Mathematical models in Newton's Principia: A new view of the ‘Newtonian Style’

Abstract

In this essay I argue against I. Bernard Cohen's influential account of Newton's methodology in the Principia: the ‘Newtonian Style’. The crux of Cohen's account is the successive adaptation of ‘mental constructs’ through comparisons with nature. In Cohen's view there is a direct dynamic between the mental constructs and physical systems. I argue that his account is essentially hypothetical‐deductive, which is at odds with Newton's rejection of the hypothetical‐deductive method. An adequate account of Newton's methodology needs to show how Newton's method proceeds differently from the hypothetical‐deductive method. In the constructive part I argue for my own account, which is model based: it focuses on how Newton constructed his models in Book I of the Principia. I will show that Newton understood Book I as an exercise in determining the mathematical consequences of certain force functions. The growing complexity of Newton's models is a result of exploring increasingly complex force functions (intra‐theoretical dynamics) rather than a successive comparison with nature (extra‐theoretical dynamics). Nature did not enter the scene here. This intra‐theoretical dynamics is related to the ‘autonomy of the models’.

Acknowledgements

The author wishes to thank James W. McAllister, the two anonymous referees, and Erik Weber for their highly useful comments and suggestions, which significantly helped to improve this essay and the ideas expressed in it.

Notes

Steffen Ducheyne is research assistant of the Funds for Scientific Research (Flanders), and is working on a Ph.D. dissertation with the Centre for Logic and Philosophy of Science in Ghent (Belgium).

Correspondence to: Steffen Ducheyne, Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, B‐9000 Ghent, Belgium. Email: [email protected].

Cohen notes that ‘the Newtonian “one‐body system” is a “system” to the extent that it is composed of two entities, even though these are not homologous, as in the case of a system of two bodies: these are a single body (or mass point) and a center of force’ (Cohen 1980: 302n). Newton remarks: ‘Up to this point I have been setting forth the motions of bodies attracted toward an immovable center, such as, however, hardly exist in the natural world’ (Newton 1726: 561).

This is an insertion made by Cohen himself.

Newton banned as much as possible all propositions which were explicitly called ‘hypotheses’ in the first edition and renamed them in the second edition under the names of ‘rule’ or ‘phenomenon’. On this matter see Crombie 1994: 1060–61; Gjertsen 1986: 466; Cohen 1999: 24.

A discontinuous motion along the sides of a polygon is reduced to a continuous motion along a smooth orbital path, letting the triangles tend to infinity and hence each of the surfaces to zero. Can a continuous force be approximated by as a limit of discontinuous impulsive force as the time interval shrinks to zero? I will not enter the discussion on the validity of Newton's procedure of taking a limit. For a defence of Newton's argument see Nauenberg 2003; for a critique see Pourciau 2003.

Hence a centripetal force is a necessary and sufficient condition for the area law. As Newton writes somewhat further: ‘Since the uniform description of areas indicates the center towards which that force is directed by which a body is most affected and by which it is drawn away from rectilinear motion and kept in orbit, why should we not in what follows use uniform description of areas as a criterion for a center about which all orbital motion takes place in free spaces?’ (Newton 1726: 449).

In the following scholium Newton notes that ‘the case of corol. 6 holds for our heavenly bodies (as our compatriots Wren, Hooke, and Halley have also found out independently)’ (Newton 1726: 452). While Newton accepted the area law as applied to the primary planets only as a very approximate empirical truth, he accepted the third Keplerian rule as empirically accurate (Wilson 1989: 91, 141, 143).

He further writes: ‘The more the law of force departs from the law there supposed, the more the bodies will perturb their mutual motions; nor can it happen that bodies will move exactly in ellipses while attracting one another according to the law here supposed, except by maintaining a fixed proportion of distances one from another. In the following cases, however, the orbits will not be very different from ellipses’ (Newton 1726, 568).

Since the motive forces are as the accelerative forces and the attracted bodies jointly (F _m = F _a.m), the absolute attractive force of body A is to the absolute attractive force of body B as the mass of body B. This can be shown as follows. By the definition of motive force, we can derive: F _a1/F _a2 = F _m1.m ₂/F _m2.m ₁ (since F _a = F _m/m). By the third law of motion both motive forces are equal. We establish: F _a1/F _a2 = m ₂/m ₁ or F _a2/F _a1 = m ₁/m ₂.

For the secondary planets Newton's application of Corollary 6 is no surprise, since he assumes that the orbits of the circumjovial planets, e.g., do ‘not differ sensibly from circles concentric with Jupiter’ (Newton 1726: 797). In the first proposition of Book III Newton is very silent about the perturbations between the sun and Jupiter. If we take a closer look at the Principia however we see that Proposition 60, Book I explains these perturbations (Harper 1990: 192).

Smith considers the laws as working hypotheses: ‘they are not testable in and of themselves, yet they are indispensable to a train of evidential reasoning. … Mediating elements of some kind are always needed to turn data into evidence’ (Smith 2001: 335). Ernan McMullin argues that the laws of motion are not open to being made ‘either more exact or liable to exceptions’. McMullin calls them constitutive principles: ‘they enable an otherwise purely formal mathematical framework [to] confront experience’ (McMullin 2001: 345). He further notes: ‘There is, still, of course the question of how well the language works in the long run, so the element of hypothesis may re‐enter’ (ibid.).

Newton seemed to think that Book I was an autonomous enterprise. He commented on Books I and II as follows: ‘In the preceding books I have presented principles of philosophy that are not, however, philosophical but strictly mathematical – that is, those on which the study of nature can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy. … It still remains for us to exhibit the system of the world from these same principles.’ (Newton 1726: 783; see also 561).

Recall that the quote in 2.1 continues as follows: ‘Let us see, therefore, what the forces are by which spherical bodies, consisting of particles that attract in the way already set forth, must act upon one another, and what sorts of motions results from such forces.’ (emphasis added).

In fact Newton when writing De Motu already realised that the true motions of celestial bodies are immensely complicated and far from being perfectly Keplerian (Smith 2002a, 152–4). Taking in account all causes ‘exceeds the force of any human mind’ (Whiteside 1974, 6: 78). This makes McMullin's picture of Newton as a defender of successive approximation hard to believe (McMullin 2001, 344). Smith notes: ‘any systematic discrepancy from the idealised theoretical motions has to be identical with a specific force – if not a gravitational force then some other generic force law. This restriction precludes inventing ad hoc forces to save the law of gravity. It hereby makes success in carrying out a program of successive approximations far from guaranteed.’ (Smith 2002a, 158).

It is only in Book III that Newton is concerned with the actual celestial motions in nature. We proceed from effects to causes (cf. the main business of natural philosophy is ‘to argue from Phaenomena without feigning Hypotheses, and to deduce Causes from Effects’). As Newton wrote at the beginning of the Principia: ‘For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces.’ (Newton 1726, 382; emphasis added).

Newton's rejection of the H‐D method can therefore be read in two ways. One could read it as a stance on the argumentative logic of science (see introduction above). One could also read Newton's rejection of hypotheses as a very specific rejection of Cartesian causal explanatory models based on non‐observable particles.

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Notes on contributors

Steffen Ducheyne

Steffen Ducheyne is research assistant of the Funds for Scientific Research (Flanders), and is working on a Ph.D. dissertation with the Centre for Logic and Philosophy of Science in Ghent (Belgium). Correspondence to: Steffen Ducheyne, Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, B‐9000 Ghent, Belgium. Email: [email protected].