It is strange that few have noticed that it can be strongly argued that the abstract concept of computability and its allied notions are a candidate for being part of the pure a priori necessary concepts for all our cognition and experience (Husserl seems to have anticipated some recursion theory in his Philosophy of Arithmetic).
We have the intimately connected triad of formal logic, arithmetic-combinatorics and computability theory. To write and check a formal proof we already are deploying computability concepts. But to investigate computability notions we need formal logic and arithmetic. Computation, proof and the sequence of the natural numbers share the ordered directed time-like quality (linear with branching possibilities). Note: we are not suggesting that computability exhausts human cognition. Also by computability we include all classes in the arithmetical and analytic hierarchies, etc. In a future post we will critique the denigratory use of the term 'mechanistic' showing it does not hold water when confronted by a serious mathematical and philosophical analysis of the use of differential equations in science. Computability theory seems very close to Kant's notion of rule and of an architectonic of reason. Church's thesis is a transcendental condition for the possibility of knowledge.
Computability has to do with prescriptive normativity (method) rather than mere general normativity (rules).
We wonder if Kant's realm of pure synthetic a priori intuition of space does not actually correspond to graph theory and combinatorics - and whether category theory, and specially higher category theory are not best viewed from this perspective (cf. simplicial sets and cubical sets). Category theoretic diagrams have a a kind of dynamic nature - at least in the way they are commonly used and visualized - which recall Kant's schematism.
See also:
https://chryssipus.blogspot.com/2023/11/the-church-turing-thesis-kripke-and-kant.html
https://chryssipus.blogspot.com/2024/01/algorithms-and-numbers.html
No comments:
Post a Comment