Randomness as a function of the observer's knowledge

In summary, the conversation discussed the concept of randomness and how it is perceived by the observer. While there is no mathematical definition of a random sequence, there is a definition of a random process. However, this definition is not applicable in most real-life cases. The closest concept to randomness in a sequence is that of a normal sequence, such as the digits of pi. There is also a notion of randomness based on computability or Kolmogorov complexity, where a sequence is considered random if it is incompressible. Ultimately, the idea that randomness is a function of the observer's knowledge was also explored.
  • #1
N123
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I found a couple of closed threads related to the definition of randomness, but my question is slightly different.

Is there a mathematical way to express the fact that randomness is the eye of the beholder?
For example, if I give a sequence of 1000 numbers such that you cannot predict the 1001th number using your existing knowledge. So it is a random sequence as far as you are concerned.
Now I tell you that in fact those were the digits 3000...3999 in the decimal expansion of pi. Now suddenly, the sequence is not random.
(I understand that we may use an absolute definition saying the sequence was never random, but the fact is that for you it was, for a while.)
Since mankind's knowledge is finite and expanding, collectively we will come across sequences which used to be random and are not random anymore. How do you say "Sequence X is random if your amount of knowledge is Y"?
 
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  • #2
A sequence is never random, because there is no mathematical definition of a random sequence, or of randomness. There is a definition of a random process, every realization of which is a sequence. But that definition is of no use in most real-life cases, because we usually only can see one realization - eg the sequence of prices of Cisco stock on the NYSE at close of trading, ever since it was first listed. The other realisations live in alternative universes, if there are such things, and can never be known by us.
The closest random-ish concept that applies to a sequence is that of a normal sequence, aka normal number. The wiki article on normal numbers is quite good. The sequence of digits of pi is conjectured to be normal, but there is no proof that they are.
Regarding the broader question of whether all randomness is simply a function of the observer's knowledge (ie epistemological), I think a good case can be made that that is true. You may be interested in this note about that issue.
 
  • #3
Not entirely helpful, but there is a notion of randomness which is based on ideas of computability or Kolmogorov complexity. Roughly speaking, a sequence is random in this sense if it is incompressible. The decompressor of choice is a universal Turing machine.

This allows for the notion of randomness to be applied not just to distributions of sequences, but to sequences themselves.
 
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1. What is randomness as a function of the observer's knowledge?

Randomness as a function of the observer's knowledge refers to the concept that the perception of randomness can be influenced by the observer's understanding and interpretation of a situation. This means that what may seem random to one person may not appear random to another, depending on their knowledge and understanding of the underlying factors.

2. How does the observer's knowledge impact the perception of randomness?

The observer's knowledge can impact the perception of randomness in several ways. Firstly, their prior knowledge and experiences can lead them to see patterns or connections where none exist, making something seem less random. Additionally, the observer's understanding of probability and statistics can also affect their perception of randomness.

3. Can randomness be objectively measured?

No, randomness cannot be objectively measured as it is a subjective concept that is influenced by the observer's knowledge and understanding. While there are statistical methods to measure randomness, they are based on assumptions and can only provide a probabilistic measure.

4. How does the concept of randomness tie into scientific research?

The concept of randomness is crucial in scientific research as it helps us understand and analyze data. Randomness allows us to test hypotheses and make inferences about a population based on a sample. It also helps us identify patterns and relationships that may exist in the data.

5. Can randomness be controlled or manipulated?

No, randomness cannot be controlled or manipulated as it is inherently unpredictable. However, certain factors can influence the perception of randomness, such as the observer's knowledge and understanding, as well as the context in which the randomness is being observed.

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