Fexpr (John Shutt)

> Number theory/abstract algebra, which is the study of natural numbers among other things, is a form of set theory.

Let S be the set {a,b} and let T be the set {c,d}. There are two bijections S -> T between these two sets {(a,c),(b,d)} and {(a,d), (b,c)}. These two sets are isomorphic to one another because they have the same cardinality.

The isomorphism classes of sets are cardinalities which are themselves the natural numbers 0,1,2,3,... and infinity. In combinatorics we can just focus on the study of these isomorphism classes rather then sets themselves.

We can enumerate binary relations: ([],[0],[1],[[0 0],[0 0]],[[0 0],[0 1]],[[0 0],[1 0]],[[0 0],[1 1]],[[0 1],[0 0]],[[0 1],[0 1]],[[0 1],[1 0]],[[0 1],[1 1]]). Logical disjunction [[0,1],[1 1]] is equal to 10 in this enumeration.

Using the enumeration of binary relations the entire field of study of graph theory can be described in terms of natural numbers without any need for sets. As a result, graph theory is considered to be a branch of combinatorics. We only need to have sets mathematically when dealing with cardinalities like aleph one that emerge from uncountable sets.

Since analytic number theory deals with the uncountable set of real numbers and abstract algebra deals with operations on sets including the real numbers, these two fields are indeed founded on set theory. Algebraic combinatorics and combinatorial number theory are the branches of these two fields that focus on countable sets.

> And set theory…

I'm reminded of the Zen koan where the students cannot tell whether the flag is moving in the wind or the wind is moving around the flag. The master tells them that the mind, neither the flag nor the wind, is what is moving.

Both sides of the interpretative coin are needed. We must simultaneously understand that QM is epistemic and also that QM is contextual. The author asks,

> How do probabilities get into QM?

Because knowledge of physical systems is always limited and QM is epistemic, all quantities we manipulate in QM are probabilistic. Further, the Kochen-Specker Theorem requires that some measurements be not fully determined by past history, as a matter of living in at least three dimensions of space. Finally, basic linear logic allows us to replicate most of QM's effects with macrostates.

While Schrödinger hated this and used his famous cat thought-experiment to try to refute it, children learn about linear logic and probabilities in macrostates when they are given random toys or packs of trading cards, and today we recognize that it really is possible to condition relatively large differences in macrostate observations on single entangled or otherwise-prepared microstates.

> In the [Everett] realist approach the history of the world is endlessly splitting;

Eh, kind of? But it's also endlessly joining. We wouldn't just be in a "cosmic history" which is "sufficiently benign" for us, but also we're likely to be in a likely history. In fact, we're exactly as likely…

Indeed. I think this something that people are often confused about, e.g. one blogger[0] comments:

> The key to understanding the fixpoint theorem, for me anyway, was realizing that when it says FG=G, it does not mean that for every F there is a G such that by calling F with parameter G one can get back G. I mention this in the blog post, above. The equality operator in FG=G is the equivalence relation induced by the calculus rewriting relation: that is, it's the symmetric reflexive transitive closure of the rewriting step relation. When you look at the proof of the theorem, it actually works by constructing an expression G such that evaluating G produces, as an intermediate result, FG. There is no need for F to even be a function, and if F is a function it doesn't matter what, if anything, F would actually do when called, because the proof of the theorem doesn't involve calling F.

I think the name "fixpoint combinator" is kindof bad, it it was called e.g. the "recursion combinator" I think people would find it more intuitive.

[0] https://fexpr.blogspot.com/2013/07/bypassing-no-go-theorems.html?showComment=1385649732798#c4614180298503254427