Outline
Guide
Here is an outline structure (work in progress):
Concepts in the incarnation of a programming language
Assume knowledge of integers $-2,-1,0,1,2,3$ with binary operations $+,-,*$.
More on that languages syntax and examples (which are instances of relevant concepts to be discussed later)
Computation and logic (presented bottom-up)
Comment on syntax
Type theories: Simple type theory $\to$ Polymorphic type theory $\to$ Dependent type theory
Formal logic: Intuitionist logic $\to$ First order logic (here, at the latest, it might be good to introduce the non-computing language, i.e. go away from the programming language and use a formal logic, as is traditional, in a descriptive fashion)
Comment on curry-Howard correspondence
Axiomatization of category theory within dependent type theory, plus comment on first order axiomatization
Axiomatization of set theory within first order logic
Informal discussion of the very first constructions in those frameworks, respectively
Comment on how cats and set can be used to study type theory and logic itself
Mathematical core (presented top-down)
Formal definitions from the main body of this wiki. The Set entries can be thought of as defining sets which ZFC shows to be non-empty. Some category theoretical definitions use sets (e.g. when setting up a category of sets, for which we wanted to add a large cardinal axiom), but conversely the set entries are independent of those (but of course a notion of categories can and is defined within ZFC).
Physics
Eventually, make a list of mutually properly distinct physical frameworks - i.e. mathematical theories used in physics, usually coming with some state space and dynamics, see Perspective.
In the latter sense, they must be exhaustively specified,
of course, together with a concise list of their respective axioms.
Some frameworks contain black boxes that are further given insight into in more detailed theories.
Here, when it comes to physics, I tend to like to have the rougher theories first.
Now one can reference real life systems and associate mathematical models in this and that framework with them.
Develop them like this:
todo: examples of experiments
| … phenomenological thermodynamics |
todo: examples of experiments
(Phenomenological thermodynamics leads to a lot of basic chemistry)
todo: examples of experiments
| electromagnetic field theory |
Entries
Logical prerequisites
| Meta |
| On syntax |
| Symbols |
| Types, terms and programming |
| up to dependent type theory (?MLTT, etc.) |
| type of natural numbers |
| type of categories |
| informal Curry-Howard correspondence |
| Foundational temp1 |
| Formal logic |
| Intuitionistic logic (independent of type theory) |
| First order logic (i.e. adding LEM to Intuitionistic logic) |
| Axioms for set theory (e.g. GT-set-theory, and then U= type of sets) |
Mathematical core
todo: the following line of “apple pie from scratch” entries have some gaps
I have neither been able to find a good n-category basis instead of the 1-category approach I take here
nor have I formalized “up to iso” anywhere (this point is relate to the one above)
References
Sequel of
