Abstract
Early modern foundations for mechanics came in two kinds, nomic and material. I examine here the dynamical laws and pictures of matter given respectively by Newton, Leibniz, and Kant. I argue that they fall short of their foundational task, viz. to represent enough kinematic behavior; or at least to explain it. In effect, for the true foundations of classical mechanics we must look beyond Newton, Leibniz, and Kant.
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Notes
- 1.
For Newton, I use his Principia. For Leibniz, I use his Specimen dynamicum, and associated texts (see nos. 14, 16, 22–3, 31, A3–4 and E3–4 in Leibniz, Essays). For Kant, I rely on his 1786 tract, Metaphysical Foundations of Natural Science (henceforth MAN).
- 2.
Broadly speaking, F is a set that contains concepts, laws, mathematical theorems, and perhaps heuristics for problem-solving and theory buildup.
- 3.
This sense might seem idle, but it was long influential. Descartes’ 1644 Principles of Philosophy advocated for reducing physics (optics, magnetism, heat flow, even physiology and earth science) to a mechanics of matter in motion (specifically, of action by contact via collision and ether pressure). And so did Hobbes, in De Corpore of 1655. Centuries later, Hertz urged: “All physicists agree that the problem of physics consists in tracing the phenomena of nature back to the simple laws of mechanics” (Hertz, Mechanics, xxi). Halfway between these termini was Fischer’s program of a mechanische Physik, whose influence extended to France (see his Physique mécanique).
- 4.
“Newton’s principles suffice for solving every mechanical problem we encounter in practice, whether in statics or dynamics. We need not appeal to any new principle for that. If we run into obstacles, they are always just mathematical. Not difficulties with the principles” (Mach, Mechanik, 239; my emphasis). Kuhn counted Newton’s Principia as a paradigm—the exemplary achievement of classical mechanics—and claimed that it “served for a time implicitly to define the legitimate problems and methods of a research field for succeeding generations of practitioners” (Kuhn, Structure, 10; my emphasis).
- 5.
Note that, if these framing assumptions were true, they would make short work of Leibniz, Kant, and anyone who diverged from the Newtonian program above. If mechanics post 1700 was in fact as the Mach-Kuhn has it, then attempts to supplant it (as Leibniz tried, with vis viva) or to correct it (as Kant tried, with foundation K) must appear as doomed to fail, or at least seriously misguided. I thank Katherine Brading for enlightening discussion of these broader points.
- 6.
Decades ago Truesdell had pointed out that differential equations—the key representational device of modern mechanics—are absent from Newton’s tract (Truesdell, Essays, 90). We may wonder if Newton would have even recognized the need for them in mechanics. I thank George E. Smith for stimulating discussion on this topic.
- 7.
The claims in this section depend on research carried out in Brading & Stan, Philosophical Mechanics, chapters 8–12. For verification, the reader is invited to consult them.
- 8.
Other types of extended-body motion (e.g. plasticity, fracture, hysteresis, creep, and brittleness) had to wait until the twentieth century for their mathematization.
- 9.
b is the net body force, T the stress, or internal force, 𝜌 the mass density, and the acceleration (the second derivative of the position vector X). An early version of this law is in Cauchy, “Sur les equations.”
- 10.
H is the net impressed torque, and L the angular momentum. The earliest expression of this law is Euler, “De motu in superficiebus,” § 48.
- 11.
See especially Truesdell, Rational Continuum Mechanics, 64ff.
- 12.
Generally, a constraint is a limit on how a particle or a body is allowed to move. Some constraints are external to the body. E.g. an inclined plane, which prevents the body from moving straight down (under the force of gravity). Other constraints are internal to the body. E.g. rigidity, which prevents the body’s component points from changing their relative distances.
- 13.
The reason is that, in general, the physical basis that secures the constraints—e.g., forces (if they are forces), their specific laws, and mechanisms of action—is not known in advance. It is not given at the outset of building the theory of mechanics. However, to apply Newton’s second law, that required knowledge must be available at the outset. The law really says that ∑f = ma, viz. the actual acceleration is the result of all the forces acting at that point. Absent knowledge of some forces, the law becomes inapplicable. “[T]he most widespread mistake about Newton’s three laws of motion is that they alone sufficed for all problems in classical mechanics.”—Smith, “Newton’s Principia,” § 5.
- 14.
For the origin of the idea that this is really Lagrange’s law, together with a lucid explanation of its role in solving constraints, see Papastavridis, Analytical Mechanics.
- 15.
See, for instance, Lagrange, Mechanique, 53ff. F is any actual, impressed force on a mass i in the system; 𝛿f is a virtual displacement that F would cause in i. And, −miẍi are so-called ‘reverse effective forces’ (or also ‘kinetic reactions’), viz. fictitious forces supposed equal and opposite to the particle’s effective acceleration ẍi; and 𝛿x a virtual displacement in their direction. Finally, 𝝀i is a Lagrange multiplier, and 𝛿L a virtual displacement compatible with the constraint given by 𝝀i.
- 16.
By a mechanical system I mean one or more masses, point sized or extended.
- 17.
The modern name for this quantity is a ‘Lagrange multiplier.’
- 18.
This capsule of Lagrange’s result is perforce terse, hence hard to follow, understandably. For a longer, more lucid explanation, see Brading & Stan, Philosophical Mechanics, chapter 11.
- 19.
The evidence for my claim is Hamel’s extensive treatise Theoretische Mechanik, which, from Lagrange’s law above, derives equilibrium conditions and equations of motion for all the species of body treated by then (viz. rigid, flexible, fluid, and elastic).
- 20.
External forces originate—they are exerted by sources—outside the bounding surface S of an extended body. Examples: gravity and magnetic forces. Internal forces are exerted below S (inside the extended body), due to the body’s parts acting on one another. Examples: pressure in a fluid, and stress in an elastic solid.
- 21.
Interpreted geometrically, a vector is an arrow-like object with a length (size) and a direction. It was used to represent the action of an impressed force on a mass point; e.g. the velocity increment (acceleration) in f = ma. A tensor is analogous to a bundle of 9 arrows, or vectors; see next footnote.
- 22.
A tensor-like force acts on a small volume of matter to compress, stretch, or twist it. Cauchy called it ‘pressure or tension,’ to indicate that it does more than just translate a point over a small distance (which vectorial forces do). We call is ‘stress.’
- 23.
Namely, only when the net resultant (of all the external forces) pass through the body’s mass center. If it does not, the resultant induces motion effects (e.g. precession) that Newton’s second law cannot predict.
- 24.
Again, for a fuller account and history, cf. Brading & Stan, Philosophical Mechanics, chapter 10.
- 25.
Newton did not have the term ‘derivative. He just had a proto-version of it, which he called a ‘fluxion.’ That Newtonian concept overlaps with our modern notion of rate of change (of a variable quantity in respect to another, e.g. dx/dt or even dr/dx). But, it cannot capture the idea of a partial rate of change, which a derivative like 𝜕f/𝜕x expresses.
- 26.
For the first example, cf. Propositions 39–41 of Newton’s Principia. For the second, see the final section of Daniel Bernoulli’s Hydrodynamica (1738).
- 27.
In our terms, they showed that, if interaction forces in a system are given by (monogenic) potentials, then Conservation of Vis Viva is a ‘first integral of motion,’ i.e. a quantity conserved over a finite stretch of time. For additional discussion and historical details, see Brading & Stan, Philosophical Mechanics, chapters 8 and 11.
- 28.
That formula is known as the Euler Equation for a perfect fluid. For details of its derivation, see Darrigol, Worlds of Flow, chapter 1; and Brading & Stan, Philosophical Mechanics, chapter 10.
- 29.
Why it is not enough: Euler’s Equation determines just the change of velocity at a point; but when a fluid moves, there is mass flow as well—the density at that point changes over time. This latter change is what the Continuity Equation (or Conservation of Mass) describes exactly: 𝜕𝜌/𝜕t + 𝜕𝜌v/𝜕xi = 0. In words, in a volume element, the mass density at an instant equals the mass in it at the previous instant, plus the rate of mass flow across the volume’s bounding surface.
- 30.
Navier, “Lois des fluids,” 252.
- 31.
In Kant, the Equality of Action and Reaction first shows up in a 1758 paper on collision theory. In MAN, he again applies it to impact, and extends its range (without explanation) to action-at-a-distance forces too; cf. Stan, “Kant’s third law” and Friedman, Kant’s Construction.
- 32.
Outcomes of collision range between two limit cases. One is inelastic impact, the other is elastic collision. To infer the outcome of each case, another premise (beside the law of action-reaction) is needed. For inelastic collision, that premise is Zero Relative Speed (viz. that the two bodies move together after impact). For elastic collision, it is the conservation of kinetic energy, or vis viva.
- 33.
Newton models first the orbit that results if the force on a particle P is directed to a point fixed in space. Then he supposes that force to emanate from another particle that is itself in motion. He proves that, relative to the mass center of these two particles, then the orbit of P is likewise elliptical.
- 34.
Separately, Watkins, Kant on Laws, and Stan, “Evidence and explanation,” have noted that not even Newton’s second law—the basis for the equation of motion of all free mass points, though not extended bodies—is to be found in Kant’s foundation. Here I am just explaining the force of that alarming conclusion.
- 35.
In natural philosophy, Newton’s standard of evidence was ‘deduction from phenomena.’ The above theory of matter did not clear his standard, and so he offered it (in Query 31 of his Opticks) not as considered doctrine, but as an (initially plausible) proposal for further research.
- 36.
Often, they couch this answer in terms of grounding as explanation. For examples and critical discussion, see Stan, “Evidence and explanation.”
- 37.
Stresses are a more general species of impressed force, and their gradients (as in the first Euler-Cauchy law) are impressed forces, like gravity. In Lagrange’s law, both the applied and also the reverse effective forces are kinds of impressed force.
- 38.
Kant in fact called his treatise a “metaphysics of corporeal nature” (see MAN, 13). A study of early-modern matter theories from this vantage point is Holden, Architecture.
- 39.
Look again at the Euler-Cauchy (13.1a) law above. On the kinematic side, it relies on the quantity 𝜌, viz. mass density. That property obtains only in continuous matter. Discrete particles do not have mass density; they have just mass, m. But m does not show up in the for fluids and elastics (e.g., it is not in the Navier-Stokes equation).
- 40.
Already ancient statics—the science of the five ‘simple machines,’ later with the inclined plane as a sixth—was a theory of rigid bodies: those ‘machines’ were all supposed undeformable. On the ‘science of machines’ in the eighteenth century, especially in France, see Chatzis, “Mécanique rationnelle.”
- 41.
See, for instance, Wilson, Physics Avoidance.
- 42.
Kant, MAN, 13 (4: 478).
- 43.
My message here dovetails with lessons that Mark Wilson has long tried to teach us, e.g. in the perceptive and rewarding studies assembled in his Physics Avoidance.
- 44.
On this point, see again Wilson, Physics Avoidance.
- 45.
For Newton and Leibniz, the floodgates of research on that topic opened after Reichenbach, “Bewegungslehre.” For Kant, it began with Cohen, Kants Theorie, and Cassirer, Erkenntnisproblem; then it continued through Friedman, Kant’s Construction.
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Stan, M. (2023). Beyond Newton, Leibniz and Kant: Insufficient Foundations, 1687–1786. In: Lefèvre, W. (eds) Between Leibniz, Newton, and Kant. Boston Studies in the Philosophy and History of Science, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-031-34340-7_13
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