Salaam to Dr. Choudhary and David and everyone
Axioms are used with a set of systemic objects e.g. predicates Booleans and inferences to compose larger and larger statements from smaller atomic finite set of Axioms, the latter as smallest atomic unit of truth. And then focus on examining if these larger and more complex statements are true or false.
Alfred North Whitehead studied the history of European philosophy, its achievements and failures, and endeavoured to replace the concept of axioms with something else, that did not require to be true or false.
Example in our times is the axiomatized theory of Evolution. It is now known the processes of evolution shaped life, therefore the theory of Evolution is proved to be correct. However it is also well known that not all processes of life were evolved, and there is a problem! The latter means theory of Evolution is incorrect and therefore false i.e. disproved!
Then the proponents of theory of Evolution argue by all means possible, at times on thin ice, to refute the lack of evolved processes in life, to make sure the axiomatic theory of Evolution is 100% intact insofar as 100% truth, thus we plunge into classical European failure of philosophical thought which we all know as Dark Ages.
Categoreal Scheme does not require atomic statements i.e. axioms to be either true or false.
Categoreal Scheme provides a blueprint, for some entity, to be described in most general possible forms and ways.
An actual entity is then the instantiation of a particular category, which describes quite well to some gradation of competence.
The thinker of Categoreal Scheme ‘restricts‘ the thoughts to the encapsulated blueprint within the category and investigates which actual entities fit the blueprint of the category and which do not, as opposed to verify truth and veracity of statements.
Restriction of thoughts are then accompanied by imaginative leap which are something other than thoughts and rational, and thus imagine to expand the boundaries of mentation limited by the restriction of thoughts to the blueprint of the category.
Example: Category of Car:
ii. Contains the energy for motion
iii. Has cavity and openings (doors) to carry load
iv. Roves on particular devices pressing against the ground (wheels)
Car is not true or false! Its blueprints are not concerned about truth.
However if we restrict our thoughts to this Category of Car we quickly notice that there are many instantiations of this category e.g. GMC truck, BMW and so on.
And yet we also discover that there are entities that do not exactly instatiate the category of car e.g. motorcycle or sail-boat. We need to find those misfits other Categoreal Schemes. Motorcycle or sail-boat does not make Category of Car false! BMW does not make the Category of Car true!
Example: Category of Evolution:
i. Every organism has a predecessor
ii. Every organism mutates to a new version slightly different
iii. Every organism mutates innovations due to environmental pressures
iv. Every organism loses innovations of iii if no longer required
v. ii – iv are gradual
We restrict our thoughts to viruses and clearly see them as instantiation of Category of Evolution. There is no truth or falsehood about the Category itself or about the virus, it is about instantiation and fitting the blueprint.
We loosen up the restrictions of our thoughts to plants and birds and see clearly that that they are instantiation of Category of Evolution.
However there is no truth or falsehood about any of it, God included. Category of Evolution does not make Category of God false, and vice-a-versa.
Then we restrict our thoughts to flowering plants and we find out that flowers do not fit the blueprint of the Category of Evolution!
This lack of instantiation does not have any truth or falsehood about the Category of Evolution. Perhaps flowering plants require a new Categoreal Scheme to fit better blueprints.
Imagination: Categoreal Scheme is free of meaning and application if the restriction of thoughts to its blueprint is not billowed at by the torrents of imagination! It is the very tumults of an imaginative mind that fuels the meaning and progress of Categoreal Scheme.
Alfred North Whitehead, may his voyage be eased and beautiful, innovated/promoted Categoreal Scheme to do away with absurd fallacies and even more absurd proofs! Hence Dark Ages which is upon us in so called modern times.