# Algebraic Logic, Algebraic Foundations of Mathematics, and Graph Theory

I greatly apologize in advance if this is not a well formed question, and feel free to redirect this in any way.

So I have begun reading Logic as Algebra by Halmos and Givant. In their book, they provide an algebraic development of mathematical logic. I haven't gotten very far, but I do see how they change the concepts of connectives to binary operations, inference rules as a function, etc. This brings me to my first question. Can algebra be reinstated as some foundations of mathematics, much like set theory? Could I very well just assume some (albeit nowhere near as elegant) axioms equivalent to ZFC in terms of algebra and develop mathematics from there? Furthermore, I would very much like to continue reading on algebraic logic, but from my own search, the literature isn't entirely as dense as it is in other subjects. One of the few that I have seen is An Algebraic Introduction to Mathematical Logic by Barnes. Is this a good book?

My last (but not least) question is based off of the spirit of the first two. I am gaining interest into graph theory, and I was wondering if there was any analogue of a graph-theoretic development of logic as with the algebraic one I mentioned above. If so, (and assuming the answer to the algebraic foundation question above is yes) could you also formulate an axiomatic development of mathematics using graph theory? These last few questions on graph theory are based on a potential project I would like to work on, so I would appreciate any help in the matter.

• Do you know "category theory"? Especially nice categories called topoi can be used as a foundation for mathematics. – Stefan Perko Sep 26 '15 at 19:15
• I am sceptical about the merits of non-standard approaches, until you have a good grasp of the standard approaches. There is a vast amount of maths to cover and not much time to cover it! Although some areas are a mess, on the whole the standard approaches are standard because they are easy to follow. Also you are likely to lack "judgment" and "taste" until you have covered a fair amount of standard maths. – almagest Sep 26 '15 at 19:18
• No, I haven't looked into category Theory, bu I hoped to eventually. I haven't heard good things about it. – Sprinkle Sep 26 '15 at 20:37

Following almagest's comment and assuming I understood correctly your question I would say that the answer to all of your questions is yes there is.

The Elementary Theory of the Category of Sets (or ETCS shortly) is a first order theory developed by Bill Lawvere.

The objects of this theory are functions and sets but instead of having a membership relation as primitive concept we have a bunch of operations (source, target, compositions and identities).

These operations are subjected to some axioms expressed via equations which basically say that the objects of the theory (sets and functions) form a very specific kind of category (namely a topos with natural number object and few other things, if a remember correctly).

The whole theory is an equational theory so it is algebraic (or quasi-algebraic to be exact, because the composition operation is not a total operation like in ordinary algebra but still...).

Depending on what you mean by "axiomatic development using graph theory" ETCS could address the other part of question as well: that is because categories are basically graphs with a composition operation on the edges (a sort of graph-algebra).

If you want to find out more about ETCS I suggest you to take a look to the following links where you can find some nice introductions on the subject:

I hope this helps.