by David Orrell
2. Powell 1998, p. 172.
3. Abraham 1994, p. 92.
4. Wood 2003.
5. The gas might have been the narcotic ethylene, released through faults in the rock. Roach 2001.
6. Cornford 1969, p. 35.
7. Iamblichus 1918, p. 6.
8. Ibid.
9. His followers often suggested that Pythagoras was Apollo in human form. See Strohmeir and Westbrook 1999 for a general history of Pythagoras. The exact dates for the life of Pythagoras are not known.See Guthrie 1962, p. 173.
10. Iamblichus 1918, p. 49.
11. Ibid., p. 50.
12. Strohmeir and Westbrook 1999, p. 49.
13. Iamblichus 1918, p. 50.
14. Ibid., p. 11.
15. Ibid., p. 21.
16. Miller 2006.
17. From Bogen 1975, quoted in Edwards 1999, p. 37.
18. Ibid.; Guthrie 1962, p. 252.
19. Iamblichus 1918, p. 79.
20. Midgley 1985, p. 98. This is a shortened version of her table.
21. Guthrie 1962, p. 458.
22. Iamblichus, 1918, p. 59.
23. Ibid., p. 32.
24. To see that the square root of 2 cannot be written as a rational number A/B, suppose first that it can, so (A/B)2=2. We can assume that A and B do not have a common factor, otherwise this can be divided out. Multiplying each side of the equation by B2 gives A2=2B2. The number A2 is even, since it is a multiple of 2. Therefore A must also be even (because if it was odd, the square would be odd). Since A is even, we can write A=2C for some C. So substituting into A2=2B2 gives (2C)2=2B2. Therefore 4C2=2B2, or dividing each side by 2, 2C2=B2.Since the left-hand side is even, it follows that B2 is even, and therefore that B is even (if it was odd, the square would be odd). So A and B are both even and have a common factor of 2. But this contradicts the assumption that A and B do not have a common factor. The only way out is to conclude that the initial assumption is wrong; in other words, the square root of 2 cannot be written in the form A/B.
25. The dart would have to be infinitely thin to pick out exactly one number. In mathematical terms, the measure of the irrational numbers in the line segment from 0 to 1 is 1, while the measure of the rationals is 0.
26. See Guthrie 1962, pp. 176–77.
27. Koestler 1968, p. 25.
28. Plato, Apology. Translated by Benjamin Jowett. Online at http://classics.mit.edu. Seaton 2004.
29. Field 1956. Guthrie also writes that “Plato’s debt to the Pythagoreans is obvious.” Guthrie 1978, p. 216.
30. Thucydides 1910. The disease may not correspond to a modern one.
31. It was believed in ancient times that plague was caused by Apollo’s arrows. For this reason, St. Sebastian, who is usually pictured with his body run through by arrows, was one of the saints most often invoked. Ferguson 1954.
32. Eudoxus was called a Pythagorean. He was an associate or friend of Plato, but probably not a student. Guthrie 1978, p. 447.
33. Wood 2003, p. 136.
34. Quoted in Peterson 1993, p. 37.
35. Quoted in Van der Waerden 1974.
36. Ibid., p. 48.
37. Reagan 1989.
38. Quoted in Gill et al. 1981, p. 1.
39. Aristotle, On the Heavens, 306a. Online at http://classics.mit.edu.
2 ⊳ LET THERE BE LIGHT
TYCHO BRAHE AND THE MODEL MAKERS
1. Watson 1968, p. vii.
2. Quoted in Zöllner and Nathan 2003, p. 104.
3. Koestler 1968, p. 192.
4. Tycho Brahe, Astronomiae instauratae mechanica (1598). The Smithsonian Institution has the text and figures online in its digital library at http://www.sil.si.edu.
5. Quoted in Christianson 2000, p. 52.
6. Image from Tycho Brahe, Astronomiae instauratae mechanica (1598).
7. Iamblichus 1918, p. 42. From Pope’s 1720 translation of The Iliad, Book 16, “The Death of Patroclus”:
A Dardan youth there was, well known to fame, From Panthus sprung, Euphorbus was his name; Famed for the manage of the foaming horse, Skill’d in the dart, and matchless in the course
8. See illustration in Christianson 2000, p. 119. From another of his poems: You, Ptolemy, Alfonso, Copernicus, I gave a hand: you slipped, but I held fast.The motion of the stars you could not grasp As I have done. In truth, my work was great: New pillars raised for heaven’s sparkling dome. Quoted in Christianson 2000, p. 216.
9. Connor 2004, p. 79.
10. Quoted in Koestler 1968, p. 242.
11. Deacon 1968.
12. Iamblichus 1918, p. 17.
13. From Mysterium Cosmographicum, by J. Kepler, 1621.
14. Ridley 2001, p. 105. The full title is A Forerunner to Cosmographical Treatises, containing the Cosmic Mystery of the admirable proportions between the Heavenly Orbits and the true and proper reasons for their Numbers, Magnitudes and Periodic Motions, by Johannes Kepler, Mathematicus of the Illustrious Estates of Styria, Tübingen, anno 1596.
15. Iamblichus 1918, p. 45.
16. Koestler 1968, p. 265.
17. Letter to D. Fabricius, 18 December 1604. Quoted in ibid., p. 330.
18. Vincenzo Galilei, 1581. Dialogo della musica antica e moderne (A dialogue on ancient and modern music).
19. Koestler 1968, p. 388.
20. Quoted in Gribbin 2002, p. 56.
21. Connor 2004, p. 360.
22. Quoted in Machamer 1998, pp. 64–65.
23. Galilei 1967, pp. 58–59.
24. The course of the plague through London was dramatized in Defoe’s 1722 A Journal of the Plague Year.
25. Brumfiel 2004.
26. Wertheim 1995, p. 124.
27. Keller 1985, p. 53.
28. Griffiths 2004, p. 167. Gender here refers to abstract properties, like yin and yang; individuals, of course, have a mix of such characteristics.
29. Iamblichus 1918, p. 36.
30. The solution is obtained by integration and gives x = –gt2/2 + v0t + x0, v = –gt + v0, where x0 is the initial position and v0 is the initial velocity of the stone. The plot of x versus t is a parabola, the plot of v a straight line.
31. Gribbin 2002, p. 111.
32. The difference between velocity and speed is that velocity has a direction (it is a vector), while speed is a magnitude. Here, velocity is a negative number because it is directed down, but speed is never negative.
3 ⊳ DIVIDE AND CONQUER
THE GOSPEL OF DETERMINISTIC SCIENCE
1. Descartes 1960, p. 50.
2. Pierre Simon Laplace, Essai philosophique sur les probabilités, 1814.Reproduced in the Oeuvres complètes de Laplace, vol. 11 (1886).
3. Quoted in Koestler 1968.
4. Bragg 1999, p. 163.
5. Darwin 1905, p. 1.
6. Ibid., p. 39.
7. Quoted in Hirsch 1984.
8. Ibid.
9. Wertheim 1995, p. 120.
10. Quoted in Sherden 1998, p. 203.
11. Griffiths 2004.
12. Latitude could be determined at sea from the stars, but since they rotate from east to west, knowing the exact time was an essential part of the deduction. Jardine 1999, p. 160.
13. For example, the acceleration of a falling stone is given by –g. The effect of wind resistance could be modelled by adding a term such as –0.1v, so the acceleration is reduced in magnitude as speed increases.The plot of either of these equations as a function of v gives a straight line: the first is just a constant, so gives a horizontal line, while the second has a slope. Wind-tunnel studies, however, have shown that a better model has a term in v2 rather than v. The plot of such a function gives a parabola.
14. Yorke and Li 1975.
15. Quoted in Wertheim 1995, p. 250.
16. Capra 1983.
17. Dawkins 1976.
18. Brooks 2002, p. 173.
19. Bohm 1974, pp. 127–28. Quoted in Midgley 1985, p. 50.
20. Quoted in Sherden 1998, p. 225.
21. In 1939, though, Einstein
had written to President Roosevelt to encourage him to build the bomb before the Germans. See Saul 1992, p. 305.
22. See, for example, http://mathworld.wolfram.com/Life.html.
23. Wolfram 2002, pp. 715–17.
24. Ibid., p. 381.
25. Ibid., p. 741.
26. Coupland 2002, p. 122.
27. Quoted in Peterson 1993, p. 96.
28. Greenblatt 2004, p. 305.
29. Quoted in Capra 1996, p. 19.
30. For a history of the role of women in science, see Wertheim 1995.
4 ⊳ RED SKY AT NIGHT
PREDICTING THE WEATHER
1. Frisinger 1977, p. 55.
2. Moran et al. 1997, p. 110.
3. Ibid., p. 135.
4. Cox 2002, p. 80.
5. Ibid., p. 81.
6. Gillham 2001, p. 149.
7. Ibid., p. 76.
8. Bjerknes 1904.
9. Cox 2002, p. 161.
10. Walker 1918.
11. Walker’s statistical methods were initially met by “extreme skepticism” by the meteorological community. Katz 2002.
12. Abraham 1994, p. 210.
13. Cox 2002, p. 203.
14. Ibid., p. 202.
15. Manabe et al. 1965.
16. Deacon 1968, p. 242.
17. Edwards 2000.
18. National Academy of Sciences, Committee on Atmospheric Sciences Panel on Weather and Climate Modification 1966, pp. 65–67.
19. The shift map, discussed in Chapter 3, has both fixed points, which
are mapped to themselves, and periodic cycles. The point 0, for example, is mapped to 0 in perpetuity, so it is a fixed point. However, any small perturbation will double in size repeatedly under the map, so 0 is not a stable point and not an attractor. Similarly, the point ID="300"> is mapped to ID="301"> , which is mapped back to ID="302"> , so this forms a periodic loop, which is again unstable and not an attractor.
20. The RMS has a number of mathematical and statistical properties that make it easier to work with than the average. See glossary for definitions of technical terms.
21. Wolfram 2002, p. 381. See Appendix II.
22. The metric was 500 mb, discussed below.
23. Toth et al. 1996. As Wolfram wrote, “It is usually assumed that—like in the Lorenz equations—the phenomenon of chaos must make forecasts that are based on even slightly different initial measurements eventually diverge exponentially.” Wolfram 2002, p. 1177.
24. As Lorenz put it, “now we had an excuse.” Quoted in Gleick 1987, p.18. See also Calder 2003, pp. 137–38.
25. Toth et al. 1996.
26. For information on the Earth Simulator Center, see http://wwwes.jamstec.go.jp/esc/eng/.
27. See http://www.oso.noaa.gov/goes/.
28. Model output cannot be taken literally (see Roebber et al. 2004).Human forecasters can increase skill by 10 to 15 percent (see Kerr 2004a).
29. Estimate is from the reinsurer Swiss Re. Anonymous 2005f.
30. Rabier et al. 1996.
31. See, for example, Kerr 2004a. Statements such as “Today’s three-day forecasts are as good as one-day forecasts were in 1981” (Anonymous 1999b) refer to the 500 mb metric.
32. There is always some turbulence at higher altitudes owing to clouds or large-scale flows like the jet stream, but it is much smaller than at lower altitudes.
33. Assuming that the forecast is free or of negligible cost. Roulston et al. 2003.
34. Results will vary depending upon the area. For an unbiased discussion, see Sherden 1998.
35. Bosart 2003; Ebert et al. 2003; Roebber et al. 2004. The National Weather Service in the U.S. publishes statistics online: see, for example, “Annual HPC -vs- NWP Guidance Threat Scores 1993–2004” at http://www.hpc.ncep.noaa.gov/html/hpcverif.shtml.
36. Kerr 2004a; Roebber et al. 2004.
37. Naïve forecasts can do quite well too. See Sherden 1998.
38. See http://www.nhc.noaa.gov/verification/ and http://www.nhc.noaa.gov/HAW2/english/forecast/errors.shtml. Predicting the number of storms in a given season isn’t much easier. On May 16, 2005, NOAA’s prediction for the 2005 Atlantic hurricane season (which began on June 1) was for twelve to fifteen tropical storms. On August 21, halfway through the season, the range was updated to eighteen to twenty-one.
39.The final tally was twenty-eight. They attributed the increase to naturally occurring, multi-decadal variability. (See http://www.noaanews.noaa.gov/stories2005/s2438.htm.) 39. Emanuel 2005; Webster et al. 2005.
40. Chagnon 2000.
41. Kerr 2004.
42. As Toth et al. (1996) put it, “The ensemble strategy will work only if the models are good enough that model-related errors do not dominate the final error fields.” A similar assumption was made at ECMWF: “From its inception, the EPS [Ensemble Prediction System] has been based on the premise that medium-range forecast errors are predominately associated with uncertainties in initial conditions.” Buizza et al.2000.
43. As Mandelbrot noted in the context of financial prediction, “If you are going to use probability to model a financial market, then you had better use the right kind of probability.” Mandelbrot and Hudson 2004, p. 105. The same applies for weather models. Parameters can be varied using a Bayesian approach, which accounts for prior (subjective) beliefs. However, if the model is structurally deficient, there need be no correct value or range of values. The parameter is a property of the model, not of the system. Oreskes et al. 1994; Smith 2000; Orrell 2005a.
44. If a distribution is desired, random errors can be added based on past forecast errors. See, for example, Roulston et al. 2003, figure 7, panels c) and d); and Jewson et al. 2002. One tool for measuring the accuracy of probabilistic forecasts is the Brier score. See Rosenthal 2005, p. 240 for a simple description.
45. Sherden 1998.
46. From a collection of historical American documents at http://www.ku.edu/carrie/docs/amdocs_index.html.
47. Sherden 1998, p. 14.
48. This was pointed out by Lenny Smith. See Matthews 2004.
49. Holton 1992, p. 117.
50. Wolfram 2002, p. 997.
51. Moran et al. 1997, p. 125.
52. Algae may play a role in regulating cloud cover. See Charlson et al.1987.
53. Jaenicke 2005.
54. Roebber et al. 2004.
55. Thelen and Smith 1994, p. xix. Quoted in Moore 2002, p. 152.
56. Toth et al. 1996.
57. Anonymous reviewer, 2000. Commenting on Orrell et al. 2001.
58. See Junger 1997.
59. Lenny’s LSE website is http://stats.lse.ac.uk/smith/.
60. Smith 1996, 1997, 2000; Gilmour 1998. See also Danforth and Yorke 2005.
61. Gilmour 1998 looked at shadowing in a weather-related context.
62. The drift technique was proposed in Orrell 2001, and is developed in Orrell 2005a, Orrell 2005b. The method can be used to separate out the contributions to total forecast error of model error and observation error. The observation error in this case corresponded only to the small truncation error that occurs when translating from the high-resolution model to the low-resolution model.
63. Orrell et al. 2001. Model convergence is not in itself sufficient for prediction accuracy, since the models could be converging to the wrong thing, but if they have not converged, it is an obvious indicator of a problem.
64. Ibid. See also Matthews 2001. There are probably particular situations where the climate system is sensitive to initial conditions but the net overall effect is small. See also Wolfram 2002, p. 1177. The effect of the data-assimilation procedure is to reduce the contribution of both the apparent forecast errors and the initial-condition errors to the drift calculation (Orrell 2005c).
65. Pearson 1905.
66. This is from an e-mail dated 24 Nov. 2000.
67. Errico et al. 2002.
68. Lorenz 1982; Simmons et al. 1995.
69. Orrell 2002. The effect is like an accounting trick in finance, in whi
ch apparently rapid growth is conjured up by transferring money from other accounts. Wolfram also mentions the problem: “Attempts are sometimes made to detect sensitive dependence directly by watching whether a system can do different things after it appears to return to almost exactly the same state. But the problem is that it is hard to be sure that the system really is in the same state—and that there are not all sorts of large differences that do not happen to have been observed.” Wolfram 2002, notes to p. 972.
70. Neumann and Flohn 1987. Quoted in Sherden 1998, p. 18.
71. The technique is known as 4DVAR. See Cohn 1997. Orrell 2005b discusses the effect on forecast errors. See also Houtekamer et al. 2005.
72. Which is why, prior to 2000 when I was writing my thesis, it was viewed as uncharted territory. Papers that appeared to show that model error is small included Downton and Bell 1988, Richardson 1997, and Harrison et al. 1999. These showed that on certain occasions, different models gave quite similar forecasts. But this does not imply that models are right, only that they are wrong in similar ways. More recent papers claiming an enhanced role for chaos are based on lagged forecasts (Simmons and Hollingsworth 2002; Hamill et al. 2006), but this method makes the model appear more chaotic than it really is. A team from the Meteorological Service of Canada performed data assimilation experiments which accounted for model error: “Our results seem to support the paradigm, proposed by Orrell et al. (2001), in which model error is the main error source for the first few days of a forecast.” Houtekamer et al. 2005.
73. Kosko 1993, pp. 41–42.
74. Wolfram 2002, notes to p. 971.
75. Ibid., p. 381.
76. As science writer Nigel Calder put it, “The butterfly served as a scapegoat for 40 years.” Calder 2003, p. 137.
77. At higher resolutions, “errors in the physical parameterizations and previously negligible effects both become increasingly important.”Roebber et al., 2004.
5 ⊳ IT’S IN THE GENES
PREDICTING OUR HEALTH
1. Sequence obtained from www.ensembl.org.
2. Quoted in Keller 2000, p. 84.
3. Galton 2001.
4. Galton 1871.
5. Galton 1869.
6. Galton 1898.
7. Galton 1876. Gillham 2001, p. 211.
