avatar_hypergraph_rewriting VS path_semantics

Also, symbolic distinction might play a role in understanding consciousness in relation to Wolfram models. If the hypothesis and underlying assumptions are true, then this could be a major scientific breakthrough. Wolfram models correspond to theorem proving with IPL, so there might be a corresponding interpretation of physics associated with PSI. See https://github.com/advancedresearch/avatar_hypergraph_rewriting for more information. Here is the paper about the hypothesis: https://github.com/advancedresearch/path_semantics/blob/master/papers-wip2/the-symbolic-distinction-hypothesis-of-consciousness.pdf

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

In topology, there is something called "point free theorem proving" style, which is almost usable in Coq or other provers like Lean, or even Cubical Type Theory. The problem is composition of normal paths. In e.g. Coq, you need a solution before you can compose. In order for normal paths to be completely painless, one use an "imaginary inverse" (paper). It is relatively easy to build your own language using an imaginary inverse to do point free theorem proving, but it also turns out that many proofs don't need types. However, adding full point free theorem proving to dependent types makes type checking undecidable.

Here is a transcript of the talk I held for the Category Theory Study Group on Applied Category Theory Server: https://github.com/advancedresearch/path_semantics/blob/master/lectures/introduction-to-path-semantics/transcript-of-introduction-to-path-semantics.pdf

Also, symbolic distinction might play a role in understanding consciousness in relation to Wolfram models. If the hypothesis and underlying assumptions are true, then this could be a major scientific breakthrough. Wolfram models correspond to theorem proving with IPL, so there might be a corresponding interpretation of physics associated with PSI. See https://github.com/advancedresearch/avatar_hypergraph_rewriting for more information. Here is the paper about the hypothesis: https://github.com/advancedresearch/path_semantics/blob/master/papers-wip2/the-symbolic-distinction-hypothesis-of-consciousness.pdf

I had a similar feeling and recently came across a project that feels like it’s closer to the core as you suggest: https://github.com/advancedresearch/path_semantics