This question is far more difficult that it appears. I suppose (using dewdney's suggestion) that one way you can measure the randomness of numbers is look for replication. It seems plausible that the more a number is replicated, the more that it is not random. Of course, there are many variables the come into play here (sample size, # of tests, etc.).
I could be wrong, but the reading gave me the impression that random numbers is an incomplete problem. There is no way to create truly random numbers. We would think that simply be generating numbers in our heads or other such things (opening a book to any page # ) would be random, but in fact we are introducing biases. (For me, my favorite number comes to mind then small two digit numbers and if I were to open a book, I'd open it right smack in the middle. )
Even computer programs cannot generate truly random numbers, since they function through the use of an algorithm that will eventually repeat itself. (I may be wrong, but I think that the probability of repeating itself is greater with an algorithm then by using one of those books of random numbers).
Dewdney talks about lottery machines and makes it sound like even those aren't random. Even though it seems like you can't get anymore random then by power-ball lottery, Dewdney has to rain on my parade. I suppose that little things, such as how long you wait to pick the next ball, what order the balls are dropped, the air pressure, etc.., could effect the outcome of the numbers picked.
So, we now ask ourselves, does it really matter if something is truly random? For what purpose could we need a truly random numbers that semi-random numbers will not suffice?
So I opted for the reading in the internet in lieu of Dewdney and after finding a few papers that were completely over my head, I was pleasantly surprised to find one that was at least comprehensible (Randomness and Mathematical Proof). What I learned from reading it was this: random numbers cannot be determined by origin. Just because you come up with numbers by flipping a coin or tossing a die (methods considered to be "random") does not mean that the resulting numbers are random themselves. Instead, Gregory J. Chaitin's conclusion was that random numbers should be defined by their "incompressibility." If you have a number, or series of number, that has no apparent logic to it, then you will not be able to represent it in a simplified form (by a formula). These numbers are truly "random."
The web page was down this morning when I went to read the additional material Jeff posted, so I didn't get a chance to read it. I did read the chapter on Random Numbers in Dewdney, and if I remember correctly, the question was something like Do random numbers really exist?
So, here goes an answer. I think that there are numbers which can be generated by methods which are satisfiably random, but that the status of the number itself is fixed in a sense outside of the process of selection. I guess a true random number would be one that changed at it's own will. Imagine if the constants we use in various equations where considered random in this sense--we'd never be able to really acurately calculate anything. There goes science out the window. But that was a digression, I do think randomness is limited by infinity. I mean this in the sense that in choosing a random number, you really ought to generate numbers in the tens just as often as the 10,000,000,000,000,000...assuming that these numbers are just that much closer to infinity, but equally possible in a random generation, and at the least represent numbers in the set of reals for which we might find ourselves arriving randomly. So, I don't really know if a truely random number could ever really exist because we don't really randomly choose any number in ranges of great magnitude like this; we stick too close to zero to experience randomness any more purely. Randomness seems like a simple adjective of quality, subject to gradient interpretation, and not a fixed description.
Having said that, I think it is interesting to see how the idea of randomness has infused both art and music. Think of the Dadist poets who cut up newspapers in a random fashion, drawing words out of hats to make poems. I'd say they achieved a level of randomness in their artistic expression of the world. And, in music, composers of aleatoric music, music designed to capture formlessness by using random notes placed together (again, chosen randomly) without a set system or ideal of musical composition, like composer John Cage. You would be surprised how unified these random artworks tend to sound in our minds, how equally resonant the Dionysian is to the Apollonian, though they are based on different principles (chaos/order, etc.). It is difficult to grasp randomness, by definition, but it goes unchallenged as a representative of the sublime in the artistic world, appreciated to varying degrees.
Given that there seemed to be some question in chapter 8 of Dewdney in terms of what random actually is, and as to whether you can ever really know for certain if a given sequence is indeed random, I don't really know how to answer the question for today. The best the Chaitin-Kolmogoroff theory seemed to be able to offer was a measure of the _degree_ of randomness in sequences produced by programs, one means of measuring being to form a ratio of program length to string length.
I suppose some good criteria to determine whether a set of numbers is random would involve checking to make sure there aren't any patterns or numbers that show up an excessive number of times. I suppose that a random number would be one with no association to any choosing process or anything like that: if I suddenly shout out "83!" there is at least an inherent like or dislike for the numbers 8 and 3 in my head, and I chose them in some kind of correlation between that and what's going on in my head. I'm not entirely sure what Jeff's looking for in this question. I hope my answer is at least helpful in some way . . .
It seems to me that the main criteria is how soon a sequence starts repeating itself. If a sequence is truly random, it will have no pattern. Also, the length of the formula used to determine the sequence it important.
To prove that a number is random, it must be shown that no program significantly shorter than it will regenerate it. There are different ways of testing randomness that produce varying degrees of randomness. The aforementioned test is referred to as comparing the ratio of the program length to the string. The longer that a string goes without repeating itself, the more random the number is.
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