introductory books then? you're not going to be able to get much "big picture" material at this stage, at least not very useful one
Tao's Analysis I is pretty good to introduce one to formal mathematics and to train you into how you're expected to work from a book. his Analysis II continues the trend but includes some chapters of material that is not so crucial.
Hoffman & Kunze's Linear Algebra is a beautiful jewel. the exercises are spot on and will probably develop for you the intuition you need on what a vector space is
for probability, is it a rigorous course based on measure theory and sigma algebras and stuff? If so "Measure Theory and Probability Theory" by Athreya & Lahiri is pretty good. otherwise, ignore and ask your professor for a good reference.
same for graph theory, there's a great Springer tome on that but I don't know if it's a rigorous, hard-math class or an introductory topic, so your milage may vary and I'd ask the prof
Hunter Jenkins
If by math you mean proving a new theorem or inventing a new branch of Teichmuller theory then I agree with you, symbol manipulation without intuition isn't going to get you anywhere.
But when you're [math]learning[/math] group theory or probability theory or some other subfield of math (or checking someone else's work), you're there to follow the path of mathematicians who have gone through the rigor for you, not to go wandering off into the jungle of informal reasoning. This is not to say that there's nothing valuable in the jungle, but it's not something you should be exploring in your first pass through the subject (precisely because it is too easy to get attached to bad intuition, and it is much harder to shake that off if you have a wrong mental model that contradicts the formal definition, see e.g. pic related).
I suppose you could argue that some bare minimum of intuition and motivating examples of what groups are or why probability spaces have the axioms they do, but if you don't even have that then why are you studying the subject formally in the first place?
Landon Howard
Right? Usually after a few weeks of doing something it clicks and I hate my teachers for not explaining it like that before
Henry Sanchez
That's what common core is attempting to do.
Mason Johnson
>you could argue that some bare minimum of intuition [...] * you could argue that it is necessary to have a bare minimum of intuition
I don't hate intuition on principle. Just the sloppy intuition that leads to faulty reasoning. #notallintuition
Brandon Cruz
>if you don't even have that then why are you studying the subject formally in the first place?
we studied group actions on the riemann sphere for weeks before I noticed my friend hadn't realized we could build all riemann surfaces with the quotients as coverings. I told him and his mind was blown and he instantly understood what we were doing and why. what was going through his mind before? who knows, but without motivation, you're forced to power through the material and it's not pretty
you need the right ideas, intuition and direction. if you lack all of them you can still power through the rigor in the book, as you said the path is laid out, but it's like taking a walk through the jungle on the safe path and just looking down at the floor the whole time instead of watching the trees you're supposed to be seeing. in the end the manipulations and proofs are forgotten and only the pictures and ideas remain
Matthew Gonzalez
Syntax and symbolism effectively make higher math a secret club open only to the initiated. When the field is inaccessible to the vast majority of people simply due to artificial abstraction and not an inability to intuit what is happening, mathematicians get a superiority complex and stroking their e-peens is EXACTLY what happens most of the time
Lucas Lee
What is this? I thought it was meant to be the Neumann ordinals but 3 don't fit.
Adam Young
>artificial abstraction you're a fucking imbecile the language of math is literally the simplest it can possibly be. if you have a simpler way to explain an idea or define a concept it WILL be taken in and used instead of the current one you would know this if you did any math instead of >HURR WHY DOES MATH NEEDS PEOPLE TO STUDY MATH TO UNDERSTAND HURRRRR retarded baboon. no one in this thread questions whether there IS intuition (there is), but HOW to obtain it (it's tricky).
John Hall
I'd posit that your friend lacked the intuition on quotient groups in the first place. The reductionist part of myself says that he lacks intuition on quotients in general, but that would probably be unfair since it takes a certain kind of autist to invest their time thinking about equality, equivalences and isomorphisms.
In any case we're in agreement that the best mathematicians need both rigor and intuition, I just happen to believe that the rest of us who can't achieve both should not be sacrificing the former for the latter.
You got me, those aren't numbers in the usual sense (unless you count surreal numbers and accept the axiom of global choice), they're the von Neumann universes and are used to provide models of set theory (pic lifted from en.wikipedia.org/wiki/Von_Neumann_universe). A consequence of the axiom of foundation (one of the standard ZFC axioms) is that every set -- and by extension every object mathematicians would ever want to consider -- is contained in one of these universes, which means that every mathematical object is made up of boxes containing boxes containing ... boxes containing nothing. In my opinion this is one of those theorems that is not only unintuitive but an example of intuition that is [math]wrong[/math], in the sense that it should be a reason to move away from set-theoretic foundations, or at least review that axiom and see if ZFC can be salvaged.
