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Quote:Many scientific fields are concerned with randomness:
Algorithmic probability
Chaos theory
Cryptography
Game theory
Information theory
Pattern recognition
Probability theory
Quantum mechanics
Statistics
Statistical mechanics
Quote:The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling, but later in connection with situations of interest in physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation, it is necessary to have a large supply of random numbers or means to generate them on demand.
Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness)—this means that random strings are those that cannot be compressed. Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf, Ray Solomonoff, and Gregory Chaitin.
In mathematics, there must be an infinite expansion of information for randomness to exist. This can best be seen by using the binary number system. If one has a random sequence of numbers each of which consists of only three bits, then each number can have only eight possible values:
000, 001, 010, 011, 100, 101, 110, 111
Therefore, as the random sequence progresses, it must recycle through the values it previously used. In order to increase the information space, another bit may be added to each possible number, giving 16 possible values from which to pick a random number. It could be said that the random four-bit number sequence is more random than the three-bit one. This suggests that in order to have true randomness, there must be an infinite expansion of the information space.
Randomness is said to occur in numbers such as log (2) and Pi. The decimal digits of Pi constitute an infinite sequence and "never repeat in a cyclical fashion"[5]. "Numbers like pi are also thought to be "normal," which means that their digits are random in a certain statistical sense."
Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.[6]
It is impossible, with computer software, to generate truly random sequences that do not repeat. This is because, in order to ensure that there is no regular repetition of any sequence, the computer would have to store all the sequences that the program has already produced. These requirements would soon mean that there would be no more storage available, and more numbers could not be produced.
I would suggest you read the wiki semi-article of 'Randomness versus unpredictability' in the source citation. An interesting question for determinists: if I send the command of [pick number=1-100] to a computer... is not the choice of the computer completely random? The circumstances have not changed at all if I do so again... and therefore the output can be different every time with the same input. Therefore: randomness.
