What is a randomness. Does it exist?

[Aleksey]

Help!
I can not catch meaning of:
1. random event
2. random number
3. random process

 

[mathman]

Is your question a mathematics or a physics question? In mathematics, all these terms are defined, usually starting with the Kolmogoroff axioms. For physics, it is generally a more complex problem.

 

[SW VandeCarr]

What is a randomness. Does it exist?

Randomness means lacking any discernible pattern. However this definition can be misleading. Is the sequence ...193589... random? It's a meaningless question because once it's written down, it's no longer random, It exists with probability 1.

What's the probability of a random process generating this six digit sequence assuming each digit from 0 to 9 has an equal probability of 0.1? Of course it's $$10^{-6}$$. How do I know this? Because I said the process was random. In this sense, randomness is self defining. If we take a process to be random, we can expect it to follow a probability distribution (if one can be defined). Experimentally, processes we think of as random do tend follow these theoretical distributions.

So randomness turns out to be a theoretical concept. No one can say whether or not truly random process exist in nature. By the way, that six digit sequence is in the first 30 digits of the decimal expansion of $$\pi$$ which is a completely determined infinite sequence.

Still confused? Don't feel bad. What one can say about a random process is that, if it's random, we may have a poor chance of guessing the outcome of one iteration of the process, but a good chance of guessing the distribution of outcomes over many iterations.

 

[Aleksey]

Is your question a mathematics or a physics question? In mathematics, all these terms are defined, usually starting with the Kolmogoroff axioms. For physics, it is generally a more complex problem.

Math. According to Kolmogorov a random event is a subset of the set of elementary events. Why call such a subset as random event. What a reason?

 

[Aleksey]

Randomness means lacking any discernible pattern. However this definition can be misleading. Is the sequence ...193589... random? It's a meaningless question because once it's written down, it's no longer random, It exists with probability 1.

What's the probability of a random process generating this six digit sequence assuming each digit from 0 to 9 has an equal probability of 0.1? Of course it's $$10^{-6}$$. How do I know this? Because I said the process was random. In this sense, randomness is self defining. If we take a process to be random, we can expect it to follow a probability distribution (if one can be defined). Experimentally, processes we think of as random do tend follow these theoretical distributions.

So randomness turns out to be a theoretical concept. No one can say whether or not truly random process exist in nature. By the way, that six digit sequence is in the first 30 digits of the decimal expansion of $$\pi$$ which is a completely determined infinite sequence.

Still confused? Don't feel bad. What one can say about a random process is that, if it's random, we may have a poor chance of guessing the outcome of one iteration of the process, but a good chance of guessing the distribution of outcomes over many iterations.

Thanks. "No one can say whether or not truly random process exist in nature". Why do we use this term? What is the reason?

 

[FlexGunship]

Thanks. "No one can say whether or not truly random process exist in nature". Why do we use this term? What is the reason?

Pattern recognition is retroactive. Therefore, until a pattern reveals itself, the perception is of randomness. For your consideration: given only a single musical note, could you tell if it came from a "random note generator" or from a famous symphony?

Events (or numbers) out of context appear to be random. Perhaps it's useful to think of random as a state; a wave function that is collapsed once it's entities are organized in a pattern.

 

[SW VandeCarr]

Thanks. "No one can say whether or not truly random process exist in nature". Why do we use this term? What is the reason?

Because it is a description of our uncertainty; our inability to predict specific outcomes. It may be our ignorance, technical limitations or, in the case of some outcomes at the quantum level, an innate feature of nature at this scale. Quantum mechanics is formulated in terms of probabilities, but it's still unknown whether this is a technical necessity given our current level of knowledge (Einstein) or if it's really like that (Bohr).

Regarding Kolmogorov's (K) defintion, it essentially has to do with the efficiency of algorithms. If an algorithm can calculate/generate a sequence, it is not random unless the algorithm itself needs to be longer than sequence it's calculating. In that case, the algorithm is simply reproducing an arbitrary sequence, symbol by symbol plus a start and stop instruction. K defined such an arbitrary sequence as random. However the more recent view is that once a sequence is known, it is no longer random. Randomness inherently involves uncertainty. If the sequence is already fully embedded in the algorithm, there is no uncertainty. Therefore no algorithmically generated sequence can be random.

 

[mathman]

Math. According to Kolmogorov a random event is a subset of the set of elementary events. Why call such a subset as random event. What a reason?

Kolmogoroff was trying to give probability theory a firm mathematical foundation. His approach was to use the ideas of measure theory with the restriction that the total measure is 1.

 

[SW VandeCarr]

Math. According to Kolmogorov a random event is a subset of the set of elementary events. Why call such a subset as random event. What a reason?

The Kolmogorov (K) defintion of probabilty that mathman alluded to should not be confused with K randomness, which is what I'm talking about. A random variable is a function which maps a probability value from the probability space [0,1] to an event space. If you look up the definition of a random variable, you'll find that the concept of randomness itself is not defined.

K randomness does define randomness in terms of algorithms, but as I said, this definition is not consistent with current view of entropy and information. You might also look up the concept of "surprise" or "suprisal" as it applies to entropy/information measure. This is the definition that is in current use and is the probability of correctly guessing the outcome of one or a specified number of iterations of a stochastic process.

 

[Aleksey]

Because it is a description of our uncertainty; our inability to predict specific outcomes. It may be our ignorance, technical limitations or, in the case of some outcomes at the quantum level, an innate feature of nature at this scale. Quantum mechanics is formulated in terms of probabilities, but it's still unknown whether this is a technical necessity given our current level of knowledge (Einstein) or if it's really like that (Bohr).

Regarding Kolmogorov's (K) defintion, it essentially has to do with the efficiency of algorithms. If an algorithm can calculate/generate a sequence, it is not random unless the algorithm itself needs to be longer than sequence it's calculating. In that case, the algorithm is simply reproducing an arbitrary sequence, symbol by symbol plus a start and stop instruction. K defined such an arbitrary sequence as random. However the more recent view is that once a sequence is known, it is no longer random. Randomness inherently involves uncertainty. If the sequence is already fully embedded in the algorithm, there is no uncertainty. Therefore no algorithmically generated sequence can be random.

I think the same. There are no randomness but uncertainty. Thanks

 

[SW VandeCarr]

I think the same. There are no randomness but uncertainty. Thanks

Well, I didn't say exactly that. At the quantum level we simply don't know if outcomes are truly random. Current theory treats such outcomes as such and nuclear decay seems to be truly random as to the timing. This is the basis of our most reliably random generators. However algorithmically based "random" generators are considered pseudorandom at best. Uncertainty involves what we do not know or are not able to know due to the limits on the precision of measurement.

 

[Aleksey]

Well, I didn't say exactly that. At the quantum level we simply don't know if outcomes are truly random. Current theory treats such outcomes as such and nuclear decay seems to be truly random as to the timing. This is the basis of our most reliably random generators. However algorithmically based "random" generators are considered pseudorandom at best. Uncertainty involves what we do not know or are not able to know due to the limits on the precision of measurement.

So what is the difference between uncertainty and randomness?
I'm scared. Still, there are pseudo-random?

 

[SW VandeCarr]

So what is the difference between uncertainty and randomness?
I'm scared. Still, there are pseudo-random?

We can also be uncertain about processes for which we have no useful model. These processes do not have the appearance of randomness (defined as no discernible pattern) but they are not understood, such as human consciousness and other complex processes.

Pseudorandom simply refers to the output of random number generators which use algorithms. These outputs have the appearance of randomness as does the decimal expansion of pi.

 

[Aleksey]

We can also be uncertain about processes for which we have no useful model. These processes do not have the appearance of randomness (defined as no discernible pattern) but they are not understood, such as human consciousness and other complex processes.

Pseudorandom simply refers to the output of random number generators which use algorithms. These outputs have the appearance of randomness as does the decimal expansion of pi.

Well. But the distinctiveness depends on the point of view. This is a subjective concept.
It turns out that randomness does not exist?

 

[Phrak]

Let's put some meat on this.

1) Can a random function be defined in terms of elementary functions?

2) Can the value of a random function be defined without reference an idealized physical system such as an idealized roulette wheel?

3) Forget for a moment that Kolmogorov's 3 axioms have anything to do with probability. What functions satisfy the axioms?

 

[Aleksey]

Let's put some meat on this.

1) Can a random function be defined in terms of elementary functions?

2) Can the value of a random function be defined without reference an idealized physical system such as an idealized roulette wheel?

3) Forget for a moment that Kolmogorov's 3 axioms have anything to do with probability. What functions satisfy the axioms?

If I knew what means randomness, i would have answered your question

 

[axm7473]

I would have to agree that the concept of randomness is quite a difficult one to grasp. If I assume others are somewhat like myself on this topic, it would seem that the idea of "randomness" has an intuitive meaning, but where the difficulty arises is in the analysis of the subject.

To begin with, I would like to object to the proposition that a perception of pattern provides as a logical means of refuting randomness within a system. As suggested earlier, any perceived sequence of numbers can be shown to have more than one pattern. In fact, the basic conception of patterns existing independently of one's self identifies a perception independent of perspective.
To avoid a discussion on the philosophy of phenomenology, it seems that in analyzing the concept of randomness, structure must be interpreted in respect to the presence of some observer restricted to some perspective of the system.

By adopting this schemata , the naturally occurring patterns of a system can be interpreted as the resulting structure produced by and individual connecting familiar objects restricted to a perspective of the system. To put more simply, natural patterns can be interpreted as actually being algorithms produced by individuals that, by construction, coincide with the familiarities perceived through a perspective of the system.

For a simple example, (1,3,5) is a sequence of three numbers that obtains a pattern of increasing by 2 on each iteration of the sequence. I could use this structure to approximate the future values of the sequence, but this also accepts the notion that such an approximation may fail due to the synthetic nature of the structure. Instead of increasing by 2 where the sequence began at 1, the sequence could be simply all prime numbers greater than 2 with the exception of 1.

Other problems must be considered in addition to those I have initially discussed.

 

[SW VandeCarr]

I would have to agree that the concept of randomness is quite a difficult one to grasp. If I assume others are somewhat like myself on this topic, it would seem that the idea of "randomness" has an intuitive meaning, but where the difficulty arises is in the analysis of the subject.

To begin with, I would like to object to the proposition that a perception of pattern provides as a logical means of refuting randomness within a system.

To put more simply, natural patterns can be interpreted as actually being algorithms produced by individuals that, by construction, coincide with the familiarities perceived through a perspective of the system.

For a simple example, (1,3,5) is a sequence of three numbers that obtains a pattern of increasing by 2 on each iteration of the sequence. I could use this structure to approximate the future values of the sequence, but this also accepts the notion that such an approximation may fail due to the synthetic nature of the structure. Instead of increasing by 2 where the sequence began at 1, the sequence could be simply all prime numbers greater than 2 with the exception of 1.

Other problems must be considered in addition to those I have initially discussed.

I have already addressed the issue of "pattern" in defining randomess, While it is used in code breaking and other applications, it is not a rigorous defintion. To identify if a sequence is random you need to know how it is generated. I won't reiterate my previous posts on this thread and it's relation to probability or to my posts regarding entropy and information in another thread (see my response to doro's post "How can we prove a text is written by a human being?" in this forum; last post Dec 7)

The concept "surprise' as it relates to information and uncertainty generally has relagated algorithmically generated randomness to the more appropriate label of "pseudorandomness." At present, nuclear decay remains one accepted basis for random number generators. But a theoretical random number generator will generate sequences like 77777777with a known probability, and it's the assumption of randomness (apparent or true) that allows probabilistic reasoning to work.

 

[BigBugBuzz]

Well, I didn't say exactly that. At the quantum level we simply don't know if outcomes are truly random. Current theory treats such outcomes as such and nuclear decay seems to be truly random as to the timing. This is the basis of our most reliably random generators. However algorithmically based "random" generators are considered pseudorandom at best. Uncertainty involves what we do not know or are not able to know due to the limits on the precision of measurement.

Interesting. Does that measuring apparatus include our mind?

 

[JeremyHorne]

As to the comment by SW VandeCarr, I see that we may have a type theory problem, here. On one hand, we have an algorithm that generates a sequence, such as found in pi, and then the result, the exactness which we cannot predict. One might liken in a loose way the algorithm - division - to a metalanguage, and the result, and object language. This is why algorithms are regarded as pseudo-random numbers generators; the algorithm is known but the result isn't. A more reputable example of "randomness" - really the same as entropy in some contexts - is Brownian movement. However, we can use the metalanguage-object language application here, as well. We know how to set up a situation in which Brownian movement can occur, along with measurement schemes, but we surely cannot predict the exact outcome. Yet, stochastic processes in thermodynamics might assist us here.

As to philosophically whether there is randomness in the universe, I would vote "no" in terms of logical analysis. This is based on extended Cartesian subdivision until we reach Planck area, where we reach a binary world, albeit with quantum processes. (See Wheeler, etc. - it from bit, etc. - Gravitation) Whenever you have at least two of anything in the most basic universe - such as Planck area in terms of not Planck area - a dialectic, you have a relationship - something in terms of what it is not, and this is the most primitive form of order. That is, all it takes is two elements to establish order, and there is no randomness. It is our interpretation of that relationship - or our inability to give one - that muddies the waters. See http://home.earthlink.net/~jhorne18 for a further logical explication of this view in "The General Theory and Method of Binary Logical Operations".