BlueHat: Randomness is a human limitation. In case of a coin being tossed, a human cannot easily predict the exact outcome. Hence he calls it as a random outcome. But if all the inputs in the act of tossing the coins (viz. the angle of orientation, the power of flipping, friction, etc, etc) is calculated, then we can conclude the exact outcome of the toss. This means that randomness is a limitation of knowledge and not limitation of physical world. Yes, I am aware of chaos theory. But is it not just a limitation of OUR knowledge rather than being a fundamental problem with the physical world?

BlackHat: Consider atomic decay.  This has been hashed out elsewhere on Wiki, I think.

BlueHat: Atomic decay is still a case where ''we'' cannot predict when an atom is going to decay.  That is still ''our'' limitation. 

BlackHat: No, it is a '''physical''' limitation, not restricted to humans.  According to atomic decay theory, it is not possible to predict atomic decay, even if you know everything there is to know about the present.

BlueHat: How could you possibly know that without somehow getting ''outside'' physical reality and taking a look at it?

WhiteHat: Your position is essentially the "Hidden Variable Theory", which basically says that such quantum "randomness" is merely due to some hidden variables we do not know, implying that we could pre-determine the outcome if we know those variables.  According to http://www.counterbalance.net/ghc-obs/hidvar-body.html -- " [The Aspect experiment showed that] Hidden-variables theories, with their underlying determinism, must be non-local, maintaining the existence of instantaneous causal relations between physically separated entities."

GreenHat: I thought this happened all the time with quantum mechanics... No?

WhiteHat: No, this implies faster-than-light communication, which most theorists try to avoid.

BlackHat: For all practical purposes, many things '''are''' random. If there is a great machine somewhere that has the answers, we can't get to it, so its existence is moot.

BlueHat:  There is no particular reason to assert that randomness is a human limitation.  There is also no supporting evidence to support that assertion.  More importantly, randomness is a very useful abstraction in mathematics.  Furthermore, there is good physical evidence offering support for physical theories that suggest randomness in physical systems.  It is a very useful abstraction; we should keep it.

Contributors: VhIndukumar, AndyPierce, and others
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As above, randomness is a useful concept.

However, when defined sufficiently precisely so as to be useful in technical contexts (math, statistics, physics, computer science) it tends to strongly contradict "natural" intuitions on the subject. Furthermore, it is resistent to being 100% nailed down technically, as is also the case with the related topics of determinism/cause and effect and free will. (Naive intuitions about cause and effect are simply false in modern physics, free will has never been even been rigorously defined in a cognitive science setting, etc). These are difficult topics.

One commonly (but not universally) agreed truism is that there is no such thing as random '''numbers''' (since any specific number obviously can only assume a single value, etc), only random '''sources'''.

Even so, radioactive decay is usually considered the prototypical example of a true source of randomness, however it is not '''completely''' random. It is influenced by ambient effects, as is obvious in extreme conditions such as neutron stars, where tera-Tesla magnetic fields, giga-temperatures, mega-gravity, etc, sharply change the statistical properties of beta decay in all isotopes. Even neglecting that, it is essentially impossible to '''measure''' random events without introducing systematic bias from the measuring equipment.

Quoted from "Randomness Everywhere: Computably Enumerable Reals and Incompleteness", http://www.dc.uba.ar/people/materias/azar/CursoCalude.pdf :
* ''...we start by looking at random binary sequences. "I am convinced that the vast majority of my readers, and in fact the vast majority of scientists and even nonscientists, are convinced that they know what 'random' is. A toss of a coin is random; so is a mutation, and so is the emission of an alpha particle...Simple, isn't it?" said Kac in [33].''
* ''Well, no! Kac knew very well that randomness could be called many things, but not simple, and in fact his essay shows that randomness is complicated, and it can be described in more than one way, even by mathematicians and scientists.''
** (Mark Kac was one of the foremost mathematical statistics researchers of the twentieth century.)
* ''According to B. Efron (cited in Kolata [35]) "There have been heroic efforts to understand randomness. Randomness is not an easy concept to define. Books on probability theory do not even attempt to define it. It's like the concept of a point in geometry books.''
* ''Beltrami [2] remarked: "The subject of probability begins by assuming that some mechanism of uncertainty is at work giving rise to what is called randomness, but it is not necessary to distinguish between chance that occurs because of some hidden order that may exist and chance that is the result of blind lawlessness. This mechanism, figuratively speaking, churns out a succession of events, each individually unpredictable, or it conspires to produce an unforeseeable outcome each time a large ensemble of possibilities is sampled."''
* ''In an extreme sense there is no such notion as 'true randomness'. Indeed, any sequence has some kind of regularity; for example, van der Waerden discovered a '''universal''' nontrivial property shared by all sequences:''
* ''Theorem 10 In every binary sequence at least one of the two symbols must occur in arithmetical progressions of every length.''

There are an arbitrarily large number of statistical tests that have been or could be devised to test "randomness" of sequences, but none are foolproof for all purposes, although many are sufficient for some given pragmatic purpose (specifically including measures of Kolmogorov-Chaitin complexity).

Mark Kac wrote "From the purely operational viewpoint, however, the concept of randomness is so elusive as to cease to be viable." ("What is random?", Marginalia, American Scientist 71(4):405-406, 1983) 

It is nonetheless an essential concept in many disciplines. The moral here is to be extremely cautious about assuming that one fully understands the subject -- and to beware of fallacies that crop up very easily. -- DougMerritt
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See EveryThingIsMath, EverythingIsRelative, HolyWar