Reading Mycielski’s elegant 1981 take on constructive nonstandard analysis, it struck me that we’re missing a category for “numbers on the fuzzy boundary between finite and infinite.”
Parafinity!
Constructive Nonstandard Analysis FTW
Constructive nonstandard analysis is one of those branches of math/science that seems like it should be a whole flourishing industry receiving substantial government funding, occupying dozens to hundreds of PhD students, and serving as the foundation of a whole bunch of applied science and engineering.
Instead it’s a super-obscure little corner that almost nobody has heard of.
To simplify just a little (errr…), constructive nonstandard analysis lets one do calculus and other forms of math that normally require infinity by creating “nonstandard numbers” that are not infinite but just “so super big they are infinite for all practical purposes.” So given a specific number like 1000, one can build a potential-infinity inf_1000 that is so big it can’t be touched by anything you could build from tractable combinations of 1000 and its subsets, nor from inf_999 or below.
Taking say 1/ inf_1000 gives one insanely small “potential infintesimals” which one can use in place of the “dx” and such in ordinary differential and integral calculus.
Overall, after you sort through the model theory and formal logic, it turns out you can do everything you need to do in calculus and other continuous math without any actual infinities, just with some “very very big” and “very very small” potential infinities — and it all comes out pretty elegantly, without the messy error terms and such one gets from discretizing calculus math the ways one does in, say, numerical analysis.
Once you get used to it, this sort of approach to analysis math makes the conventional approach seem weirdly bizarre and excessive — like, why introduce the continuum with all those confusing semi-incoherent uncomputable numbers, the Axiom of Choice and yadda yadda, when one can actually do all the practical stuff elegantly with simple assumptions about “super big numbers” aka “potential infinities.”
The only complexity in the constructive nonstandard approach is that, instead of looking at objectively defined infinities and infinitesimals, one has to fix a finite reference point and say “too big for anything of size 1000 to understand” or “too small for anything of size googolplex to understand”, etc.
But in this light the typical infinity which is "Way way too big for any system at all too understand" seems a bit absurdly objectivist and excessive -- i.e. how can we ever empirically distinguish "way too big for me to understand" from "way too big for ANYONE to understand" and why would we want to reflexively assume the latter esp. when it leads to wacky conundrums.
Parafinity Awakens
So… while working through the math embodying these ideas, what I started wondering about is the logic of "Just barely too big for me to understand" and "So big I can only slightly sorta understand it"
And conversely, "So small I can just barely perceive it"
It feels to me that this fringe of conception/perception may important for the process of self-transcendence in open-ended cognitive systems. It’s by sensing things on this fringe that we know what direction to move in, so that after we move some of what was barely comprehensible/perceptible becomes more fully so...
What I’d like is to say something like: The number x is parafinite to system A, if in a chosen paraconsistent logic it is both true and false that x is bigger than anything A can construct.
One can generalize this idea beyond numbers as well.
Let’s say: X is paraconstructable using axioms A if there is significant evidence supporting X being constructible via applying A, but also significant evidence otherwise (e.g. perhaps there is a partial construction, and some similar constructions have succeeded and some have failed (giving ambiguous probabilistic evidence) … we can represent this situation paraconsistently using e.g. Constructible Duality logic.
Parafinite numbers emerge as a special case: Given a finite number c, pinf_c is p-parafinite if there is evidence probabilistically supporting the notion that pinf_c is p-infinite and also evidence probabilistically supporting the notion that pinf_c is finite
The Creator in the Simulation Hypothesis is parafinite.
What happens when one uses paraconsistent logic to derive analogues of differential and integral calculus using parafinite numbers in place of constructive-nonstandard infinities and infinitesimals… for now I will leave to the reader as an exercise! (I did some back-of-the-envelope scribblings but … well … I lost the envelope…)
It seems to me that you're assuming a Platonist mathematics, in which the fact that we use the word "infinity" implies there is an actual Infinity that "exists" out there somewhere. This is mistaken. Infinity is a concept used in analysis. Think of it not as a number, but as a symbol. Real analysis is a logic, not arithmetic; and the term "infinity" in it doesn't need to fit into the category of "number" or "operator"; it just needs to be defined in a way that doesn't render the logic inconsistent. When you then apply the theorems you proved in analysis on values which include "infinity" terms, "infinity" there isn't an atomic symbol with a referent out there somewhere; it's more like a macro, which would expand into the entire body of constraints imposed by its use according to the logic of analysis.
At least, that sounds more plausible to me.