SimonF:
Since I'm the only one arguing the opposite position, I figure that I should comment on this. I got through section 3 (after which he had stated his main thesis, and presented his evidence) before losing interest.
Basically, he points to quantum theory and a class of Newtonian examples as his evidence against deterministic time evolution. Quantum theory, however, still evolves deterministically -- probabilities do enter the theory in a non-classical way, but this doesn't have to do with how things evolve in time (the probabilities have to do with how things are measured in quantum theory) -- the state of a closed system at t1 still evolves to a unique state at t2. Deterministic evolution is still there.
The Newtonian examples are basically special points in certain solution spaces that are hand-built to have certain pathological features. The terminology people use is that such solutions "are of measure zero" -- i.e. that you can never actually encounter this kind of phenomenon in real life, because they require unrealistic idealizations of infinite precision in order to see the pathological effects. An analogous example of this kind of thing are two infinitely rigid masses which collide and bounce off each other at time T. What is their velocity at precisely time T? OMG it's undefined! But this does not really signify a breakdown of the general properties of Newtonian physics -- it's just a pathological property of an unphysical system, combined with asking a question which would require infinite precision to test (something never possible in reality). The "counterexample" of determinism he points to is just this type of thing. It is not a physical effect that could ever be observed -- it "occupies zero volume" in the space of possibilities.
Anyway, for these reasons I don't take his thesis very seriously. The deterministic evolution in both Newtonian and quantum mechanics is quite manifest, even in the basic postulates of the theory, from which the mathematical construction follows. To the extent that he's arguing against deterministic evolution as a fundamental component of physics, he's trying to build an argument out of vapor. I do agree that deterministic evolution isn't a logical necessity for physics, but it just turns out that nature seems quite clearly to be doing it that way.