How to Learn Advanced Mathematics Without Heading to University

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#81

Post by bitL » Mon, Sep 05, 2016, 6:58 PM UTC

yodsanklai wrote: post #73

bitL wrote: post #62

Earlier quoted context omitted.

The main problem of contemporary mathematics is that it is unbelievably obfuscated to most people, unnecessarily so. So even if you a have super-simple thing, mathematicians invented ways how to completely obfuscate meaning (often unfortunately in order to achieve prestige and being considered elite as a form of intellectual pride). Imagine Dirichlet's box principle, a thing that a 5-year old should understand; now l…

> contemporary mathematics is that it is unbelievably obfuscated to most people, unnecessarily so I disagree. I think maths are intrinsically complex. Some results may have intuitive geometric interpretations but if you want to understand the whole edifice, there's no shortcut, you have to absorb tons of theories. Take probability theory and statistics, you can always see it a set of recipes, but if you really want t…

Yes, to an extent. When you actually study history of mathematics, you find many ideas swept under the rug as they are for whatever reason making some people uncomfortable. Simple example is a material implication, precisely handling false antecedents in binary logic (90% of population finds it weird as it doesn't correspond to their thought processes). The problem of its adoption was solved by waiting for logicians that didn't accept it to die. Arguably, this very logical connective is the cause of Goedel's incompleteness problem and some logics that reject it such as Relevance logic get to almost complete systems but are way more complicated (though also way more logical to lay persons and arguably more similar to how humans think). There is a reason why medicine doesn't use mathematical logic and rather is based on counter-factuals.

So you can compare current mathematics to be like a certain programming language. Let's say it's like FORTRAN. There might be C++ for the same concepts, there might be Python, Smalltalk, Prolog or Haskell for the same concepts, but everything you read is in FORTRAN. And very few people like or are capable reading FORTRAN.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#82

Post by svanderbleek » Mon, Sep 05, 2016, 6:58 PM UTC

hal9000xp wrote: post #60

As I a software developer with no degree (working since 2009 in 3 countries), I can share my experience of attempts to learn advanced math. In 2010, I was very interested in foundations of mathematics, an extremely abstract math branches: https://en.wikipedia.org/wiki/Foundations_of_mathematics In particular I spent huge amount of time on: https://en.wikipedia.org/wiki/Nicolas_Bourbaki (Set theory) https://en.wikiped…

This approach is what took my math skills to the next level. I would also frequently skip exercises and forget material that I learned previously. Since I started adding exercise books to my study routine I have seen enormous gains in knowledge retention and my ability to build on concepts already learned. It also doesn't have to be Olympiad style, there are also Math Circles and the general Problems in {Area} book, like one of my favorites Sequences, Combinations, Limits.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#83

Post by lumberjack » Mon, Sep 05, 2016, 7:01 PM UTC

This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own.

I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll get a basic understanding of the topic. You won't be an expert by any means. That will require more exposure, more time.

The lectures themselves are not that useful, I find. The lecturers are mostly useful in guiding you along, telling you which aspects of the theory to focus on and weeding through the study material to deliver you the best bits. The problem sets are indispensable. Exams make sure you actually know the basics in depth instead of just knowing about them.

My advice: enrol part-time, take one class at a time, catch up on the lectures and do the problem sets and the homework over the weekend.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#84

Post by FabHK » Mon, Sep 05, 2016, 7:02 PM UTC

bitL wrote: post #62

yodsanklai wrote: post #17

> How to Learn Advanced Mathematics Without Heading to University I wonder if it's even possible. Learning maths requires much work, time and dedication. Doing so alone must be very difficult. There are several things universities provide that are hard to replicate alone: a degree, which gives you access to a job, motivation, learning environment, and "peace of mind". What I mean by peace of mind is that, when you're…

The main problem of contemporary mathematics is that it is unbelievably obfuscated to most people, unnecessarily so. So even if you a have super-simple thing, mathematicians invented ways how to completely obfuscate meaning (often unfortunately in order to achieve prestige and being considered elite as a form of intellectual pride). Imagine Dirichlet's box principle, a thing that a 5-year old should understand; now l…

Just because something seems obvious does not mean that it is.

Famously, for example, Bertrand Russell and Alfred Whitehead prove in Volume II of their Principia Mathematica, using theorem 54.43 from page 379, Volume I, that 1+1=2 (adding that "the above proposition is occasionally useful.")

Now, that is clearly obvious to everyone, and yet what Russell and Whitehead achieved in the intervening 400+ pages was more than just obfuscation.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#85

Post by heimatau » Mon, Sep 05, 2016, 7:06 PM UTC

mungoid wrote: post #71

heimatau wrote: post #56

Earlier quoted context omitted.

I'm glad you had success but...let's not measure your outlier experience with the rest of the world. Especially in Adv Mathematics. I'd point you and other HNs to Srinivasa Ramanujan. He is self taught but...he was wrong [1]. He had a brilliant mind but...due to being self taught, he made some critical mistakes. Being self taught can easily lead the learner to some critical mistakes. Eventually, they may be corrected…

I agree 100% with you and, most of my situation was because I lean a bit too far in the "against the grain" category. Because of that, I definitely made it more difficult for myself and would regularly lose drive to continue because I felt "I'm not getting it, I suck. Why can't I learn this the normal way?" >Being self taught can easily lead the learner to some critical mistakes. ... I really glad you brought that up…

> There have been countless times that I was working on some formula which looked good to me, and even had correct results (some of the time), only to find that it was completely backwards when someone else looked at.

I think many people forget that THIS is what a Scientist is. Someone subjected to their peers. This humble way of looking at things (that our work isn't accepted until it's verified/peer-reviewed), is our way of life. It's a shame to me that the current culture has a massive backlog of research, without peer review.

I'm grateful for your reply as it will give others insight into the 'less trodden path' of trying things yourself. It worked for you, so that should motive others. And hopefully I added to the conversation to encourage others to seek out peers/mentors, since that will accelerate their learning.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#86

Post by lumberjack » Mon, Sep 05, 2016, 7:07 PM UTC

mathgenius wrote: post #14

This stuff is brutally difficult to learn from books. Sigh. Maybe in the years since I studied this as an undergraduate things have changed with youtube and so-on. But there is nothing quite like talking to a real mathematician. One minute you are asking a question about some little thing you are stuck on, and the next minute the master is levitating and bending spoons! That's when you start to feel the real depth be…

I guess it varies from person to person. I don't get much from lectures personally. Just go there to force myself to go through some proofs of major theorems cause otherwise I know I wouldn't.

I don't recall ever asking a lecturer anything. Whenever I'm stuck I like to double down and stare the sheet until I do some progress.

And unlike you, I most definitely don't feel like a genius in class lol.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#87

Post by FabHK » Mon, Sep 05, 2016, 7:09 PM UTC

elorant wrote: post #19

yodsanklai wrote: post #17

> How to Learn Advanced Mathematics Without Heading to University I wonder if it's even possible. Learning maths requires much work, time and dedication. Doing so alone must be very difficult. There are several things universities provide that are hard to replicate alone: a degree, which gives you access to a job, motivation, learning environment, and "peace of mind". What I mean by peace of mind is that, when you're…

You could learn math as a hobby, not because you're looking for a job. At least that's what I did. I like them for what they are and the immense satisfaction I get once I realize how a certain theory works. Other than that I don't expect any material reward from that knowledge.

Wonderful book, btw, for maths as a hobby is "Proofs from THE BOOK" [1].

One of Erdős's quirky notions was THE BOOK, in which God collected the most elegant and wonderful mathematical proofs. He said "You don't have to believe in God, but you should believe in THE BOOK."

The book above collects some wonderful proofs that could have made it into THE BOOK.

[1] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#88

Post by FabHK » Mon, Sep 05, 2016, 7:12 PM UTC

tgb wrote: post #15

biofox wrote: post #3

I have found the Chicago undergraduate mathematics bibliography useful for directing self-study: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm Previously discussed on HN: https://news.ycombinator.com/item?id=9927909

As a math grad student, I can say that this Chicago list has primarily books that mathematicians know. The one posted here has primarily books that I am unfamiliar with. If one were to follow that one, other people trained in math would have a hard to judging what you've done (it's common to say things like "I've learned algebra at the level of Dummit and Foote" but this only works if people know the book you're refe…

> Perhaps the author of this post has selected those that are more appropriate [for autodidacts]

The Springer SUMS series (which appears on this list a couple of times) is very nice for autodidacts, so your conjecture could well be correct.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#89

Post by bpchaps » Mon, Sep 05, 2016, 7:14 PM UTC

lumberjack wrote: post #83

This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own. I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll…

Sometimes the issue isn't time - it's money and access to schooling. Just because you've had the access, doesn't mean others do - and there are many, many who don't.

Re: How to Learn Advanced Mathematics Without Heading to University – Part 3

#90

Post by bitL » Mon, Sep 05, 2016, 7:14 PM UTC

FabHK wrote: post #84

bitL wrote: post #62

Earlier quoted context omitted.

The main problem of contemporary mathematics is that it is unbelievably obfuscated to most people, unnecessarily so. So even if you a have super-simple thing, mathematicians invented ways how to completely obfuscate meaning (often unfortunately in order to achieve prestige and being considered elite as a form of intellectual pride). Imagine Dirichlet's box principle, a thing that a 5-year old should understand; now l…

Just because something seems obvious does not mean that it is. Famously, for example, Bertrand Russell and Alfred Whitehead prove in Volume II of their Principia Mathematica, using theorem 54.43 from page 379, Volume I, that 1+1=2 (adding that "the above proposition is occasionally useful.") Now, that is clearly obvious to everyone, and yet what Russell and Whitehead achieved in the intervening 400+ pages was more th…

I am actually suggesting that current mathematical language is not sufficient to describe real world and the language itself has self-imposed structural problems preventing it from achieving higher precision in describing the real world in fewer symbols. Now with computers doing all the menial work we should be able to tackle on the challenge of improving the mathematical language itself stuck with over-simplistic mental models so popular at the beginning of 20th century.

Try to use mathematics to describe an artistic work. Or even precise muscular movement of a human arm in a ballet in its wholeness. Good luck with that!