Strange Loops

Mathematicians have always prided them selves on being poised half-way between the arts and the sciences. On the one hand, mathematical theorems share with artistic works the features of beauty and unexpectedness – qualities which general readers may have been enabled by Hardy’s Mathematician’s Apology to perceive in this simple but profound theorem: a prime number that leaves the remainder I when divided by 4 can be expressed as the sum of two squares in one and only one way. Mathematical proofs, from those of Euclid onwards, as Hardy again illustrated through one of Euclid’s best, may brilliantly combine the artistic merit of economy of means with these features of beauty and unexpectedness. Yet, above all, it is the imaginativeness that great mathematicians showed as they went beyond the numbers to create far more complex logical structures – where profound insights were nevertheless used to discern and demonstrate powerful simplicities of interconnection – that links them to the great artists with their gifts for imaginative invention.

On the other hand, many of the greatest mathematicians such as Archimedes, Newton and Gauss have combined with these artistic gifts a very different type of genius: one enabling them to use mathematics to solve practical problems in the physical world, and to carry further the massive growth in knowledge of the physical sciences. These successes were achieved, above all, by mathematicians who were able to acquire keen insight into the experimental method. Collaborating with mathematically-minded experimental physicists and observational astronomers, they progressively uncovered an extraordinary fact: that all those indefinitely repeatable regularities in observation and experiment which we call physical laws take a mathematical form. Every physical law, as it is probed more and more deeply, needs the numerical and logical structures of mathematics to express it with precision. Of course, physicists themselves created their own new imaginative inventions: ‘physical ideas’ of great subtlety which must be grasped in depth by anyone who seeks’to apply mathematics effectively in that field. Nevertheless, after experiment and imaginative thought have generated these powerful physical ideas, deep mathematical investigation has added its contribution to establishing that the laws governing them are of an inherently mathematical form.

The dual claims of mathematics, as the framework for natural laws and as an important field of imaginative expression, increasingly attracted a special sort of prestige. Here, it was felt, poised between the arts and the sciences, lay one of the most characteristic and powerful faculties of the human mind. Naturally, mathematicians themselves were among the most forward in arguing that to study the mind’s mathematical modes of operation could be a matter of very special significance.

By the First World War, such analysis seemed already to suggest that mathematics and pure deductive logic were essentially the same faculty. Not only did mathematics necessarily have a structural form consisting of theorems and proofs, and leading back ultimately, by a chain of deductive logical argument, to a very small number of axioms: but, also, logic itself could be expressed in a mathematical formalism (the ‘propositional calculus’ of Peano, with its own axioms); and, conversely, number, and all the other protagonists of the mathematical scene, could be defined, as Frege showed, in terms of purely logical concepts.

It was natural, then, for the logically deductive mode of thought to inherit the special prestige already attracted by mathematics. To Bertrand Russell, for example, the establishment of logic’s invincibility as a deductive weapon seemed an aim of supreme importance. In the atmosphere of excessive hopes generated by Russell’s early successes in modifying the rules of logic (through the Theory of Types) to avoid a whole class of difficulties typified by the classical Epimenides paradox, Kurt Gödel’s demolition of Russell’s aim in a conclusive argument known as Gödel’s Theorem could be seen as a spectacular event.

Gödel’s Theorem is equally impressive as a piece of mathematics. The imaginative powers he used to build up its proof illustrate perfectly the fact that a mathematician’s skills extend far beyond formal logic. This is no mystical contention, contrasting the ‘reductionism’ of seeing a proof as a single long chain of deductively logical steps with the ‘holism’ of identifying a higher entity in the chain as a whole. It is concerned, rather, with recognising how a great step forward in mathematics involves the mind’s ability to create structures of a completely new kind through an imaginative activity closely akin to that of the great artist.

The primary purpose of Gödel, Escher, Bach is to explore such kinships of the imagination, while also exploiting Douglas Hofstadter’s own artistic imagination to the maximum possible extent, so as to make all the essential ideas underlying Gödel’s achievement accessible to readers without specialised mathematical knowledge. Previous popular introductions to the beauty of mathematics have been like Hardy’s, which used a splendidly clear and interesting style of writing but required no other artistic aids because Hardy attempted to explain only very simple proofs. In mathematics, Hardy’s book scaled, as it were, the Chiltern Hills of haute vulgarisation, but Hofstadter attempts the Himalayas.

His primary aim is an exploration of relationships between a wide variety of artistic and mathematical imaginations, embracing an exposition of all the essential ideas used by Gödel in his Theorem and its proof, and here he is outstandingly successful. His method is in a ubiquitously acknowledged debt, not only to Escher’s art and Bach’s music, but also to a literary precursor: Lewis Carroll, whose minor works (even more than his master-pieces) used long witty dialogues sprinkled with ingenuity to illustrate many aspects of what we now call the propositional calculus and its famous paradoxes. These are paradoxes of contradiction, as when the statement ‘I am lying,’ if true, must be a false statement, but, if false, must be true. All such paradoxes involve what Hofstadter calls a ‘strange loop’, as when an argument carried to its logical conclusion denies its premise. Analogous strange loops occur also in Escher’s paradoxes of perspective – as when an apparently continuous descent retrieves the starting point – and in Bach’s explorations of the paradox of completing an enharmonic cycle.

Besides paradoxes of contradiction there are paradoxes of infinity like Zeno’s of Achilles and the Tortoise; a paradox resolved by the discovery that an infinite series of time-intervals can add up to a finite sum. Later on, Carroll expounded a quite different paradox of logic by means of a witty dialogue between the Tortoise and Achilles. Hofstadter uses these two characters, and a number of others of his own invention, in a sequence of 20 still more ingenious dramatic pieces exhibiting the most lively wit in such a way that each of them suggests some recondite idea to be applied in the immediately ensuing mathematical discussion. This makes his book a ‘braid’ which alternates passages of clearly-written logical exposition with intricately-structured jeux d’esprit.

Among the great composers, Bach is arguably the most mathematical, and surely the one most beloved by mathematicians. Although the well-tempered keyboard which he so effectively popularised did make one ‘strange loop’ possible (the enharmonic modulation), Bach’s work also has a second, a ubiquitous, presence in Hofstadter’s book. The author’s admiration of the composer led him to give each of the 21 witty dramatic pieces in his ‘braid’ a title close to that of one of Bach’s works (the original Carroll dialogue between Achilles and the Tortoise, for example, being appropriately named ‘Two-Part Invention’).

For a much smaller number of pieces, the parallelism goes much closer than title alone and involves an intricate similarity of form. The dialogue with the most thoroughgoing Bach-like form is ‘Crab Canon’, where Achilles replies to the Tortoise’s sequence of remarks with, essentially, the same sequence of remarks in reverse order (cancrizans), and yet the whole dialogue makes excellent sense. Other dramatic pieces are fugues: at least in the sense that the same words (normally with different meanings) are used by each character when he first appears; and, in some cases, in a few other respects too.

The Goldberg Variations dialogue is used to remind the reader of simple statements in number theory which no one has succeeded in proving true or false: for example, the Goldbach conjecture that every even number is the sum of two primes. Goldbach, of course, had nothing to do with either Goldberg or Bach, but the author’s association of the three names is typical of the sort of word play he loves. Mathematicians had always assumed that the power of mathematics would inevitably increase until one day the Goldbach conjecture would be either proved or disproved. Ever since Gödel, however, we know that this need not be the case. His Theorem establishes the existence of statements in number theory (although it does not show that Goldbach’s conjecture is one) which can never be proved or disproved.

The same dialogue includes a puzzle which is a real triumph for the author’s method of making puns and acrostics work to establish mathematically significant ideas in the reader’s mind. The puzzle is as follows: in this list of great mathematicians,

De Morgan,
Abel,
Boole,
Brouwer,
Sierpinski,
Weierstrass,

‘subtract 1 from the diagonal, to find Bach in Leipzig.’ The diagonal letters are Dboups, and the letters preceding each of those are Cantor, which, of course, was J.S.B.’s Leipzig post. Yet the reader soon recognises that the true reference is to yet another great mathematician, Georg Cantor, to whose most brilliant discovery he has just been introduced.

Cantor uncovered the biggest paradox of infinity, that ‘the continuum is uncountable.’ For example, the continuum of all positive numbers less than 1 consists of all the infinite decimals with just zero in front of the decimal point. If there were any way of arranging them in a numbered sequence (1st, 2nd, 3rd, 4th and so on, a ‘counted’ sequence all the way to infinity), we could see that the sequence, even though extended indefinitely, could not include all such infinite decimals by Cantor’s diagonal construction: make a decimal from the 1st digit of the 1st number, the 2nd of the 2nd, the 3rd of the 3rd, the 4th of the 4th, and so on. Then increase all the digits by 1 (changing 9, however, to 0). The new number cannot have been in the original list because every number in that list (say, the 395th) has one of its digits (in that case, its 395th digit) differing by 1 from that of the new number.

Cantor had established, then, that one infinite set (the continuum) can be more numerous than another infinite set (the integers) in the precise sense that it would never be possible to establish any one-to-one relationship between them. From Hofstadter’s point of view, however, it is even more important that Cantor’s diagonal construction had, later on, many other crucial applications in mathematical logic, including one in the proof of Gödel’s Theorem itself.

Not only music, visual art and literature are related in this book to the ideas of Gödel’s Theorem. The author suggests that the philosophy of Zen (which he punningly relates to that of Zeno) contains an analogue to Gödel’s undecidable propositions in its concept of ‘unanswerable questions’ – questions to which the Zen sage Mumon would respond with the meaningless monosyllable MU. Hofstadter’s most brilliant combination of a drawing à la Escher and a fugue à la Bach is used to stigmatise as unanswerable the conflict between holism and reductionism. The drawing embodies a vast MU where the uprights and cross-piece of the enormous M each consists of the word HOLISM, wherein, however, every straight or curved line in each letter is replaced by the straight or curved word REDUCTIONISM; and where the curve of the enormous U consists of the word REDUCTIONISM, wherein every straight or curved line in each letter is replaced by the straight or curved word HOLISM. The first three voices of the fugue discover the meaning of the diagram as, respectively, MU, holism and reductionism, and pursue the discussion of those two irreconcilable philosophical standpoints; and then the fourth voice springs on the student of the drawing an exciting surprise which it would be unfair for a reviewer to reveal.

Coding is another fundamental idea in the proof of Gödel’s Theorem, and, once more, the author pursues to the fullest extent analogies with other disciplines. The book succeeds in being interesting in its discussions both of linguistic coding (and translation) and of the genetic code, Indeed, Gödel’s work is related by Hofstadter to linguistics and to biology quite as much as to musical and visual art.

Visual art returns to the fore in relation to another recurring theme: that of ‘self-reference’, in the sense that, for example, the paradoxical statement ‘I am lying’ is a statement about that statement itself. More advanced work in mathematical logic, including the proof of Gödel’s Theorem, develops the concept of self-reference in a considerably more refined (and mathematical) way. Several analogies are found in Escher’s work, as where one hand uses a pencil to draw another hand, which reciprocally, in another ‘strange loop’, is found to be drawing the first, Biology, too, is adduced in this context, since the living cell incorporates a coded self-reference in its double helix of DNA. Stimulating analogies are drawn, furthermore, between self-reference and the cell’s fundamental property of self-replication.

The first 16 of this book’s 20 chapters complete what I have ventured to call this book’s ‘primary’ aim (which Hofstadter admits to have been the work’s original aim): to use every sort of analogy in art and music, literature and language, philosophy and life science, to illuminate, and be illuminated by, the ideas underlying Gödel’s Theorem and its proof. In that primary aim, the book seems to me brilliantly successful. Furthermore, its very success seems to have established the aim as worthwhile. Forty years on, Gödel’s demolition of the excessive pretensions and expectations of some logicians and mathematicians may no doubt appear a bit less shattering than at first because the modern mind distrusts systems that purport to be able to answer all questions. Nevertheless, the fact that for one such system Gödel went beyond sceptical disbelief to rigorous disproof, by means of an ingenious chain of reasoning which Hofstadter has made to appear still more beautiful and unexpected by the richness of his analogies to other areas of beautiful and unexpected human activity, seems in retrospect to have justified such celebration; and, even, to have justified the author’s claims regarding Gödel’s kinship with Escher and Bach, as well as such alliterative devices as calling this ‘braided’ book in its golden cover, expounding an eternal theorem, an ‘Eternal Golden Braid’. Hofstadter’s pair of three-dimensional artifacts, which, illuminated from three mutually perpendicular directions, give the shadows G, E and B and E, G and B respectively, make an inspired and appropriate cover illustration.

How might the book best have ended? One possibility would have been to sketch the further development of mathematical work on undecidability since Gödel. It could be claimed that this ultimately improved upon the quite artificial, though ingenious, construction of the proposition whose undecidability Gödel had demonstrated. That was achieved when Cohen proved the undecidability of a proposition of immense interest to mathematicians: not, actually, the Goldbach Conjecture, or the conjecture misnamed Fermat’s Last Theorem, but the Continuum Hypothesis, yet another conjecture on which an immense amount of work by mathematicians had been done. The Continuum Hypothesis states that the continuum contains no set which is both more numerous than the integers (that is, uncountable)and less numerous than the whole continuum (in the sense that a one-to-one between it and the whole continuum does not exist). Cohen’s proof of the undecidability of this important question gave the whole issue of undecidability a new mathematical relevance.

Hofstadter prefers, however, not to mention this line of development. Rather, he ends his book by moving in Chapters 17 to 20 into an extended discussion (which, for me, he completely fails to relate significantly to Gödel’s Theorem) of yet another area of biology: the neurosciences. Unlike the genetic code, which, after two decades of immensely complicated experimental work, is rather well established and can be expounded by mathematicians, the neurosciences are a field where a vast amount of even more complicated experimental work is now in progress all over the world and where any kind of synthesis remains extremely remote. Any opinions which the author can throw out in this field are necessarily tentative and even ephemeral, nullifying the Golden Braid’s claims to a subject-matter that, taken as a whole, could be viewed as Eternal.

The trouble is that, with all Douglas Hofstadter’s gifts in the fields of logic and pure mathematics, and his wide artistic sympathies and abilities and other far-ranging interests, he lacks the essential quality of those most effective within the scientific applications of mathematics: namely, that clear insight into experimental method, coupled with a determination to probe relationships between raw experimental data and mathematical theories, which has characterised all the great developers of theories. In the neurosciences, for example, Hofstadter is too much attracted towards elegant a-priori constructs like the ‘Church-Turing hypothesis’ for which there is absolutely no neurobiological evidence.

This is the hypothesis which can be caricatured as the statement that brains act like computers. It is made easier fox the author to believe by the outdated, simplistic view of the individual neuron (almost like a logic gate in a computer) which he describes on page 340. A few pages later, the book suggests that he is acquainted with some of the long-established evidence for nerve cells of greater complexity, but he seems unaware of the much more recently accumulated data of the 1970s indicating highly complex behaviour of individual neurons. Equally, he ignores the immense chemical intricacies of neuro-transmission and neuro-reception processes. Furthermore, copious evidence from psychological research (or common observation) that the vast majority of the actual functions of the central nervous system are not at all computer-like in character is largely disregarded.

Here the pure mathematician’s emphasis on logical deduction and logical structure, and his general lack of interest in how perceptions are processed by the brain of either an experimental scientist, or a dog, leads him to ignore some of the biggest difficulties in his approach. By contrast, other difficulties in the huge area of brain activity concerned with emotional response are recognised by this artistically sensitive author, although they are minimised.

A reviewer may especially need to register disagreement, on a subject of such great importance, with an author who, by the excellence of his first 16 chapters, will necessarily have won a measure of general confidence from many readers. The disagreement is important because of its major implications for the forecasting of possible futures. In broad terms, these implications concern the nature of developments that can be expected in the area of ‘advanced automation’.

One extremely reasonable expectation regarding the future in this field is admirably illustrated by the famous computer program of Winograd reproduced between Hofstadter’s Chapters 17 and 18. Winograd’s work on linguistics pointed to the conclusion, which this program illustrated well, that the accurate parsing of, and response to, language requires a computer program to use an in-built model of the so-called ‘universe of discourse’, to which sentences refer. Programs for such tasks of advanced automation are successful whenever the universe of discourse is relatively small, as in the ‘table-top’ world of Winograd’s program, or as in the ‘factory-floor’ world of modern industrial robots.

It might be thought that future large increases in the power of computers would proportionately increase the managable size of a program’s universe of discourse. This is a mistaken expectation, however, as a result of a phenomenon called the ‘combinatorial explosion’: the time taken for all the searches of the universe of discourse, needed in advanced automation programs, goes up immeasurably more steeply than in mere proportion to the size of the universe of discourse. Thus the next generation of computers, with their enormously increased power, may perhaps only double the size of universe of discourse which advanced automation programs can handle.

Automation has, of course, reached very advanced stages within limited universes of discourse. Among many good examples which are given here is the now relatively familiar efficacy of computer typesetting. This allows an author to do all the work involved in setting his own type, with results, as in the case of the present book, of quite unusual accuracy (except, perhaps, on line 6 of page 311) and aesthetic charm.

By contrast, the important issue on which readers of this book might be dangerously misled by all its charm and eager persuasiveness is whether computer programs can be given advanced automation capabilities within an immensely complex and variegated universe of discourse such as that perceived by our own senses, or those of a dog. On that subject, I remain unshaken in my conclusion, voiced six years ago in a proposition I defended during a Royal Instisution debate: the general-purpose robot is a mirage.