I'm testing the assertion that human beings (leaving out other life forms for simplicity) can and do cause effects, that is, changes in the physical world, which a) have arbitrarily-low probability and which b) cannot be explained, even in principle, by the action of natural law.
In support of a), flipping a fair coin once per Planck time (~5.39e-44s) without human involvement ("in the nonliving universe"), the average time required to cause 202 heads to appear in a row is 2.2e9 years, or about 1.5 times the age of the universe. In contrast to this maximally-efficient process in the nonliving universe, the average time required for a human being to cause 202 heads to appear in a row is about three minutes by placing them down as heads.
In support of b), effects in the nonliving universe always have a distribution of measured values, but these can always be characterized statistically and are intrinsically repeatable. For every effect in the nonliving universe, the mean of these values (the expectation value) follows a law that can in principle be expressed mathematically. That is, the universe is deterministic. (Effects of chaotic processes may not be expressible by us, but that's the result of our limited resources: they're still deterministic. Quantum effects also follow this direction and are statistically predictable.) Humans can produce outcomes whose value and distribution are not predictable by any physical law: you can model and predict with great accuracy the acoustical characteristics of a drop of water impacting a surface, but no law exists by which you can predict what I'm going to say next. Similarly, you can always predict with good accuracy the normal distribution of marbles dropped into the center top of a pegboard array, but you cannot ever predict under any physical law how I will choose to distribute those same marbles at the bottom of the array if you hand me the bag.
Great comments below; I'll edit to attempt to adjust the question based on these. (I hope this is the right protocol. I haven't figured out the @ referrals yet--apologies for not notifying commenters.) Mauro, good, good point, but then I am forced to ask how to account for free will under natural law. Pe, likewise very helpful, but for my purposes extends outside naturalism, which I'm trying to stay within. Cort, outstanding thinking: but for me, information always presupposes a question that this information answers. The most basic question would be something like, "what pattern is desired?" but the problem then becomes that this "desired" pattern simply specifies an arbitrarily improbable outcome, and I've just recast the problem in different terms. I'm doing my best to move your point forward, so do let me know your thoughts on this. On your idea of atomic coin flipping, I'd just wonder how many heads in a row we'd have to place before a single human being has outdone the entire universe at the atomic level (maybe your 300, but in any case probably fewer than we'd expect) and the question becomes all the more pointed. Again, help me clarify and think well as you have been.
If we humans are the product of natural law's operation, how have we so thoroughly escaped, apparently, its limitations in this fashion?
Another edit to try to better state what I'm comparing. The problem is not just that my descriptions are insufficiently precise, it's also that this is a hard matter to "grok" as I mentioned in another comment. I am trying to avoid any premises but the assumption of the raw cause-effect character of naturalistic materialism, and I believe those apply equally under that view, without reservation or modification, to both situations. Under naturalism, then, natural laws of cause and effect completely govern the process by which our "perfect" coin-flipping machine functions. Natural laws of cause and effect also completely govern every aspect of the development, existence and operation of our human being in question. This includes everything from the initial random assembly of Earth's first self-replicating molecule through every stage of non-teleological evolution, speciation and ultimately the arrival of our person holding our coin ready to take action on it. The sense that one cause-effect chain may not be fairly compared with the other is strong throughout all the comments and answers. (And for strength of reaction, you should have seen the answer that was deleted!) The difference is so great between the two situations and environments that it completely invalidates my question, people say. These two things simply may not validly be compared with each other.
But why am I in principle prohibited from comparing two purely naturalistic cause-effect processes at the most basic level of causation and asking for some explanation for the stupendous difference in their performance in reaching a specified result, that of 202 heads in a row? I'm not bothered or offended by the assertions that this is not allowed (in fact, as I said, I expect it), I just don't understand the reasons for the assertions.
It would be one thing to say that an extremely nonspecialized, inefficient, inapplicable process can't be fairly compared with and asked to achieve what a highly-specialized, highly-efficient process can perform, but that is exactly the opposite of the case here, I think. So it seems all the more important to explore why the evidently simpler, maximally-efficient, purpose-built hypothetical (nonliving) process cannot be expected, in the entire lifetime of the universe, to perform the task that the second, highly-complex, wildly-diversified, completely-nonspecialized (human) process accomplishes with an ease that approaches the idiotic in simplicity and facility.
Again, to tap into the earlier comments briefly, if we invoke free will, how have nature's laws (and they alone) created this thing and what is the cause-effect chain involved? And with all due respect to the ability to track the spectrum of my audible sounds, the meaning is conveyed by words whose choice and sequence will remain completely unpredictable by those naturalistic laws. Further, in the case of needing to do the impossible task of tracking every atom's behavior in my brain to be able to predict or replicate what I might say, then if it's so impossible, how can I do it? How can I do it? (That was for practical demonstration, not for emphasis :-)
It's long past time for me to say again that all these comments are absolutely wonderful, helpful contributions to my efforts to really get to the bottom of this. (If my responses ever sound rude or impolite, please tell me, as it will never be what I intend.) I have tried hard in many venues to raise this point for discussion to no avail. I hope people will continue to respond to the points so that I can understand what I'm getting wrong if that can be grasped. And in particular, if this seems nonsensical in any aspect, please stretch your patience as far as you can with me. I can only offer that beneath these possibly-nonsensical considerations there may be something I can discover. Your patience, especially if I'm not grasping your point, is key to that result.
Velraen, thank you. I'm a bit confused on some of your points, so let me see if I can provide capsule descriptions of my needs for clarity. (I'll edit the post to make these points broadly available.) This may involve some expansion on my original points, but I think everything will still lie within the scope of my original question.
I have intended to present living and nonliving entities as exactly the same kind of entity: both strictly natural phenomena produced purely and entirely by natural law. That's essential to my question. So the premise you see and report in my question is hopefully not actually there. What is however distinctly and clearly true, it seems to me, is that living beings act very differently than all other processes and phenomena--they produce wildly-different results than nonliving phenomena produce.
I understand your engagement of evolution to describe us and all other living beings, which is exactly what I want to consider as well, but I don't know what evolution means other than the action of natural law of the same sort as all other natural law. So again, I ask, why such a different outcome of natural law's action in this narrow band of the physical universe we call "living beings"? The result of that process is not just the ability to follow rules, or to put everything in order: as I indicated in my question, it is the ability to create arbitrary results. 202 heads up, 101 up and 101 down, or absolutely any pattern of 202 coins: this means we can produce a pattern of coins that corresponds exactly with a 202-bit digital word that provides StackExchange with my desired avatar image. So my abilities have nothing to do with orderliness: they have everything to do with absolutely whatever in the entire world I choose to make them have something to do with. Why, if natural law created and governs me along with the rest of the entire universe, does the nonliving component of that universe evidently never do anything in this class of results? Why does every nonliving process produce exactly the expectation value every single time, with minor variations?
Beyond my point about order above, your example of the assembly of a crystal makes my point beautifully. Crystals form because the laws of nature cause that configuration of atoms to be the lowest-energy state. A crystal's order is necessary. While I may not have the mechanical ability to create that long-range order, that's just a matter of scale. We humans create large-scale crystals all the time: they're called office buildings, and we make all kinds of arbitrary varieties, whenever we want. That's a bigger deal than 202 heads, I suggest. Besides, if scale is significant, we're getting there: the IBM folks some time back placed individual atoms on the surface of a crystal to spell out their company name.
I see your point about my proposal for a naturalistic equivalent to our coin flipping, but again, I'm confused. I still find no nonliving process that produces the arbitrary prespecified results we as living beings produce all the time when we perform virtually any activity that produces a noticeable effect on the world. Would nonliving processes ever create a single simple clay brick with its correct proportions and shape? I propose the likelihood of this is completely negligible. Then what about a wall built of many such bricks? A fortiori, it's even less likely. Then a building composed of those walls? Even worse for the nonliving processes. And I haven't gotten to the aggregate structures in a city. So the argument that somehow we're just doing the same thing that all other natural nonliving processes do seems, not to be too strong, a little hard to accept given the dumbfoundingly extreme difference in the behaviors of living and nonliving processes. Again, if we invoke evolution, I'd be concerned to avoid begging the question; how then does that account for these stupendous differences if it's exactly the same physics? We need to be able to explain this difference well, not in generalizations but in specific invoked physics and chemistry, I think. The extreme extent of the anomaly between the effects of living and nonliving phenomena demands that level of detail. And that is again the question I began with.
As a bit of dessert, let me return again to the matter of determinism as applied to our mental processes, apropos your third comment. If the two components, innate and acquired, make my and every human's behavior predictable, this surely is a falsifiable theory, no? Except that when I ask the proponents of that theory to demonstrate it and predict my behavior (or anyone's), it never seems to get demonstrated. In fact, it appears that our entire civilization is fraught with trouble predicting what we individuals will do. That inability is a main factor in all sorts of good and bad situations like crime, interpersonal conflict, your favorite television program's plot, even surprise birthday parties. So the evidence that human behavior is not predictable fills the life of every human being on a daily basis. Sometimes the determinist response is that we just can't get enough insight into the brain, or we don't have sufficient control over the environment, or some other limitation. It ends up reminding me of Samuel Beckett's Waiting for Godot: after enough possible explanations the suspicion eventually arises that it's actually not going to happen because it's not reality, just as our daily experience testifies so well.
I hope this helps with some more concrete descriptions of the nature of the problem I'm trying to address, and my need to get some concrete insights. As before, I'd beg your patience with the strength of my statements; it's fueled by the visible, unavoidable, persistent evidence of this extremely deep differentiation that confronts one constantly.
This all still falls under the category of attempting to defend the validity of my question. Since I continue to think it's valid, and at least to my mind feel there are good arguments for its validity, I may stand by after this for responses and thoughts from those (few? any?) who accept its validity.