Addendum : I don’t know how relevant it is to the discussion, but the whole “page for d10” scheme reminded me of another RPG numbers scheme. Some RPGs use d100 to generate percentage chances, but since actual dice with a hundred facets are hard to come by, and those that do exist aren’t really practical anyway, the usual way to roll a d100 is to roll two d10s : one for the tens, one for the units. It is customary to announce which die is which before rolling (e.g. the red is tens, the white is units).
During one sessions I played, one player didn’t announce firsthand which of his dice was which. He just rolled two dice, and announced numbers. The GM was puzzled, because while this player didn’t always use the same die for tens, he wasn’t dishonest either (he didn’t use the common player trick of “whichever die rolled higher was the tens on this particular roll, I just didn’t announce it :)”) - sometimes he rolled good numbers, sometimes bad, but there was no immediately discernable pattern.
Near the end of the session, the GM finally called him up on it, out of sheer curiosity. Did he roll the white for tens on one roll, then the white for units on the next one ? Was there some weird yet consistent system (roleplayers can be funny about how to make a die “lucky” ). Turns out he just rolled the dice, and whichever die ended up on the left was the tens, without any predetermination.
I dunno why, but I found that very clever (and used that method myself from then on).
Only the fact that they match those formulas. Is there anything in any digit sequence which suggests extrapolation in any more compelling way?
Well, we couldn’t call it “the algorithm for calculating the ratio of a circle’s circumference to its diameter”, obviously, but we might come to consider it all the same. Perhaps we’d have our interest aroused by looking at the sum of 1/1^2 + 1/2^2 + 1/3^2 + …, just because it’s such a natural series to think about. Or of 1 - 1/3 + 1/5 - 1/7 + … . Or by thinking about the period of the differential equation f ’ = f in the complex plane.
Sure, if there was no significance to a string of digits, we’d never bother looking for algorithms to calculate it in the first place. The very fact that we bother imbues it with significance. But that’s just empty tautology.
Anyway, yes. You are right that it’s not possible to come finitarily to know of an infinite digit sequence that it is “random” in the computability-based sense.
But so what? This is a trivial fact. No nontrivial “long-term” property about infinite digit sequences is finitarily determinable. It’s not possible to come finitarily to know of an infinite sequence of bits that it does not end in a trail of all zeros. It’s not possible to come finitarily to know of an infinite sequence of real numbers that it is always increasing. It’s not possible to come finitarily to know of an infinite sequence of words that it doesn’t repeat any word infinitely often. But those are all still interesting properties…
There are easy examples in the real-world of where we have thrown up our hands and said, on inductive grounds, that certain things cannot be predicted by any rule. E.g., look at the acceptance of quantum mechanical phenomenon as random. Sure, it could turn out that we were misled, and there actually are rules which allow one to predict such things better than chance, but we weren’t prevented from at least inductively concluding otherwise, which is the best we could hope for with any non-finitary property.
Well sure, if the digit sequence is 1010101010… or something. But that sequence doesn’t exhibit statistical randomness. For a sequence that is random as far as statistical tests are concerned, I don’t think there would be anything in the digits themselves to suggest what the pattern is. That was kind of my point.
My point was just that without knowing the significance of a string of digits, we have very little chance of finding the formula that predicts them. I mean, if I give you the digits of pi (which obviously have significance to me) and say “Is this a random sequence of numbers?” then unless you also know that it’s pi, you’re in all likelihood not going to be able to show that they aren’t random.
Sure, but as you say, if you came up with some algorithm to predict the N+1st digit from the first N, and you tested it for a whole lot of values of N and it was right every time, you could use inductive reasoning to conclude that the algorithm generates the sequence. But if the sequence is statistically random, I don’t even think you’re in general going to be able to come up with an algorithm that we can inductively conclude matches the sequence. I mean, you could look at the first 100 digits of pi and make an algorithm specifically to match them, but it’s still quite probably going to fail for many values of N greater than 100 – unless you used some knoweldge of what pi is besides just the list of digits in coming up with the algorithm.
Now whether this observation is worth the amount of time we’ve spent talking about it, I’m not so sure.
What does “statistical randomness” mean? “314159…” fails the “Don’t keep matching the digits of π” test; is that not as much a statistical test of randomness as anything else?
What’s the difference between the way in which “10101010…” matches a simple pattern and the way in which “314159…” matches a simple pattern?
You keep saying no one could possibly see “314159…” and come to realize “Oh, that matches the output of this short, simple program. Let’s see if it keeps matching.” without some prior knowledge of the “significance” of π (presumably, this means knowledge that these digits describe half the ratio between a circumference and radius). But why not? This is just a blanket assertion. It may even be true (as a contingent fact about human history, though not as a necessary or mathematical fact of any kind), but you’ve given no argument for it.
Eh, whatever. I don’t really know why I’ve been arguing against you (tim314), at least so far as this goes. Just been in an argumentative mood, I suppose. Sorry; consider this matter dropped (if you like).
[Though I still claim that “314159…” fails statistical tests for randomness just as much as “10101010…” does, the only difference being which particular tests are failed.]
But the examples you use simply show humans taking advantage of the statistical nature of certain processes, not actually generating a random series themselves. Anyway, though, so far my initial intuition that humans ought to be incapable of true randomness seems to be shared by the majority of responders, especially Chronos’ scissor-paper-stone example seems to show this rather well, though not necessarily conclusively – it may well be the case that initial, ‘random’ moves are discarded out of superseding superstitions and ignorance regarding the nature of randomness… Like in typing out a string of random numbers, you think of one number, then discard it because that number already occurs twice in the preceding string. So it might be that underlying randomness exists, and just gets distorted by ‘higher-level’ thought processes, yet nevertheless plays a role in decision-making; how to test for something like that, though, I have no idea.
Yes, there would be a natural aptitude for pattern-recognition to overcome (since true randomness can include patterns. 11111 is just as random as 39642, and 10 11 12 13 14 15 can be part of a random string). But I don’t think it would be impossible to achieve near-perfect randomness should one actively train to do so. Since there is no incentive for such training, I suppose we’ll never know for certain.
). Turns out he just rolled the dice, and whichever die ended up on the left was the tens, without any predetermination.