Lucidity and science II: From acausality illusions and free will to final theories, mathematics, and music

Interdisciplinary Science Reviews

Volumn 22, Issue 4, 1997, Pages 285-303

  McIntyre, Michael E ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0348139699     PISSN: 03080188     EISSN: None     Source Type: Journal    
DOI: 10.1179/isr.1997.22.4.285     Document Type: Article

References (71)

1. M. E. MCINTYRE: 'Lucidity and science. I: Writing skills and the pattern perception hypothesis', Interdisc. Sci. Rev., 1997, 22, 199-216.
2. A. FRANKLIN: 'Are there really electrons? Experiment and reality', Phys. Today, 1997, 50, (10), 26-33. One of the many centenary celebrations, and historical discussions, of what may reasonably be called the discovery of the electron.
3. 89 acting as 'free open reading frames (cassettes)', or, in the language of this essay, versatile model fitters. Such capabilities make the development and spread of resistance to any antibiotic almost inevitable, and underline the irresponsibility of the profligate use of antibiotics. Davies reminds us that The breakdown of public health measures due to famine, poverty, and war contributes to the dissemination of resistance, ... compounded by the growth of international travel ...'
4. S. H. SCHNEIDER: 'Laboratory earth: the planetary gamble we can't afford to lose'; 1996, London, Weidenfeld and Nicolson/New York, HarperCollins (Science Masters series), 184 pp. This gives a good lay person's explanation of some of the reasons for the large uncertainty about future environmental change - including what is simplistically called 'climate change' - and about what its early warning signals might be. For instance, there are straightforward uncertainties about the numerical magnitudes of critical parameters, concerning cloud droplet sizes for instance, and the rates and modes of transport of water, carbon dioxide, and other chemicals and pollutants in the atmosphere-ocean-cryosphere-biosphere system and the concomitant biological adaptations (e.g. Pfiesteria piscicida). Another, more subtle, reason is cogently argued in Ref. 76. Some of the oft-repeated claims, implicit or explicit, to 'infallible prior knowledge' of what's ahead, despite the large uncertainties - and despite the possibility of continuous, unstoppable sea-level rise for two or more centuries - are discussed in Schneider's book. More of them are collected and discussed in R. GELBSPAN: 'The heat is on: the high stakes battle over earth's threatened climate'; 1997, New York, Addison Wesley, 278 pp. For a sufficient example of an implicit claim to prior knowledge, see p. 46.
5. S. H. SCHNEIDER: 'Laboratory earth: the planetary gamble we can't afford to lose'; 1996, London, Weidenfeld and Nicolson/New York, HarperCollins (Science Masters series), 184 pp. This gives a good lay person's explanation of some of the reasons for the large uncertainty about future environmental change - including what is simplistically called 'climate change' - and about what its early warning signals might be. For instance, there are straightforward uncertainties about the numerical magnitudes of critical parameters, concerning cloud droplet sizes for instance, and the rates and modes of transport of water, carbon dioxide, and other chemicals and pollutants in the atmosphere-ocean-cryosphere- biosphere system and the concomitant biological adaptations (e.g. Pfiesteria piscicida). Another, more subtle, reason is cogently argued in Ref. 76. Some of the oft-repeated claims, implicit or explicit, to 'infallible prior knowledge' of what's ahead, despite the large uncertainties - and despite the possibility of continuous, unstoppable sea-level rise for two or more centuries - are discussed in Schneider's book. More of them are collected and discussed in R. GELBSPAN: 'The heat is on: the high stakes battle over earth's threatened climate'; 1997, New York, Addison Wesley, 278 pp. For a sufficient example of an implicit claim to prior knowledge, see p. 46.
6. 114 implies still greater uncertainty than one might gather from Ref. 75, especially about early warning signals. Palmer's paper uses simple but apt analogies to bring out these points.
7. D. C. MARR: 'Vision: a computational investigation into the human representation and processing of visual information'; 1982. San Francisco, Freeman, 397 pp. See also Note 5 of Part I.
8. M. HUNT: 'The story of psychology', 1993, New York. Doubleday, 762 pp. More detail in Note 17 of Part I.
9. 108 (PC .wav) recordings, for download in binary mode. The visual animations include two walking lights demonstrations, in MPEG files named lucidity-walking-lights1.mpg and lucidity-walking-lights2.mpg, and a rotating Necker cube demonstration in lucidity-necker.mpg. See also Notes 8 and 31 of Part I.
10. A. S. BREGMAN: 'Auditory scene analysis: the perceptual organization of sound'; 1990, Cambridge, MA, MIT Press, 773 pp. Further notes under Note 22 in Part I; on grouping in music, see also Refs. 6 and 23 of Part I.
11. 80 of related phenomena like the 'duplex perception of speech' and the 'McGurk effect'; and see also Refs. 82-84 and 92. All these studies clearly suggest not only that perception works by model fitting, but also that different sets of sensory data - for instance auditory and visual - are fitted to the same internal model. Also Note 40 of Part I.
12. R. SEKULER, A. B. SEKULER, and R. LAU: 'Sound alters visual motion perception', Nature, 1997, 385, 308. Another multisensory psychophysical experiment, with a clearcut result that is easy to make sense of if we assume that visual and auditory data are fitted to the same internal model. Experiments were controlled against a popular alternative explanation in terms of attention switching.
13. N. K. LOGOTHETIS, D. A. LEOPOLD, and D. L. SHEINBERG: 'What is rivalling during binocular rivalry?', Nature, 1996, 380, 621-624; see also p. 587. Here 'binocular' replaces 'multisensory', but with the same implications. These new experiments add powerfully to the evidence for model fitting by showing that the percept, i.e. the internal model, chosen by the visual system is insensitive to changes in the route taken by the incoming sensory data. Included are experiments that present conflicting stimuli to the two eyes and, at intervals of one-third of a second, swap which stimulus goes into which eye. The resulting pairs of percepts exhibit bistability not at intervals of one-third of a second but at longer, randomly variable time intervals, qualitatively like Necker cube flipping. This firmly rules out hypotheses that assume 1-1 mapping, like monocular dominance or fatigue.
14. G. PELLEGRINO, E. LÁDAVAS, and A. SARNÉ: 'Seeing where your hands are', Nature, 1997, 388, 730. The results of this experiment, on a perceptual phenomenon called 'cross-modal extinction' resulting from certain kinds of brain damage, become intelligible if we hypothesise that the subject's brain was fitting two modes of sensory data, visual and tactile in this case, to a single internal model of his own hands and their spatial positions.
15. D. HOFSTADTER and co-workers: 'Fluid concepts and creative analogies: computer models of the fundamental mechanisms of thought'; 1995, New York, BasicBooks, 518 pp.
16. F. CRICK: 'The astonishing hypothesis'; 1994. London, New York, Simon and Schuster, 317 pp. See caveat in Note 32 of Part I.
17. S. KAUFFMAN: 'At home in the universe: the search for laws of self-organization and complexity'; 1995, New York, London, Viking Penguin, 321 pp. Also Notes 1-4 of Part I.
18. I am grateful to Max Perutz for kindly vetting my statements about the immune system. He prefers to call the immune system's internal models 'templates' - because the model information resides in the shapes of 'antibody' molecules that complement or fit the incoming material hence 'recognise' it and clamp on to it - and to say that the system works by, in his words, 'constructing, by a stochastic mechanism, many millions of different templates, one or other of which may happen to be complementary to any particular chemical compound or viral or bacterial component'. Once the particular compound or component is thus recognised, 'the cell carrying that template on its surface is stimulated to proliferate and make multiple copies of that template' (M. F. PERUTZ: personal communication). Such proliferation, of a template pattern selected, in a very Darwinian sense, by its goodness of fit to the incoming material, is the main mechanism whereby the immune system can be said to 'fit an internal model' to the incoming data. The system's repertoire of possible models - its 'many millions of different templates' - is constructed stochastically, by a kind of 'genetic roulette', from pre-existing, permutable model components or, so to speak, permutable building blocks, specified by permutable DNA segments. See Notes 89 and 126 for more detail, including recent work on how the system iteratively improves the goodness of fit.
19. 88 Recent progress in understanding how the system iteratively improves the goodness of fit is reported in Science, 1997, 276, 1658-1659 and references therein, particularly G. J. WEDEMAYER, P. A. PATTEN, L. H. WANG, P. G. SCHULTZ, and R. C. STEVENS: 'Structural insights into the evolution of an antibody combining site', Science, 1997, 276, 1665-1669. An ingenious experiment has illustrated in great detail, in one case, just how far the fit can be tightened through 'the well understood process of antibody maturation'.
20. 88 Recent progress in understanding how the system iteratively improves the goodness of fit is reported in Science, 1997, 276, 1658-1659 and references therein, particularly G. J. WEDEMAYER, P. A. PATTEN, L. H. WANG, P. G. SCHULTZ, and R. C. STEVENS: 'Structural insights into the evolution of an antibody combining site', Science, 1997, 276, 1665-1669. An ingenious experiment has illustrated in great detail, in one case, just how far the fit can be tightened through 'the well understood process of antibody maturation'.
21. 88 Recent progress in understanding how the system iteratively improves the goodness of fit is reported in Science, 1997, 276, 1658-1659 and references therein, particularly G. J. WEDEMAYER, P. A. PATTEN, L. H. WANG, P. G. SCHULTZ, and R. C. STEVENS: 'Structural insights into the evolution of an antibody combining site', Science, 1997, 276, 1665-1669. An ingenious experiment has illustrated in great detail, in one case, just how far the fit can be tightened through 'the well understood process of antibody maturation'.
22. 88 Recent progress in understanding how the system iteratively improves the goodness of fit is reported in Science, 1997, 276, 1658-1659 and references therein, particularly G. J. WEDEMAYER, P. A. PATTEN, L. H. WANG, P. G. SCHULTZ, and R. C. STEVENS: 'Structural insights into the evolution of an antibody combining site', Science, 1997, 276, 1665-1669. An ingenious experiment has illustrated in great detail, in one case, just how far the fit can be tightened through 'the well understood process of antibody maturation'.
23. 88 Recent progress in understanding how the system iteratively improves the goodness of fit is reported in Science, 1997, 276, 1658-1659 and references therein, particularly G. J. WEDEMAYER, P. A. PATTEN, L. H. WANG, P. G. SCHULTZ, and R. C. STEVENS: 'Structural insights into the evolution of an antibody combining site', Science, 1997, 276, 1665-1669. An ingenious experiment has illustrated in great detail, in one case, just how far the fit can be tightened through 'the well understood process of antibody maturation'.
24. C. SAGAN: 'The demon-haunted world: science as a candle in the dark'; 1996, London, Hodder Headline/ Random House, 436 pp. Chapter 6 gives a clear summary of the evidence concerning the perceptual phenomenon called hallucination. See also Refs. 91 and 92.
25. C. D. FRITH and P. FLETCHER: 'Voices from nowhere', Critical Quarterly, 1995, 37, (2), 72-83. This brief review, written by a research neuropsychologist and a research psychiatrist, gives a readable and well-referenced discussion of the auditory hallucinations reported by people suffering from schizophrenia. These occur in 'about 65 per cent' of cases, and are experienced as wholly real, often distressingly so. See also Ref. 111.
26. O. SACKS: 'An anthropologist on Mars: seven paradoxical tales'; 1995, New York, Alfred Knopf, 330 pp. See also Note 38 of Part I, including remarks about the permanent adaptive visual hallucination of the patient 'Greg F', who was blind yet 'watched television'.
27. Again see the later section 'On epistemology' etc.; and to get a more detailed idea of the 'difficulties in the philosophy of science' see for instance A. F. CHALMERS: 'What is this thing called science? An assessment of the nature and status of science and its methods', 2nd edn; 1982, Milton Keynes, Open University Press, 179 pp. This summarises recent thinking among historians and philosophers of science, carrying on from naive inductivism to Popper. Lakatos, Kuhn, and Feyerabend. In Chalmers' own admirably honest words, 'We start off confused and end up confused on a higher level' (p. xix).
28. See the historical remarks on pp. 18-29 of Ref. 92, also the chapter 'To see and not see', about a personal tragedy that could have been avoided, almost certainly, had those involved had the slightest inkling of how perception works. For another such case, see Ref. 117.
29. E.g. J. L. MILLER and A. M. LIBERMAN: 'Some effects of later occurring information on the perception of stop consonant and semivowel', Perception Psychophys., 1979, 25, 457-465. Here the acausality illusion is referred to as an 'after going effect'. Other terms, such as 'backward referral in time', can be found in the literature. I have not yet come across the simpler and more straightforward term 'acausality illusion'.
30. R. PENROSE: 'Shadows of the mind: a search for the missing science of consciousness'; 1994, Oxford, Oxford University Press, 457 pp. Also Note 53 of Part I, and the Appendix to Part II, the present article.
31. Ref. 96, pp. 385-387, gives a brief discussion of the neurological experiments of Benjamin Libet and the controversies they have triggered: more extensive discussions and debates, with large bibliographies, can be traced through Refs. 99, 100, and 111. There have even been suggestions to the effect that physical causality principles be abandoned.
32. N. HUMPHREY: 'A history of the mind'; 1992, London, Chatto & Windus, Vintage, 230 pp. An important reference on the nature of perception and consciousness. See also Note 33 of Part I.
33. D. C. DENNETT: 'Consciousness explained'; 1991, London, Penguin, 511 pp. This insightful book by a well known philosopher discusses several examples of what I am calling acausality illusions, including the important Grey Walter slide projector experiment, apparently not published elsewhere. On the consciousness debate itself, see also Ref. 98, and D. C. DENNETT: 'Kinds of minds: towards an understanding of consciousness'; 1996, London, Weidenfeld and Nicolson New York, HarperCollins (Science Masters series), 184 pp.
34. D. C. DENNETT: 'Consciousness explained'; 1991, London, Penguin, 511 pp. This insightful book by a well known philosopher discusses several examples of what I am calling acausality illusions, including the important Grey Walter slide projector experiment, apparently not published elsewhere. On the consciousness debate itself, see also Ref. 98, and D. C. DENNETT: 'Kinds of minds: towards an understanding of consciousness'; 1996, London, Weidenfeld and Nicolson New York, HarperCollins (Science Masters series), 184 pp.
35. D. C. DENNETT AND M. KINSBOURNE: 'Time and the observer: the where and when of consciousness in the brain', Behav. Brain Sci., 1992, 15, 183-247. See also correspondence in Behav. Brain Sci., 1995, 18, 810-811.
36. D. C. DENNETT AND M. KINSBOURNE: 'Time and the observer: the where and when of consciousness in the brain', Behav. Brain Sci., 1992, 15, 183-247. See also correspondence in Behav. Brain Sci., 1995, 18, 810-811.
37. 86 Chap. 9, p. 112. It is typical for the activity observed in this way to continue for '400 msec or longer' (C. D. FRITH: personal communication). In Crick's examples, the ensemble averaged signals last around half a second, each showing one peak near 200 msec and another near 300-400 msec. Crick states that a prominent peak near 300-400 ms is 'fairly common', and 'usually correlates with something that is surprising', or attention grabbing or eye catching, like visual 'pop out' (p. 111 and caption to Fig. 35). Psycholinguistic experiments reveal similar timespans, corresponding to two or three syllables of speech, for pre-conscious processes in speech perception. Examples include the work of Swinney, Seidenberg, Tanenhaus, and others summarised in Ref. 104, p. 143 and in pp. 210-211 of S. PINKER: 'The language instinct: the new science of language and the mind'; 1994, London, Allen Lane (Penguin), 494 pp. The same goes for music perception, as in the simple examples described in the text.
38. 79 in the PC .wav files lucidity-radiopips100.wav and lucidity-radiopips200 .wav, giving the 100 and 200 ms cases respectively (low quality, each just over 100 kbyte), and in the MIDI file lucidity-radiopips.midi (just over 200 byte), giving both cases together. In musical language, the effects in the 200 and 100 ms cases could be described as a leisurely 'dotted rhythm' and 'grace note' respectively.
39. S. HAWKINS: 'Arguments for a nonsegmental view of speech perception', in Proc. Int. Cong. on 'Phonetic science', Symp. on 'Dynamic, nonsegmental approaches to phonetics', (ed. K. Elenius and P. Branderud), Vol. 3, 18-25; 1995, Stockholm, University of Stockholm and Royal Institute of Technology (KTH), ISBN 91 71708367.
40. R. JACKENDOFF: 'Languages of the mind: essays on mental representation'; 1992, Cambridge, MA, MIT Press, 200 pp., ISBN 0 262 10047 9. See the allusion to 'temporal anomaly' on p. 141, and the hypothesis of 'parallel multiple analysis' on pp. 140-145, the latter first published in Music Perception, 1991, 9, 199-229.
41. R. JACKENDOFF: 'Languages of the mind: essays on mental representation'; 1992, Cambridge, MA, MIT Press, 200 pp., ISBN 0 262 10047 9. See the allusion to 'temporal anomaly' on p. 141, and the hypothesis of 'parallel multiple analysis' on pp. 140-145, the latter first published in Music Perception, 1991, 9, 199-229.
42. 79 in the MPEG file lucidity-apparent-motion.mpg.
43. 99 namely, 150 ms on and 50 ms off (cf. 200 ms on and 200 ms off, as used in the demonstration described in the text). We clearly saw a red disc departing and a green disc arriving, but no definite perceived time for the colour change.
44. U. LE GUIN: 'The dispossessed', Chap. 6; 1974, London, Gollancz/1996, London, HarperCollins, 319 pp.
45. 79 for download in binary mode, as audio PC (.wav) files at sampling rates 22 and 44 kHz (standard compact disc quality). The recordings use a standard high quality computer driven sound module. Though unable to match the delicacy and subtlety of the best human performances, they are accurate realisations of the written scores and sufficient illustrations of the points under discussion. The files, in the order mentioned in the text, are named lucidity-piano1..., lucidity-orch1..., lucidity-orch2... (the 1-1 mapping version), lucidity-piano2..., and lucidity-orch3... (the last two corresponding to Fig. 4), where the dots denote a suffix like -22m.wav or -44s.wav indicating the sample rate and whether mono or stereo, -22m.wav files being the smallest, about 0·7 Mbyte each. The corresponding written scores are in the graphics files lucidity-piano12.gif, lucidity-orchl.gif, lucidity-orch2.gif, and lucidity-orch3.gif. I am grateful to Ben Finn of Sibelius Software (http://www.sibelius-software.com/) and Jeffrey Ginn of Ginn Music (http://www.ginn-mus. demon.co.uk/) for lending their professional expertise with the audio recordings, and to the composers Alexander Goehr, Robin Holloway, Virginia Seay Ploeser, Yuval Shay-El, Roderick Skeaping, and Hugh Wood for advice on the scoring. (They all agree that an accompaniment designed to preserve the musical sense of the piano solos would have to change harmony at, not after, the time of the arrow.)
46. 79 for download in binary mode, as audio PC (.wav) files at sampling rates 22 and 44 kHz (standard compact disc quality). The recordings use a standard high quality computer driven sound module. Though unable to match the delicacy and subtlety of the best human performances, they are accurate realisations of the written scores and sufficient illustrations of the points under discussion. The files, in the order mentioned in the text, are named lucidity-piano1..., lucidity-orch1..., lucidity-orch2... (the 1-1 mapping version), lucidity-piano2..., and lucidity-orch3... (the last two corresponding to Fig. 4), where the dots denote a suffix like -22m.wav or -44s.wav indicating the sample rate and whether mono or stereo, -22m.wav files being the smallest, about 0·7 Mbyte each. The corresponding written scores are in the graphics files lucidity-piano12.gif, lucidity-orchl.gif, lucidity-orch2.gif, and lucidity-orch3.gif. I am grateful to Ben Finn of Sibelius Software (http://www.sibelius-software.com/) and Jeffrey Ginn of Ginn Music (http://www.ginn-mus. demon.co.uk/) for lending their professional expertise with the audio recordings, and to the composers Alexander Goehr, Robin Holloway, Virginia Seay Ploeser, Yuval Shay-El, Roderick Skeaping, and Hugh Wood for advice on the scoring. (They all agree that an accompaniment designed to preserve the musical sense of the piano solos would have to change harmony at, not after, the time of the arrow.)
47. T. G. BEVER, M. F. GARRETT, and R. HURTIG: 'The interaction of perceptual processes and ambiguous sentences', Memory and Cognition, 1973, 1, 277-286.
48. 127 P. MEDAWAR: 'Science and the sanctity of life', in 'Pluto's republic', 311-323; 1982, Oxford, Oxford University Press, and K. E. DAVIES, A. J. CLARKE, and P. S. HARPER: 'The genetic revolution and medicine in the 21st century', Europ. Rev., 1997, 5, 39-54 ('Is having the wrong genes going to become an acceptable line of defence in a court of law in cases of violence? Fortunately, ... the biological basis of behaviour ... is unimaginably complex, and is open to modification by numerous environmental influences.').
49. 127 P. MEDAWAR: 'Science and the sanctity of life', in 'Pluto's republic', 311-323; 1982, Oxford, Oxford University Press, and K. E. DAVIES, A. J. CLARKE, and P. S. HARPER: 'The genetic revolution and medicine in the 21st century', Europ. Rev., 1997, 5, 39-54 ('Is having the wrong genes going to become an acceptable line of defence in a court of law in cases of violence? Fortunately, ... the biological basis of behaviour ... is unimaginably complex, and is open to modification by numerous environmental influences.').
50. 127 P. MEDAWAR: 'Science and the sanctity of life', in 'Pluto's republic', 311-323; 1982, Oxford, Oxford University Press, and K. E. DAVIES, A. J. CLARKE, and P. S. HARPER: 'The genetic revolution and medicine in the 21st century', Europ. Rev., 1997, 5, 39-54 ('Is having the wrong genes going to become an acceptable line of defence in a court of law in cases of violence? Fortunately, ... the biological basis of behaviour ... is unimaginably complex, and is open to modification by numerous environmental influences.').
51. S. A. SPENCE:'Free will in the light of neuropsychiatry', Philos. Psychiat. Psychol., 1996, 3, 75-90; see also pp. 91-100 for commentaries by C. Frith, B. Libet, G. L. Stephens, and reply by Spence.
52. 127
53. H. BONDI: 'Assumption and myth in physical theory'; 1967, Cambridge, Cambridge University Press, 88 pp.
54. I. STEWART: 'Does God play dice?', 2nd edn; 1997, London, Penguin, 401 pp. Along with Ref. 112, this gives an insightful view of how science works as a model fitting process, and why mathematics is relevant: 'To criticise mathematics for its abstraction is to miss the point entirely' (p. 363). My remarks in the section 'On epistemology' in the text and in the Appendix are, in effect, an elaboration of this point.
55. S. WEINBERG: 'Dreams of a Final Theory - the search for the fundamental laws of nature'; 1993, London, Vintage Books, 260 pp.
56. D. DEUTSCH: 'The fabric of reality'; 1997, London, Allen Lane, 390 pp.
57. 92 and similarly tragic. To pursue the analogy drawn in Notes 88, 89, and 126, one might say that a visual system not exposed to data is, apart from the enormous extra complexity, like an immune system not exposed to invading material.
58. The mathematician Leopold Kronecker is reputed to have said, 'God made the integers: all else is the work of man'. He must have been unaware of illusory contours.
59. Mozart admitted this himself: see H. MERSMANN and M. M. BOZMANN: 'Letters of Wolfgang Amadeus Mozart'; 1928, 1972, New York, Dover, 278 pp. Relevant quotes can be found, for instance, on pp. 40 and 221.
60. The creation of anything worth creating, in the arts and sciences equally, always seems to involve an intricate interplay between conscious and unconscious construction. This is well described on pp. 191-196 of Ref. 121. See also 'brain as a committee', p. 292 above. There are the celebrated 'eureka moments': 'illumination, which can happen in a fraction of a second, is the emergence of the creative idea into the conscious', and there is the less celebrated, but arduous, preparation for such moments, impossible without 'an intense conscious curiosity about the subject a craving to exercise the mind on it, quite like physical hunger', lasting for many years. There is the need to find ways, different for different individuals, of 'giving the subconscious every chance'; see also Note 34 of Part I. There is the 'devastating experience' of losing the curiosity and the drive to undertake such arduous labour. All this should be required reading for science policymakers and bureaucrats.
61. J. E. LITTLEWOOD: 'A mathematician's miscellany'; 1953. paperback reissue as 'Littlewood's miscellany', with further material (ed. B. Bollobás: 1986, Cambridge University Press, 200 pp). See pp. 5 and 89 for comments on the relations between prime numbers and complex functions. See also the vivid description, on pp. 249-256 of Ref. 96, of how complex numbers, a key stage in the journey toward complex functions, were discovered, against all the odds, by Gerolamo Cardano in the sixteenth century.
62. J. E. LITTLEWOOD: 'A mathematician's miscellany'; 1953. paperback reissue as 'Littlewood's miscellany', with further material (ed. B. Bollobás: 1986, Cambridge University Press, 200 pp). See pp. 5 and 89 for comments on the relations between prime numbers and complex functions. See also the vivid description, on pp. 249-256 of Ref. 96, of how complex numbers, a key stage in the journey toward complex functions, were discovered, against all the odds, by Gerolamo Cardano in the sixteenth century.
63. n has no positive integer solutions (x, y, z) when n is an integer greater than 2, has challenged mathematicians ever since Pierre de Fermat stated it well over 300 years ago. The chapter on it in Ref. 121, pp. 74-79, sketches a view from the 1920s; the theorem was not proven until the 1990s. The proof draws on deep and wide-ranging concepts, far beyond anything Fermat could have known, and was discovered by the ex-Cambridge mathematicians Andrew Wiles and Richard Taylor. Readers interested in a brief commentary by an expert in the field may consult J. H. COATES: 'The work of Andrew Wiles', Not. Am. Math. Soc., 1996, 43, (7), 760-763. A more substantial survey for a nonspecialist but mathematically literate reader, explaining some of the key ideas but stopping short of the full, and very formidable, technicalities, is that of K. RIBET: 'Galois representions and modular forms', Bull. Am. Math. Soc., 1995, 32, 375-402. Full technical details are given in a 139-page survey of the problem and its history by H. DARMON, F. DIAMOND, and R. TAYLOR: 'Fermat's last theorem', in 'Elliptic curves, modular forms, and Fermat's last theorem', 2nd edn, (ed. J. H. Coates and S.-T. Yau), 2-140; 1997, International Press. 1 am grateful to John Coates for providing these references.
64. n has no positive integer solutions (x, y, z) when n is an integer greater than 2, has challenged mathematicians ever since Pierre de Fermat stated it well over 300 years ago. The chapter on it in Ref. 121, pp. 74-79, sketches a view from the 1920s; the theorem was not proven until the 1990s. The proof draws on deep and wide-ranging concepts, far beyond anything Fermat could have known, and was discovered by the ex-Cambridge mathematicians Andrew Wiles and Richard Taylor. Readers interested in a brief commentary by an expert in the field may consult J. H. COATES: 'The work of Andrew Wiles', Not. Am. Math. Soc., 1996, 43, (7), 760-763. A more substantial survey for a nonspecialist but mathematically literate reader, explaining some of the key ideas but stopping short of the full, and very formidable, technicalities, is that of K. RIBET: 'Galois representions and modular forms', Bull. Am. Math. Soc., 1995, 32, 375-402. Full technical details are given in a 139-page survey of the problem and its history by H. DARMON, F. DIAMOND, and R. TAYLOR: 'Fermat's last theorem', in 'Elliptic curves, modular forms, and Fermat's last theorem', 2nd edn, (ed. J. H. Coates and S.-T. Yau), 2-140; 1997, International Press. 1 am grateful to John Coates for providing these references.
65. n has no positive integer solutions (x, y, z) when n is an integer greater than 2, has challenged mathematicians ever since Pierre de Fermat stated it well over 300 years ago. The chapter on it in Ref. 121, pp. 74-79, sketches a view from the 1920s; the theorem was not proven until the 1990s. The proof draws on deep and wide-ranging concepts, far beyond anything Fermat could have known, and was discovered by the ex-Cambridge mathematicians Andrew Wiles and Richard Taylor. Readers interested in a brief commentary by an expert in the field may consult J. H. COATES: 'The work of Andrew Wiles', Not. Am. Math. Soc., 1996, 43, (7), 760-763. A more substantial survey for a nonspecialist but mathematically literate reader, explaining some of the key ideas but stopping short of the full, and very formidable, technicalities, is that of K. RIBET: 'Galois representions and modular forms', Bull. Am. Math. Soc., 1995, 32, 375-402. Full technical details are given in a 139-page survey of the problem and its history by H. DARMON, F. DIAMOND, and R. TAYLOR: 'Fermat's last theorem', in 'Elliptic curves, modular forms, and Fermat's last theorem', 2nd edn, (ed. J. H. Coates and S.-T. Yau), 2-140; 1997, International Press. 1 am grateful to John Coates for providing these references.
66. E.g. work on 'genetic algorithms' at the Santa Fe Institute and elsewhere: also e.g. B. A. HUBERMAN and T. H. LUKOSE: 'An economic approach to hard computational problems', Science, 1997, 275, 51-54. 'Portfolio' or risk-spreading algorithms make tradeoffs between speed and algorithmic unsoundness.
67. P. W. SHOR: 'Algorithms for quantum computation: discrete logarithms and factoring', Proc. 35th Ann. Symp. on Foundations of Computer Science, 124-134; 1994, New York, IEEE Press.
68. T. N. PALMER: 'A local deterministic model of quantum spin measurement', Proc. Roy. Soc. London, 1995, A 451, 585-608. The idea is to replace quantum non-locality by algorithmic non-computability arising from 'riddled basins' of attractors in phase space. For an excellent lay person's description see Ref. 114, 2nd edn, pp. 348-356.
69. 88,89 when and where to look for an invader, and part of what stops it looking for and attacking its owner's own tissues. See also, for example, earlier correspondence and controversy in Science, 1996, 272, 1405-1408. Such an attention-directing role of more ancient parts of the system seems to me to deepen the partial analogy between immune system function and brain function.
70. 88,89 when and where to look for an invader, and part of what stops it looking for and attacking its owner's own tissues. See also, for example, earlier correspondence and controversy in Science, 1996, 272, 1405-1408. Such an attention-directing role of more ancient parts of the system seems to me to deepen the partial analogy between immune system function and brain function.
71. C. WILLS: 'The runaway brain'; 1994, London, HarperCollins, 358 pp. See also Note 13 of Part I.