What is the study of systems (like logic, maths etc) called?

Question

Before I begin, what I mean by 'systems' is what I've dubbed 'axiomatic systems', those which act as the starting point for all knowledge, for which I know of three: Maths, Logic, and Set Theory. I'll just call them systems from now on, though.

I'll first begin with my intuitions behind this question. Most of you here will be familiar with logic, being that it's the 'tools of Philosophy'. Whenever you do logic, you're working within a system. One can't just 'do' logic, for logic is the activity within the structure of a system. So the reason you can't just do logic is for the same reason you can't drive without a car.

So, such systems have to be literally designed; its axioms must be discovered, it's properties must be studied, its values and operations must be declared before use, etc.

I've been looking for this answer for quite a while, and in searching the depths of the internet the best I can find are papers that partially (and very briefly) study some of the aspects of the analysis and creation of systems. What I really want, however, is an academic, and rigorous field of study that gives an exhaustive account of how such systems are created. I think the reason why am having trouble is because am using 'systems' in a very technical way, and I have no other way to really express what I mean other than "systems".

Quick note: There are fields out there looking at the use of systems (maths, logic, set theory), how they can be combined, how there used in computer programming, but none which seems to study the actual systems themselves.

Oh, by the way, am not sure what to tag this as, so sorry if I've missused the tags I have provided.

Answer 1

The field of study that you are looking for is metalogic which is a field of philosophy (or, arguably it is at the intersection of philosophy and math as the regular study of logic is. For the sake of the argument, I'll stick with saying its in philosophy, but the point remains the same no matter what university department you'd find it in.) Here is a quote from the wikipedia article that explicitly states what you are looking for:

The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory.

Those two fields, which are part of mathematical logic, study exactly what you are talking about per your comment

What I really want, however, is an academic, and rigorous field of study that gives an exhaustive account of how such systems are created.

Your use of the term 'system' is correct and as such you are asking for the field of study which rigorously defines what a formal system is. That is the field of metalogic. Logic is the field of study that uses formal systems, metalogic is the field of study that assertains these system's properties. For instance, Gödel's completeness and incompleteness theorems, Tarski's undefinability theorem, Lindström's theorem, all of these theorems are results of metalogic. They don't just show one statement that exists in, say, propositional calculus. They are meta results that apply to a vast number of, if not all, formal systems.

Now of course there are formal systems outside of just zeroth, first, second order logic and so on. There are the formal systems that are studied in formal linguistics and computer science (which really are just generally part of computability theory). These formal systems obey the same exact metalogical results that something like, say, first order logic obeys. Lambda Calculus is another great example of a formal system that has been studied greatly. In fact, the paper were it was introduced was already stating a meta-theorem about itself!

If by "how they are created" you also mean to include the anthropological background, I would suggest looking into neuroscience. It is a pretty decently supported fact that the human mind tries to organize its thoughts in a somewhat logical way, although of course that is not always the case.

You would probably be very interested in Douglas Hofstadter's book Gödel, Escher, Bach as it deals with exactly what a formal system is, how we use them and a lot of metalogical theorems about them in a non-technical and intuitive way.

Answer 2

There's a field called General Systems theory. https://en.wikipedia.org/wiki/Systems_theory

Here's the classic text in the field. https://www.amazon.com/Introduction-General-Systems-Thinking-Anniversary/dp/0932633498