Do logicians prove theorems for the sake of proving theorems like mathematicians do (as in do logic for logic’s sake)? Are there structures in logic akin to groups, fields, rings etc. in mathematics? There’s a number of famous outstanding problems in mathematics, for instance the P=NP question. Are there similar problems in logic? Where does logic end and mathematics start?
Note that P=NP? is a famous problem in Computer Science not in Mathematics. Different field, but common mistake. (So you can Google yourself crazy finding lots of examples.) Hardly any Mathematician cares about the answer. Virtually all Computer Scientists do.
It’s a problem in computability theory, surely? I thought that both mathematicians and computer scientists were interested in it?
My fiance is a mathmatician whose concentraition is logic. I will alert him of this thread… But I mostly wanted to point out the fact that some mathmaticians are logicians, for what it’s worth, so I don’t think there’s one big line separating the two.
I believe that an important distinction between propositional logic and (say) axiomatic set theory is that Gödel’s incompleteness theorem doesn’t apply to the former; in other words, every statement in predicate logic can be shown to be either true or false from the axioms. This makes it somewhat less “interesting” to try to prove stuff.
No one proves theorems for the sake of proving theorems. Mathematicians, logicians, computer scientists and others who deal with theorem-proof structures all operate by finding something that interests them and studying it. Proving theorems about your interests is a good way to study them.
Certainly. Models/interpretations and theories are the most common.
Yes, but I don’t know any off the top of my head. Logic gets very esoteric very quickly.
The answer, of course, depends on whether you ask a logician or a mathematician. There are some people who study both, and I assume they have a third answer.
<dons offficial logian hat>
Logic is a branch of mathematics. We prove theorems just like any other mathematician. What defines a given branch of mathematics is not usually cut and dry, but it is usually the subject matter of the questions asked, and to a lesser degree, the techniques used to answer questions (prove theorems).
Logic, like any branch of whatever, can be further subdivided into branches. These include, but are not limited to, set theory, proof theory, model theory, and computability theory. I’m sure there are others…
Now then, I study what is called descriptive set theory. It is a branch of set theory.
I really only know about set theory and model theory, but this is what I know:
Structures? Yes, set theoroticians study sets. Descriptive set theorists study Borel sets. Set theorists who study lagre cardinals study large cardinals. Set theorists who study forcing study what can be ‘forced’ about various models of set theory under certain conditions. Model theoriticians study models. Shelah studies everything.
Big famous problems? Well, not famous, but… Many mathematicians know about Cantor’s continuum hypothesis (CH). It was proven by Godel (using ‘constructible sets’) that CH is consistent with the axioms of set theory, and it was proven by Paul Cohen (using a technique called ‘forcing’) that the negation of CH is also consistent with the axioms of set theory. So is CH true? The axioms of set theory aren’t strong enough to tell us.
There are also philosophers who are called logicians. I don’t know what they study. (not an insult, just my ignorance)
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You’re very loose about what “logic” is in your mind. I’ll echo Jamaika that logic is part of mathematics. I’d say the closest overarching field that you could be referring to would be topos theory, and in that case a topos is a structure like you’re thinking of. A topos then has an “internal logic”, and to each coherent notion of logic there are topoi with that internal logic.
Why is logic a branch of mathematics and mathematics not a branch of logic?
I’m having difficulty even making sense of the question before I can answer it. Logic can be studied as a mathematical structure, just like groups can. How would you argue that “mathematics is a branch of logic”?
What do you understand by the word ‘logic’?