This paper by Alexandre Korolev arguing that Norton’s Dome isn’t a kosher case of indeterminism in Newtonian physics appeared on my RSS reader recently. I was very curious about Korolev’s paper, but found it to be a letdown. In fact, Norton addresses most of Korolev’s objections in his 2006 update to the manuscript.
After some lengthy and (in my opinion) somewhat unnecessary refinements of the original thought experiment, Korolev gets to one of his main points, that the Dome fails to satisfy something called a Lipschitz condition. The significance of this is that failure to satisfy the Lipschitz condition violates the time reversibility of trajectories. It doesn’t strictly imply time irreversibility for any given trajectory; only that time irreversible solutions to the equations of motion exist. Korolev seems to think that because the Lipschitz condition allows for time-irreversible solutions, the Lipschitz condition must be an ‘implicit assumption within Newtonian mechanics’. Since Norton’s Dome violates the Lipschitz condition, it isn’t (by Korolev’s reasoning) really a Newtonian system, and hence it has ‘no serious metaphysical import’.
Korolev’s second main line of objection is that the Dome example works only if the dome is infinitely rigid, i.e. it’s shape is completely unchanged by the point mass. When I talked about the Dome to some physics undergrads I had the same complaint about idealizations — no real dome, made of molecules, can be perfectly rigid, etc. And there’s also quantum mechanics, which implies the point mass cannot be exactly poised at the top of the Dome. Well, we can ignore quantum mechanics because indeterminism simply isn’t an interesting find for a quantum mechanical system; we already know quantum mechanics is indeterministic. Norton deals with the point about the rigid idealization well (and Korolev doesn’t address it, even though his paper seems to have been published later than Norton’s update). There is no internal principle of Newtonian mechanics that would lead us to rule out such an idealization. Furthermore, it is acknowledged that we are talking about an idealized model and not a real system. (Otherwise, why even bother with Newtonian systems? Let’s go straight to quantum.) So it seems the only way we can think that this idealization is impermissible is by already having some external reason for not wanting to allow idealizations that violate time irreversibility and/or determinism. But then that would no longer be a reason to think that the Dome is not a Newtonian system, for nothing internal to Newtonian mechanics prohibits it.
I think Korolev’s first objection (that the failure to satisfy the Lipschitz condition allows for time irreversible trajectories) can be dealt with the same way — by seeing that his objection arises from an external demand put on Newtonian mechanics. If it so happens that Newton’s Laws allow time irreversible trajectories, then why should we declare such trajectories un-Newtonian, as opposed to simply accepting that there exist very special conditions under which Newtonian mechanics can be time irreversible? (Norton covers more ground in his 2006 update — pondering if we should add a ’4th Law’ to Newton’s Laws in order to save Newtonian determinism from the Dome. I largely agree with his reasons for concluding that the Dome is a legit Newtonian system.)
Keep in mind that Norton originally intended the Dome to have the following ‘metaphysical import’:
Even quite simple Newtonian systems can harbor uncaused events and ones for which the theory cannot even supply probabilities. Because of such systems, ordinary Newtonian mechanics cannot license a principle or law of causality.
He just needed a simple Newtonian system. No extra assumptions of time reversibility or correspondence with real world systems. Seems to me the Dome can have the metaphysical import it was originally intended to have.
Alexandre Korolev (2007). Indeterminism, Asymptotic Reasoning, and Time Irreversibility in Classical Physics Philosophy of Science, 74 (5), 943-956 DOI: 10.1086/525635
Norton, John D. (2006). The Dome: An Unexpectedly Simple Failure of Determinism. In [2006] Philosophy of Science Assoc. 20th Biennial Mtg (Vancouver): PSA 2006 Symposia.
Norton, J. (2003). Causation as folk science. Philosopher’s Imprint 3 (4), 1-22.
Posted by Ponder Stibbons 
