Abstract
In previous papers,1 I successfully introduced constructive mathematics into theoretical physics. Unlike classical mathematics, (Cauchy — Weier-strass — Dedekind) constructive mathematics does not make use of actual infinity. It extends rational numbers to a denumerable field of real numbers, which, although more restricted than the field of classical mathematics, includes all common real numbers, and moreover introduces some subtle differences which are extremely relevant for a student of the foundations of physics. Historically, constructive mathematics did not gain the attention of mathematicians until 19672, although its formal beginnings date from the first years of our century. It substantially fulfills Duhem’s wish for an “approximation mathematics,” Poincaré’s idea of “physical continuum,” Weyl’s call for that “elementary mathematics” which is useful only to physics, and Bridgman’s hints at an “operationalist mathematics.”
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A. Drago, Constructive mathematics and Carathéodory’s thermodynamics, Lett. N. Cimento 34 (1982) 52–56.
A Drago, Dimensional Analysis and Constructive Mathematics, Lett.N.Cim.37 (1983) 409–12
A. Drago, Relevance of Constructive Mathematics, to Theoretical Physics, in E. Agazzi et al. (eds.), Logica e filosofia della scienza, oggi, CLUEB, Bologna, 1986, vol. II,267–272.
E. Bishop, Foundations of Constructive Mathematics, McGraw-Hill, New York, 1967. For introductory literature.
see: D.S. Bridges, F. Richman, What is constructive Mathematics? Math. Intell. 6 (1984) 29–38.
A. Robinson, Non-standard Analysis, North-Holland, Amsterdam, 1960, ch.X.
E. Beth, Natuurphilosophie, Görinchen, 1948, and The Foundations of Mathematics, Harper, New York, 1970, ch. I, § 2.
N.I. Lobatchevsky, Geometrische Untersuchungen zur Theorie der Parallel-linien, Berlin, 1840 (Engl. transi. by Halsted, Austin, 1893). Geometry is included in the list because since Gödels theorem one may question the Hilbert’s view; actually, Lobachevsky meant geometry as a theory about real bodies. For the remainder of the theories, see: A. Drago, C. Mordillo, Perchè l’ordine logico dei principi della termodinamica inverte l’ordine di scoperta storica?, Atti IV Congr.Naz.St.Fisica, Como, 1983, 113–121.
G. Berkeley first advocated an “instrumentalistic” view of scientific theories. Many authors claimed an organization opposed to the Aristotelian one: Nagel speaks of “instrumentalist theories,” Hintikka of “goal-oriented process,” Lakatos of “empiricist theories.” The best illustration of the two organizations of a theory is sketched by A. Rapoport in R.L. Ackoff, Scientific Method, Wiley, 1962, pp. vii-viii. choice between actual infinity and potential infinity contributed to fashion the scientific theories which different scientists would offer. The second premise is a consequence of giving historical relevance to the Jacobin program of recasting the foundation of all science (C.C. Gillispie, The Encyclopédie and the Jacobin Philosophy of Science: a Study in Ideas and Consequences, in M. Clagett (ed.), Critical Problems in History of Science, U. Wisconsin P., London, 1969, 255–289) and/or to the “odd” way in which S. Carnot and L. Carnot built their theories: L. Carnot’s geometry and mechanics, and S. Carnot’s thermodynamics present a problematic organization.
Together with modern chemistry and with L. Carnot’s new foundations of differential calculus — both of which are theories with problematic organization — constitute a new foundation for the whole of contemporary science. All in all, an accurate historical investigation of both the birth of modern science and the philosphy of science during the French Revolution can make apparent the two basic options of a physical theory. The two premises mentioned clearly show the few stereotyped statements which Koyré uses again and again to characterize the passage from ancient to modern science (A. Drago, La storia del concetto di spazio come rivelatrice delle due scelte fondamentali di una teoria fysica. II, A. Koyré e la metafysica della scienza moderna, in F. Bevilacqua (ed.), Atti VII Congr.Naz.St.Fisica, Padova (1986), 1987, 119–123)
As well as Galilei’s historical case (A. Coppola, A. Drago: Galilei ha inventato il principio d’inerzia?, in S. D’Agostino, S. Petruccioli (eds.), Atti V Congr.Naz.St.Fisica, roma, 1985, 319–327). In the study of the historical development of modern science, the next most relevant problem facing the historian is that of giving a full interpretation of Newton’s scientific work. The present paper is aimed at accomplishing this task.
A. Drago, A. Coppola, op.cit. n.7.
A. Koyré, Études d’histoire de la pensée scientifique, Gallimard, Paris, 1966, p. 188.
A. Koyré, op. cit. n. 7, ch. 4.
G. Berkeley, The Analyst, London, 1734.
A. Drago: Storia del cocetto di spazio come rivelatrice delle scelte fondamatali della teoria fisica, I; Atti VII Congr. Naz. St. Fisica, Trieste, 1985, (in press); (Engl, abstract in Not.SIF suppl. 1986, p.38).
Let us note that the momentum of such new scientific theories generated new philosophical attitudes (mainly, positivism); but this fact did not greatly contribute to the acceptance of the theories by the scientific community.
It should be noted that the metaphysical content of two such options supports the traditional claim by physicists that metaphysics is irrelevant to clarify theoretical physics. However, some recent philosophers of science (Husserl, Burtt, Koyré, Feyerabend) did rightly suggest that a rigid metaphysics underlies modern science. The following historical appraisal of the whole scientific development by R. Burtt seems to me particularly penetrating: “Metaphysics, they tended more and more to avoid, so far as they could avoid it; so far as not, it became an instrument for their further mathematical conquest of the world.” See R. Burtt, The Metaphysical Foundations of Modern Physical Science (1924), Routledge and Kegan, London, 1972, p,. 303.
A. Drago, Una definizione precisa di incommesurabilità delle teorie scientifiche, Atti VII Congr. Naz. St. Fisica, 1986, Padova (1987), pp. 125–129, (Eng. abstract in Not.SIF suppl., 1986, p. 212). A. Drago, Koyré e la metafisica della scienza moderna, ibidem, pp. 119–123.
This characterization of a “paradigm” differs from that offered by the structuralists because their “core” does not take any of the two basic options into account. Thus, their theory-nets would here become much more complex and split up. Instead, the present characterization is similar to the broad quadruple division of scientific theories suggested by V. Tonini, Le scelte della scienza, Nuova Universale Studium Roma, 1977, and to the four types of abstract productions relative to the four types of societies suggested by J. Galtung, Ideology and Methodology, Eijlers, Copenhaven, 1976, ch. I.
T.S. Kuhn, The Structure of Scientific Revolutions, U. of Chicago P., 1962. In fact one could re-read this work while identifying Kuhn’s paradigm with Newton’s model.
A. Drago, see the papers listed in footnote 15, as well as A. Drago and S.D. Manno, La meccanica di Lazare Carnot è una alternative alla meccanica newtoniana migliore di quella di Mach, Atti VII Congr. Naz. St. Fisica, 1986, Padova (1987), pp. 137–142.
C.C. Gillispie, Lazare Carnot Savant, Princeton U.P., Princeton, 1971. A. Drago and S.D. Manno, op. cit. n.18. One more example of a list of physical concepts at radical variance owing to the incommensurability of two physical theories is studied in A. Drago and V. Guerriera, Il primo caso di incommensurabilità tra teorie fisiche: ottica car-tesiana vs. ottica newtoniana, in F. Bevilacqua (ed.), op. cit. n. 7, 131–136.
The famous Einstein confessed to being an opportunist on the metaphysical level.
L. Carnot, Principes fondamentaux de l’équilibre et du mouvement, Crapelet, Paris, 1803. A detailed study of its principles is to be found in A. Drago and S.D. Manno, Le ipotesi fondamentali della meccanica secondo L. Carnot (submitted to Epistemologia).
I.N. Lobachevsky, New Principles of Geometry, (Russian), 1835–38; (Engl. transl. Neomonic Ser. no. 5, J. Hopkins Press, Austin, 1897).
I.N. Lobachevsky, Geometrische Untersuchungen…, op. cit. n. 5.
Already H. Poincaré had perceived the distinction between the last two cases and called them resp. the “mathematical continuum” and the “physical continuum.” (H. Poincaré, La Science et l’Hypothèse, Paris, 1907, ch. 3.
A. Coppola, A. Drago, Storia dei principi della dinamica alla luce della matematica cos-truttiva, Atti IV Congr. Naz. St. Fiosica, Como, 1983, 122–127.
Such result agrees again with Berkeley’s criticism which thus reveals itself to be the most penetrating of all times. In this respect, see K.R. Popper, Conjectures and Refutations, Rougledge and Kegan, London, 1963, ch. 6. Although disagreeing with Berkeley, Popper declared himself “struck for its (Berkeley’s philosophy of science) modernity” (p. 171). Only recently did some philosophers reconsider the problem of absolute vs. relational space; it is noteworthy that they reached results similar to that of the present paper.
See K.L. Manders, On the Space-Time Ontology of Physical Theories, Phil. Sci. 49, (1982) 575–590
B. Mundy, Relational Theories of Euclidean Space and Minkowsky’s Spacetime, Phil. Sci. 50, (1983) 205–226.
P.T. Landsberg, Thermodynamics, Interscience, New York, 1961. It is worth noting that constructive mathematics offers a principle linking discontinuity to undecidability, which is very useful for the analysis of physical concepts
see O. Aberth, Introduction to Computable Analysis, McGraw-Hill, New York, 1980, p. 48.
A. Drago, Constructive Mathematics…, op. cit. n.1.
O. Aberth, op. cit. n. 27, ch. 12.
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Drago, A. (1988). A Characterization of the Newtonian Paradigm. In: Scheurer, P.B., Debrock, G. (eds) Newton’s Scientific and Philosophical Legacy. Archives Internationales D’Histoire des Idées / International Archives of the History of Ideas, vol 123. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2809-1_16
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