What is randomness?
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What is randomness?
First part. With a number such as π, let's say the sequence has been computed to a billion places. Now let's say the digits are known to be uniformly distributed. I'm saying that based on the study of π which is known to have a random sequence, it has a uniform distribution of its digits (up to a billion places). Steve is arguing that suppose for the next billion places, the distribution becomes weighted towards some of the digits which could happen. Then what does randomness mean?
Second part: can it be proven for any irrational number that the distribution of the digits can be equally weighted?
PhilX
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Re: What is randomness?
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Re: What is randomness?
Don't know about chaos theory to say, but I can say that a pattern shows up on three irrational numbers with respect to the digits. Steve maintains a different pattern may result if the sequences are extended.Spiral Out wrote:Doesn't one aspect of chaos theory play into this where if a seemingly random sequence of numbers is taken far enough into the sequence then a pattern will always emerge?
PhilX
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Re: What is randomness?
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Re: What is randomness?
I was wondering whether it ever could be proved for any transcendental and/or irrational numbers. And I was also proposing that the digits could actually be weighted towards one (or more) particular digit(s) and still be random.
Spiral Out:
Possibly you're thinking of monkeys and typewriters and the works of a certain Warwickshire playwright?...if a seemingly random sequence of numbers is taken far enough into the sequence then a pattern will always emerge?
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Re: What is randomness?
I think you're referring to the Law of Large Numbers (LLN) in probability theory?Steve3007 wrote:I was wondering whether it ever could be proved for any transcendental and/or irrational numbers. And I was also proposing that the digits could actually be weighted towards one (or more) particular digit(s) and still be random.
Not a clue of what you're talking about there. Sorry.Steve3007 wrote:Possibly you're thinking of monkeys and typewriters and the works of a certain Warwickshire playwright?
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Re: What is randomness?
Complexity Theory (to which Chaos Theory belongs) demonstrates the emergence of such patterns out of chaos only for dynamical processes. The sqare root of 2, π, or any irrational number per se, are not dynamical processes; they are specific results from a non-iterative computation. Patterns (generally known as attractors in the theory) emerge from iterative executions of an algorithm. (In Nature, as opposed to abstract mathematics, the algorithms often entail feedback, that is, where the output becomes input to the algorithm's next iteration). There is no reason to believe that configuring the values of π's digits into a graphic representation would yield evidence of attractors, as complexity theory defines them.Doesn't one aspect of chaos theory play into this where if a seemingly random sequence of numbers is taken far enough into the sequence then a pattern will always emerge?
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Re: What is randomness?
No, I don't think so. My understanding is that that is more relevant to the emergence of predictability from random experimental results - e.g. the basis on which the laws of thermodynamics emerge as reliable deterministic macroscopic laws from consideration of very large numbers of (effectively) random microscopic events.I think you're referring to the Law of Large Numbers (LLN) in probability theory?
Calculating the value of π is not an empirical experiment. It is the ratio of the circumference to the diameter of a perfectly Euclidean circle and is calculated by consideration of the abstract mathematical properties of that abstract mathematical object. In that sense it is fundamentally different from, say, Planck's constant. That's why, in another thread related to this one, I speculated about the fact that if some kind of message were built into the value of π (billions of places downstream from the decimal point) then the entity that designed that message would be doing something much more fundamentally weird than "merely" building a message into the physical laws of the universe.
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Re: What is randomness?
Interesting concept, but I think such a message would more likely be found in the value of Φ (phi - 1.6180339) rather than π (pi - 3.14159), since Φ is a ratio (1.6180339:1) found ubiquitously throughout nature.Steve3007 wrote:I speculated about the fact that if some kind of message were built into the value of π (billions of places downstream from the decimal point) then the entity that designed that message would be doing something much more fundamentally weird than "merely" building a message into the physical laws of the universe.
Phi seems a more likely source of such a coded message.
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Re: What is randomness?
So if there was some kind of message hidden somewhere in the extremely high precision calculation of either of them, it would perhaps still be a property of pure mathematics? Or would it?
I don't know. This is starting to make my head hurt.
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Re: What is randomness?
Cheers,
enegue
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Re: What is randomness?
If the knowledge isn't there then it cannot be assumed with absolute certainty that there is any definite causal order to the associated phenomenon. Randomness may be only apparent as you've suggested, or it may be an actual state. It cannot be known.enegue wrote:Randomness doesn't exist. It's a pseudonym for "chaos", and describes a state in which the motion of a body, or bodies, is unpredictable because knowledge of the forces in play is incomplete.
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Re: What is randomness?
So, you do what any good scientist would do, you make one of the possibilities an assumption and you go about collecting data to prove or falsify it.Spiral Out wrote:enegue said:
Randomness doesn't exist. It's a pseudonym for "chaos", and describes a state in which the motion of a body, or bodies, is unpredictable because knowledge of the forces in play is incomplete.
If the knowledge isn't there then it cannot be assumed with absolute certainty that there is any definite causal order to the associated phenomenon. Randomness may be only apparent as you've suggested, or it may be an actual state. It cannot be known.
I am convinced that my incomplete knowledge of the forces in play has been sufficiently proven, which is why I can make such a bold claim. However, I understand how some people might believe their knowledge is so complete that they can show how things spontaneously occur without cause, or that others might not care enough to go to the bother of finding out.
Of course, the attitude that your knowledge is incomplete is the only one of those three positions that will move you to want to investigate and discover more about the forces in play.
Cheers,
enegue
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Re: What is randomness?
Just to feed into that, he's a little factoid that I remember from Uni: There was a very beautifully constructed chaotic pendulum in the foyer of the physics/electronics department. On a sign next to it the claim was that it was so sensitive to the details of initial conditions that, even when considered as an entirely deterministic classical mechanical object, it was calculated that the gravitational pull of a single raindrop a mile away would be enough, within 20 minutes, to completely change its motion in comparison to what it would have been if the raindrop were not there.
If this is true it suggests that even chaotic classical mechanical systems might be regarded as being fundamentally random. It's impossible, in principle and practice, to measure anything with 100% accuracy, even in a classical world. You can get arbitrarily close to 100% accuracy, but never reach it. So, if the fine dependence of the system can be shown to always be ahead of your never-ending race towards the unattainable goal of 100% accuracy, then I guess the system could be shown to always be unpredictable.
Just a thought.
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Re: What is randomness?
Is that what good scientists do? They make an assumption based on an unjustified possibility and then prove or falsify it with an inherent lack of knowledge?enegue wrote:So, you do what any good scientist would do, you make one of the possibilities an assumption and you go about collecting data to prove or falsify it.
If you cannot possibly know the causes for any given phenomenon then you also cannot assume the assumed causes have any specific order or direct causal effect, but merely an unknown degree of (also assumed) influence, or not.
If you don't know then you don't know. Assumptions are ****.
Honesty.
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