OK Mafialligator. Sorry for the delayed response.
First I'd like to note that you addressed almost none of my points -- specifically those that dismantled your "key" (your words) example of the water droplet and its behavior -- and think I have therefore earned the right to point out that it's ironic that it is I who supposedly gave no argument other than "uh no you're just wrong." ;-)
Anyway. There are several further points I'd like to raise. I'll start with a big and important one.
So, you argue that randomness DOES necessarily lead to order, via probability theory. Now I actually referred already to this kind of "order," and I'd argue it's more disorder. But I'll get to that. Deferring technical issues and other arguments, let us assume for now, for the sake of argument, that this really is order. A primary problem here is that you are NOT allowed to assume, without proof or argument, that the order takes a particular form.
Let's put this another way. Your argument, if it were true, would prove far too much. If the laws of physics as we have them actually followed from an assumption of utter randomness, then we would be able to show that, and in fact, we wouldn't have to do any more experiments. Wonder what the mass of the Higgs boson is? Easy! Don't build a multi-billion dollar supercollider. Just deduce it from principles of randomness, using probability.
After all, in the example you mention -- tossing a coin - the reason we're able to know about the 50% coin toss rate, irrespective of what we might be observing locally in time, is by computing it from first principles by assuming randomness (a fair coin). Unless you can do the same for the specific order you believe will continue in this case, you are certainly not allowed to assume that some SPECIFIC form of order (for example, the Schroedinger equation) is the result of probability and not of fluctuation.
Let's take an example. Say you have Bob, who knows nothing about probability, except that he's been told that in very large or infinite numbers, it leads to order from randomness. You give Bob a fair coin, tell him it's fair, and tell him to toss it 10,000 times. Now, say Bob tosses it the first 300 times, and -- surprisingly enough -- gets 75 repetitions of the pattern HTTH. "Well, well," says Bob. "I guess I see now that probability really does create order from randomness. I can now assume that I will keep getting the patter HTTH for every four tosses I make from here on out."
Bob has just committed a fallacy, the same one as you -- he is assuming that, because SOME KIND of order should emerge from a coin toss, the SPECIFIC order that he is seeing should continue. In fact, it shouldn't. The odds are 15/16 that the pattern will be broken on the very next quartet of throws.
So, while probability does allow impressive calculations of what may appear to be regular properties in large amounts of random data, it is not a license, in the absence of such a calculation, to just assume that any old order we see in a partial data set must in fact be global and due to probability. Quite the contrary. Probability shows this to be fallacious.
Next I would like to turn to the question of what kind of "order" it really is that probability gives us. A convenient place to start here will be with the water droplet you brought up. As I asked before -- what happens if you heat the droplet? Well, what happens is that it completely decoheres. It evaporates into more and more smaller and smaller droplets, precisely because you are enabling the random aspects of its motion (the random degrees of freedom, to be more technical) to overcome the ordered degrees of freedom. It turns from a liquid, which is relatively ordered, into a gas, which is relatively disordered.
Now, a gas does exhibit the specific type of "order" you mention probability as giving you -- for a large gas (which almost all gasses are, having on the order of 10^23 particles), we can be pretty sure that the velocities adhere somewhat well to a specific distribution of velocities; so for the momenta, spins, etc. as well. Yet the gas is actually disordered -- far more than the water droplet.
Are you familiar with statistical mechanics and the second law of thermodynamics? In this branch of physics, entropy is defined as the (log of) the number of states compatible with macroscopic information. Entropy always increases, which is just to say that the state becomes more and more probable. By the time you have reached the most probable state -- the 50/50 heads/tails state, in your example -- the system is completely disordered, we know as little about it as possible, and it is useless for doing anything. The heat death of the universe is a classic example. If reached (cosmologists differ on whether it will be), it would be a completely uniform, uninteresting, homogeneous gas with no other properties at all. Oh, we'd have the order you speak of, all right -- things would be distributed exactly as probability says they must. But there would be no order, no predictability except static nothing -- AT EVERY SCALE -- nothing.
THIS is what probability generically predicts: order should quickly degenerate to complete, thorough chaos and homogeneity at every possible scale. The continuing existence of your chair is an insult to the "order" that probability guarantees.
So, to finish this subtheme, the type of regularity probability gives is completely wrong for what you need for induction.
I'll now introduce a digression: what DO you want for induction? It turns out that the kind of order you want is something called low Algorithmic randomness, or sometimes, low Kolmogorov randomness. Fix some computer / language. Say linux with g++. The algorithmic randomness of a string of text is defined to be the length of the shortest program that would generate it.
So let's consider coin tosses again. Say we see HTHTHTHTHTHTHTHT, etc., going on for millions of characters. There's a very short program that would generate that. Similarly for HHHHHHHHHHHHH.... On the other hand, most of your 50/50 solutions would NOT have any program even a few characters shorter than themselves that would generate them. This is trivial to prove. The upshot is that the VAST, VAST majority of long strings have very HIGH Kolmogorov randomness, and, conversely, that almost none of them can be predicted to any extent at all given partial information (say, from the first million characters). For example, of the million-bit strings, fewer than 1000 can be compressed by even 10 bits (to 999,990 bits). As the length of the strings goes up, the percentage that can be compressed 0.1% goes down. As you get to infinity, it goes to zero.
And it is exactly this property, compressibility, that the existence of scientific law and regularity implies. So the upshot (of this and other arguments) is that almost no random universe -- 0% of them -- can be predicted, at all, on any scale whatever.
While you talk of generalities and vague order-from-randomness that just happens to be the order you so desperately need, here is an actual demonstration that it doesn't happen: you don't get scientific law or induction in a random universe, with even positive probability.
Moving on, I'll close with a few minor points. None of these is that important in light of what we've already said, but I'll mention them for completeness.
First, your infinite-universe claim. It's actually very controversial in cosmology whether the universe is infinite or not. Certainly you can't just assume it is, in general. And even if it is, you can't conclude order RIGHT HERE ON EARTH from the fact that there are billions of particles somewhere else. So when you say,
"And when you're talking about the behaviour of the universe you're always talking about infinite numbers."
That's not really true. If I'm talking about whether your chair will keep existing, I'm just talking about the particles in your chair, and maybe in your room. Not all the other particles in the universe at all. So I don't have infinitely many to think about at all. In fact, of course, talking about "particles" as things with identity and behavior already assumes quite a lot of order.
Not that it matters. Infinity only makes it worse for you, not better (see above), but it's absolutely terrible in either case. Still, I just thought I'd point out this problem with your argument.
Next: your whole scenario actually sneaks in a lot of order already. In speaking of a universe, you assume a space of a fixed number of dimensions that will keep acting like one. You assume these particles will go on behaving in a particular way. Etc.
Finally, you say the following, speaking of infinite coin tosses:
"It will ALWAYS be exactly 50/50. Always, without fail. No exceptions. Ever."
This is not actually quite true. It's true that it will be 50/50 with 100% probability, but there's nothing to keep it, for example, from being all heads. The problem is that 0 probability is no longer the same as impossible once you're talking about infinitely many things. Something can be 0 probability and still possible. In fact, this is the case with an ordered/inductive universe in the space of all universes. It IS possible; it's just also 0 probability.
A minor point, but worth mentioning.
Anyway, thanks again for the response. As you see, probability doesn't help you at all in rescuing induction.
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