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.