, a few months after Martin died.  These are open to all, anywhere on the planet.  Anyone can host one or attend one---a few are strictly private but that’s not common.  They’ve happened on all continents and at both poles. They take place in and around October 21, which was Martin’s birthday.  We’d like this to go viral as well as global, especially with Martin’s centennial approaching in 2014.  There should be thousands of them each year, celebrating the triumphs of the human mind with cool stuff Martin made us think about:  recreational mathematics, brain teasers, logical loopholes, rationality, skepticism, optical illusions, puzzles mechanical and cerebral, the mathematics of M.C. Escher and Mandelbrot, origami, the list goes on.  The  www.celebrationofmind.org webpage has terrific resources, check out Vi Hart’s series of four hexaflexagon videos from last year to get some ideas of the fun and depth which lies just below the surface of innocent-sounding explorations.  Please also follow @WWMGT on Twitter!

There’s another celebration I’d like your readers to know about: the theme of the next Mathematics Awareness Month in the USA is “Mathematics, Magic and Mystery,” named after Martin’s groundbreaking book from 1956.  So in April 2014, American school children and students of all ages will be exposed to some cool stuff reflecting the amazing mathematical legacy Martin left behind.

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

Life can be short---just ask Steve Sigur or Kirsty MacColl---and it’s certainly finite.  Follow your dreams while doing something worthwhile.  Share your toys. Cherish your family and friends.  And as Nick Lowe said four decades ago, what’s so funny about peace, love and understanding?
Oh, you said a math-oriented audience… same advice, plus or minus epsilon.
[-- A splendid note to leave on!]

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

Are those some wonderful responses or WHAT!!... THANKS so much for participating here Colm, and just like Martin Gardner may you keep contributing to us for many many years to come! ...or, as they say, 'Luck o' the Irish to you!'

Please do check out Colm's Webpages and book.

(p.s.: I haven't had a chance to check all the hyperlinks above, so do let me know if you find broken/misused ones.)


ADDENDUM:  want to attach a voice to a name and learn still more about Dr. Mulcahy (with some nice detail on his card magic and book, and still more on Gardner)... Sol Lederman now has up a great podcast interview with Colm on his "Inspired By Math" series:

http://wildaboutmath.com/2013/09/21/colm-mulcahy-inspired-by-math-32/

Mathematics… Not Immune

Somewhat oddball topic today… just felt like airing it (bit of a rant on skepticism....):

I'm  sometimes amused by 'scientists' on the Web calling themselves "skeptics" only to find that they're aiming most of their fire at what I can only call "low hanging fruit": creationists who think the Earth is less than 10,000 years old, psychics who bend spoons, UFO abductees who've taken trips to the planet Kazaar (or some such), etc. etc. They often proudly call themselves "evidence-based" scientists… but don't seem to acknowledge that scientific "evidence" itself can be highly subjective, manipulated, and tainted fare... requiring skepticism itself.
What I've always wanted to see is more "skeptics" turning their keen eyes on the likes of the Journal of the American Medical Association, New England Journal of Medicine, CELL, Nature, Science, etc. Most "science" is so poorly done it doesn't even see the light of publication, but even research that does make its way into such 'premier' publications (let alone lesser ones) often escapes the scrutiny it deserves, receiving a sort of 'free ride' once published (the data, methods, underlying premises/assumptions, and of course conclusions, never being adequately challenged, nor replicated; and don't assume 3 peer reviewers have done their job either!). Most who have been heavily involved in research, IF they're honest about it, will admit that journal articles, as composed, often don't accurately reflect the actual work, as carried out. But with the digital age upon us, the needed sort of ongoing, ever-watchful skepticism is finally emerging.

Dr. Ivan Oransky has made a job of keeping track of journal "retractions" (often, but not always, for malfeasance) from research journals, with his wonderful, must-read "Retraction Watch" blog (sub headed: "Tracking retractions as a window into the scientific process"). And a recent 'Peer Review Congress' meeting in Chicago, well-tweeted (hashtag #pcr7) by Oransky and others, shined light on many of the important research issues that don't get voiced enough… and these are NOT new, but have been around for a long, long, long time, just finally getting the attention deserved. Anyone who has followed this area will be familiar with the name "John Ioannidis" as one of the most-vocal of those who've questioned the reliability/validity of much research. There are also now a number of twitterers (some better than others) focusing on biomedical/research skepticism, though 140 characters isn't always much to work with!
My sense is that a majority of the retractions that Oransky reports on involve the biomedical sciences, but even mathematics papers or journals occasionally appear among his subjects (sometimes almost laughably). So math, the 'queen of the sciences,' is not immune from sloppiness (or even fraud). Check these out:

  http://retractionwatch.wordpress.com/?s=mathematics

WHY the heck do I mention all of this now? Well, nothing earth-shattering, but in preparing the John Casti quote for the post that I did last Sunday on Math-Frolic (thought it would make a nice Sunday meditation), I came across information that disturbed me. Casti (who focuses on "complexity" theory) has long been one of my favorite science writers/thinkers; he is both a good explainer, and a thought-provoking one; I've read him for decades. So I was surprised/disappointed to learn from the internet that he'd been caught extensively plagiarizing material in the past -- multiple times... that doesn't make his material any less interesting or important to me (he plagiarizes from good people ;-), but it does throw a troubling ethical shroud over that material. Here are 3 web links that address Casti's sins:

http://www.nytimes.com/2002/03/09/books/connections-plagiarism-that-doesn-t-add-up.html
http://www.ams.org/notices/200206/commentary.pdf
http://onlinelibrary.wiley.com/doi/10.1002/cplx.20050/pdf

Last year, budding science-writer phenom Jonah Lehrer was disgraced and humiliated when he was found to have plagiarized, and even fabricated, passages in some of his best-selling books. His actions (and Casti's) are an affront to the many high-quality science communicators who choose to write books the old-fashioned way… by actually generating their own words ;-)  Casti's lapses may not be as grievous as Lehrer's, but it will now be difficult for me to read him with quite the same respect I did previously -- his repeated actions being such an unexpected/disheartening bit of chicanery to learn about.
Moreover, it harkened back to my prior MathTango post, wherein I wrote about learning that some observers suspect that Daniel Tammet (renowned autistic savant, and another person whose books I've enjoyed) may not be a genuine savant, but only a memory expert who uses his mental skills/tricks to portray himself as he does -- again, disappointing to learn that this is even a possibility. Sigh....  I consider myself a fairly critical reader, yet was caught unaware by these controversies around Casti and Tammet. (While I'm on this whole subject it may even be pertinent to note that the post prior to the Tammet post was a review of the new book, "Magnificent Mistakes In Mathematics," coincidentally, yet another focus on flaws-of-a-sort in science/math.)

On the Web, there are plenty of individuals, with or without some math credentials, who stake out math ideas that aren't credible (Mark Chu-Carroll of "Good Math, Bad Math" occasionally writes entertainingly about them, and their math "crackpottery"): http://scientopia.org/blogs/goodmath/?s=crankery ). But to discover shenanigans going on at a more professional level and/or in journals, is more disconcerting and unsettling… though I s'pose no field is immune from flim-flam artists of all sorts, math/science included.

Luckily, pure math really is probably more resistant to outright fraud, or even unchecked sloppiness, than most fields. In fact when I typed "math" plus "fraud" into Google most of the examples arising were, not too surprisingly, in regard to economics or finance. The other area that popped up, again not too surprising upon reflection, was statistics, which often gets used (though not necessarily deliberately) incorrectly to argue some point.

Anyway, I write all of this as a way of saying that real skepticism needs (unfortunately) to cut across all boundaries -- I'd dare say epidemiology can be critiqued almost as easily as astrology! Don't aim doubt and critical faculties at just the naive, the non-empirical, the 'low-hanging fruit'… but at the 'evidence-based,' the peer-reviewed, and occasionally even the mathematical as well. I applaud Oransky and others for bringing a critical eye to "the process of science" and trying to keep scientists not just skeptical, but honest as well. Science succeeds best through its vigilant, self-scrutinizing, self-correcting functions, which, for a variety of reasons, too often get left on the sidelines... occasionally even in mathematics.

In sum, Margaret Mead was famous for saying: "Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has."
I'm tempted to parody that by saying, 'Never doubt that self-skepticism, close scrutiny, and doubt aren't key driving forces behind scientific progress… indeed, they always have been.'

Numbers, Mnemonism, Savantism, Oh My

Well, this is a bit disconcerting….

NPR recently ran an interview with author/autistic savant Daniel Tammet:

http://www.npr.org/2013/08/11/206660281/the-beauty-and-calm-of-thinking-in-numbers

I've enjoyed all three of Tammet's books. My favorite was his second one, "Embracing The Wide Sky," while the current one, "Thinking In Numbers," is a bit less consistent (from essay to essay), though still pleasant... couple of reviews here:

http://tinyurl.com/8llkr35
http://tinyurl.com/myz48jw

When I first became aware of Tammet, years ago, he struck me as a bit peculiar, especially his ability to introspectively and articulately describe his own mental processes (unusual for savants); he had an almost 'Uri Geller' style about himself, but, as he was studied by psychological experts in the field, I fully bought into his storyline. He has traveled widely and appeared extensively on television promoting his books, and there are plenty of examples of Tammet on YouTube as well:

 http://tinyurl.com/lx9oby6
(and even more of course on Google about him: http://tinyurl.com/mteo7sc )

...I've never read Joshua Foer's bestselling "Moonwalking With Einstein," but have always seen excellent reviews of it. So I was surprised to read in the comments to the above NPR piece that apparently in that volume Foer expressed doubt about Tammet's genuineness, believing he may just be a highly skilled mnemonist (memory expert) who passes himself off as a synaesthete and savant (turns out Tammet's real name is Daniel Corney though he had it legally changed over a decade ago, and Foer marshals evidence that Tammet isn't always candid about how he succeeds at what he does).

It's hard to know where the truth lies (especially since 'savantism' itself can be a bit hard to define), but the more I surfed around the Web the more suspicious Daniel's talents (or his personal portrayal of them) seemed. Even though many find Foer's skepticism unconvincing and continue to defend Tammet, others, with knowledge of memory training, do not. One of the most lengthy discussions comes at this forum site:

http://mnemotechnics.org/forums/daniel-tammet-840.html

The above has 6 pages (or 163 responses) of comments to sift through about Daniel Tammet (I haven't read them all myself, but they're not very encouraging for the authenticity of the standard Tammet-savant view). It's enough to make one a tad more leery of those neuroscience "experts" who study and report on savants (a bit reminiscent of scientists who studied and were taken in by Uri Geller decades ago) -- is the reality ever as incredible as portrayed by all the hype?

Still, savantism, including extraordinary mathematical abilities, remains a fascinating subject area, but perhaps one where a focus on Daniel Tammet is not entirely appropriate (of course the tricks of mnemonists are an interesting topic in their own right, just a different topic). Perhaps tellingly, Tammet himself, has long voiced a belief that his talents are not so rare or extraordinary, but rather are a function common to human brains, just not readily accessible to most people... that would indeed make sense if Foer's contention of extensive mnemonist training was the underlying mechanism.

Anyway, sorry if this is all old news for some of you, and you were already aware of the so-called "debunking" of Tammet; it was news to me that, given my occasional interest in savantism, seemed important to share, especially since Tammet is currently getting attention here in the States where his "Thinking In Numbers" volume was only recently widely distributed (it was available through the UK a year ago). I still enjoy the volume... but with at least a bit more of a grain of salt. And I should add, Tammet remains an interesting fellow, whether it be as a true savant, OR as someone who has successfully pulled-the-wool over the eyes of trained specialists (getting a lot of free travel and attention in the process)!


...As long as I'm mentioning books, this might be a good time to point out that Edward Frenkel's "Love and Math: The Heart of Hidden Reality" is due for release in about a month, and I suspect it may possess some of the same deep, haunting joy of math that Tammet expresses, but coming from a true mathematician. From the Amazon description:

"In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space."

Some more about Frenkel here:

http://math.berkeley.edu/~frenkel/Frenkel-Love-for-Math.pdf

New Math Book, Full of Mistakes

…and delightfully so!!

-- Review of "Magnificent Mistakes In Mathematics" -- by Alfred Posamentier and Ingmar Lehmann

Any volume from Alfred Posamentier is to be looked forward to, and the latest one, "Magnificent Mistakes In Mathematics" is no exception (written again with Ingmar Lehmann); a fairly quick and entertaining read for typical math buffs, with a focus not often found in math books… on famous math errors.

The book endeavors to demonstrate that even the precise, empirical field of mathematics has its share of mistakes made by prominent, knowledgeable practitioners in the road to progress. And they note that the very need to examine and explain math 'flaws' is a good thing, often leading to whole new ideas/concepts.
 
The book starts right off putting the reader at ease by highlighting "noteworthy mistakes by famous mathematicians," including such accomplished figures as Pythagorus, Galileo, Fermat, Leibniz, Euler, Poincare, and several others. It's as if to say 'if the remarkable Euler blundered why should YOU dread making math mistakes.'  Many of these errors are well-known, but still interesting, or in some instances even humorous (for example a blackboard mistake by Enrico Fermi that ended up saved for posterity on an Italian postage stamp). Interestingly, at the end of the chapter it is mentioned that the also extraordinary Carl Friedrich Gauss was not known to have made mistakes in his published material.

Chapter 2 embarks on the progressive journey through mathematics with a look at mistakes in arithmetic. This may be the least interesting, or most mundane of the chapters, and is followed by chapters that delve, in order, into algebra, geometry, and probability and statistics errors; a seeming natural progression from the more abstract to the more applied or real-life-type examples. 
Geometry mistakes, of course are often a matter of misperception or interpretation (moreso perhaps than algebra, where mis-computation may be more frequent). In the geometry realm I was very surprised that the volume leaves out one of my very favorite 'mistakes,' which goes around the internet from time-to-time, demonstrating that pi=4: http://www.bestwtf.com/2010/11/explaining-why-pi-is-4.html The sort of error involved, "misleading limits," is documented in the book with other classic examples, but still, the circle-inscribed-in-a-square paradox is so good it ought not be missed.

Of course probability and statistics are perhaps the cause of more slippery mathematical mistakes than any other area, even among professional mathematicians. It is often famously told that even the great Paul Erdös initially had difficulty with the 'Monty Hall problem,' so it is a good chapter to end the book with. Many deceptive probability conundrums of recent times are now pretty much classic, and continue to elicit great debate when heard for the first time.

There are LOTS of types of examples used through the book, demonstrating how varied the sources of math mistakes can be. Having said that, there is also sometimes redundancy to the many examples employed for any one sort of error; but using multiple examples to make a point is not necessarily a bad thing.

For the professional or broadly-read mathematician this won't likely be a highly substantive or weighty math read. The bulk of examples in the volume are well-known, but what is new is bringing them altogether under one cover-to-cover format… an entire book focused contrarily not on what math does right, but on where it may go wrong. I think this somewhat unique approach makes the volume a worthwhile, entertaining addition to one's math bookshelf, and it may be particularly useful to secondary school teachers, providing a lot of grist for instructive, thoughtful examples in the classroom. As the authors repeatedly note, there is a LOT to learn from mathematical mistakes.

In short, a thumbs-up for this volume! Posamentier seems to produce a new book almost every year, and each one simply leaves me wondering what will he come up with next.

By the way, if you missed it, Posamentier did a great "Inspired By Math" podcast interview with Sol Lederman late last year here:

http://wildaboutmath.com/2012/12/16/alfred-posamentier-inspired-by-math-13/

A Book, A Cryptographer, and a Fun Guy

I'd normally post this over at Math-Frolic, but am so past-due to get a post up here... well, here goes:


1) If you're a fan of Mark Chu-Carroll's "Good Math, Bad Math" blog (one of the oldest and most popular general math blogs on the Web) you'll be happy to know his recent book  "Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation" is available through Amazon:

http://tinyurl.com/l6b5jkm 

Here's what Amazon begins by saying about it:

"Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you."

[Haven't read it myself, but assume from Mark's blog writing, it is good.]

2) I've previously mentioned my joy with "The New York Times Book of Mathematics" anthology, and its last chapter is composed entirely of wonderful profiles of accomplished mathematicians... mathematicians fascinate me as much as mathematics itself.  So I'll reach from there again today for this link to Gina Kolata's 1994 (and still hugely interesting) portrait of Leonard Adleman, the "A" in RSA encryption:

http://tinyurl.com/mmh8agy

It starts off thusly:

"There is nothing to look at in Dr. Leonard Adleman's office at the University of Southern California, no clue that the office is even occupied. There are no pictures of his wife or of his three daughters, no cartoons or mementos -- just a computer, two chairs, a desk and a blackboard. And that is fine with Dr. Adleman. For although he is an active faculty member at the university, although he is a devoted husband and father, his is a life of the mind.
"It is a life that is nourished by deep philosophical questions and the overarching beauty of mathematics. It is a life that involves days, weeks, months of pure thought, alone in his equally barren office at his home in Northridge, Calif., 30 miles from the campus. And it is a life that has led Dr. Adleman to play a central role in some of the most surprising, and provocative, discoveries in theoretical computer science."

...and just gets better and better from there.

3) Finally, a blast from the past… if you're ~55 or over a trip down memory lane….
Re-reading an old Jeremy Bernstein volume ("Cranks, Quarks, and the Cosmos") recently, I came across his chapter on Tom Lehrer (unfortunately I can't find a free full copy of it on the internet). Lehrer was a popular, clever, irreverent singer/satirist of the '50s, '60s and 70's before he largely retired from the scene (he was, by his own admission, a huge fan of Danny Kaye, whose style he very much emulated).
During his heyday, I honestly only found his material mildly amusing (there were LOTS of satirists around in those days), and it was only years later that I discovered his background (…which made him a far more interesting figure to me!)
So for any who don't know, Lehrer entered Harvard at the age of 15 and majored in mathematics, graduating at 19; a year later he got his Masters degree (and worked on, but never completed PhD. work). He taught for awhile at Harvard, MIT, and Wellesley. Joining the Army in 1955, Lehrer worked a couple years for some outfit called… drrrrrrumroll… the NSA!

If you're unaware of Lehrer you can still find his work on YouTube:

http://tinyurl.com/m3n43zv

(He's 85 now and his own website is here:  http://www.tomlehrer.org/ )

Anyway, Chu-Carroll and Adleman may well be fun guys as well for all I know, but Lehrer took math and fun to the level of turning it into a living. Nice work if you can get it (actually, a lot of his satire wasn't math-related).

Here's one sample of Lehrer's 'mathematical' work ("Lobachevsky"), to give you his flavor: