Thanks to Jason1990 for his very interesting Newtonian response. But is it obviously acceptable to claim that the "real" Newtonian mechanics works with straight-line trajectories, with difference equations rather than differential equations, and so forth? Are we supposed to think that Newton believed that the real trajectories of particles are in fact kinky polygon-sides with many non-differentiable points? Ermmm ... I doubt it. And regardless of whether Newton himself thought any such thing, it is *not* what people have meant when they talk about "Newtonian mechanics" since about 1800, which is the theory that the Dome example is targeted at.

By the way, the author of the paper under discussion, referred to here repeatedly as "the guy", is John Norton, a professor of History and Philosophy of Science at the University of Pittsburgh. It is the top department of its kind in the world, and Norton is one of the top philosophers of science in the world, period; to see him referred to as "that guy", as if he was a half-bright interloper with no idea of what real science is like, is amusing. Imagine stumbling into a thread in a philosophy-discussion forum where the discussants tear apart an article by "this guy - oh, what was his name, T'Hooft or something?" - you get the idea.

PhilGuy

I'm sure he didn't think that. The idea he's getting at is that you solve the equation as if the trajectories were polygon-shaped, ie. discrete with timestep ∆t, say, and then find the solution in the limit of ∆t going to zero. What the author has done is take the limit of ∆t going to zero before finding any solutions when he differentiates h(r) using standard rules of differentiation. He then finds the solutions to this problem r''=sqrt(r), which can permit higher order (and totally unphysical) solutions.

Philosophers need a thicker skin.

Also people don't hesitate to criticize (even ridicule) some of t'Hooft's ideas.

It amuses me greatly that googling "newtonian indeterminism" yields this thread as the 4th hit. This is of course what we owe the pleasure of the bump to.

EDIT: Also, odds PhilGuy = John Norton?

At least 99%. His ratios of putting himself over:saying anything nominally on topic and obfuscation:argument are pretty impressive even compared to other crackpots. The dome is garbage. If taking a known issue in one field (Lipschitz continuity + DiffEqs), putting it on a different background, and claiming that the issue is then an issue of the background, and not the original field, is the realm of "the top philosophers of science in the world", then they need to aim higher.

Early dome papers talk about Newton's polygons and later papers leave that out- coincidence?. No, it's obvious that translating Newton's actual work leads to limits of difference equations (which he intentionally obfuscates in his post, asking if Newton thought polygons were the truth, when Newton himself says that limits of polygons are the truth, derf derf, nice one Mr. Leading Philosopher of Science in the World). The guy knows the dome is complete quackery, but keeps using it to get himself over instead of just acknowledging it as a cute physical example of Lipschitz (dis)continuity.

True odds: zero - sorry! But he's a friend.

Whether the dome example is quackery or not, Norton isn't making his reputation on it - he was already a top guy long ago (try googling "John Norton and Einstein" and see what you get). And you may be interested to know that he's taken tons of criticism from other philosophers, for the dome example, and on a wide variety of grounds. Have at him! But do use his name once in a while.

PhilGuy

Also, keep in mind that we are just a bunch of degenerate gamblers.

It is obvious, if one looks at Newton's original formulation, that direct applications of his laws of motion are to made on polygonal paths.

It is obvious that Newton did not believe this, since he provided us with a recipe for dealing with curved paths. The recipe is to apply his laws to obtain polygonal approximations, then take a limit. Rigorously following his recipe leads to at most one solution in any given problem.

This is a fair point. So then the moral of the Dome example is this: scientists, since about 1800, have been somewhat careless in their use of Newtonian mechanics. Their carelessness is justified by the fact that it has little if any practical consequences in real-life applications, and it greatly simplifies calculations. But in certain contrived examples, we can see how their carelessness leads to absurdities.

I think that is a fantastic story and is worth telling. I like the Dome example because it illustrates this very clearly, and vindicates the mathematician who preaches the importance of rigor and is sometimes derided for it by his counterparts in the physics community.

But this has nothing to do with the philosophy of determinism or Bayesian probability.