What is the difference between category theory, group theory, and set theory?
If they aren't fucking synonyms I'm going to beat up a math professor (I would have said shoot up a math class but I don't want the FBI breaking down my door. Now that I've said that let me say I plan to neither beat up a math professor nor shoot up a math class.) Because that shit is ridiculous.
Other urls found in this thread:
mathematics.pitt.edu
mathematics.pitt.edu
twitter.com
>cs major detected
category theory studies objects and arrows between them
group theory studies symmetries
set theory studies sets
They are completely different.
That being said your professor probably just wanted you to shut up and stop asking pedantic questions so he could teach the class and go home.
wtf context?
What types of classes do I learn this shit in? I've had zero exposure to it.
>set theory
you should start learning this in high school when you learn about functions
>category theory
most likely learn about this in some abstract algebra class, i don't think most people take classes purely on category theory
>group theory
in a group theory class
is there something you're explicitly confused about? you just seem like you found 3 random subjects
Excellent post, fellow citizen. I would very much like to learn more facts, such as your name, academic institution and firearm dealer.
I'm not a math major but I can give you a rough idea
Category theory is the study of categories, which is to say, groupings of logics that remain consistent across all fields of math. Basically it is a way of categorizing identities and patterns that share a common truth among every discipline.
Group theory is the study of algebraic groups which are basically systems that fulfill the 4 group axioms (invertibility, identity, closure and associativity)
Set theory is the study of sets, which are just collections of objects. Those objects can be real numbers, algebraic structures, anything really. So in a sense set theory is very similar to category theory but not the same.
>you should start learning this in high school when you learn about functions
Not that I can remember. Is it simple all the way through?
Only time I ever came across a set is once in a digital logic class.
>most likely learn about this in some abstract algebra class, i don't think most people take classes purely on category theory
Okay, I didn't know this.
>in a group theory class
As far as I can tell my school doesn't have anything explicitly named group theory.
>is there something you're explicitly confused about? you just seem like you found 3 random subjects
It's just everyone is always talking about them here and I have literally never learned about any of them.
this post is a little too rough
>groupings of logics that remain consistent across all fields of math
category theory makes use of these but it's not specifically studying things that remain consistent, just objects and maps between them
>algebraic groups
this is a specifically different thing than a group
>Those objects can be real numbers, algebraic structures, anything really.
>anything really
except things like 'sets that contain themselves'
Might be different in the US or other countries, but in the UK you generally won't study "proper" set theory (ZFC etc.) until undergrad at uni, after learning fundamental analysis/algebra etc. (of course stuff like cardinals are introduced much earlier than ZFC).
Group theory is introduced in some syllabuses in secondary school (UK), it is covered in detail in first/second year courses at most unis.
Category theory will again probably be taught after the fundamentals in undergrad, although I don't this (or a full module/course on proper set theory) will be a compulsory module at any uni given that areas such as analysis, algebra, topology etc. are much more popular.
As for the differences, group theory is the study of particular structures (called groups) and is in the algebra area of mathematics. It's very useful directly and indirectly in many areas of mathematics, and is usually built assuming a certain set theory is true (i.e. axioms of ZFC holding in most cases).
Set theory is generally used as a way to explain the foundations of mathematics from certain axioms, I'm quite interested in this area although it's not one of the most popular areas of study.
Category theory is the study of objects and morphisms (arrows) and the idea is to formalise mathematical structure. It is generally studied (from what I understand, this is nowhere near my area of expertise) with set theory (say ZFC) as a base but can be studied without underlying set theory (you'll have to research this yourself, again not my area of expertise).
by 'basic set theory' i would include things like functions, whether they're injective or sujective or bijective, things like that
things are easy enough for finite sets but eventually you want to study infinite sets and you need to look at cardinalities which concerns the sort of 'relative size' of infinite sets (there's infinitely many infinities...)
you can keep going further and further with this, set theory gets very complicated eventually
>As far as I can tell my school doesn't have anything explicitly named group theory.
probably a second or third year class for most math majors
>It's just everyone is always talking about them here and I have literally never learned about any of them.
groups and symmetry by mark armstrong is a very basic and friendly text
the wikipedia pages for set theory and category theory are probably good enough places to start but the motivation for category theory will probably be lost on you without some background in algebra
Also you have probably touched on "naive" set theory in high school (at least indirectly) and indeed perhaps in undergrad before some kind of foundations module (or a full on proper set theory module). Don't get this confused with set theory, as pedantic as this post sounds.
Thanks.
>Probably a second or third year class for most math majors
Thing is I might want to pursue a math minor, so maybe I'm thinking about too advanced topics. Honestly I really don't know which classes I would want to take considering I only have room for a relatively small number of classes since it's just a minor. I should probably talk to someone from the math department.
Upon further research my school actually has a set theory class...at the graduate level. Category and group theory are mentioned in the descriptions of several graduate courses. But nothing in undergrad.
Like I said I'm not a math major so I won't pretend that I know as much as some of you guys on this board. I like reading about different types of math though and just wanted to give the guy a crude idea of what they are.
Thanks for clarifying though, I hope I didn't confuse the guy.
Can you show us a list of the classes your school offers? The graduate set theory class will be "proper" set theory (probably ZFC and such), in the UK set/category theory will be offered at good unis in undergrad (most likely later years) but I've heard in the US a lot of topics don't come up till postgrad (this might be complete BS so ignore me if this isn't true).
However, any good maths uni will have a lot of undergraduate group theory modules.
As for what to take for fundamental maths, take some kind of "foundations" class (basic stuff like natural/integer//rational/real/complex numbers, functions and qualities like injection/surjection/bijection, basic sets, basic number theory etc.) then some basic analysis, linear algebra, abstract algebra (this should include some basic group theory), geometry and motion, some kind of basic differential equations module and some kind of basic probability/statistics module. This covers the basics of most common maths you'll find in undergrand. Stuff like topology, category theory and proper set theory are usually taken after learning these topics to some degree.
I've got class right now, I will come back with a list. Thank you again for the help my friend.
You ever been to an american high school? They didn't teach us anything about sets other than set notation.
that's what i meant by 'basic set theory' in a later post
the term set theory is extremely vague on its own without any other context, could be anything from injective/surjective functions to cardinalities to forcing
Not sure if you're still here, but here you go.
mathematics.pitt.edu
mathematics.pitt.edu
Only looking at undergraduate courses because I don't think you'd be able to take any graduate courses without the prerequisites.
The courses 280, 290, 400, 413, 420, 430 and 450 look like a good start, however these require 031, 200, 220 and 230 (seems you can skip 031 with a placement score of 61 or higher). From the descriptions it looks like 031, 220, 220 and 230 aren't "university" maths, but the other ones seem to cover some of the topics a 1st year maths course at university would cover.
Well, I will have taken 280 and 290 since my major requires them, and I have already taken all of the calculus courses, so prereqs are not a problem. If I am remember minor requirements correctly i will only have to take 1 more of the classes under 1000, and then 6 credits of 1000 level courses. What I choose from those seems like it depends on whatever course I take from the ones you listed, since those serve as prereqs at this level, along with calc, diffeq, and linalg, which I have covered.
Do you know of any resources where I can read about these subjects to gauge which path I should take? I suppose I could just read the Wikipedia articles.
Not sure if just reading the wikipedia article would help, I'd search e.g. stackexchange for entry level books in each of the topics I mentioned earlier and then search google for pdf files of the books. Read each book for a while and see what you think.
>my school doesn't have anything explicitly named group theory
BAIL
A
I
L
group theory is part of an algebra course
gallian is a common text - artin also
Group theory way of representing roots of high order polynomials 19th century. Set theory the way we use it now* popularized during mid 20th century by bourbaki although originally used to prove uniqueness theorem for trig series 19th cent got appropriated by logicians into new notation for predicate logic which became standard formalism for proofs. Category theory inspired by bourbaki "was an attempt to make specific the idea of a natural transformation" prolly quoting wrong. spoke of in bourbaki text. Now can be used to talk generally about... Anything where the transitive prop holds. Appropriated by CS bec of generality and bec of usefulness of composition notation in the more general fed of function and data used in CS compared to classical math. Funny enough also popularized in alg geom by Alexander Shapiro despite his hatred of computers he really liked avoid calculation and stay at a very high level which is more of how CS people prefer to do math. Cat theory prob most useful in CS than in classical math but ill prolly get flamed for saying so.
Will do. Thanks again.
OP here, same school? Are you talking about the abstract algebra course?
yes modern/abstract algebra