Abstract
Science: where the applicability of mathematics, particularly in empirical science, is discussed and the effectiveness of mathematics in science, including the heuristic effectiveness, explained.
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Notes
- 1.
See da Silva n.d.
- 2.
Wigner 1960.
- 3.
- 4.
Crisis §9.
- 5.
Crisis, §9 h.
- 6.
- 7.
Husserl refers explicitly to a correspondence between the system of perceptions (objectified sensations) and its mathematical representative: “[…] the specifically sensible qualities […] that we experience on bodies given to intuition are intimately connected according to a rule, in a particular manner, to the forms that belong to them according to their essence” (Crisis §9c). Still, according to him, the perceptual world has “its double in the realm of forms, in a way that any change in the manifold of contents [the perceptual world, my note] has a causally induced copy in the sphere of forms.” Ibid.
- 8.
da Silva 2010.
- 9.
Of course, scientific theories can refer to levels of empirical reality that are, in practice, inaccessible to experience, but that can be experienced indirectly via chains of physical causation. It remains always open the possibility in principle of directly accessing them.
- 10.
Crisis, §14.
- 11.
Husserl conducted an extensive investigation of the constitution of internal, subjective time in his essay On the Phenomenology of the Consciousness of Internal Time (Husserl 1990). The important thing to be noticed is that subjective time, as any object of consciousness, is constituted, not simply given.
- 12.
“But logical concepts are not concepts extracted from the simplicity of the intuitive; they grow by an activity proper to reason: the formation of ideas, the exact formation of concepts; for example, by this idealization that, in opposition to vague empirical lines and curves, produces the geometrical line, the geometrical circle”. Ibid.
- 13.
“If it is necessary that all experienced qualities have their right to objectivity, this is only possible to the extent that they indicate something mathematical.” Annex I to §9 of Crisis.
- 14.
“[A] meticulous intentional analysis, strictly free of prejudices and in absolute evidence […] does not deprive in the least of its sense the natural conception of the world, that of daily life and also of the exact science of Nature, but proceeds to the lecture of what is effectively and properly contained in this sense.” Appendix I to §9 of Crisis.
- 15.
“[…] the general hypothesis according to which empirical Nature is experienced as an approximation of the mathematically ideal Nature.” Appendix IV to §12 of Crisis.
- 16.
“The general mathematical legality is definite in the sense that it has the form of a finite number of fundamental mathematical laws (the axioms) in which all laws are included, in a purely deductive manner, as consequences”. Ibid.
- 17.
Ibid.
- 18.
Ibid.
- 19.
“The science of nature does not deal with nature itself, but with nature as man considers and describes it”, in W. Heisenberg, Discussione sulla fisica moderna. Turin: Enaudi, 1959.
- 20.
Conservation of energy does have empirical support. It is nonetheless conceivable that a principle is adopted that has no or almost no evidential support. The principle of least action or conservation of parity come to mind.
- 21.
Jeans 1981, p. 155.
- 22.
“Analogies in Nature” (1856), apud Longair 2003, p. 88.
- 23.
Lawden 1995, p. 27.
- 24.
Steiner 1989, p. 458.
- 25.
Id. Ibid. p. 458.
- 26.
In fact, Ampere’s law, as Maxwell clearly recognized, was valid only for closed circuits [my note].
- 27.
These words are Maxwell’s own; he however is not referring to the existence of the displacement current itself, but to the fact that, like “real”, conduction currents, displacement currents can also produce magnetic effects [my note].
- 28.
See Pauli 2000, pp. 109–10, where the justification of the concept of displacement current is based in analogous manipulations. But, notice, Pauli is not committed to, and does not claim historical accuracy.
- 29.
Id. Ibid. p. 114.
- 30.
I follow Longair 2003, pp. 88–98.
- 31.
See his “On Physical Lines of Force” of 1861–2.
- 32.
Remember, in those days the idea that electromagnetic action was due to the presence of mathematical fields was still in the future. However, far-fetched as it is, Maxwell’s mechanism at least offers a picture of physical reality that fields do not. The introduction of the concept of field in science (to which Faraday and Maxwell contributed) reinforced the Platonist trend in science initiated with Galileo and other scientists of the seventieth century.
- 33.
Scientific Papers, vol. 1, p. 486, apud Longair 2003, p. 102.
- 34.
One cannot avoid thinking of electrons and positrons as positive and negative integers, and consider the similarities between the strategies that led to the discovery of, respectively, positrons and negative numbers.
- 35.
See Kumar 2010, pp. 101–02.
- 36.
On seeing Balmer’s formula, M. Kumar tells (op. cit. p. 102), Bohr immediately saw what caused spectral lines, it was electrons “jumping” between different allowed orbits.
- 37.
Paty 1988.
- 38.
I suppose we can understand this relation of analogy as formal analogy.
- 39.
Lautman 1977, p. 281.
- 40.
Brunschvicg 1972, p. 569.
- 41.
D’Espagnat 1979, p. 12.
- 42.
Patty 1988, p. 323.
- 43.
Ibid. pp. 324–5.
- 44.
Ibid. pp. 327–8.
- 45.
Ibid. p. 328.
- 46.
Ibid. p. 331.
- 47.
Ibid. p. 332.
- 48.
Ibid. p. 333. For Paty, however, not all theoretical previsions follow the same pattern. There is the case of the neutrino, originally a mathematical hypothesis (if only in disguise), but there is also the case of de Broglie’s wave-particle duality hypothesis, from the beginning a physical hypothesis, even though mediated by mathematical considerations (the theory of relativity in de Broglie’s case).
- 49.
Ibid. p. 335.
- 50.
Ibid. p. 339.
- 51.
“Then the very idea of facts prevailing independently of observation becomes dubious.” (Weyl 1963, p. 258).
- 52.
“The principle of causality holds for the temporal change of the wave state, but must be dropped as far as the relation between wave and quantum states is concerned” (Weyl 1963, p. 263). From this perspective, the formation of ideas in quantum mechanics must obviously rely less on physical and more on formal or mathematical models and analogies. See “L’influence des images méthaphysiques du monde sur le développement des idées fondamentales dans la physique, particulièrement chez Louis de Broglie”, in de Broglie 1994, pp. 103–114.
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da Silva, J.J. (2017). Science. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_8
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