Editor's Note: Douglas Hofstadter gave permission to Scientific American to post this essay in light of the death of Martin Gardner, who wrote the magazine's "Mathematical Games" column for 25 years and published more than 70 books. Gardner died May 22, at 95.
I've been trying to reconstruct how I first encountered Martin Gardner. It may have happened in 1959, when at age 14 I happened to visit the home of a boy a couple of years older than myself, who I thought was extremely smart (and indeed he was—he later became a well-known mathematician on the Princeton faculty). While scanning his bookshelves, I noticed a Dover paperback with the curious title Fads and Fallacies in the Name of Science. I pulled it out and my curiosity was further aroused by the front cover, which mentioned such things as flying saucers, human gullibility, strange cults, pseudoscience, and so on. I had of course heard of things like telepathy, ESP, and such, but didn't know what to make of them. Though they seemed a bit far-fetched, they also appealed to my romantic nature. The year before, I had even half-convinced myself that I could discover my romantic fate by spinning a top and seeing where it fell on a marked board; I also enjoyed the thought that maybe, just maybe, the first initial of the girl I would someday marry was revealed by reciting the alphabet as I twisted an apple stem and stopping at that letter when the stem broke off. Why not? At that tender and rather gullible age, I had never devoted much thought to the demarcation line between sense and nonsense, science and silliness.
But in this book, somebody—clearly somebody very intelligent—was tearing one oddball belief system after another to shreds in a lucid, acerbic, yet at the same time humorous way. This "Martin Gardner" person was wielding common sense as a surgeon wields a knife—and occasionally twisting the knife with glee. It was probably the first time I had realized that systematic and critical thinking could extend beyond such precise domains as math and physics, and could demolish ideas in far hazier fields with great power. It was also the first time I had realized how very many crazy belief systems there are out there in the world, and how important it is to recognize this fact and to combat them.
Something about that book impressed me deeply. When I bought my own copy and read the whole thing, my ideas about the so-called "paranormal" were forever changed. In his preface to the second edition, Martin observed that, of the many readers whose ire had been aroused by the first edition and who wrote to tell him so, most focused their venom on the one chapter that attacked their pet belief system, although, curiously enough, thinking the rest of the book excellent. This wry little remark said volumes about human nature, the nature of truth, and the nature of objectivity. I've never forgotten it.
Fads and Fallacies may have been my introduction to Martin Gardner, but another avenue strikes me as equally plausible. My family had subscribed to Scientific American forever, it seemed, and I may well have encountered the monthly "Mathematical Games" column in browsing through an issue. In any case, I vividly remember the day when my mother brought home for me The Scientific American Book of Mathematical Puzzles & Diversions—Martin's first collection of his columns. Reading through that book was a mind-opening experience for me, a teenager deeply in love with math.
Martin had a magical touch in writing about math. His column's title was very humble and in a sense misleading. The word "games," with its lightweight flavor, did not even hint at the depth of the issues that the column dealt with. Theoretically, it was about "recreational math," which sounds frivolous, but in fact the column was about beauty and profundity in math—and in many other fields as well. In each column Martin managed to put his finger on some little-known but profound issue, and to present it in such a clear (and often droll) fashion that its importance just grabbed me and infected me like a virus. After reading a Gardner column, I would often walk around for days with the ideas swimming through my head, like an incredibly catchy melody.
Just to mention a few, there were the Soma cube; the inductive game of Eleusis; 4-D tic-tac-toe; nontransitive dice; the unexpected-hanging paradox; Newcomb's paradox; nonconservation of parity; machines that learn; thinking machines; squaring the square; fallacies in mathematical reasoning; cryptography; knot theory; Nim and its variants; properties of e and pi; random walks; tesseracts; hyperspheres; Fibonacci numbers; Lucas numbers; Catalan numbers; Pascal's triangle; numerological nonsense; Carl Hempel's "all crows are black" paradox; Nelson Goodman's bleen-and-grue puzzle; palindromes; Möbius strips; Klein bottles; spirals; helices; cycloids; conic sections; M. C. Escher; communication with extraterrestrial intelligence; infinity; isomorphisms; graph theory; pseudo-random numbers; curves of constant breadth; superellipses; 4-D space; Zeno's paradoxes and supertasks; the 3n + 1 problem; the game of Life; Fermat's last theorem; mathematical music; magic squares; the four-color problem; time reversal and time travel; Gödel's theorem; topological games; knotted holes—and on and on and on it goes. In truth, I have just scratched the surface!
Lurking behind it all, it seems to me, was Martin's passion for paradox. I would say that more than anything, this passion gave and continues to give to Martin his virtually unerring sense for what is important.
Paradox comes in many forms. Special relativity violates our intuitions about time and space at such a basic level that, even though it is far from being self-contradictory, it seems like absolute nonsense when one first encounters it. Four-dimensional geometry, likewise, posits an unvisualizable space—within which one then gaily plays! The chaos of the prime numbers is almost a paradox—at least it is a deep mystery, and it can provoke and inspire thoughts for as long as one wishes to think about it. Cutting a Möbius strip down the middle and not getting two pieces is as crazy an experience as one could ever wish to have!
How can it possibly be that "All crows are black" and "All nonblack things are noncrows"—two sentences that logically are exactly equivalent—have completely different meanings when it comes to verifying them by citing examples? How can a sentence be readable backward as well as forward? That's nonsense! What is the unfathomable secret to the power of music, and do the never-ending patterns upon patterns of mathematics have anything to do with it? Why are the fundamental laws of nature symmetrical—or are they? What would happen if time started to go backward, or left and right were suddenly switched? Why does a mirror seem to reverse left and right but not up and down? If infinity has no end at all, how can there be different sizes of infinity? How could a tabletop roll completely smoothly on a set of logs all of which have different cross-sections, none of which is circular? It all makes no sense. It simply makes no sense. At least at first sight.
Again, I could go on and on, but the point is that Martin's style is intimately bound up with paradox and mystery—the clear exposition of paradox, but just as much, the clear resolution of mystery whenever possible.
Martin's columns and writings radiate a profound exuberance in the constant novelty of human thought. What comes through, even if it's never explicitly expressed, is a kind of informal version of Gödel's theorem for human thinking—a sense that creative minds will always one-up the pedestrian expectations generated by unimaginative, logic-bound thinking. There is an exultation in the breakout from expected patterns, the violation of seemingly ironclad laws, the making of wildly unexpected connections, the revelation that two seemingly identical properties are really quite different, and the counterexamples that make it all blindingly clear (at least for a moment—then you forget how it worked!)…. If nothing else, reading Martin Gardner should convince you that the human mind's pathways of finding truths are as diverse and unpredictable as the pathways of evolution itself.
Not surprisingly, mixed in with this celebration of the unpredictable is a love for humor and oddities, which are certainly close to the antipodes of pure logic (as is, in a sense, Gödel's theorem). "Mathematical Games" was always bringing to its readers such diverting items as poems written without the letter "e," limericks with the wrong number of lines, astonishing anagrams and numerical coincidences, droll poems on science and math, endless new varieties of wordplay, paradoxical pictures and sculptures, surprises of figure–ground play, and so on. And in April issues, Martin would occasionally try to pull hoaxes on his own loyal readers, to test their gullibility.
Even those columns simply labeled "Nine Problems" always had something classical and fine about them. Never was a problem a mere exercise without a deeper point. Every one of them contained a lesson in how to think. Once again, common sense was being put to the test in ever-novel ways.
I remember that when I was a teenager, every time I discovered an issue of Scientific American in our family's mailbox I would instantly flip it open to the "Mathematical Games" column, filled with excitement and curiosity to see what amazing new set of ideas would be discussed. Some of my friends felt exactly the same way—and many years later I came to understand that there were thousands of such people spread all around the world—mathematicians, physicists, philosophers, computer scientists, and on and on—who thought of Martin Gardner's column not as merely a feature of that great magazine Scientific American, but as its very heart and soul.
* * *
In the early 1970's, when I was a graduate student in elementary particle physics, I went through a period of deep crisis. I found myself utterly unable to relate to any of the ideas I was exposed to through readings and seminars. The more I thought about my future, the gloomier it looked. There came a point where I could see no future for myself other than teaching math or physics in a high school or junior college. Though I loved teaching, this would mean foregoing research, foregoing the pursuit of novel ideas, foregoing creativity. And this didn't seem at all right for me.
After all, despite my deep discouragement, some part of me was convinced that I had a creative mind, even a mind with something of the Martin Gardner spirit. I had always resonated deeply with the paradoxes that he offered, and more than once I had come up on my own with something that he later published in his column, submitted by somebody else. In my files I had amassed a sizable collection of quirks and oddities and mathematical surprises, and in my greatest flights of fancy, I would imagine being offered the chance of writing the "Mathematical Games" column myself, if Gardner should ever retire. But it felt like an utter pipe dream. How would the higher-ups at Scientific American ever come to think of me, a lowly physics graduate student, as a worthy successor to Martin Gardner?
And yet, just a few years later, that exact dream did come to pass. I never did reconcile with myself with particle physics, but instead moved into solid-state theory, and got my Ph.D. in it. However, as soon as I had done so, I slid out of physics and into artificial intelligence, sensing that my deepest interests were actually in unraveling how the mind works. Those interests, interacting with many of my other interests, gave rise to my first book, Gödel, Escher, Bach: an Eternal Golden Braid, to which Martin devoted a whole column in Scientific American and on which he lavished praise. I was and am enormously indebted to Martin for that act, which probably more than anything else made my book known and launched it on a pathway of success. It was quite a thrill to be touted in the very column that I had so greatly admired and had followed so faithfully for so many years.
Only a year or so later, Martin decided to stop writing his column in order to have more free time. Could someone be found to carry on the "Mathematical Games" spirit? I believe it was Martin himself who suggested to Scientific American's editor and publisher, Dennis Flanagan and Gerard Piel, that I might be a plausible person to consider. When Flanagan and Piel approached me with this thought, I was both overwhelmed and frightened. I had in the meantime become a professor of computer science and was seriously engaged in artificial intelligence research. How could I continue to do my research and also do justice to the column that Martin Gardner had created, which by then had turned into an international institution?
After churning it over in my mind for several weeks, I finally decided to risk it, because it simply was too good an opportunity to turn down. I knew that I would forever regret it if I failed to grasp the very chance that I had once yearned for and that had seemed beyond my wildest dreams. Moreover, in a letter to me, Dennis Flanagan said explicitly that I should feel free in following Martin's lead in writing about anything under the sun that interested me—that it need not concern either mathematics or games, that it could be scientific, literary, artistic, what-have-I. Talk about carte blanche! It was almost too good to be true.
Nonetheless, I felt trepidation about being placed precisely in Martin Gardner's shoes. Even though I felt we were in some ways kindred spirits, I also recognized that our interests and skills were not by any means identical. It would give readers a false set of expectations if I simply adopted Martin's mantle, and it would put a heavy set of burdens on me. So I tried to preserve the spirit but not the letter of Martin's column by making the title of my own column an anagram of his title. I took the 17 letters in "Mathematical Games" and scrambled them to make "Metamagical Themas". By this gesture, I tried implicitly to let readers know not to expect me to be Martin Gardner, but that I would try to carry on some of his marvelous spirit.
In order to prepare for writing my column, I felt it would be a good idea to meet Martin Gardner, for he was the person whose mantle I was taking over, the person who had set the magical tone of the column for some 25 years. And Martin in turn was very glad to welcome me and give me some friendly "coaching". And thus, sometime in the fall of 1980, I visited him in his home in Hastings-on-Hudson, in New York State, and met his extremely intelligent, extremely generous, but no-nonsense wife Charlotte. I will never forget the bell up on the third floor of their house, where Martin had his office and where he and I sat and talked for hours. Down on the first floor, Charlotte tugged a string and the bell rang to summon Martin and me down to lunch—and no dawdling was allowed. After lunch Martin and I climbed the stairs to regain his office, and at some point, while were talking about the challenges of writing a monthly column with so many readers of such brilliance, and also about our many overlapping interests, the phone rang. When Martin answered it, I heard him talking with someone first about mathematical logic and then about Zen Buddhism and Taoism. To my amazement, this turned out to be another long-time hero of mine, Raymond Smullyan, whose book Theory of Formal Systems had had a big impact on me as a teenager, and who had just finished writing a new book called The Tao is Silent. Smullyan's multiplicity of interests really struck me, and all the more so when I found out from Martin that Smullyan was also a top-notch magician and an excellent pianist. I realized then that Martin was a kind of hub for communication among an amazing set of sparkling intellects in all sorts of disciplines.
Some of the brilliant people whom Martin linked together were the statistician/magician Persi Diaconis, the mathematician/juggler Ron Graham, the Uri Geller–debunker and psychologist Ray Hyman, the mathematician/logician/musician/magician Raymond Smullyan, the magician and pseudoscience-fighter James Randi—and there were many, many others. It was humbling to realize that this very unassuming, gentle man sitting across the table from me was so admired by, and so central to, so many people who were gifted with both extreme intelligence and extreme creativity.
Well, soon my column began, and I had a grand time for some three years, writing on all sorts of topics, some that I knew Martin himself would surely have written about had he not retired (such as Rubik's cube), others that I knew were quite unlike his own intellectual directions (such as sexism in language). But the burden on me grew greater and greater, and eventually it became clear that I would be unable to continue to produce columns at a monthly pace. My column thus wound to a close in 1983, which freed me up to return to my research. Luckily, just as Martin had done, I was able to take my pieces and convert them into a book. I had had a three-year affair with a dream, and then it broke up—but it certainly was great while it lasted. I have since run into quite a few people who, just like me at one time, would have given their eyeteeth to be in Martin Gardner's shoes. I had been really lucky!
* * *
Since retiring from the "Mathematical Games" column, Martin has continued with full vigor to pursue all of his far-flung interests. In fact, I cannot understand how he finds the time to read and write as many diverse books and articles as he does. In the past 12 years his writings have deeply enriched the mathematics community, the anti-pseudoscience community, the magic community, the wordplay community, the philosophy community, the science fiction community, and on and on.
Even though Martin is very self-deprecating and would disagree with this, I feel that an intellect like his is a treasure. Many of today's most influential mathematicians and physicists, magicians and philosophers, writers and computer scientists, owe their direction to Martin Gardner. They may not even be aware of how big a role he played in their development. After all, influence is often transitive—if A influences B and then B influences C, A may thereby have a large influence on C, yet C may never have even heard of A!
There should be, it seems to me, be a prestigious national or international prize for writing about scientific ideas. As everybody knows, human civilization relies on science and technology more than at any time in the past, and that reliance can only increase. Yet the worldwide ignorance of and disdain for science, mathematics and precise thinking in general is appalling. Because of this tragic situation, people like Martin are precious purveyors of precious knowledge. I hesitate to use the word "popularizer," because to me, that word has always carried a connotation of being subordinate, derivative and secondary—in short, of lesser importance than the originators of scientific ideas. And in many instances, this is undoubtedly the case. But in Martin's case, it is emphatically not the case. His approach and his ways of combining ideas are truly unique and truly creative, and, if I dare say so, what Martin Gardner has done is of far greater originality than work that has won many people Nobel Prizes. Simultaneously achieving both depth and breadth is almost unheard of in today's scientific world, but Martin Gardner is an exception, and it is a delight and a privilege to celebrate here his many achievements. Just as Martin's writings have inspired me for decades, so they will undoubtedly continue to inspire other people for many decades to come.
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Douglas Hofstadter is the Pulitzer Prize
– winning author of Gödel, Escher, Bach: an Eternal Golden Braid, and holds the position of College of Arts and Sciences Distinguished Professor of Cognitive Science at Indiana University Bloomington.