Re: Gödel’s Lost Letter And P=NPMounting or Solving Open Problems : Programs Not Programs

The following primer on differential logic uses the cactus graph syntax to represent propositions (boolean functions f : \mathbb{B}^n \to \mathbb{B}) and the operators on propositions that arise in developing the subject of differential propositional calculus.

The cactus graph syntax for propositional calculus is based on minimal negation operators.

Re: Language LoreThe Pragmatistic Force of Analogy in Language Structure

Some work I started on Aristotle, Peirce, Inquiry, Analogy, etc.

Re: Azimuth • Quantropy (Part 3)

Another approach to discrete dynamics is by way of differential logic, where we take the pragmatic or topological attitude of ignoring all the differences that don’t make a difference.

If p is a proposition, a boolean variable, that characterizes — holds true of and only of the points in — a region P of the relevant state space X, then \mathrm{d}p is a proposition that characterizes the equivalence class of paths that cross the boundary of P in either direction in the relevant interval of time.

The crux of the matter is getting the proper definition of a tangent functor. There are a few expositions at these places:

Re: Gyre & Gimble • Abstract Objects

The transformation on logical relations that C.S. Peirce described as “hypostatic abstraction” is key to understanding the realm of abstract objects, in particular, mathematical objects. It is an example of a reflective operation, one that links a relation of a given arity to a relation of the next higher arity, and it appears to be involved in many, if not all, acts of reflective practice, formally, if not always consciously.

Here is a blog post I wrote on the subject.

Re: Gyre & Gimble • Representations

The concept of representation in general is one of the things that Charles Sanders Peirce developed his theory of triadic sign relations to cover, borrowing the name “semiotics” (in various spellings) from Locke and reconstituting a number of earlier traditions within the framework of mathematical relation theory.

Our usual notions of representation in mathematics are a special case of what he had in mind, since they refer to functions and functions are dyadic relations. They fall very roughly into the category of analogical or iconic representations.

I’m not really sure these comments are getting through, so I’ll break off here and try to post a few links later.

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