Comment by pilgrim0
2 years ago
Correct me if I’m hallucinating, anybody. Doesn’t a foundation have to change and adapt in relation to what is grounded in it? If the foundation is to be all encompassing, to abstract enough to allow and absolutely account for every possible superstructure that it may ever ground, would not the foundation itself be contentless, inexpressive? “Silence contains all sound”, and “Every moment has an infinite amount of time” are constructs that came to my mind as isomorphic to the questioning. As mathematical structures gets finer and more intricate, purely on the basis of necessity (survival of the mathematician, mankind and mathematics itself) and experimentation, so their foundations must undergo at least some change? Even though such changes are purely virtual, and of interpretative nature? The same ideology, but “taken as” something else, thus, ultimately, subjective? There also seems to emerge an issue of ordering, of a demand for primality and derivation, which screams paradox. Could all of mathematics fit within itself, being itself employed to bootstrap its own, with no second system? A sort of ontological auto-genesis? Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class? Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?
In mathematics, the roof holds up the building, not the foundation. Since humans use mathematics a lot, we design foundations to our specific needs. It is not the building we are worried about, we just want better foundations to create better tools.
Not only are we going to treat mathematics as subjective, but also having formal theories that reason about different notions of subjectivity. https://crates.io/crates/joker_calculus
> Could our conception of paradox be itself primal, and perhaps, in some plane, could it be something ranking higher, of first-class?
Yes! Paradoxes are statements of the form `false^a` in exponential propositions. https://crates.io/crates/hooo
> Also, I’ve been thinking, recently, on the role of time in structures. There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?
Path semantical quality models this relation, where you have different "moments" in time which each are spaces for normal logical reasoning. Between these moments, there are ways to propagate quality, which is a partial equivalence. https://github.com/advancedresearch/path_semantics
Doesn’t a foundation have to change and adapt in relation to what is grounded in it?
No. An infinite number of possible papers about number theory could be written without having to change ZFC.
There can’t possibly be any structure whatsoever without time, or, more concretely, at least the memory of events, recollecting distinctive and contrasting entropic signatures. So, mathematics manifesting as, of, and for structure, wouldn’t it require, first and foremost, a treatment from physics? Regular or meta?
ZFC stands as a counterexample.
Building on ZFC, we can build mathematical structures that represent the universe with time. Not the other way around.