What is Randomness?

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I have only studied mathematics, so I don't know how, say, physicists, or engineers, approach randomness. I've have never encountered any rigorous definition of what randomness actually is, very surprising given the typical mathematician's fanaticism for rigor.

As far as I know:
statistics/axiomatic probability theory - support set (of samples) omega on a Lebesque measure of probability P, with event set epsilon. I find this to be like saying that an apple is red, an apple is sweet, an apple is a good source of nutrients etc. - explaining every last detail of an apple without actually saying what an apple is.

Thermodynamics/chaos - High entropy systems. I don't know if chaos has "entropy" in its nomenclature but it's got parallel concepts.

cellular automata - Complexity science where a large, patternless, seemingly disorganized behavior can emerge from iteration of a few very simple rules - the thing is that if we were to only see the output of the system after many, many iterations, we would think that the system is random, but it turns out that the system is entirely deterministic. (For an example, see Rule 30 here) Also related to chaotic behaviour. Randomness isn't actually directly addressed in cellular automata, but the analogy is interesting.

Kolmogorov Complexity - So far, the closest I've found to a definition of randomness, because it's the only field of mathematics I've encountered that actually provides one:

A string of bits is random if and only if it is shorter than any computer program (or universal description language) that can produce that string (in short a random string is incompressible.)

For example, "abababababababababababababababababababab" (40 bits long) can be described as "ab 20 times" (11 bits including spaces). A sequence like "jk2hydnqaowicrnhwobvt4n2562qw1" (me hammering the keyboard any-old-how, 30 bits long) can only be described and understood as "jk2hydnqaowicrnhwobvt4n2562qw1", therefore this sequence is incompressible and random.

Although it's pretty far off from the conventional understanding of randomness, not particularly easy to understand and not very useful anyway...

Random Trivia[pun intended]
- The last 2 digits of the normal distribution table (with 4 d.p.) appear to be randomly distributed, although we know for sure that the normal distribution is pretty much the same everywhere.
- The digits of pi also appear to be random, once again although we also know that any simulation of pi will yield the exact same result.
- The same can be said to apply to almost all real numbers, so the real numbers are (kolmorogov) random almost everywhere.

So, especially from people with a more practically-centered background: what is randomness?

It depends on the application. Random is whatever gives you the behaviour you need. True randomness isn't easy, but for most applications also not necessary.

If all you want is select a "random" tip from a collection when starting an application, pretty much anything will do.(because it isn't really important if the generator is able to provide random sequences)
If you're working with heuristic algorithms, you might think you need good random number generators, but that may not be true either.(I was once working on a genetic algorithm and thought it'd be good to use a better rng than the one provided by C; then I found a paper that indicated that this won't really change much)

I like the Kolmogorov Complexity idea of compressibility (which ultimately boils down to repetitive patterns) and think I would probably use something like that to evaluate the randomness of a generator if I ever had to. I believe it's hard to do though, since truly random numbers can look remarkably unrandom.

Kolomogorov complexity comes closest. What cannot be compressed is random.

ruveyn

I think there are basically a few different concepts in the word 'random'.

One is whether something (data or whatever) is more or less patternless, which I think corresponds to your Kolmogorov Complexity.

Another is whether something is predictable or not, for example, in one sense the digits of pi are predictable, because we can calculate them and predict that what we calculated will come to pass. (And lo and behold, it does.) But on the other hand, they aren't predictable, because other than actually calculating out pi, we can't predict what the 10,000th digit will be.

Another is whether or not something is known. If I flip a coin while your back is turned, the coin flip is determined, but you still don't know whether it's heads or tails.

The formula for calculating pi is very compressed. You can write it one one side of a page of paper.

Being forced to calculate the digits of pi seriatum is inconvenient but the algorithm is finite and quite compressed.

And there is a formula for the n th digit of pi.

See: http://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml
and
http://en.wikipedia.org/wiki/Bailey%E2% ... fe_formula

who cares if it is hexadecimal digits?

ruveyn

I would say that, with respect to the physical universe, random means intrinsically non-deterministic. Put another way, if it is impossible to predict the future state(s) of something regardless of any amount of knowledge of the current state then that thing may be said to be random.

Decades ago, I read a definition of "random" just about the opposite of "that which cannot compressed".

The concept of "random" is self-contradictory, and IMO, "randomness" is not a good word to use in Quantum Mechanics, as it will soon become recognized as a God again.

"Random" has a vague, and controversial, definition in Quantum Mechanics. "Random" has self-contradictory definitions in mathematics, and "Random" has more of an ignorance/religious definitional usage in physics. Some mathematical definitions of "random" numbers and sequences are already established and recognized as self-contradictory:
http://mathoverflow.net/questions/69773 ... dont-exist

This goes through cycles of from being very controversial, to more or less silently accepted.

I'm more with old school views of Einstein's opinion of "God does not play dice".

Then, The Great God of Randomness may return to former stature, despite that it is often claimed "God does not [and can't] play dice with the Universe":
http://www.google.com/search?q=Skinner+ ... =bks&tbo=1

Tadzio

This whole thing about predictability, I'd say more to do with algorithmic complexity. E.g. the BBP algorithm is polynomial-deterministic. Which is why I mentioned originally the KC definition of randomness doesn't quite tally with the lay definition of randomness.



The way I see it: mathematics and computer science does not overly concern itself with the eventual future state, more the prediction of all possible future states (entropy, phase states...). Traditional probability is a secondary matter of assigning a lebesgue measure to each state.

yes: but also, 'random' is always used as the opposite of, or spectrum extreme from, another word ('patterned', 'predictable', 'known') & these pairs are really how the complete concept exists in language.

however, it is perfectly feasible to look for triadicities, or sets of three things, which also define a concept. for instance, the sort of patterned energy that defines life has as its upper bound, excessive motion; at its lower, insufficient. likewise one can conceive of a similar degree of order that is incompatible with the presence of this other order (e. g., 'cancer'...)--Blake was a significant thinker when he pointed out the difference between 'contraries' (like this) & 'opposites'-- so on, for 'random/predictable' & 'known/unknown'.

we simply don't possess enough terms in ordinary language to describe all of it (which is why there is mathematics). it was only very recently that the phenomenon of finding more of a quality when you had a reason to look for it, gained the name 'apophenia'. that would be another parallel to 'random'. the Lojban word CUNSO ('random') reserves its third argument for the probability distribution (though no one has yet used it like that): with this word, it would be possible to name a kind of scattering that clustered, say, at either extreme rather than around a norm.