I see in literature very different descriptions of what is a deterministic system such as:
"... a system in which no randomness is involved in the development of future states of the system...>>>"
I have this clear but rather broad question that might be answered by different opinions and view points. However, my question is really not targeting an intuitive or phiplosophical answer, and I beg you for view points with a strength of mathematical foundation:
May one explain based on a formal mathematical description, what needs to be exactly fulfilled in order to justify that an infinite sequence of positive integers $\langle x_i \rangle$, along its forward progression (that means vs. increasing $i$), is purely deterministic with no randomness involved?
$i \in \Bbb N$ and $x \in \Bbb N$.
A posterior
Following the first comments, let me carefully conjecture the following that might be correct and possibly helping to a rather concrete and qualified answer to the above question by the community:
- Any such sequence (abovementioned) can be described in some way by some probability distribution that involves randomness. Here probability means the probability that certain integer occurs at a certain location $i$ of the sequence.
- Taking such probability distribution into account, any such sequence can be described by either a linear or non-linear master equation, that can represent the stochastic model for that sequence.
- This stochastic model of such sequence encapsulates substantial information about the interaction (association) of the elements $x_i$ of the sequence (along progression) as well as the memory ($x_{i-1},x_{i-2},x_{i-3},\dots$) that is required for predicting $x_{i+1},x_{i+2},x_{i+3},\dots$ (determinism)
- If the first, second and higher moments and their interactions would allow, there might be a possibility to connect the stochastic model (that includes randomness) to a deterministic model.
- While any such sequence can be described in some way by some probability distribution that involves randomness, not every such sequence must have a deterministic description and
- dependent on the moments and the way they interact, it is not always possible to derive from the stochastic model a deterministic model.
"Stochastic" or "probabilistic" or "random" in this context means there is a probability distribution assigned to it. The probability that the sequence has thus-and-such behavior is this number.
"Deterministic" simply means that it does not.
The word "random", like the word "infinity" actually has a variety of different uses in mathematics, the relationship between which is often not defined in a logically rigorous way. For example, Kolmogorov-Chaitin randomness is a different thing from what I write about above.