Dear Flavio,
you've presented an eminently readable and well-reasoned overview of the notion that classical physics can be interpreted in an indeterministic way. I think the point of view you suggest is a very intriguing one---with the 'orthodox' interpretation of classical mechanics being akin to a hidden variable interpretation, with real numbers obviously forever beyond experimental access.
From a larger point of view, I think this sort of project constitutes an important check on the intuition that, apart from quantum mechanics, which is 'weird' anyways, our physical theories simply tell us how stuff works. But they don't, and not just since the advent of QM: there is always an issue of interpretation, of matching the formalism to the world, so it's a good thing to point out that even if that interpretation appears obvious, one can suggest alternatives.
I also particularly appreciate the brief rundown of the history of indeterminism in classical mechanics; I was not aware of any dissent with the 'received view' of classical mechanics as a deterministic 'clockwork'. Although of course, such ideas go back to antiquity: Lucretius, in his poem De Rerum Natura, attributes to Epicurus the view that atoms sometimes 'swerve' in an unpredictable way while falling through the void, holding this swerve (the 'clinamen') as being responsible for any sort of creative coming-into-being whatsoever.
You're also right in perceiving some kinship between your approach to indeterminism in classical mechanics, and mine to quantum mechanics---or at least, our visualizations thereof: the figure I use to explain the origin of the uncertainty principle (see slide 9 in this presentation) is almost exactly your Fig. 2! :)
I've got a few questions, though. First of all, your main argument for indeterminism seems to be that the finiteness of information within a certain volume of (phase-)space entails that certain quantities are not perfectly determined. You appeal to the Bekenstein bound to substantiate this; however, this strikes me as a bit circular---after all, the Bekenstein bound is based on quantum considerations, and hence, already comes from a theory which we know can be interpreted indeterministically. What other reasons are there for believing in the finiteness of information?
Furthermore, even if I agree (as I actually do) that the amount of information is finite, it seems that one could 'save the phenomena' by just moving to the computable reals instead---one might then imagine basing classical physics on computable analysis, which also evades the somewhat troublesome use of---in your interpretation---unphysical limits in the definition of key quantities, as happens in ordinary real analysis applied in classical mechanics. Also, the computable reals lie dense in the real numbers, so to any---actually every---degree of accuracy, you can always assume that a given quantity is provided by a fixed, computable real number. Do you see any issues with this approach? (It's obviously fine, however, to just consider this as one possible interpretation of classical mechanics, you however having chosen another.)
Additionally, I'm not quite clear on how your requirement of intersubjectivity can be guaranteed, if certain quantities have a fundamentally random value until they're measured. For one, doesn't this imply a kind of nonlocal influence? If I measure the electron's charge to an unprecedented degree of precision in my lab, does that force the value that Joe Q Alien measures in the Andromeda galaxy to spontaneously agree with my measurement? Of course, if we're talking Newtonian mechanics, there is no obvious issue with this, as for instance gravitational influences are already taken to be instantaneous; but it would apparently make it harder to reconcile this view with relativistic theories, like Maxwellian electromagnetism. Then again, this sort of nonlocality would seem to be of the same kind as that in Bohmian mechanics, in the sense that one could not use it to actually transmit information.
Anyway, so much for my initial thoughts on your fine essay. I wish you the best of luck in this contest, and look forward to discussing with you!
Cheers
Jochen
