I don't accept strict determinism, but I also don't think the strong anthropic principle (SAP) cuts much ice here. The SAP is often deployed to deflate the shock people feel that we exist in an otherwise lifeless, hostile cosmos. We only notice the delicate balance of features which make the emergence of sentient beings like ourselves suitable just because the cosmos has that balance! If it lacked that balance, we 'd not be here to marvel. As I say, though, I don't think the SAP cuts much ice against Mary's observation, since the only empirically grounded basis for initial conditions we have is the world we in fact inhabit. As such, determinism, coupled with our existence, means the world is necessarily personally sentient, and that sounds a lot like ID, or something more. The tortuous spread of time in which it took for us to 'get here' is just an anthropocentric illusion: the universe was 'getting to' our level (and perhaps beyond) from the very beginning.
Indeed, on strict determinism, the present is but a holographic function of the very first instant, and thus the first instant contained within itself, as in the mind of God, all subsequent potentialities of its own immanent necessity. Only if we have some empirical basis for saying there could have been a "different world altogether" (i.e. absolutely different initial conditions) could we say we might not have existed in it. But then, if the actual world's initial conditions could have been different, then the actual world (i.e. its most basic set of conditions for being) is radically contingent. What, then, of strict and total determinism? Is not the multiverse a mad grab for contingency by otherwise narrow-mindedly deterministic folks. Look at David Lewis' modal hyperactualism: physicalist Platonism.
Even if Larry grants there is epistemic indeterminacy, due to our ignorance of an otherwise complete determinism, he has no coherent basis for distinguishing between our epistemic uncertainty (U:e) and the ontological necessity of the world (N:w). "When you don't know every input," Larry concedes, "you can represents the range of results probabalisically." Yet, he maintains, "that is stochasticism via ignorance, as opposed to inherent stochasticism." The problem is that, since N:w causes U:e without any metaphysical 'slack' (on determinism), if the latter is genuinely stochastic, and yet is genuinely continuous with the underlying physical world, then the physical world itself generates genuine stochasticty (i.e. in us, if nowhere else).
Further, we should make Larry familiar with D. M. MacKay's arguments about the inherent unpredictability of self-knowledge. Briefly stated, MacKay argues that even if we at time t knew every possible 'input' about ourselves at time t+1, we could not predict our action at time t+n, since at t+1 our knowledge that we will certainly do A or not-A would recursively influence our total epistemic state at t+1 and force us to recalculate what we-at-t+1-with-A-certainty versus we-at-t+1-with-not-A-certainty would do. Interestingly, no one, not even an omniscient calculator (OC) who knew as much about us possible would be able to assert a prediction of our action (OC[A]), since, first, OC[A] would be a certainty only if OC told us our action and we in fact bore its truth out (but then the recursive self-prediction problems arise), and, second, the truthmaker of OC[A] (even if OC whispered it in secret) would be true only when we in fact did A. Since there is always a logical possibility I will do not-A, OC's prediction depends on my doing A, not vice versa. Indeed, OC[A] is not a scientific prediction if it is not in principle falsifiable. As such, once again, we see how science is inherently non-deterministic.
Now, to be more precise, the distinction between my two points about OC's prediction is this: The first point means that OC[A] would support determinism iff OC's assertion of OC[A] to me could not even logically influence me to do not-A. However, once I know OC[A], I know that I know OC[A] and my knowledge of OC[A] (k:OC[A]) becomes a new factor which OC factor in his prediction of my action. So in order for OC to prove to me that I am subject to complete determinism, he must announce his prediction so I may witness how I invariably comply with it. Once he tells me my future, however, he is logically one step behind the epistemic state k:OC[A] to which OC[A] applies. OC must then recalculate OC[*] to include what OC[A] did not, namely k:OC[A]. The upshot is that no one can ever prove to me that I am a deterministic system.
Second, if determinism is true and could be mapped onto a complete knowledge of the world, there would be no logical space for "prediction". Prediction is an assertion about a state of affairs (SOA) which will arise with a certain probability. A necessary effect cannot be anymore predicted than one can "predict" the sum of 2 and 2 or that with which X is identical. The combination of determinism and Laplacian omniscience removes the secondary causal efficacy of anything O being examined, since "what O does" at time t is nothing more than what "the world prior to t" is. On Laplacian determinism, distinct entities are just illusory epiphenomena of the encompassing total causal SOA. As such, there is nothing to predict "beyond" SOA at t (SOA(t)), only descriptions to be made of SOA(t), since a complete account of SOA(t) will be symmetrically identical with and logically inclusive of SOA at any time (SOA(*)). If that were true, though, there would be no "me" about whom to make predictions. Only if I contribute something ontologically distinct to the SOA can predictions be made about my effects. In which case, however, no prediction is justified by a reference to SOA prior to my effects, but rather all such predictions hinge on the actuality of my effects. The truthmaker, therefore, for OC[A] is not SOA at time t(OC[A]), since an assertion about that SOA would predate (viz. not include) the occurence of A. Therefore, any putative OC[A] would have SOA:t(OC[A])-1, not SOA:t(OC[A]), as its truthmaker.
I have written about this problem before in a post titled "Reporting live". The gist of that post was this:
Determinism entails that there is such a "report" on all things at every instant, since at every instant the world necessarily is the way it is without an indeterminate remainder. … However, if the way the world is necessarily entails a report on all things, then the way the world is at any instant would have to include a report about the way of the world just subsequent to the report's existence. … We can easily spin this inherent indeterminacy to infinity, but in that case we have a W which is indeterminately true not in just two ways, but in an infinitude of possible states of affairs. Consequence for determinism? A state of affairs comprised of an infinite number of possible states of affairs is indeterminate in potentially infinite ways. So determinism is false in the actual world. It can't be a determinately true R in W that R(W(x,y...n)), since R(W(x,y...n) would have to include itself as a determinate truth in W(x,y...n), whereupon W(x,y...n) is no longer determinately and singularly W(x,y...n), but is W(R(W(x,y...n)))."
On top of all this, there is the matter of the inherent physical underdetermination of theoretical explanation. Briefly, since any physical SOA can be subsumed to innumerable competing formal explanations, no physical SOA perfectly and exclusively exemplifies a single, determinate formal operation. On the other hand, we know we perform determinate formal operations. Ergo, we know we are not merely physical SOAs. The physical is formally indeterminate, but formal truth is not. As such, formal truth is not purely physical and there is no basis for total physical determinism. I have, of course, written about these determinate vs. indeterminate lines of reasoning at length before. I will add all this into the book I am writing about determinism and free will, Deo volente.