octelcogopod
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I really enjoyed this post. The first thing that comes to mind (especially regarding the superset/subset question) is that it would seem logical that by abstracting reality into shapes these abstractions in the mind would in fact be a superset. If they would be a subset it implies to me that there is some aspect which will never be reachable and imaginable since they would either have to be equal to, or "above" reality to encompass reality, right?apeiron said:This ontological confusion is then compounded by the usual epistemological one - mistaking the map for the terrain. The desire is to deal with the substance~form dichotomy (materiality and its boundary states) by assigning reality and unreality to an epistemological division. So the world (being out there) is real, the maths (being in our heads) is unreal. Yet clearly this does not work because the maths is still really out there in some sense - as the boundaries, the limits, the constraints. The maths is more than just a potential fiction, a social construction due to restricted cognitive technologies.
The way I sort out this nest of confusion is first to accept the epistemological division (I think the "modelling relations" crowd in theoretical biology - Rosen, Pattee, Salthe - do the best job here). So nature, reality, is a constructed view. Both our notions about its materials and its laws, its substance and its form, are "in our heads" and justified by a modelling relation (so it is a process with its own purpose, its own needs, not some dispassionate god's eye view).
So reality, as far as we can know it, is our invention. That applies to our mathematical ideas about it, but also our "physical impressions" too. It is all a map.
However on the whole, it is a very good map - as it has developed within the self-refining tradition of metaphysical abduction, scientific induction and logico-mathematical deduction (Peirce's pragmatic triad!).
And then the ontological bit of the story. We can see that limits only actually exist in the sense that they are the boundaries to what exists. They are how far a process of development can go in some direction before asymptotically tending to a limit. So in fact they are the ontically unreal. The boundary remains always infinitesimally just beyond where reality can reach (so as to be able to be seen to enclose it fully).
Yet boundaries also have a real causality. At least if you are a process thinker, a systems science, you believe that there is such a thing as downward causation and even final cause. So forms can act as constraints that actually shape materiality. Maths exists "out there" as something real in the sense that there are forms (of the kind maths can describe) which have a causal role in the realm of the real.
I guess we have to ask the question then whether the set of forms that humans can imagine is a superset or a subset of those that reality can express. Sometimes it seems our inventions are more fertile - we can elaborate to create more imagined things than can actually exist. Other times, that maths is in fact quite impoverished. It is a pretty crude map of the terrain. More subtle things are going on than we have captured so far.
It could be the barrier is simply the fact that the abstracted models are not actually physical(in the normal physical definition), but in that sense the models and map could be identical to reality? In my mind it becomes a bit confusing after awhile I must admit.
. But from our human point of view, we can see the rules are arbitrary. Or "axiomatic" only in the sense that the rules must be such that we can derive pleasure or diversion from their existence.