# Space and time involution

J. C. C. McKinsey, A. C. Sugar and P. Suppes (hereafter MSS) wrote “Axiomatic foundations of classical particle mechanics”, (Journal of Rational Mechanics and Analysis, v.2 (1953) p.253-272), which is also described in Suppes’ Introduction to Logic (Van Nostrand, New York, 1957), pp.291-322 (see here). It is only a partial axiomatization of Newtonian mechanics but is sufficient to present an involution of mechanics below.

An involution in mathematics is a function that is its own inverse, which means if it is repeated the output is the input, that is, f(f(x)) = x. The involution here is the interchange of spatial and temporal quantities along with the inversion of mass. We start with MSS:

MSS system has six primitive notions: P, T, m, s, f, and g. P and T are sets, m is a real-valued unary function defined on P, s and g are vector-valued functions defined on the Cartesian product P × T, and f is a vector-valued function defined on the Cartesian product P × P × T. Intuitively, P corresponds to the set of particles and T is to be physically interpreted as a set of real numbers measuring elapsed times (in terms of some unit of time, and measured from some origin of time). m(p) is to be interpreted as the numerical value of the mass of p ∈ P. sp(t), where tT, is a 3-dimensional vector which is to be physically interpreted as the position of particle p at timepoint t. f (p, q, t), where p, qP, corresponds to the internal force that particle q exerts over p, at timepoint t. And finally, the function g(p, t) is to be understood as the external force acting on particle p at timepoint t. (Anna & Maia p,9)

We define MSS´ by the following involution: interchange P and Q; p and q; T and S; m and n=1/m; s and t; f and k; g and ; particle and tempicle; timepoint and placepoint; mass and vass. Then the explanation is:

MSS´ system has six primitive notions: Q, S, n, t, k, and . Q and S are sets, m is a real-valued unary function defined on Q, t and  are vector-valued functions defined on the Cartesian product Q × S, and k is a vector-valued function defined on the Cartesian product Q × Q × S. Intuitively, Q corresponds to the set of tempicles and S is to be physically interpreted as a set of real numbers measuring scalar space (in terms of some unit of length, and measured from some origin placepoint). n(q) is to be interpreted as the numerical value of the vass of qQ. tq(s), where sS, is a 3-dimensional vector which is to be physically interpreted as the position of tempicle q at placepoint s. k(q, p, s), where q, pQ, corresponds to the internal release that tempicle p exerts over q, at placepoint s. And finally, the function (q, s) is to be understood as the external release acting on tempicle q at placepoint s.

The corresponding axioms are as follows:

A1 Q is a non-empty, finite set.
A2 S is an interval of real numbers.
A3 If qQ and sS, then tq(s) is a 3-dimensional vector (tq(s) ∈ℜ³) such that d²tq(s)/ds² exists.
A4 If qQ, then n(q) is a positive real number.
A5 If p, qQ and sS, then k(p, q, s) = −k(q; p; s).
A6 If p, qQ and sS, then tq(s) × k(p, q, s) = –tp(s) × k(q, p, s).
A7 If p, qQ and sS, then n(q) d²tq(s)/dt² = ΣpQ k(p, q, s) + ℓ(q, s).

These axioms generate a dual to Newtonian mechanics. A5 corresponds to a weak dual version of Newton’s Third Law: to every release there is always a counter-release. A6 and A5, correspond to the strong dual version of Newton’s Third Law. A6 establishes that the direction of release and counter-release is the direction of the line defined by the coordinates of tempicles p and q. A7 corresponds to the dual of Newton’s Second Law.

MSS show that mass is independent of the remaining primitive notions of their system. Because of this, its dual could be defined differently. It was thought best to take the inverse of mass, called vass, for the involution.