Information about a continuum

I experience a universe via information-caputure and -processing systems in which the information is discretely encoded. It's not just the machines that are so; my brains neural structure operates in waays that convey information by discrete transitions of various kinds; there may be a continuum element to these transitions (their timing, at least) but there is also a discrete element to it. Our artificial information-processing systems are designed to work in ways that an entirely discrete model fully and faithfully describes (although the arrival of quantum computing might change that, when it comes); our brains can probably be well-approximated by a simulation running on those, albeit possibly requiring vastly more processing power than would fit inside a human skull, to do the simulation accurately enough to fully and faithfully model a human mind. Even without such a model of my mind's hardware, I think my thouhgts and communicate with others in discretised forms (in languages made of words and symbols, or in discrete concepts within my mind); so my information about the world has discrete elements in its nature.

It remains that the models I have to describe the universe all, also, contain some element of the continuous. Measurements that we use to establish the continuum parameters in our models will have discrete aspects and will get us finite amounts of information. Our models do allow us to chose representation (units, co-ordinates) such that (at least many of) the quantities in our system, and the ratios between these, are expressible finitely – in contrast with the infinity needed for precise expression of almost any actual continuum variable. In so doing, we factor out some of the continuum nature from a model, reducing more of it to discrete forms. Even so, the model's meaning is most clearly understood in terms of the continuum form of expression, with the factoring out then being expressed in terms of that model to enable us to map between the model's dscription and how we use our results of experiments.

Our models (that involve continua) need the means to describe discrete processes and need to enable us to encode, finitely, the (finite) information we have from experience; and our ability to extrapolate from knowledge of the present to information about the future requires the model to be capable of describing causal relations between information about one part of a system (our past and present) and information about other parts (those whose nature we infer from the information we have). By virtue of its continuum aspects, the model is apt to have a diversity of choices of its continuum parameters for which the model would behave in ways that match our finite amount of information; each bit of our data only halves the range of options open to the continuum model, which has a continuum infinity of options; a finite number of halvings still leaves a continuum infinity open, though we may be able to encode some of the continuity of variation in terms of equivalences derived from the model's encoding of how we make measurements. (As example, consider the phase variation in QM's state vectors; it's non-observable, although relative phase between two sub-systems may be observable.)

We thus select models that describe discrete information about a continuum model in a continuum-diversity of states, all sharing some discrete properties (expressing our experience). The model encodes those properties, like the states themselves and our information about them, in terms of model parameters and relations among their values.

At least in so far as the universe is random (or our model characterises any of its behaviour as random), our model needs to have some way of representing the relative probability of diverse outcomes from any given initial state. In so far as our information comes predominantly from the past (rather than the present), its contribution to our knowedge of present state comes via our model's application of its causal reasoning to past-derived state information, to obtain information about present and future state; the model's causal process includes any randomness involved, incorporating randomness into the past's contribution to our prior knowledge of the present, which we filter with present information to deliver our information about future state.

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