## An apple pie from scratch

 An apple pie from scratch $\blacktriangleright$ Outline

### Guide

$$\require{AMScd} \begin{CD} {\large\hbar} @>{\large{!}}>> {\large{*}} \\ @V{{\large{m}}}VV @VV{{\large\top}}V \\ {\large\heartsuit} @>>{\large{\chi}}> {\large{\Omega}} \end{CD}$$

#### What?

As a working physicist, one regularly learns new mathematical structures and then doesn't know how they fit into the bigger picture or how it's related/generalization/special case of things they know. The axiomsofchoice.org graph aims are sorting things out and already provides the definitions.

Now this An apple pie from scratch monograph is a linear “that's how it all comes together” documenting the emerging hierarchy of mathematical structures. In contrast to individual entries, the text can reflect on interplay and on the “philosophy behind mathematical constructions”. It ranges from propositional logic to statistical physics and field theory. On the way, with reference to the formal definition, it also introduces and reflects colloquial physics/math language and (hopefully) can thereby aid communication.

#### How?

• presentation of mathematical concepts in a top-down way:

The focus is on how “mathematical data” can be set up in set theory and I also use category theory for “structural characterizations”.

(As foundational paradigms, both set- and category theory have strong merits and disadvantages. While it's possible to develop a version of categories, functors and natural transformations in set theory and a version of sets in category theory, I rather use both paradigms early on. As a side note, it's also possible to axiomatize categories in logic directly and also to specify sets in type theory, but that's pretty nonstandard and more difficult. In discussion sections, I will often use categorcal language to to characterize sets a the category of sets.)

• Be self-contained

I start out with an more or less informal explanation of formal languages and foundational theories and common axioms. The structure of the presentation is summarized below. The primitive notions in this wiki, discussed in the first block, are summarized in Domain of discourse.

• Use a “two models” principle:

For each abstract definition, I'll try to exemplify it with two different structures.

• Convey information in different visual language languages:

- proper cat'ish drawing (see diagram at the top of the page)

- drawing where you see terms, circled (the type universe is circled as well)

- typey syntax

- FOL/SOL