Indeterminism is generic among systems with infinitely many degrees of freedom.

1. Indeterminismin systems with infinitely and finitely many degrees of freedomJohn D. NortonDepartment of History and Philosophy of ScienceUniversity of Pittsburgh

2. Indeterminism is generic amongsystems withinfinitely many degrees of freedom. Source: Appendix to Norton, “Approximation and Idealization…”

3. The mechanism that generates pathologies system of infinitely many coupled components 0 2 3 4 5 1 Enforced by embedding in still larger solution. Solution that manifests pathological behavior for a few components, e.g. spontaneous excitation …and so on indefinitely. Enforced by embedding in larger solution.

4. Masses and Springs Motions governed by d2xn/dt2 = (xn+1 – xn) - (xn – xn-1) Expected solution xn(t) = 0 for all n, all t with initial conditions dxn(0)/dt = xn(0) = 0 for all n

5. Masses and Springs Motions governed by d2xn/dt2 = (xn+1 – xn) - (xn – xn-1) Unexpected solution with same initial conditions x1(t) = x2(t) = (1/t) exp (-1/t) Non-analytic functions needed to ensure initial conditions preserved. Solve for remaining variables iteratively x3 = d2x2/dt2 + 2x2 - x1 dx3/dt = d3x2/dt3 + 2dx2/dt - dx1/dt x4 = d2x3/dt2 + 2x3 – x2 dx4/dt = d3x3/dt3 + 2dx3/dt – dx2/dt etc.

6. Indeterminism is exceptional amongsystems withfinitely many degrees of freedom.

7. The mass experiences an outward directed force field F = (d/dr) potential energy = (d/dr) gh = r1/2. The motion of the mass is governed by Newton’s “F=ma”: d2r/dt2 = r1/2. The Arrangement • A unit mass sits at the apex of a dome over which it can slide frictionless. The dome is symmetrical about the origin r=0 of radial coordinates inscribed on its surface. Its shape is given by the (negative) height function h(r) = (2/3g)r3/2.

8. Possible motions: None • r(t) = 0 • solves Newton’s equation of motion since • d2r/dt2 = d2(0)/dt2 = 0 = r1/2.

9. r(t) = 0, for t≤T and r(t) = (1/144)(t–T)4, for t≥T solves Newton’s equation of motion d2r/dt2 = r1/2. For t≤T, d2r/dt2 = d2(0)/dt2 = 0 = r1/2. For t≥T d2r/dt2= (d2 /dt2) (1/144)(t–T)4 = 4 x 3 x (1/144) (t–T)2 = (1/12) (t–T)2 = [(1/144)(t–T)4]1/2 = r 1/2 Possible motions: Spontaneous Acceleration • The mass remains at rest until some arbitrary time T, whereupon it accelerates in some arbitrary direction.

10. The computation again For t≤T, d2r/dt2 = d2(0)/dt2 = 0 = r1/2. For t≥T d2r/dt2= (d2 /dt2) (1/144)(t–T)4 = 4 x 3 x (1/144) (t–T)2 = (1/12) (t–T)2 = [(1/144)(t–T)4]1/2 = r 1/2

11. Without Calculus • Imagine the mass projected from the edge. • Close…

12. Without Calculus • Imagine the mass projected from the edge. • Closer…

13. Now consider the time reversal of this process. Without Calculus Spontaneous motion! BUT there is a loophole. Spontaneous motion fails for a hemispherical dome. How can the thought experiment fail in that case? • Imagine the mass projected from the edge. • BINGO!

14. What should we think of this?