about the relationship between reason and consequence.
1 It has been traced back to
Nicholas of Autrecourt (1298–1369) (Wöhler 2006, p. 163). Leibniz (1677, 1695,
1989, pp. 201, 441, 443) changed it to a metaphysical law for motion. Mayer (1842)
exploited it to advance the energy conservation law, although in too much an
absolute sense (Helmholtz 1884, p. 70). Typical modern statements are these:
From given Equal, Equal follows. (Newton, n.d., see also 1999, Book III, 2nd Rule)
La symétrie caractéristique d’un phénomène est la symétrie maxima compatible avec
l’existence du phénomène. Un phénomène peut exister dans un milieu qui possède sa
symétrie caractéristique ou celle d’un des intergroupes de sa symétrie caractéristique.
. . .C’est la dissymétrie qui crée le phénomène. (Curie 1894, p. 400)
2
If an ensemble of causes is invariant with respect to any transformation, the ensemble of their
effects is invariant with respect to the same transformation. (Renaud 1935, as quoted in
Rosen 1982, p. 26)
The symmetry group of the cause is a subgroup of the symmetry group of the effect. Or less
precisely: The effect is at least as symmetric as the cause. Equivalent states of a cause are
mapped to (i.e., are correlated with) equivalent states of its effect. . . Also less precisely:
Equivalent causes are associated with equivalent effects. (Rosen 2005, pp. 308f.)
In agreement with Helmholtz (1884, p. 69), I will use the rule “aequat causa
effectum” rather heuristically and concentrate myself on the benefit of Newton’s
(1999, Axioms) and Euler’s (1750, Ch. 7) notions of (stationary) state, cf (Enders
2008).
To be specifically, let us consider the Newtonian model potential ~1/r as “cause”
of the “effect” Kepler orbits. Now, while the “cause” is spherically symmetric, the
“effect” is obviously not spherically symmetric, see Fig. 4.1. Does this difference
contradict and thus disprove that rule? In this contribution, this contradiction will be
shown to be seemingly only, because the initial conditions have not been taken into
account.
Generally speaking, this issue is not new. Zee (1999, pp. 13f.) explains,
It is crucial to distinguish between the symmetry of physical laws and the symmetry imposed
by a specific situation... This distinction ... was one of Newton’s great intellectual achievements, and it enabled physics as we know it to take shape.
In his Nobel lecture, Wigner (1963, p. 7) defined “law” and “situation” as
follows.
The regularities in the phenomena which physical science endeavors to uncover are called
the laws of nature... The elements of the behavior which are not specified by the laws of
1 Cf https://de.wikipedia.org/wiki/Aequat_causa_effectum.—For the similar maxims “The same
causes will always produce the same effects” and “like causes produce like effects,” see Maxwell
(1877, para. 19).
2 For broad reviews, see e.g., Conway et al. (2008), Rosen (2008).
38
P. Enders
1 It has been traced back to
Nicholas of Autrecourt (1298–1369) (Wöhler 2006, p. 163). Leibniz (1677, 1695,
1989, pp. 201, 441, 443) changed it to a metaphysical law for motion. Mayer (1842)
exploited it to advance the energy conservation law, although in too much an
absolute sense (Helmholtz 1884, p. 70). Typical modern statements are these:
From given Equal, Equal follows. (Newton, n.d., see also 1999, Book III, 2nd Rule)
La symétrie caractéristique d’un phénomène est la symétrie maxima compatible avec
l’existence du phénomène. Un phénomène peut exister dans un milieu qui possède sa
symétrie caractéristique ou celle d’un des intergroupes de sa symétrie caractéristique.
. . .C’est la dissymétrie qui crée le phénomène. (Curie 1894, p. 400)
2
If an ensemble of causes is invariant with respect to any transformation, the ensemble of their
effects is invariant with respect to the same transformation. (Renaud 1935, as quoted in
Rosen 1982, p. 26)
The symmetry group of the cause is a subgroup of the symmetry group of the effect. Or less
precisely: The effect is at least as symmetric as the cause. Equivalent states of a cause are
mapped to (i.e., are correlated with) equivalent states of its effect. . . Also less precisely:
Equivalent causes are associated with equivalent effects. (Rosen 2005, pp. 308f.)
In agreement with Helmholtz (1884, p. 69), I will use the rule “aequat causa
effectum” rather heuristically and concentrate myself on the benefit of Newton’s
(1999, Axioms) and Euler’s (1750, Ch. 7) notions of (stationary) state, cf (Enders
2008).
To be specifically, let us consider the Newtonian model potential ~1/r as “cause”
of the “effect” Kepler orbits. Now, while the “cause” is spherically symmetric, the
“effect” is obviously not spherically symmetric, see Fig. 4.1. Does this difference
contradict and thus disprove that rule? In this contribution, this contradiction will be
shown to be seemingly only, because the initial conditions have not been taken into
account.
Generally speaking, this issue is not new. Zee (1999, pp. 13f.) explains,
It is crucial to distinguish between the symmetry of physical laws and the symmetry imposed
by a specific situation... This distinction ... was one of Newton’s great intellectual achievements, and it enabled physics as we know it to take shape.
In his Nobel lecture, Wigner (1963, p. 7) defined “law” and “situation” as
follows.
The regularities in the phenomena which physical science endeavors to uncover are called
the laws of nature... The elements of the behavior which are not specified by the laws of
1 Cf https://de.wikipedia.org/wiki/Aequat_causa_effectum.—For the similar maxims “The same
causes will always produce the same effects” and “like causes produce like effects,” see Maxwell
(1877, para. 19).
2 For broad reviews, see e.g., Conway et al. (2008), Rosen (2008).
38
P. Enders