# Axiomatics

## On the Possibility of Basic Concepts

### TODO Excerpt from Blanché

### What is “Basic”

#### vs. “Foundations”

### Categories

Ala Kant and Aristotle

### E.g., some potential candidates

- togetherness
- belonging
- collection
- symmetry
- self
- other
- alterity
- relatedness
- space
- time
- order
- conjunction
- disjunction
- inference
- negation
- affirmation
- being
- nothingness

### Problematization

Given some set of concepts, how do we determine which ones are *basic*?

#### Suspected solution

Constructivist Inferrentialism

##### TODO Excerpt from Paul Hertz’s *Axiom Systems*

Also elaborations from http://www.academia.edu/2313465/Paul_Hertz_and_the_Origins_of_Structural_Reasoning

##### TODO Excerpt from Martin-Löf’s *Constructive Mathematics and Computer Programming*

#### Non-foundationalism

This is not a project of foundations: we needn’t claim that our basic
concepts are **the** basic concepts; they needn’t transcendent force. This is
an exercise in *axiomatics*: we are searching for a minimal set of concepts
which suffices for the construction of all others (that we happen to want to
make use of).

#### E.g.,

##### Sample concepts under concern

- togetherness
- relation
- conjunction
- symmetry
- sameness

##### Genealogy

I suspect there is no single correct constructive genealogy, but there is at least one (the flat genealogy where all concepts are root ancestors). And we can guess at something like this:

- relation
- sameness
- symmetry

- togetherness
- conjunction

- sameness

### re: Type Theory

As Martin-Löf says explicitly, ITT is explicitly following in the Kantian tradition of trying to deduce categories of the understanding from judgments.

Our aim is to extend this beyond judgment/cognition, to other forms of thought such as inquiry, instruction, speculation, and poetization.