Abstract
In a famous 1960 paper, Wigner discussed “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I suggest that the effectiveness of mathematics in producing successful new theories and surprising discoveries is even more unreasonable than Wigner claimed. In this paper, I present several historical case studies to support the claim that mathematics is often responsible for instigating scientific revolutions. However, that does not mean that mathematics is always the key to the universe, and other cases where mathematization was not successful are discussed in order to problematize a naïve Platonism.
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Notes
The “constant” b would theoretically increase at high densities to 4 times its value at low densities, so if its value were determined from measurements at low density, the equation would give bad results at high densities. The a/V² term does not have any clear meaning in terms of interatomic forces; while it is sometimes incorrectly assumed to be derived from an attractive force inversely proportional to the 7th power of the distance between two molecules (hence the name “van der Waals force”), it was pointed out by Boltzmann and others that it actually implies a small constant force acting over an infinite distance. For further discussion and references see Klein (1974, pp. 28–47), Kipnis et al. (1996), Brush (1976, Chapter 7), Boltzmann (1995).
An earlier conjecture involves more complicated mathematics, but also invokes physical arguments such as the desire to keep the hypothetical “celestial spheres” (cf. title of Copernicus’ book De Revolutionibus Orbium Caelestium where the Latin orbis can mean either sphere or circle) from interfering with each other. See Swerdlow (1973), Swerdlow and Neugebauer (1984).
In the traditional view, the anomaly that led Copernicus to reject the geocentric system was the need to add more and more epicycles—even “epicycles on epicycles”—in order to accommodate more accurate observations. This view is now discredited; see Gingerich (1975). In fact, Copernicus himself admitted that the Ptolemaic theory was “consistent with the numerical data” but was “neither sufficiently absolute nor sufficiently pleasing to the mind.” The general claim that Copernicus asserted his theory on formal rather than physical grounds—that it gives “a satisfactory account of the relations of the phenomena in Space and Time, that is, of the Motions” of heavenly bodies, not the “Causes of the Motions,” was stated by William Whewell (1857, p. 29). Copernicus admits that while his postulated motions of the Earth “though they be difficult and almost inconceivable, and against the opinion of the majority,… we will make clearer than the sun, at least to those, who are not ignorant of mathematics” (ibid., pp. 30–31).
Kuhn argues that the “as if” attitude of these sixteenth-century astronomers was similar to that of their predecessors: “Ptolemy himself had never pretended that all of the circles used in the Almagest to compute planetary positions were physically real; they were useful mathematical devices, and they did not have to be any more than that” (p. 187).
“It is inconceivable, that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, in the sense of Epicurus, be essential and inherent in it. And this is one reason, why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another as a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical [i.e. scientific] matters a competent faculty of thinking, can ever fall into. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers.” Letter to Richard Bentley, 25 February 1693.
See also the translators’ comments on pp. 274–77. On the problem of translating “fingo” see Koyré (1965, p. 35).
For details see Aiton (1995, pp. 13–15).
For further discussion of this point see Gingras (2001). Maupertuis defended the Newtonian theory by arguing that Cartesian “impulsion” pushing by collisions was no more intelligible than attraction.” This argument allowed the Cartesians “to abandon the vortices [of Descartes] with a clear conscience and accept universal gravitation as a physical axiom” after its superior predictive power had been demonstrated. (Aiton 1995, pp. 19, 21).
Inaugural Lecture at Aberdeen University, 3 November 1856; see also Inaugural Lecture at Kings College London, October 1860. Maxwell (1990, pp. 429–30, 671).
Maxwell used a derivation published by John Herschel in a review of books by the Belgian statistical Adolphe Quetelet; Quetelet used the normal law to describe human physical characteristics and social behavior. See Garber et al. (1986, pp. 7–12, 226–28, 289–91). His 1866 attempt to justify the law by a consideration of molecular collisions is in Maxwell (1990, pp. 437–440).
Brush (2003, pp. 480–504). Here are two crucial stages in the reception of randomness: (1) “The weakness of the theory lies… in the fact that it leaves the duration and direction of the elementary processes to ‘chance.’ Nevertheless I am fully confident that the approach chosen is a reliable one.”—Einstein (1916, pp. 47–62). (2) “Schrödinger’s quantum mechanics yields a very definite answer on the [scattering] problem; but one does not obtain a causal relation… One only obtains an answer to the question: ‘How probable is a given effect of the collision?’… Here the whole problem of determinism arises. From the point of view of our quantum mechanics no quantity exists which in a single collision fixes its effect causally… I myself tend to the opinion that in the world of atoms, determinism has to be abandoned.”—Born (1926, pp. 863–867).
Planck (1900, pp. 237–45). Translation by D. Ter Haar in Planck (1972, see pp. 8 and 40). Planck (1920); translation by R. Jones and D. H. Williams in Planck (1960, pp. 102–14, on p. 109). This interpretation was first published by Kuhn (1978), based on a detailed analysis of the early papers of Boltzmann and Planck. Kuhn’s view was rejected or ignored for several years, but has now become the consensus of historians of physics; see Brush (2002), pp. 119–127 and references cited therein).
Wigner (1960, p. 7).
Dirac (1928, pp. 610–624, quotations from pp. 610, 611, 612, 613, 618).
Dirac (1963, pp. 45–53 on p. 53).
The technical definition of an entangled state is the following: one that cannot be written as a combination of product states. In the case of pure quantum states, this definition would reduce to the following: an entangled state is a pure state that cannot be written as a product state. (This corresponds to a partial differential equation that cannot be solved by separating the variables.) The example discussed by EPR is unnecessarily complicated; for the purpose of experimental tests, it is simpler to use, instead of position and momentum, the two possible spin states of a spin ½ particle. Using Dirac’s notation, let |0> and |1> be the two possible spin states. Then, for a system composed of two such particles, when the spin is measured for an axis in the z direction, the state can be expressed as |z = 0>|z = 1 >–|z = 1>|z = 0 > (note the antisymmetrization: the sign changes when the labels of the particles are interchanged). If you measure particle A’s spin in the z direction and find it is “up” then you know that particle B’s spin must be “down” with respect to the z direction. But if you choose to measure A’s spin in the x direction and find it is up, then B’s spin must be down in the x direction. Since the two spins are associated with non-commutating operators, they cannot both be “real” in EPR’s terminology. Quantum mechanics predicts, and experiments confirm, that the spins of A and B are correlated even if the particles are so far apart that it seems impossible that your observation of one can affect the other. (I thank Jeffrey Bub for this explanation).
Nielsen and Chuang (2000), Bub (2000). Expositions for the lay reader: Milburn (1998), Aczel (2002, 2003). The history of this subject is treated in a forthcoming article by Joan Lisa Bromberg, “The Rise of “Experimental Metaphysics in Late Twentieth Century Physics” (unpublished draft on deposit in the Quantum Optics Archive at the Center for History of Physics, American Institute of Physics, College Park, MD). For a review of the debate up to 1978 see Brush (1980).
For doubts about the objectivity of Eddington’s eclipse report see Brush (1999), pp. 184–214, note * on p. 201 and note 64 on p. 213).
Quoted by Holton (1988), pp. 279–370, on p. 255).
Einstein (1917): 142–152. English translation, “Cosmological Considerations on the General Theory of Relativity” in Einstein (1923), quotations from pp. 184, 186, 188).
Longair (1995); quotation from Ya. B. Zeldovich, Usp. Fiz. Nauk 95 (1968): 209. Longair writes that the λ-term should lead to “a number of quite specific astronomical phenomena, but there has been no convincing positive evidence that any of these has been observed” (p. 1724).
This is a different translation of Einstein’s remarks from that in the original printing of his lecture. The older version can be found as Einstein (1933a, b). The most accessible version to find the quote in this older version can be found in a reprint Einstein (1933a, p. 167). Einstein’s views on the role of mathematics in science are discussed by Jungnickel and McCormmach (1986, pp. 334–347).
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Brush, S.G. Mathematics as an Instigator of Scientific Revolutions. Sci & Educ 24, 495–513 (2015). https://doi.org/10.1007/s11191-015-9762-x
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DOI: https://doi.org/10.1007/s11191-015-9762-x