A few quick comments about **Path Deformation**, **Univalent Foundations**, and **Connection Theory**.

Various theories have taken **Category-Theoretic** notions as the basis for formalizing concepts from abstract algebra such as *Fields*, *Groups*, and *Rings*. Such formalizations place emphasis on *geometric* and *topological* notions rather than traditional *symbolic logic*.

Foundational concepts like

proof,interpretations,wff's, andmodelsnaturally arise in such systems (but without explicit recourse toFirst Order Predicate Calculus).

One concept central to this approach is *path deformation* which aligns with commonsense intuitions about modifying networks, paths, or connections between topological items.

Modifying one's route on a GPS map, for example, involves redrawing the representation of route path which may involve explicit use of this concept. Another example can be found in modifying the walls of a sand-castle. It's the mathematical property of that phenomena.

Univalent Foundations takes a specific set of findings from the considerations above to shore up certain techniques and moves made in mathematical justification and proof.

Namely, the equivalence of *identity* and *isomorphism* of mathematical structures (which has always been assumed but with little mathematical proof to undergird all such phenomena).

Read: https://www.andrew.cmu.edu/user/awodey/preprints/uapl.pdf (especially, the comments onPage 6aboutpath deformation)

Connection Theory has several core objectives:

- To formalize diagramming used in mathematics (
*representation theorem*) and mathematical proof (*proof-theoretic*). - In doing so, to address one of the several "abuses of math" identified by Steve Awodey above CMU - Departments of Philosophy, Computer Science, and Mathematics.
- Note too the tension between
*topological*diagrams of concepts like*path deformation*and the ZFC set-theoretic notion (which is language-symbolic). Here, an explicit formalization of those*topological*concepts is given*and*which is itself a standalone mathematical theory and language. - It unites those formalized diagrams with the (actually) represented mathematical objects in one language (since it can construct them).
- In this way, it's a potentially semantically-closed language system (or at least significant sub-fragments potentially are).
- The mathematical objects can be expressed via operations of Thinking Notation and not just to say some diagram that just so happens to be lexicographically isomorphic to ZFC.
- It closes the barrier between language and mathematical object since the entities that are traditionally symbols of the language are also constructed from the same system as the objects they are the names of (in line with my stated philosophical objective regarding
*Pythagoreanism*). - In other words, the distinction between
*object languages*and*meta-languages*may blur or collapse. - It also aligns with familiar concepts that are independently of great interest including:
**Homotopy***path deformation*and*retracts*,*internal logic*(**Category Theory**), etc. It also aligns with*Inner Semantics*since the language can define its own semantics procedurally.

##### post: 1/31/2021