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path_semantics
A research project in path semantics, a re-interpretation of functions for expressing mathematics
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https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... :
> * Today, Zermelo–Fraenkel set theory [ZFC], with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common* foundation of mathematics.
Foundation of mathematics: https://en.wikipedia.org/wiki/Foundations_of_mathematics
Implementation of mathematics in set theory:
> The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969 (here understood to include at least axioms of Infinity and Choice).
> What is said here applies also to two families of set theories: on the one hand, a range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like
IEEE-754 specifies that float64s have ±infinity and specify ZeroDivisionError. Symbolic CAS with MPFR needn't be limited to float64s.
HoTT in CoQ: Coq-HoTT: https://github.com/HoTT/Coq-HoTT
leanprover-community/mathlib4//
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