They are just painful to read. Who the fuck enjoys the autistic style of "definition - lemma - proof - theorem - proof"? Why do only so few authors explain intuition? All the stuff could be understood so much more efficiently if someone took the time to explain why it is defined like that or why a theorem does make sense.
And don't tell me I'm a brainlet for not figuring it out on my own. I usually do, but it just costs a lot of time. The real brainlets are those professors who write a book but don't have the literary or teaching skills to actually present the material in such a way that it will be remembered easily.
Math grad student. After years of studying math I still occasionally develop new intuition on very basic things and I wonder why nobody told me to view it like that before.
Anthony Wood
well, I think the problem is that intuition is deeply personal. the things that make one person "click" and understand the material aren't the same for anyone. sure, using the correct diagrams and pictures helps, and sure, a few motivating examples will help anyone, but it's highly subjective and what will help one person won't help another one
there's intuition to be obtained no matter the format (def-lem-thm-ex is my favorite) as long as the examples and proof techniques are on point and show the inner workings of what you're trying to explain
David Thompson
Yeah, there's not much wrong with the Def -> Lemma -> Theorem -> Example style if the proofs are actually instructive and provide intuition instead of being a perhaps elegant and short proof that doesn't offer much insight into what's really going on.
Leo Diaz
Starting simple then getting more complex with it as the reading goes on would be more interesting than just switching from Point A to Point B. We learn best from connections, so it makes sense that linking it all together from a certain point would be the best way to teach it.
Ian Moore
The issue I have with math textbooks is that for the headings the notes they use autistic purple shadowed comic-sans witch draw your eyes to them while you're trying to focus. This is only in lower level textbooks but it still bothers me.
Jacob Carter
I understand you, I'm taking a group theory class next cycle so I decided to learn it by myself last month, here's what happened:
>get a shitton of abstract algebra books >can't understand even the basics of groups >get "group theory" books >even worse, can't pass the first 10 pages >start getting upset about it, like I'm retarded or something >go to youtube >watch a superficial 15 min video about it >enlightenment - everything makes sense now >get the fucking book >completed half of it already, and I was able to prove most of the things they asked
It's a mystery for me the capacity of the mathematicians to understand a fucking book/article jumping directly to the formalization, I'm just incapable of doing this.
It fucks me up and I get upset about it, if you ask me. I promised to myself that if I become a great mathematician, I'll write books about the fields of mathematics being as intuitive as possible, giving a shitton of examples. Brainlets deserve to learn too.
Jeremiah Ortiz
What I do is I get a book, start reading and I look at the first definition. I don't even read the name of the thing they're defining on the off-chance that I get even remotely some intuition or hint of what it may resemble.
Then I get to work, and try to conject and prove every theorem I can muster, and even define new quantities that could be of interest and prove new things.
Then I read the name of the definition and start this process again, and continue like this throughout the book.
Chase Jackson
The problem is that popular textbooks are usually written by great mathematicians. And they've been doing math a very long time before they write a book. Things that are obvious to them now probably aren't obvious to beginners. But they can't remember what it was like to be a beginner and even if they could they're probably much smarter than you so it still might not make sense. There's not really a way to fix this since these books are still very useful. I'll take Lang over Dummit and Foote any day now. Maybe when I started but at this point Lang is just better since the stuff he thinks is obvious is obvious to me as well after learning it. You just need to look for books at a lower level. For example when I was learning analysis and algebra I used Rosenlicht, Gallian and Herstein. Rudin wasn't useful to me back then either. But now it's a very useful reference whenever I need to brush up on analysis I've forgotten.
Andrew Turner
if you two had a baby it might actually have a chance at being a halfway decent human being
Joshua Cooper
If you want to make math intuitive again, you've got to start by fixing the foundations.
Connor Price
My Calculus Prof hates textbooks so much that he writes his own handouts with his own practice problems. He explains and teaches things better and more intuitively in one paragraph than the Stewart book does in 2 pages. He teaches at every level from Calc I to PDE's and Analysis and I plan on taking him for every course I can.
Evan Russell
Do you have any textbook recommendations for good math books? I am currently a math undergrad and face the problem you are complaining about.
Kevin Thompson
which topics? you just need to try tons of shit and over time you get a sense of what works for you and learn how to aim for that
but I can tell you some generic good ones
Thomas Green
>well, I think the problem is that intuition is deeply personal. the things that make one person "click" and understand the material aren't the same for anyone. sure, using the correct diagrams and pictures helps, and sure, a few motivating examples will help anyone, but it's highly subjective and what will help one person won't help another one
Honestly, this is one of the more terrifying motivations of making the most of your undergraduate years. Who the fuck knows how much hidden knowledge is kept alive beyond textbooks, because no textbook or paper can reliably explain the most tricky concepts or the most elegant intuitions?
Eli Garcia
>it's an OP reveals his own brainletism by calling other people brainlets episode
Brayden Taylor
>everyone worrying about intuition Any nontrivial mathematical concept will invite a whole spectrum of interpretations, each as unfalsifiable as the other.
We do not use dry, formalist syntax out of some desire to stroke our e-peens for doing """abstract""" symbolic manipulations, but because the precision of our language entails that what we say is verifiable and consistent INDEPENDENTLY of our individual interpretations.
For example, some anons may disagree with my interpretation of math, but we can still get together, DO math, and come to a consensus with little to none of the philosophical quibbling that comes with just about every other subject.
linear algebra, statistics and probability, analysis, and graph theory are the topics I'm studying. If you know of any good books, please let me know. I like proofs, but dry exposition without any context, intuition, or connections between different branches of math is terrible.
Luke Perez
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.
Dylan Brooks
thanks. I'm going to look into these.
Connor Perry
meant for , though the fact that von Neumann universes are indexed by the ordinal numbers means that you can (if you want) define all your favorite number systems N, Z, Q, R, ... in terms of von Neumann universes instead, by making the obvious bijection a -> V_a. And that definition will be equally consistent so long as you remember to do it everywhere.
For all of the importance I've been placing on syntax ITT, it does have a habit of being awfully arbitrary sometimes.
John Jones
these days, there are tons of high quality math blogs where the authors are knowledgeable and don't feel the need to get bogged down in textbook-ese. and you say you're a grad student, so presumably your department has regular talks, seminars, etc. another thing you can try is to set an ambitious goal for yourself and then look up stuff you need to learn as you go along-I think most higher-level math textbooks are written to be used in this way, not as a linear exposition.
Logan Moore
Look up Ben Garside on youtube, he did an excellent playlist on Groups that is very intuitive and easy to understand. It makes everything seem just as elementary as middle school algebra.
Easton Rodriguez
You completely missed the point
Carter Hernandez
>Syntax and symbolism effectively make higher math a secret club open only to the initiated no. you made your retarded point very clear. it's just a stupid point. it's not the syntax. it's the fucking meaning. you study tons of years, and hard, to understand these concepts. an idiot with no knowledge of math can't just come in and start using them obviously it's not because >hurr aritificial complexity they're trying to keep my genius brain out with slang :( idiot
Grayson Stewart
>an idiot with no knowledge of math can't just come in and start using them obviously So you didn't miss the point; you actually proved me right when I said higher math is like a secret club. You assume all non-mathematicians are idiots who would be unable to understand certain concepts even if they understood the syntax. The syntax makes the math inaccessible to most people, yet when asked about a way to provide a base for intuitive understanding, you reply with FUCKING NORMIES REEEEEEEEE STAY OUT OF MY SECRET CLUB
Dominic Lee
I think you're forgetting that most textbooks are written under the assumption that they will be read by students who are enrolled in university courses, who will be provided with intuitive explanations in lectures, as well as the ability to ask for clarification on anything that hasn't quite clicked. Very few textbooks are designed to be completely self-contained courses on their respective topics.
Gavin Richardson
Because articles get peer reviewed and get rejected if they don't provide enough intuition to the reviewer, but whole textbooks get only proofread and maybe checked by the authors buddies for big mistakes and thus have little motivation to include intuition
Austin Price
read the whole convo you slayed his ass user
Caleb Howard
"Collapse of the wavefunction" is what happens to the probability when the indeterminate future suddenly becomes determined by a measurement in the present.
Mason Martin
If intuition was included in math textbooks, retards would think intuition was a valid method to prove things. And intuition will mislead you very often (think of the 20 shitposting threads this board gets every day, monty hall, zeno's paradoxes, etc.)
Jonathan Jones
t. Engineer
Cooper Carter
Unfortunately, a lot of intuition is hard to explain because it's based on a lot of previous knowledge you have, which you don't necessarily have to be aware of at the moment it 'clicks'.
Robert Wright
Whoa, thanks for the advice, although it's too late now. But sure, I'll take a look.
Aaron Gutierrez
>asked about a way to provide a base for intuitive understanding you never did this you imbecile baboon all you did was >BAWWWWWWWWWWW I DONT KNOW MATH AND DONT UNDERSTAND IT BAWWWWWWWWWW enroll in a class or open a fucking book, colossal fucking retard
Gavin Barnes
Personally, I think your point is well made but I disagree with it.
Sometimes even relatively simple concepts, like the Direct Fourier Transform need rather complicated notation to get across the complete nuance of the concept. What I am saying is that even easier topics in Mathematics are subject to nuance, making it difficult to communicate in a simple way.
Daniel Edwards
That's absolutely fair, and you have a good point. Most math beyond the really basic stuff does require some understanding of certain concepts before more complex concepts are introduced. My point was that that the fundamental aspects of some complex math can likely be communicated to someone with a lower level of understanding, even if learning the details of said complex math will require more education before it can be fully communicated. For example, since you mentioned the Discrete Fourier Transform, check out the Wikipedia article, and then check out a couple videos Khan Academy did on the subject. Khan gives a basic yet fairly intuitive (and visual) explanation while Wikipedia makes it looks like rocket surgery to someone who's never been introduced to it. Obviously those are two extreme examples, but in my experience most people would not be afraid to approach a subject, would learn faster, and would develop a deeper understanding of the subject with a "friendlier" introduction in most cases. Thanks for responding normally and not sperging out like this guy
Depends on the field imo. Most of the asymptotic analysis texts I've read have been intuition-oriented, even the ones that aren't written for engineers.
