AI & Soc (1990) 4:242-246 © 1990 Springer-Verlag London Limited

AI & SOCIETY

Reviews

"Kurt GOdel: A Mathematical Mystery", Peter Weibel and Werner Schimonovich. ( V H S ~ A L 80 minutes) This video sets out to present the life and work of one of the major mathematical figures of all time, Kurt G6del (1906--1978). In eleven sections it covers; his biography, an examination of the historical and cultural setting of his work, a background to the mathematics of his incompleteness result together with an informal exposition of it, consideration of the relevance of this proof to artificial intelligence, and the comparative result of Turing. A considerable amount of detail to pack into eighty minutes. It is not completely clear to whom this package is addressed (of which more below) but the indication is that it is intended for a reasonably educated audience, and to be itself educational. The request for me to review this video coincided with my reading Neil Postman's (1987) caustic analysis of the effect of television on culture, Amusing Ourselves to Death. Had this not been the case, I would have written a very different, and less critical review. Post Postman, I sense a certain irony in an attempt to mediate an exposition of the culture of mathematics, the prime subculture of the "Age of Exposition", through the prime media of the "Age of Show Business". This is the stated intention of the producers, when they quote Norbert Weiner, "Mathematics is a part of our cultural heritage and it is our task to introduce our fellow men to the secrets of mathematics". Specifically they wish to make the work of Kurt Grdel more accessible to a wider public. This is a laudable aim, but it also carries with it an irony. First the aim. As someone with minimal formal mathematical training, I was first introduced to the ideas of Kurt G6del through the pages of Hofstadter's (1979) book GOdeI, Escher, Bach: An Eternal Golden Braid. With hindsight I find that astonishing. Can anyone go through a contemporary formal education without at least hearing the name of Einstein and having a vague notion of his contribution to the current world view? Why then is Grdel's contribution to thought, with its crucial metasignificance for the competence of thought, not as widely appreciated? This is the man who single-handedly derailed the juggernaut of mathematical logic; who proved that there are things of which we cannot know, or that we know but cannot prove. Why has this result not impressed itself upon the mass consciousness in the

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way that relativity physics has? By this I mean not an understanding of the analysis of velocity and mass, but its "metaphorical" effect in the undermining of absolutism in politics and ethics. After Einstein, "it depends on how you see it". After Grdel, "it doesn't matter how you see it, there is something you can never know". What effect may that view have? A representative example of the television culture's history of thought is the book published to accompany Jacob Bronowski's (1973) television series "The Ascent of Man" (produced by the British Broadcasting Corporation in 1973). There, Einstein merits twenty extensive references; Grdel, none (yon Neumann gets five). Grdel's result simply does not fit in with the idea of an ascending rationality by which the mind of man (it would be man) would come to encompass all truth and control would be absolute. Maybe now its time has come. Maybe it has shaken the old confidence. Maybe post-Challenger and Chenobyl mass consciousness has come to see the old rationalist arrogance for what it is. Weibel and Sehimonovich also intend this video to " . . . make a contribution to the aesthetics of mathematics". What then is mathematics that this effort is required? For the authors, mathematics is practised by artists. (However they mean "artist" in some special sense.) "The mathematician as an artist distinguishes himself from the sculptor or musician by the rigid discipline which is necessary to become expert in the true nature of mathematics."

Surely they cannot mean moral discipline; the self-denial and application that certainly successful musicians (if not sculptors) are required to practice. Perhaps "discipline" should really be "rigour", the requirement that the production of the artist-mathematician requires infinite pains be taken with the detail of the structure. Here also, music makes an odd choice of contrast. Perhaps composers rather than performers are more like mathematicians - perhaps very like mathematicians. As Leibniz suggested, "The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." This may not be all there is to music, but it is crucial to the structure of it. The link between the structural rigour of mathematics and music is clearly made in Hofstadter's book, which is in turn referred to in the video. So what then do Weibel and Schimonovich mean? I suggest that they betray an incomplete vision of culture. For them it has something to do with "art". They think that for mathematics to make a contribution to culture, it must be identified as an artistic practice. Certainly it is creative, and the source of this creativity may also be the fount of other creative practices. 1 However, culture is not "high culture", and cultural productions are not all artistic productions. It may be important to point out that mathematics is, or may be creative in the sense appreciated by artists, but this is not a necessary precondition to taking its contribution to culture seriously. The development of a rigorous mathematics with its codification at the turn of the century, was the culmination of a striving for rationality that has a direct line of descent from the Renaissance. It is a striving that has found its expression in all spheres of Western cultural life; economic, political, medical, agricultural, technological etc. In mathematical logic, the culmination of advancing rationality

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was represented by the production of an all embracing system based on the predicate calculus. It was this citadel of rational certainty, Russell and Whitehead's "Principia Mathematica" (PM) that Gfdel chose to attack. It is a great irony that PM's Achilles' heel turned out to be similar to the flaw in set theory pointed out earlier by Russell. (That is, recursive set inclusion. The problem of "who shaves the barber who shaves all men who don't shave themselves?") G/Sdel Russell'd Russell and Whitehead by showing the indeterminacy, within the system of PM, of a proposition that asserts its own falsity (via an encoding of the proposition as a number). A limit to the advance of this sort of rationality had been drawn. This was a cultural event of massive proportions, with ramifications for the course of all aspects of the culture in which it was engendered. But these things take time to percolate through. The question is then raised, is video a good method for assisting this process? As I indicated at the start of this review, I have my doubts. Video is not culturally neutral (nor is any medium). As Postman suggests, it comes equipped with its own epistemology, which is the antithesis of the epistemology inherent in the normal media of mathematical discourse, printed text. In textual discourse, ideas are developed linearly with a comparatively leisurely pace. The overriding imperative is that ideas follow one another, in context, and in logical relationship to each other. The reader has constantly the question, "is this true?" in mind. In video or television, the context is not set by the propositional content of ideas, but of the medium itself. The viewer is not concerned with the development of an argument, but the unfolding of a narrative. The imminent question is "Is this entertaining?" Of course videos can be produced using the tenets of textual presentation, but it often amounts to little more than talking heads, and is usually aimed at a specialist audience. This video, whilst intended to be pre-eminently educational, is dearly intended for more general appreciation, and hence attempts to abjure this style. Postman suggests that " . . . there are three commandments that form the philosophy of the education which television offers . . . . Thou shalt have no prerequisites . . . Thou shalt induce no perplexity . . . Thou shalt avoid exposition like the ten plagues visited upon Egypt." Postman is not advocating these rules (he profoundly questions the educational validity of television), but suggesting that failure to follow them will make for an unsuccessful programme. It may be useful to examine Weibel and Schimonovieh's production in the light of these tenets. Are there any prerequisites for viewing this video? There certainly are! If you had not heard of Kurt Gfdel previously, and had not got some idea of his work, then much of the content would have remained the mystery referred to in the English language title. References to the Vienna Circle, and the influence of its members on the development of Gfdel's thought are presented in some detail. Gfdel's work is historically located within the development of mathematics (with the aid of a graphic containing pictures of Aristotle and Hilbert sprouting right pointing arrows, stopped by railway buffers emanating from a picture of Gfdel). Clearly then, this video is not aimed at the mathematically unsophisticated. It makes a good case for the influence of the very special cultural milieu that Gfdel inhabited. However, this is a case that can only be appreciated by those with some familiarity with the ideas and their historical significance.

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I was certainly perplexed at a number of points in the presentation. The fact that the original language of the video was German, and that only voice has been translated, may account for some problems in interpreting diagrams. (The dubbing of the voice-over is appalling. The sound studio was not even soundproof, and the passing of nearby lorries is recorded. Plainly the speakers are not professionally trained, and announce incidentals, such as the address of G6del's parents fiat, with the same breathless earnestness as details of his thought.) However, some of the expositions of proofs, which are considerably simplified and presented with animated images, are unsatisfying. Magic steps appeared to be made, which would be guaranteed to perplex anyone not familiar with the topic and able to keep up with the breathless pace of the presentation. The "Party Problem" ("How many people at a minimum must I invite to a party so that either three of them know each other or three others are not mutually known to each other?") is presented as an example of an actual problem that is not susceptible to a formal solution. It sounded compelling, and I had not heard of it before, but I am still perplexed by the description of the problem. (I had to wind the tape back a couple of times to take it in.) The avoidance of exposition was dearly an impossibility if an attempt was to be made to present G6del's work to the uninitiated. Certainly it has been kept to a minimum in terms of time during the presentation. I have already referred to the frenetic pace with which expository sections are given. The whole thing works much better when in narrative mode, giving G6del's biographical details, presenting anecdotes from friends and relations, however even here the medium has its own imperatives. The producers seem to have been obsessed with the idea that each sentence of the dialogue should have its own visual image. The result is technological over-kill. The full resources of a sophisticated image manipulation system have been directed to this end. What is at first visually arresting, for example an early sequence in which successive photographs of G6del are faded one on top of the other, eventually becomes vertiginous, as images are multiplied and superimposed, the sides of buildings become screens for views of more images, G6del's head becomes the centre of a fairground ride... This video attempts too much, and attempts that which is inappropriate for this medium. (More specifically the genre of "Television Documentary" rather than video per se.) Had it been directed at mathematical sophisticates, and concentrated on G6del's biography, it may well have been the " . . . informative entertainment and entertaining information" that an incongruous talking robot wishes us in the opening sequence. There were many interesting leads that I would have enjoyed following up (e.g., the suggestion that G6del's work on Einstein's field equations led to an interest in demonology). It may even have had the intended cultural result, and made a few narrow technologists aware of the historical and cultural backcloth to their concerns. However, its desire to be entertaining is its undoing as an introduction to the culture of mathematics. It is simply yet another introduction to the culture of entertainment. Of course, without being entertaining it would not be viable for general consumption. Its form must defeat its substance - a paradox which G6del himself might have appreciated.

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The final irony; Kurt G6del died of a paradox. Convinced that those around him were attempting to poison him, he refused food and died, " . . . weighing 68 pounds, curled in the foetal position". Those who live by the s w o r d . . . Richard N. Griffiths Brighton Polytechnic, Brighton, England

Note 1 For insight into this process in mathematics, see the astonishing paper by Emil Post (1965) "Absolutely unsolvable problems and relatively undecidable propositions: Account of an anticipation". An appendix gives extracts from Post's notes and diary during the writing of this paper, where vivid images and impressions of a visual nature are recorded. For example, " . . . dark clouds pierced by flashes of lightning accompanied by rolling thunder." Incidentally, the paper gives details of Post's attempt to explore the same territory as G-fdel and Turing.

References Bronowski, Jacob (1973). "The Ascent of Man". Book Club Associates by arrangement with the British Broadcasting Corporation. Hofstadter, Douglas (1979). Grdel, Escher, Bach: An Eternal Golden Braid. Basic Books Inc. Post, Emil (1965). In Davis, M. (ed.) The Undecidable. Raven Press. Postman, Neil (1987). Amusing Ourselves to Death. Methuen London Ltd.