Predictive success, partial truth and Duhemian realism

Abstract

According to a defense of scientific realism known as the “divide et impera move”, mature scientific theories enjoying predictive success are partially true. This paper investigates a paradigmatic historical case: the prediction, based on Fresnel’s wave theory of light, that a bright spot should figure in the shadow of a disc. Two different derivations of this prediction have been given by both Poisson and Fresnel. I argue that the details of these derivations highlight two problems of indispensability arguments, which state that only the indispensable constituents of this success are worthy of belief and retained through theory-change. The first problem is that, contrary to a common claim, Fresnel’s integrals are not needed to predict the bright spot phenomenon. The second problem is that the hypotheses shared by to these two derivations include problematic idealizations. I claim that this example leads us to be skeptical about which aspects of our current theories are worthy of belief.

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Notes

  1. 1.

    Note that the bright spot phenomenon was not the only one predicted by Poisson. The dark spot is more complex than the bright spot, because the diffraction of white light through a circular aperture produces diffraction rings of different colors surrounding the dark spot.

  2. 2.

    I give in the appendix a modern derivation of the bright spot prediction based on Fresnel’s integral.

  3. 3.

    The case of an irregular circular screen was solved in the 1980s with the help of computer simulations (see (Harvey and Forgham 1984)). Even if Poisson was an outstanding mathematician, he would not have “easily” solved the case of an irregular circular screen. Moreover, these simulations show that the bright spot phenomenon is very sensitive to small-scale deviations from the ideal circular case. A regular sinusoidal corrugation of the circular shape of amplitude 100 \(\upmu \)m edge almost completely removes the central bright spot. It is therefore necessary to assume that the screen is perfectly circular, and not only roughly circular.

  4. 4.

    It is true that supporters of a corpuscular theory of light (such as Laplace, Biot or Poisson) did reformulate their own theory to take diffraction phenomena into account. But, to my knowledge, they offered no derivation of the bright spot phenomenon in the framework of their reviewed corpuscular theory.

  5. 5.

    This distinction between three kinds of novelty has been discussed in the context of the theory of confirmation. In the context of realism, Mario Alai argues that a predicted fact is novel enough to fuel the no-miracle argument if it is a priori improbable and heterogeneous with the old empirical data, i.e. “not inferable from these data by a standard generalization procedure” (Alai 2014, p. 310). On that account, Fresnel’s prediction should also be considered as novel: the bright spot is a priori improbable without a wave theory of light, and it cannot be extrapolated from other diffraction data such as the ones presented by Fresnel in his memoir.

  6. 6.

    In Fresnel’s proof, the geometrical model F5 implies that we add up the “absolute velocity of ethereal particles” of the rays coming from one ring with the oscillation in opposite phase of two contiguous rings: the only ring having just one adjacent ring is the smallest, that is why half of its oscillation is not canceled and is found at the center of the shadow.

  7. 7.

    Modern Quantum Physics has taught us that the same entity could be considered as a discrete particle or a continuous wave. However, these two accounts are still incompatible and mutually exclusive. As shown in the next subsection, Fresnel’s wave theory of light was considered by its contemporary as compatible with an amended version of the corpuscular theory of light. But this claim would have been inconsistent if the hypothesis that light really propagates as a continuous wave had not been replaced by the assumption that light is composed of rays.

  8. 8.

    “We can give to the axis of polarization of light’s molecules a conic oscillatory movement around the axis of translation. [...] This conception seems necessitated by a class of phenomena in which the light’s molecules experience an ordinary refraction by a crystal, not only in a specific position of the principal section, but also left and right from this position” (Biot 1816, p. 284).

  9. 9.

    Laymon’s treatment of idealizations in physics is similar: “In compressed slogan form, the view proposed is this: a theory is confirmed if it can be shown that it is possible to show that more accurate but still idealized or approximate descriptions will lead to improved experimental fit; a theory is disconfirmed when it can be shown that such improvement is impossible” (Laymon 1982, p. 114).

  10. 10.

    Fresnel’s theory is restricted to “near field diffraction”.

  11. 11.

    In another paper, Vickers suggests that a modern analysis based on Maxwell’s equations can show that it is possible to identify “idle wheels within Kirchhoff’s original boundary conditions”, but “that does not mean that what remains is ‘working’ [i.e. deductively indispensable for the prediction]” (Vickers 2013, p. 207). Therefore, a reformulation of an idealization compatible with our present state of knowledge such as R may not be sufficient to circumscribe the part of this idealization responsible for its success. This opacity of the partial truth of idealizations is discussed in Sect. 5.

  12. 12.

    This case is an interesting one for Psillos’ explanationist defense of realism, because in a Newtonian framework, there is no genuine explanation (or prediction) of the fact that all planets move in the same direction. Applying the explanatory criterion, we should then consider that Descartes’ vortex model is real (even if it was latter discarded) because no alternative hypothesis explains this well-founded phenomenon.

  13. 13.

    I describe this stance as “skeptical” because if we cannot circumscribe which parts of our best theories are true and reflect the basic furniture of nature, then the no-miracle argument is not a sufficient basis for a metaphysical knowledge of unobservables. This view is compatible with Saatsi’s version of realism (Saatsi 2015). The “minimal realism” defended by Saatsi claims that there are some theoretical progresses through theory-change, but states that we cannot have any theoretical knowledge of unobservables. Yet, if we could circumscribe the true parts of present theories, this would grant us theoretical knowledge of these parts. Therefore, it seems that in order to separate theoretical knowledge from theoretical progress, Saatsi should accept that the criterion to circumscribe the true parts of a theory is only a retrospective one, and therefore cannot be applied to our present theories.

  14. 14.

    Lemaître’s prediction of the relation between the distance and the redshift of galaxies is not temporally novel but can be considered as a use-novel prediction for two reasons:

    • The equation used by Lemaître to derive this relation has not been designed to accommodate this relation. It was independently discovered by the Russian physicist Alexander Friedmann in 1922, and Friedmann had no interest in the radial velocity of galaxies. His paper is a purely mathematical exercise: it is not even sure that he was aware of the systematic redshifs of galaxies (see (Kragh and Smith 2003) for details). Therefore, the relation between radial velocities and distance is not indispensable to derive the Friedmann-Lemaître equation, which is then used to derive the radial/velocity relation.

    • De Sitter’s derivation of this relation predicts that there should be some “systematic relation” between the redshift of galaxies and their distance (de Sitter 1917, p. 236). But Lemaître’s prediction implies that the relation is proportional, which has been observed by Hubble (Hubble 1929).

  15. 15.

    Another example from relativistic cosmology is the prediction of the cosmological background radiation which was derived first by Alpher and Herman in 1948 on the basis of an isotropic model of the universe (Alpher and Herman 1948), and then by Dicke and Peebles in 1965 on the basis of a non-isotropic but cyclic model of the universe (Dicke et al. 1965, p. 415). The prediction of the existence and position of Neptune was also derived from different working hypotheses by Adam in 1843 and Le Verrier in 1846.

  16. 16.

    I am thankful to an anonymous reviewer’s suggestion for this example.

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Acknowledgements

I would like to thank Kevin Buton-Maquet, Gladys Kostyrka, Timothy Lyons, Peter Vickers, Pierre Wagner, two anonymous reviewers, the organizers of the conference New Thinking on Scientific Realism and the organizers of the conference Scientific Realism and The Challenge from the History of Science for their precious advice and comments.

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Correspondence to Gauvain Leconte.

Appendix: The derivation of the bright spot from Fresnel’s integral

Appendix: The derivation of the bright spot from Fresnel’s integral

This appendix shows how to derive the bright spot prediction form Fresnel’s integral with mathematical tools available at the beginning of the nineteenth century. This derivation can be found in a classical textbook of optics (Moeller 2007, p. 133). Let’s apply Kirchhoff–Fresnel’s integral to compute the amplitude u of the light wave at the center of the shadow of a circular screen:

$$\begin{aligned} u = A2\pi \rho _{0} \int _{\sqrt{a^{2}+\rho _{0}^{2}}}^{\infty } \left( \frac{1}{\rho ^{2}}\right) e^{ik2\rho }d\rho \end{aligned}$$
(1)

In this formula, u is the amplitude of the wave at the center of the shadow, A the amplitude of the incident wave at the source and a is the radius of the circular object. \(\rho \) is the distance between the diffracted wave and the center of the shadow, \(\rho _{0}\) is the distance between the center of the circular object and the center of the shadow (see Fig. 1). Then, \(\sqrt{a^{2}+\rho _{0}^{2}}\) is the distance between the edge of the circular object and the center of its shadow. Thus, the integral (1) represents the contribution of all the diffracted light wave at the “optical center” of the screen, i.e. the center of the shadow.

Fig. 1
figure1

Diffraction by a circular screen. S source of light, O center of the shadow

Integration by parts of (1) yields:

$$\begin{aligned} u = \left[ \left( \frac{1}{\rho ^{2}}\right) \left( \frac{1}{2ik}\right) e^{ik2\rho }\right] ^{\infty }_{\sqrt{a^{2}+\rho _{0}^{2}}} + (1/ik) \int _{\sqrt{a^{2}+\rho _{0}^{2}}}^{\infty } \left( \frac{1}{\rho ^{3}}\right) e^{ik2\rho }d\rho \end{aligned}$$
(2)

Neglecting the right part as null gives:

$$\begin{aligned} u = \left[ \left( \frac{1}{\rho ^{2}}\right) \left( \frac{1}{2ik}\right) e^{ik2\rho }\right] ^{\infty }_{\sqrt{a^{2}+\rho _{0}^{2}}} \end{aligned}$$
(3)

To solve this equation, we have to assume that at infinity, the light wave is not diffracted, i.e. that rays infinitely far away from the screen have a null contribution to the constructive interference at the center of the shadow. Then, to compute the intensity, we use the relation: \(I=uu^{*}\) If we pose that: \(I_{0}=A^{2}\frac{\rho _{0}^{2}}{\sqrt{a^{2}+\rho _{0}^{2}}}\), then (3) gives:

$$\begin{aligned} I= \frac{I_{0}\lambda ^{2}}{4} \end{aligned}$$
(4)

The intensity I thus only depends on the wavelength and is proportional to \(I_{0}/4\) for a given wavelength. This results corresponds to the intensity in the absence of any obstacle, which means that the center of the shadow is illuminated as if there was no circular object.

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Leconte, G. Predictive success, partial truth and Duhemian realism. Synthese 194, 3245–3265 (2017). https://doi.org/10.1007/s11229-016-1305-8

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Keywords

  • Scientific realism
  • Partial truth
  • Novel prediction
  • Fresnel’s bright spot prediction
  • Divide et impera move
  • Pierre Duhem