Metamagical Themas Read online

  Metamagical Themas

  Questing for the essence of mind and pattern

  An Interlocked Collection of

  Literary, Scientific, and Artistic Studies

  Douglas R.Hofstadter

  Contents

  Notes on the Cover 23

  Introduction 22

  Section I: Snags and Snarls 3

  1. On Self-Referential Sentences 5

  Post Scriptum. 16

  2 Self-Referential Sentences: A Follow-Up 25

  Post Scriptum. 40

  3 On Viral Sentences and Self-Replicating Structures 49

  Post Scriptum 64

  4 Nomic: A Self-Modifying Game Based on Reflexivity in Law 70

  Initial Set of Rules of Nomic 77

  Post Scriptum. 82

  Section II. Sense and Society 89

  5 World Views in Collision: The Skeptical Inquirer versus the National Enquirer 91

  Post Scriptum 106

  6 On Number Numbness 115

  Post Scriptum. 130

  7 Changes in Default Words and Images, Engendered by Rising Consciousness 136

  Post Scriptum. 148

  8 A Person Paper on Purity in Language 159

  Post Scriptum. 164

  Section III: Sparking and Slipping 171

  9 Pattern, Poetry, and Power in the Music of Frédéric Chopin 173

  Post Scriptum. 188

  10 Parquet Deformations: A Subtle, Intricate Art Form 191

  Post Scriptum. 211

  11 Stuff and Nonsense 213

  12 Variations on a Theme as the Crux of Creativity 231

  13 Metafont, Metamathematics, and Metaphysics: Comments on .Donald Knuth's Article "The Concept of a Meta-Font". 4

  The Mathematization of Categories, and Metamathematics 4

  An Intuitive Picture of Gödel’s Theorem 5

  Completeness and Consistency 7

  A Misleading Claim for METAFONT 9

  Interpolating Between an Arbitrary Pair of Typefaces 12

  A Posteriori Knobs and the Frame Problem of Al 12

  A Total Unification of All Typefaces? 16

  The Essence of `A'-ness Is Not Geometrical 18

  Chauvinism versus Open-Mindedness: 19

  The `A' Spirit 23

  Happy Roles, Unhappy Roles, and Quirk-Notes 23

  Modularity of Roles 24

  Typographical Niches and Rival Categories 25

  The Vertical and Horizontal Problems Two Equally Important Facets of One Problem 27

  Letter and Spirit 29

  Post Scriptum. 31

  Section IV: Structure and Strangeness 43

  14 Magic Cubology 45

  15 On Crossing the Rubicon 73

  Post Scriptum 97

  16 Mathematical Chaos and Strange Attractors 108

  Post Scriptum. 129

  17 Lisp: Atoms and Lists 140

  18 Lisp: Lists and Recursion 154

  19 Lisp: Recursion and Generality 169

  Post Scriptum. 187

  20 Heisenberg's Uncertainty Principle and the Many Worlds Interpretation of Quantum Mechanics 199

  Section V Spirit and Substrate 225

  21 Review of Alan Turing: The Enigma 227

  22 A Coffeehouse Conversation on the Turing Test 236

  Post Scriptum. 255

  23 On the Seeming Paradox of Mechanizing Creativity 14

  24. Analogies and Roles in Human and Machine Thinking 35

  Post Scriptum. 49

  Post Post Scriptum. 86

  Post Post Post Scriptum. 87

  25 Who Shoves Whom Around Inside the Careenium? or, What Is the Meaning of the Word "I"? 92

  Post Scriptum. 113

  26 Waking Up from the Boolean Dream, or, Subcognition as Computation 118

  Introduction 118

  Cognition versus Perception: The 100-millisecond Dividing Line 119

  The Human Mind and Its Ability to Recognize and Reproduce Forms 121

  Toy Domains, Technical Domains, Pure Science, and Engineering 123

  AI and the True Nature of Intelligence 124

  Expert Systems versus Human Fluidity 126

  Anagrams and Epiphenomena 127

  Not Cognition, But Subcognition, Is Computational 129

  Passive Symbols and Formal Rules 131

  Active Symbols and the Ant Colony Metaphor 132

  Who Says Active Symbols Are Computational Entities? 132

  The Substrate of Active Symbols Does Not Symbolize 134

  Symbol Triggering Patterns Are the Roots of Meaning 135

  Beyond Intuitive Physics: The Centrality of Slippability 137

  Al's Goal Should Be to Bridge the Gap between Cognition and Subcognition 138

  Statistically Emergent Mentality Supersedes the Boolean Dream 139

  Post Scriptum. 140

  Section VI: Selection and Stability 151

  27 The Genetic Code: Arbitrary? 153

  Post Scriptum. 174

  28 Undercut, Flaunt, Pounce, and Mediocrity: Psychological Games with Numbers 188

  Post Scriptum. 199

  29 The Prisoner's Dilemma Computer Tournaments and the Evolution of Cooperation 202

  Post Scriptum. 216

  Section VII:Sanity and Survival 225

  30 Dilemmas for Superrational Thinkers, Leading Up to a Luring Lottery 226

  Post Scriptum. 238

  31 Irrationality Is the Square Root of All Evil 244

  Post Scriptum. 249

  32 The Tale of Happiton 255

  Post Scriptum 263

  Post Post Scriptum. 267

  Epilogue 29

  Bibliography 34

  Notes on the Cover

  A Spontaneous Essay on Whirly Art and Creativity

  The drawing on the cover is a somewhat atypical example of a non-representational form of art I devised and developed over a period of years quite a, long time ago, and which my sister Laura once rather light-heartedly dubbed "Whirly Art". The name stuck, for better or for worse. Generally speaking,. I did Whirly Art on long thin strips of paper (available in rolls for adding machines) rather than on sheets of standard format. A typical piece of Whirly Art is five or six inches high and five or six feet long. Many are ten feet long, however, and some are as much as fifteen or even twenty feet in length. The one-dimensionality of Whirly Art was deliberate, of course: I was inspired by music and drew many visual fugues and canons. The time dimension was replaced by the long space dimension. I used the narrow width of the paper to represent something like pitch (although there was no strict mapping in any sense). A "voice" would be a single line tracing out some complex shape as it progressed in "time" along the paper. Several such voices could interact, and notions of what made "good" or "bad" visual harmony' or counterpoint soon became intuitive to me.

  The curvilinear motions constituting a single voice came from a blend of alphabets. At that time (the mid-60's), I was absolutely fascinated by the many- writing systems found in and around India, exemplified by Tamil Sinhalese, Kanarese, Telugu, Bengali, Hindi, Burmese, Thai, and many others. I studied some of them quite carefully, and even invented one of my own, based on the principles that most Indian scripts follow. It was natural that the motions my hand and mind were getting accustomed to would find their way into my visual fuguing. Thus was born Whirly Art.

  Over the next, several years, I did literally thousands of pieces of Whirly Art. Each one was totally improvised-in pen-so that there was no going back, a mistake was a mistake! Alternatively, a mistake could be interpreted as a very daring move from which it would be difficult, but not impossible to recover gracefully. In other words, what seemed at first to be a disastrous mistake could turn into a joyful challenge! (I am sure that jazz improvisers will know e
xactly what I am talking about.) Sometimes, of course, I would Fail, but other times I would succeed (at least by my own standards, since I was both performer and "listener").

  Whirly Art became a (very) highly idiosyncratic language, with its own esthetic and traditions. However, traditions are made to be broken, and as soon as I spotted a tradition, I began experimenting around, violating it in various ways to see how I might move beyond my current state-how I might "jump out of the system". Style succeeded style, and I found myself paralleling the development of music. I moved from baroque Whirly Art (fugues, canons, and so forth) to "classical" Whirly Art, thence to "romantic" Whirly Art. After several years (it was now the late 60's), I

  reached the twentieth century, and found myself spiritually imitating such

  favorite composers of mine as Prokofiev and Poulenc. I did not copy any pieces specifically, but simply felt a kinship to those composers' style. Whirly Art iS not translated music, but metaphorical music.

  It is natural to wonder if I managed to jump beyond the twentieth century and make visual 21st-century music. That would have been quite a feat! Actually, in the early 70's I found that I simply was slowing down in production of Whirly .Art. It had taken me seven years to recapitulate the history of Western music! At that point, I seemed to run out of creative juices. Of course, I could still make new Whirly Art then, as I can now-but I simply was less often inclined to do so. And today, I hardly ever do any Whirly Art, although the way that I draw curvy lines and letterforms bears indelible marks of Whirly Art.

  The piece on the cover, then, is atypical because it was done on an ordinary sheet of paper and has no direction of temporal flow. Also, the really is no concept of counterpoint-in it. Still, it has something of a Whirly Art spirit. There are also seven Whirly alphabets, in the book, one on each of the title pages of the seven sections. They ;are all somewhat atypical as, well, but for slightly different reasons. Each was done on an ordinary sheet of paper but there is still always a clear flow, namely from `A' to, `Z': The real atypicality is the fact that genuine letters from a genuine alphabet

  are being used. I usually eschewed real letters, preferring-to use shapes inspired by letters-shapes more complex and, well, "whiny" than most letters, even more so than Tamil or Sinhalese letters, which are pretty, darn whiny.

  Whirly Art is, I feel, quite possibly the most creative thing I have ever done. That, of course, is my opinion. Other people may disagree. It is a fairly strange and idiosyncratic form of art, however, and cannot be instantly understood. It has its own logic, related to the logics of musical harmony

  and counterpoint, Indian alphabets, gestalt perception, and who knows what else. I've kept it all quite literally in my closet for years-rolled up and piled into many paper bags and cardboard boxes. Because of its physical awkwardness, it is hard to show to people. But Whirly Art itself, and the experience of doing it, is an absolutely central fact about my way of looking at art, music, and creativity. Practically every time I write about creativity, some part of my mind is re-enacting Whirly Art experiences In other words a lot of my convictions about creativity come from self-observation rather than from scholarly study of the manuscripts or sketches of various composers or painters or writers or scientists. Of course, I have done some of that type of scholarship too, because I am fascinated by creativity in general-but I feel that to some extent "you don't really understand it unless you've done it", and so I rely a great deal on that personal experience. I feel that way that "I know what I'm talking about."

  However, I would make a slightly stronger statement: Any two creative things that I've done seem to be, at some deep level, isomorphic. It's as if Whirly Art and mathematical discoveries and strange dialogues and little pieces of piano music and so on are all coming from a very similar core, and the same mechanisms are being exploited over and over again, only dressed up differently. Of course it's not all of the same quality: my real music-is not as good as my visual music, for instance. But because I have this conviction that the core creativity behind all these things is really the same (at least in my own case), I am trying like mad to get at, and to lay bare, that core. For. that reason I pursue ever-simpler domains in which I can feel myself doing "the same thing". In Chapter 24 of this book-in some sense the most creative Chapter, not surprisingly-I write about three of those domains, the Seek-Whence domain, the Copycat domain, and the Letter Spirit domain.

  It is the Letter Spirit domain--"gridfonts" in particular-that is currently my most intense obsession. That domain came out of a lifelong fascination with our alphabet and other writing systems. I simply boiled away what I considered to be less interesting aspects of letterforms-I boiled and boiled until I was left with what might be called the "conceptual skeletons" of letterforms. That is what gridfonts are about. People who have not shared my alphabetic fascination often underestimate at first the potential range of gridfonts, thinking that there might be a few and that's all. That is dead wrong Thee are a huge number of them, and their variety is astounding.

  As I look at the gridfonts I produce-and as I feel myself producing a gridfont I feel that what I am doing is just Whirly Art all over again, in a new and ridiculously constrained way. The same mechanisms of 'shape transformation, the same quest for grace and harmony, the same intuitions bout what works and what doesn't, the same desire to "jump out of the system"-all this is truly the same. Doing gridfonts is therefore very exciting me and provides a new proving ground for my speculations. The one advantage that gridfonts have over Whirly Art is that they are preposterously constrained. This means that the possibilities for choice can be watched much more easily. It does not mean that a choice can be explained easily, but at least it can be watched. In a way, gridfonts are allowing me re-experience the Whirly-Art period of my life, but with the advantage several years' thinking about artificial intelligence and how I would like t try to make it come about. In other words, I can now hope that perhaps I Can get a Handle-a bit of one, anyway-on w at is going on in creativity by means of computer modeling of it.

  Since I feel that in a fundamental sense, Whirly-Art creativity is no deeper, than gridfont creativity, the study of gridfont creation-more specifically, the computer modeling of gridfont creation-could reveal some things that ' I have sought for a long time. Therefore the next few years will be an important time for me-a time to see if I can really get at the essence, via modeling, of what my mind is doing when I create something that to me is , excitingly novel.

  This book, as it says on its cover and in the Introduction, deals with Mind and Pattern. To me, boiling things down to their conceptual skeletons is the royal road to truth (to mix metaphors rather horribly). I think that a lot of truth about Mind and Pattern lies waiting to be extracted in the tiny domains that I have carved out very painstakingly over the past seven years or so in Indiana. I urge you to keep these kinds of things in mind as you read this book. This "confession", coming as it does in a most unexpected place, is a very spontaneous one and probably captures as well as anything could the reason that my research is focused as it is, and the reason that I wrote this book.

  Introduction

  This book takes its title from the column I wrote in Scientific American between January 1981 and July 1983. In that two-and-a-half-year span, I produced 25 columns on quite a variety of topics. My choice of title deliberately left the focus of the column somewhat hazy, which was fine with me as well, as with Scientific American. When Dennis Flanagan, the magazine's editor, wrote to me in mid-1980 to offer me the chance to write a column in that distinguished publication, he made it clear that what was desired was a bridge between the scientific and the literary viewpoints, something he pointed out Martin Gardner had always done, despite the ostensibly limiting title of his column, "Mathematical Games". Here is how Dennis put it in his letter: -

  I might emphasize the flexible nature of the department we have been calling "Mathematical Games". As you know, under this, title, Martin has written a great deal that is neit
her mathematical nor game-like. Basically, "Mathematical Games", has been Martin's' column to talk about-anything under the sun that interests him. Indeed, in our view, the main import of the column has been to demonstrate that a modern intellectual can have a range of interests that are confined by such words as "scientific" or "literary". We hope that whoever succeeds Martin will feel free to cover his own broad range of interests, which re unlikely to be identical to Martin's.

  What a refreshingly open attitude! So I was being asked to be the successor to' Martin Gardner-but not necessarily to continue the same column, Rather than filling the same role as Martin had, I would merely occupy the same physical spot in the magazine.

  I had been offered a unique opportunity to say pretty much anything I wanted to say to a vast, ready-made audience, in a prestigious context. Carte blanche, in short. What more could I ask? Even so, I had to deliberate long' and hard about whether to take it, because I did not consider myself primarily a writer, but a thinker and researcher, and time taken in writing would surely be time taken away from research. The conservative pathway, following what was known, would have been to say no, and just do research, The adventurous pathway, exploring the new opportunity and forsaking some research, was tempting. Both were risky, since I knew that, either way I would inevitably wonder, "How would things have gone had I decided the other way?" Moreover, I had no idea how long I might write my column, since that was not stipulated. It, could go on for many years-or I could, decide it was too much for me, and quit after a year.

  In a way, I knew from the beginning that I would take the offer, I guess because I am basically more adventurous than I am conservative. But it was a little like purchasing new clothes: no matter how much you like them, you still want to see how you look in them before you buy them, so you put them on and parade around the store, looking at yourself in the mirror and asking whoever is with you what they think of it. So I talked it over with numerous people, and finally decided as I had expected: to take the offer.