Information space in mathematics
From Wikinfo
Information space, in mathematics, is the formal concept of an extension of variation.
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Time as pure novelty
To state this more precisely, one must first accept "time" as the emergence of novel variation a.k.a. "noise" a.k.a. information production. That is, time is the contradiction of the 2nd law of thermodynamics; the going away from what was priorly percieved as "equilibrium". The "rate" of time is thus proportional to the rate of production of new information.
Space as a synchronous continuum of variation
In the study of dynamical system, one often employs the concept of phase space. When two or more dynamical systems are examined together, their phase space can be combined into a single phase space, with dimension the sum of the dimensions of the phase spaces combined into one. However, to combine in this way an uncountable number of dynamical systems would result in an uncountable-dimensional phase space. Alternatively, each dynamical system's phase space can be distinguished, such that there are an uncountable set of phase spaces. Insofar as the state of each dynamical system has a non-infinitesimal amount of unique information, this set of phase spaces is an extension of variation, with information dimension at least that of the uncountable set.
Examples of information spaces
- Any system with a countable set of states, such as a digital computer, has a zero-dimensional information space.
- Dynamical systems defined by a finite set of partial differential equations have at most a one-dimensional information space.
- For each orthogonal spatial extension that cannot be removed without losing information about the system (i.e. is irreducible), the system has a dimension of information.
- Thus, for example, a finite continuous 3 dimensional space, where each point in that space is governed by a set of differential reaction-diffusion equations, including a noise term, has an information dimension of 4.
