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How to Learn Advanced Mathematics Without Heading to University

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Re: How to Learn Advanced Mathematics Without Heading to University

#71

Post by goc »

chris11 wrote: post #68
goc wrote: post #67

Earlier quoted context omitted.

To start from absolute zero, check out Suzanna Epp's Discrete Math[0]. I believe even a motivated high school student could get started with it and even finish it. If your proof-writing is shaky, the book provides a very good workout. From there it will be easy to choose the areas of discrete math to specialize. [0] http://www.amazon.com/gp/product/0495391328?keywords=susanna...

Thanks, I'll check it out. My proof-writing is definitely shaky. I can clearly see the relationship between programming on writing proofs, but I can't get immediate feedback on the validity of my mathematical proofs like I can with code.

Questions in Epp are by no means unique. If you search MSE, you'll see that every question in Epp has probably been asked and reasked about a thousand times each. That goes for subjects like Real Analysis, Abstract Algebra, Topology as well.

Re: How to Learn Advanced Mathematics Without Heading to University

#72

Post by totony »

lumberjack wrote: post #58

There is no reason to shun university. If you want to learn mathematics but do not have the time for a full time degree do it part time. You are most definitely not actually going to reach the same level of mastery of mathematics as even the most mediocre graduate by studying alone. It requires a huge amount of commitment and staying focused and discipled enough is very hard when you don't have a set goal and good co…

>it is much less productive

My annecdotal evidence differs. I miss a lot of university class because I find either the pace or the content lacking, in which case I just read the book and go to exams. It takes me way less time and I usually learn a lot more.

Re: How to Learn Advanced Mathematics Without Heading to University

#73

Post by [deleted] »

chatmasta wrote: post #30
ska wrote: post #21

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there". So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in…

This is probably the reason why there are so many more autodidact programmers than mathematicians. When programming, you know if you made an error as soon as you try to run your code. The program, by definition, must be "correct" in order to execute. The "source of correction" is the error detection built into any programming language runtime or compiler. No such "source of correction" exists for mathematics, and tha…

[deleted]

Re: How to Learn Advanced Mathematics Without Heading to University

#74

Post by theoh »

ves wrote: post #59
Joof wrote: post #56

Earlier quoted context omitted.

If anyone knows a good way to check proofs by yourself, let me know. I love it, but you pretty much have to have other mathematicians tell you when you're doing something horribly wrong. By the way, did anyone else have to take engineering maths instead and feel they were somewhat lacking?

Just check everything carefully. If you've written your proof properly every step should follow logically from previous steps, and you should be able to check if yourself without issues. The problems happen when you're not critical enough of yourself, when the proof is very large and complex, or when you don't know your base axioms and the assumptions of your theorems well enough.

Proving the security of cryptographic protocols is a counterexample to this. Sometimes basic assumptions are very difficult to catch.

Re: How to Learn Advanced Mathematics Without Heading to University

#75

Post by amai »

A single book is enough to learn mathematics:

Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide

It has a whopping 1300 pages, but it has everything you need. And if that is not enough for you get

Cahill: Physical Mathematics

This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.

Re: How to Learn Advanced Mathematics Without Heading to University

#76

Post by dalke »

chatmasta wrote: post #30
ska wrote: post #21

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there". So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in…

This is probably the reason why there are so many more autodidact programmers than mathematicians. When programming, you know if you made an error as soon as you try to run your code. The program, by definition, must be "correct" in order to execute. The "source of correction" is the error detection built into any programming language runtime or compiler. No such "source of correction" exists for mathematics, and tha…

> "The program, by definition, must be "correct" in order to execute"

This definition of 'correct' is why software may have a poor user interface and many security holes.

Your observation also assumes that the language is completely specified. In C, for example, certain constructs lead to undefined behavior. As a modification of the old warning, running your program may unexpectedly cause demons to fly out of your nose tomorrow. Execution therefore does not imply correctness.

Proof verification, btw, offer a "source of correction" for mathematics -https://en.wikipedia.org/wiki/Automated_theorem_proving#Rela... .

Re: How to Learn Advanced Mathematics Without Heading to University

#77

Post by diymaker »

amai wrote: post #75

A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need. And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.

Thanks for the suggestion. Like you mentioned though getting to the end of 1300 pages will take ages. Hope I can find that sort of time.
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