Does Mathematical Randomness Exist?

I'm interested in the mathematical consequences of randomness if randomness is defined as follows. Let's represent an infinite sequence of numbers by the function f on the natural numbers N to the set {0,1,2,3,4,5,6,7,8,9}.

Suppose we say that the infinite sequence f is random if no algorithm exists that generates the sequence f(n). For example, the digits of pi seem random but there are many elementary formulas that represent the numerical value of pi perfectly. Thus, in theory, pi can be computed to arbitrary accuracy.

Question: Can it be proven that mathematically random sequences exist with this definition?

As John von Neumann humorous said, "Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.''

The method I have specified for defining non-random numbers is clearly deterministic. But how do we know that truly random numbers exist?

 

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As far as we know, radioactive decay is a random process.

 

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I recommend a visit to the site random.org (still not enough posts to make links - annoying!)

 

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RANDOM and NOTHING cannot exist unless these paradigms are factored in the process. So says a finite universe.

 

The Indescribable Random Numbers

When I woke up this morning it occurred to me that the set of all nonrandom numbers must be countable because the set of all algorithms that generate the nonrandom numbers can be put into a countable lexicographical order. Therefore the set of all indescribable random numbers must be uncountably infinite. But if the set of all nonrandom numbers is countable, we can use Cantor's diagonalization process, which is an algorithm, and find an unlisted nonrandom number. That's a contradiction. Did I discover a new paradox in set theory?

 

Can you prove that you can enumerate the set of all algorithms that generate random numbers? Even the ones that we haven't discovered?

 

By definition, no mathematical algorithm exists that can generate a random number. So you are really asking if I can enumerate the empty set. The answer is yes.

 

Nope. Your nonrandom numbers are the countable numbers. I'll see if I can find a better reference than wikipedia, but until then http://en.wikipedia.org/wiki/Computable_number#Properties

“Properties
Although the set of real numbers is uncountable, the set of computable numbers is countable and thus almost all real numbers are not computable. The computable numbers can be counted by assigning a Gödel number to each Turing machine definition. This gives a function from the naturals to the computable reals. Although the computable numbers are an ordered field, the set of Gödel numbers corresponding to computable numbers is not itself computably enumerable, because it is not possible to effectively determine which Gödel numbers correspond to Turing machines that produce computable reals. In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Thus Cantor's diagonal argument cannot be used to produce uncountably many computable reals; at best, the reals formed from this method will be uncomputable.”​

 

What is relatively random to us, may just be a lack of knowledge on the system. I have found the fact that we cannot understand the deterministic qualities, if there are any, is that it's an indication that we cannot obtain the full information on a system, or systems.

Randomness is a concept - and like any mathematical formula, they are equally abstract.

 

There is no one countable subset of the real numbers. But I agree: my nonrandom numbers are the known computable numbers, and they are countable.

My argument is different so your assumption that I must be held to the limitations of Gödel numbers is false.

I am not claiming that uncountably many computable reals can be produced. I agree that the computable reals are countable. So why is Cantor's diagonal process not an algorithm? It clearly is.

 

Sorry, I meant computable numbers.

 

It's not a computable algorithm. You are applying the algorithm to the set of noncomputable numbers. Think about that for a bit.

 

Why are the numbers noncomputable?

I admit that I did not provide a precise definition of algorithm. Looking at the Wikipedia article on computable numbers, I see their definition: computable numbers are real numbers that can be computed to within any desired precision by a finite, terminating algorithm. I don't see anything infinite about determining the first n lexicographically ordered algorithms and their computable numbers, each to n decimal places.

I agree that computing pi to n decimal places is unbounded as n approaches infinity. But everything is finite for each finite n, even in diagonalization.

I don't think that the problem is with the algorithm. I think the logic breaks down due to the fact that the set of all indescribable real numbers to n decimal places is precisely the set of all real numbers to n decimal places. So I guess I doubt the mathematical existence of randomness in my opening post.

 

No! pi is a computable number.

Randomness is a well-defined concept. Just as I suggested to Pete I suggest that you read up on axiomatic probability theory.

 

Eugene, I believe the answer is no. Any process that generates the random string of digits to be tested IS ITSELF an algorithm, even if that algorithm is ultimately defined by first creating the Universe...

 

Green destiny: The following is a naive view of randomness

The current evidence strongly supports the belief that radioactive decay is a random process.

Physicists expect that future knowledge providing more insight into radioactive decay will not explain away the randomness: It will only push it down to a lower Quantum level.

For Example, suppose that it is discovered that when a certain Quark frazzles (new term for a newly discovered quantum event), the radioactive nucleus decays.

The above would not explain away the randomness: It would merely move the randomness from the nuclear level to the Quark level.

A future physicist might say that radioactive decay is caused by Quark frazzling, which is a random process. Hence radioactive decay, while having a cause, is essentially a random process.​

 

Dinosaur, don't call it naive please. What evidence is there that strongly supports radioactive decay being "truly" random? That it conforms to a Poisson distribution? What if, say, decay occurs only when a qualified advance wave collides with the material? Absorber Theory is not widely popular but it certainly cannot be discarded. My point is that whether or not decay is acausal is a matter of your personal interpretation of QM.

 

Are you saying that the result you perceive to some experiment is dependent on your chosen interpretation of quantum mechanics? That's quite obviously rubbish.

 

There's no way to decide whether an arbitrary program actually computes a number or not (by Rice's theorem), so this cannot be done. A simple diagonalization argument shows you that the set of (programs that compute) computable numbers is not computable, so you cannot define a sequence of programs that "hits" all, and only, the computable numbers.

 

Yes, exist..But only mathematically.You can not generate so it is random.

The Function x2-5x+6=o has two solutions x1=2,x2=3
When 2 and when 3? Is random.The two solutions are equally good.

Does not exist in nature,function that generates two different solution for the same value.Would contradict the theory of causality.
A set of cases generates an effect.Same set of causes can not generate two random effects.If a set of cases generates two different effects,it certainly is a cause that has not been taken into account and determines the effect.

An example is throwing the dice.
If we could take into account all the variables involved in this action,initial position of the dice,muscle tone, frequency of shaking the dice in his palm, skin elasticity,dice weight, distance between punch and table, table roughness, etc., etc...we could calculate the number that will appear on the dice.But this is impossible.For this is the calculation of probability.

But this has interesting implications, philosophical.
If there aren't random, and everything is cause and effect, which in turn becomes the cause for the next effect.
This means we have predestined fate?