Philosophy of concepts - can it be (gradually) expressed in type theory?

Question

Reasoning in mathematics is simple and subject to automation and discipline/system, because every concept (e.g. integer number, real number, derivative, integral, differential equation and its solution, etc.) can be expressed using some very small set of simple notions. If one considers the type theory approach to the fundamentals of mathematics, then there are only two basic types (entity and Boolean-truth) and all the other types, all the other notions and concepts are formed from those two simple types. Reasoning in mathematics is systematic because we completely know the content of the every concept. Yes, sometimes we imagine some new concepts (poetics of math) but even in such cases we manage to write down those concepts (or approximations of them) into the other concepts that can be traced to the first principles. Concepts in the mathematics are formed (or at least - can be expressed) in the bottom-up manner.

Reasoning about physics, about real world (ontology, metaphysics, nature, social world, humanities, emotions, mind, etc.) is very hard, because we can only make guesses about the eventual concepts, about the connections with other concepts and we don't know the full content of the concept, every research discover new shades of some concept, concepts are created, merged etc. And all this happens in non-rigorous manner, because we don't know the complete content of the concepts expressed in the first principles. We even don't know the first principles that can be used for the real world.

Semantics of the natural language is perfect example for efforts to discover such first principles. E.g. reading from https://edinburghuniversitypress.com/book-elements-of-formal-semantics.html one can see the table that expresses each grammatical category as the derived type that is made from just two basic types (e and t - entity and Boolean-truth):

Abstract type F-type S-type
NP→S intransitive verb ff et
NP→(NP→S) transitive verb f(f f ) e(et)
A→(NP→S) be copula f(f f ) (et)(et)
A→A adjective modifier f f (et)(et)
S→(S→S) sentence coordinator f(f f ) t(tt)
A→(A→A) adjective coordinator f(f f ) (et)((et)(et))
(NP→S)→S quantified noun phrase (f f )f (et)t
N→((NP→S)→S) determiner f((f f )f ) (et)((et)t)
(NP→S)→(N→N) relative pronoun (f f )(f f ) (et)((et)(et))

One can guess - if mathematics is the model of the real world, then we already have all the first principles, we just need more efforts to express such concepts as 'happiness according to Aristotle', 'ontology according to Hegel', 'ontology according to British encyclopedia', 'ontology according to some famous philosopher N.' (we should always take into account that well defined concepts are connected to some personality in whose inner semantic we they can be found and only from such personal concepts the conventional concepts can emerge by convention in some scientific community, legal system, etc.) using the basic notions of math.

OK, I know that my thoughts are very childish. That is why my real question is this - is there some discipline in philosophy that tries to express the content of the each concept in some basic notions, is there discipline of the philosophy that tries to uncover such basic notions and types (be they the already known mathematical notions and types or something other)? What are the names of such disciplines of philosophy? What are common terms and research themes in such disciplines? Just keywords and names? Everything other I can find further myself.

I know, that there is metaphysical ontology (as opposite to applied ontology) but I don't know the efforts to find the content of concepts and the first principles. I know that there is mereology, but it is about parts, about structures and systems, but the essence of the concept is something more that just its structural build-up. So, I am completely lost and I don't know where to search further.

p.s. Why I am asking this? Well, I have zero internal/personal drive to understand world in such basic terms. I am just trying to automate thinking/reasoning (artificial general intelligence) for applied purposes and that is why I need systematic, disciplined, extensible and automatable way of handling concepts and I am just seeking for theories that are already created for such handling of concepts. Of course, they can not give the final answers, but they can be good starting point and the system can discover further horizons itself.

I am starting to read https://www.bloomsbury.com/uk/ontology-and-metaontology-9781441191953/ final chapters can give some answers.

Answer 1

[I]s there some discipline in philosophy that tries to express the content of the each concept in some basic notions, is there discipline of the philosophy that tries to uncover such basic notions and types (be they the already known mathematical notions and types or something other)? What are the names of such disciplines of philosophy? What are common terms and research themes in such disciplines?

As usual, Conifold's encyclopedic knowledge is accurate. Let's unpack your question a little and put you on the track to understanding the broad nature of your question which has some metaphysical presuppositions built into it!

At a first glance, your mathematical formalism to express propositional content is a tool of the symbolist camp in AI. The notion is simply that thought is largely the processing of symbols that correspond to the true state of things in the world. This is the correspondence theory of truth and is typical of realist philosophies. The famous Ludwig Wittgenstein set philosophy on a different course with his Philosophical Investigations which pushed analytical philosophers to broaden notions of semantics, veracity, and language to be less formal and more natural (as in the Oxfordian tradition of ordinary language) and pragmatic.

There is an entirely separate branch in AI which relies on connectionism. It looks to do what neurons do. In philosophy, one can consider this an extension of the position of psychologism. By building systems that mimic the foundations of the brain, some philosophers are engaged in a discipline called philosophy of the mind and rely heavily on cognitive science for propositions. The famous philosopher Willard Van Orman Quine is famous for his push to use a naturalized epistemology which roughly advocated using science as a discriminant for truth about philosophical positions. The article begins:

Epistemology is concerned with the foundations of science. Conceived thus broadly, epistemology includes the study of the foundations of mathematics... the foundations of mathematics divide symmetrically into two sorts, conceptual and doctrinal. The conceptual studies are concerned with meaning...

This is why one might look to model theory, type theory, set theory, category theory, or topoi as a basis for building a symbolic conceptual system used by inference. He goes on:

Epistemology, or something like it, simply falls into place as a chapter of psychology and hence natural science.

Therefore, besides the philosophy of mind, the philosophy of language and the philosophy of psychology as well as their offspring, psycholinguistics figures heavily in understanding perception, conception, and inference.

I have zero internal/personal drive to understand world in such basic terms. I am just trying to automate thinking/reasoning (artificial general intelligence) for applied purposes and that is why I need systematic, disciplined, extensible and automatable way of handling concepts and I am just seeking for theories that are already created for such handling of concepts.

There is no widely effective conceptual reasoning system. Computers still do not largely understand anything they do. From Artificial Intelligence: Foundations of Computational Agents on page 179:

It is very important to understand that until we consider computers with perception and the ability to act in the world, the computer does not know the meaning of the symbols.

There are over 40 institutions engaged in researching the automation of concepts for creating the foundations of metaphysical systems in computers as we speak, and no one has a cookbook for this stuff, which is why people like you and I are here looking for more understanding in philosophy.

To handle concepts, you have to understand what a concept is, not only through the philosophical lens, but also from that of cognitive science, which embraces philosophy and science equally. I'd recommend the following books from my own library:

1. Cognitive Linguistics: An Introduction (Cognitive semantics dominant school in psycholinguistics as far as I can tell.
2. Foundations of Language: Brain, Meaning, Grammar, Evolution (origins of semantics and some brief musings on metaphysical implications)
3. Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought (puts forward a non-mainstream approach to philosophy called embodied realism that rejects dualism in favor of trichotomy of ontologies: physical, computational, and mental)
4. Also, two fantastic introductions to philosophy of science and epistemology are Blackwell's Companions of those topics.

These should be fertile metaphysical works which can function as primers to go along with Goertzel's work on AGI (Springer) for instance, and help go beyond Russell and Norvig's encyclopedia on narrow AI topics. You should also recognize that as an AGI thinker, standard philosophy types here will often come up to the line in science and math, but not cross over it into the very epistemology that Quine called for, cognitive science, so when posting here, do your best to phrase questions philosophically, as opposed to presuming it's an interdisciplinary forum.