I want to pursue research mathematics as a hobby, a supplement to a career (science, engineering, art), or even a career in and of itself. What particularly excites me about mathematics is how seemingly disparite formal concepts can be linked though a newly discovered theory or generalization.
In mathematics there is a lot to memorize if you intend on keeping up with the latest discoveries. Even if you are specializing in a particular branch, as most mathematicians do, the amount of theorems you must be able to recite and have available to you at any instant easily goes into the hundreds.
A lot of books exist on the various branches of modern mathematics but there are very few on best-practices for everyday working mathematicians. In the natural and social sciences, university laboratories dictate the pace and style of research. Also many guides exist for best-practices for conducting non-theoretical research. However for mathematics, approaching research-level work seems almost impossible without going the graduate school route at a decent school.
Where does someone begin who only knows fundamental definitions and a handful of theorems, with no specialization in mind yet? How do people who are tied up with other aspects of life get to the point where they are active in the mathematical community?
Bump
It is almost impossible to compete at a high level academically in mathematics as just a hobby. You would need to be a fucking genius or that one guy who lived with his mom and solved that millennium problem. Most people pursue an undergraduate degree in mathematics, then branch to a masters degree and then go to graduate school while doing a PhD. The academic path is never for the faint of heart who give up easily. Its going to be torture at times. But the more satisfying it is when you finally get it, when I finally clicks. However don't fool yourself: you will have to spend fulltime on this, and even after working hours your mind will be busy thinking about the problems. If you are hoping to compete at the edge of mathematical knowledge, you either dedicate your life to it or do not try it all.
Thanks for your comment. I recognize the math video your image comes from.
>>7743826
Mathematics is not stamp collecting of theorems and definitions. Memorization will not take you far at all, building a strong intuition will. With a strong intuition you can easily sense why / which things are true and with training and rigor you can easily prove most basic things. This, along with experience of the main results of different fields, is why mathematicians have "hundreds of theorems at their disposal".
The place to begin is a common mathematics curriculum for an undegraduate. This starts with Linear Algebra (at the level of Axler's "Linear Algebra Done Right" or Hoffman's "Linear Algebra") and Real Analysis (at the level of the great reference but lousy for self study Rudin's "Principles of Mathematical Analysis"). You can most likely self study linear algebra on your own from either of those books (after reading something like "How to prove it, a structured approach" if you don't know how to do formal proofs), and for real analysis I recommend these lectures: https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC
Do note that you should not attempt to cover the full books. Read the preface and familiarize yourself with the topics so that you know how far a one-semester class on them takes you, and do just that. This will take you a couple months at least. Once you're done, come back and ask again.
Thanks for your comments. Where does mathematical "intuition" come from - I find it hard to develop reading precise papers and fast-paced lectures. I like reading informal papers that detail a sort of "meta-mathematical reasoning" (perhaps super-mathematical would be a better term, not to be confused with metamathematics). Does the "intuition" develop in university cafeteries for example, or elsewhere?
>>7744295
it develops from solving problems
Not very prestigious I guess but OEIS has a lot of hobbyist contributions and I thought that was pretty cool.
>https://oeis.org/
>>7744295
Intuition comes with familiarity with the subjects and their history.
Mathematical concepts are developped for a reason. They usually model things that people had in mind but had difficulties working with due to a lack of proper definitions: vector spaces allow one to talk about independence, orthogonality, duality and so forth, manifolds give a reasonable notion of what a "geometric object" might be, probability provides a framework to talk about randomness, correlation, independence, distribution, etc.
Knowing the motivation for the introduction of a concept helps getting an intuitive idea of what it means.
Another very important thing is to always have a couple of examples and counterexamples for every definition.
Thinking hard about each theorem is also important: does this seem obvious and why ? why is each hypothesis important ? what happens if I drop this part of the requirements ? how could I do this differently ?
This is an important part of learning. It helps you delineate what an object can or cannot do, which is useful when you do not immediately have an intuitive feel for it.
Where do you learn about motivation behind objects? If it's fundamental to advancing th subject, it sure is hard to find material on motivation.
>>7744135
What is this fucking picture you chose, it looks like a shitty animated text to speech program interrupted by a screen cap. Did you save it thinking it was worthwhile or would be useful later? Then in this thread you thought "this is it. This is the picture I want for this reply." And you finally got to use it. It has accomplished its mission as being an utterly forgettable visual anecdote to your shitty never-give-up jerkoff speech.
>>7744764
You just broke my newfag detector.
>>7743826
>Where does someone begin who only knows fundamental definitions and a handful of theorems, with no specialization in mind yet? How do people who are tied up with other aspects of life get to the point where they are active in the mathematical community?
I want to put something out here before I write this post. While I think that it's certainly possible for a person with no formal (i.e., university) mathematical training to prove a few open questions, I think that this person would have to be especially motivated and bright to even scratch the surface of whatever topic interests them, let alone be able to prove minor open questions. That being the case, learning math with the intention to try to find something new is a noble pursuit in and of itself, even if you end up not discovering anything.
Now, if you don't know that much math, I would suggest looking through a university's pure math curriculum (e.g., MIT or University of Michigan), and reading the books that a typical undergrad would, while doing all the problems. It is then that you can decide what you want to study and move forward with it. You will find actually, that soon after introductory graduate level math textbooks, you'll find books that list open questions as well as normal theorems and propositions (here is an example http://www.amazon.com/Graded-Syzygies-Algebra-Applications-Irena/dp/0857291769/ref=sr_1_1?ie=UTF8&qid=1451114037&sr=8-1&keywords=peeva+graded+syzygies) It would not be inappropriate to try to contact the people
who wrote books that state problems you want to work on, and I recommend it. By reading more papers on the subject specifically and understanding the work that has been done already on the problems, you may be able to see places where progress could be made.
Just as an fyi, it's going to take a few years to get to the point where you're studying things at the edge.
>>7744135
he does NOT live WITH his mother, he lives in his own appartment, his mother just happens to live in the same appartment complex
>>7745152
Perelman, pls. Get that cockroach infested shithole of yours cleaned.
>>7745152
>Living in commie blocks
Far worse fate than living in your parent's suburban home in the US to be honest.
>>7744764
>He hasn't seen it yet.
I don't want to be the one to break it to him.
>>7745285
The cockroaches are his only friends ok. Leave him alone.
>>7744122
math are just conventions. to study maths in your spare time is like saying that you study laws in your spare time because you love when some people accept concepts which they qualify as more fundamental/crucial than the concepts advocated by other people before.
>>7743826
Practically speaking, the edifice of mathematics has been built up so much that it is hard to make real contributions in any field without being a professional mathematician. There is simply too much to know.
If you are really into it, I would suggest to first get the basics down, ideally to a 2nd or 3rd year undergrad student. After that you should totally focus on the area you are interested in and get familiar with it. Once you can somewhat follow recent research papers on related topics you might be able to contribute. However, keep in mind that you narrow focus on your research topic places you at a big disadvantage to professionals. You will be like a blind man in the woods looking for some kind of animal and everyone else has jeeps and drones.
>>7745299
Nice bait
>>7745299
agreed, its like studying the dictionary.
>>7743826
Memorization won't get you anywhere. It will barely get you through a bachelors in mathematics.
>>7744135
>You would need to be a fucking genius or that one guy who lived with his mom and solved that millennium problem.
Perelman was a mathematics professor and researcher. He attended talks on unpublished results on Ricci flow by another researcher working on a program to solve the Poincare conjecture. He was not just some guy who studied mathematics as a hobby and single handedly solved a millenium problem.
He also doesn't live with his mother.
>>7744295
You sound really lazy, to be honest. If you like to waste your time in "informal meta-mathematics" (aka bullshit) and don't like "fast paced" lectures, then I don't really see you putting in the effort needed to do math. This could also be because you're young maybe so that's okay. The intuition is developed with hard work. Go and learn from the books, go and watch the lectures.
It's easier with someone explaining it to you, at least during the first steps. So take full advantage of elective classes in mathematics, better still if you actually take a degree in math.
>>7745484
>>7744295
Just to elaborate. In my experience the intuition develops in several different ways. There's the "visualization" kind of moment, where you're able to have a picture of the object you're studying in your mind that lets you kind of "feel" its properties in a sense. (E.g., picturing small blob-like things with a hard boundary for compact sets).
There's also the "aha!" moment of understanding relations between things. You get to understand how things work when you understand them in relation to other things. An example could be some important unifying theorem which at first glance seems weird and formal until you are able to actually picture what it means, and then understand much more about the objects it talks about. Concepts of continuity are an early example of this, with the epsilon-delta definition, which seems like a weird scrabble of symbols until you start thinking mathematically, and then you sort of look at it in terms of small open balls and arrows.
I'd say intuition is just developed as you grow and go through each topic, and understanding it. You definitely need to do the work.
>>7745500
>(E.g., picturing small blob-like things with a hard boundary for compact sets).
>"hard" boundary
I was having trouble pinning down my intuition for that and I think you just helped me. Thank you anon.
Now if I can just pin down my intuition for Compact Hausdorff being some sort of equilibrium point between Hausdorff and Compact spaces.
>>7745526
Me again.
>>7745500
I completely agree with this anon's post and I'd like to add a warning about intuition.
Sometimes, after spending a good amount of time studying a topic, people will have an insight that suddenly makes the topic clear. However, when it comes time to try and explain the insight to another person it's possible it might just not make sense. This isn't because the person is using the wrong analogy or the wrong touchy-feely description (like "hard" used to describe the boundary for compact sets above) or whatever. Instead it's often because the person hearing the analogy hasn't built up the abstract scaffolding to understand the insight yet.
This might sound like I'm lying and only saying this because I want to hoard all my key insights to myself but that is absolutely not the case.
By far my favorite example of this problem occurs with Monads. People say that Monads are cursed because once a person reaches that "aha!" moment they instantly lose the ability to explain monads to anybody else. As a result there's an overabundance of tutorials, explanations, analogies, etc.. for monads out on the internet but the trend is that people who don't understand monads find them incomprehensible while people that do understand them find them intuitive.
Here is a far better write up of this phenomena (and it doesn't just apply to monads) that also makes an attempt of explaining why it occurs.
https://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/
Here is some more info on the monad curse.
https://www.youtube.com/watch?v=dkZFtimgAcM
http://stackoverflow.com/questions/19544794/what-is-crockfords-law?answertab=active#tab-top
Intuition is good. However, be prepared to face situations where an insight is incomprehensible to you until you put in enough work.
>>7743826
Within dogmatic mathematics even advanced amateurs stand no chance to contribute anything substantial, unless they find a suitable niche right at the foundations of math where they are able to find/develop new concepts and solutions for old problems -> which then will lead to a paradigm shift dogmatic mathematicians are so afraid of, because it would show they are just confined robots.
I think, such a person would be more philosopher than mathematician, because it would mean finding/creating something completely new.
Chances are, most of the people trying to achieve that are just self entitled morons that lack the basics for advanced research, unless they are able to solve a real world problem where the solutions is proof of concept itself.
Bomp. Any good methods to memorize last name pronunciations? Is there a book of mathematician's names and their pronunciations?
>>7746677
Just go to wikipedia
https://en.wikipedia.org/wiki/Main_Page and look those mathematicians up
All last names have explained pronunciation and audio files on how to pronounce their names
e.g.
https://upload.wikimedia.org/wikipedia/en/b/bb/LeonhardEulerByDrsDotChRadio.ogg
https://en.wikipedia.org/wiki/Leonhard_Euler
