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

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

#71

Post by mungoid »

heimatau wrote: post #56
mungoid wrote: post #47

Earlier quoted context omitted.

I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…

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. Its essentially like learning to program versus learning to program correctly. I cant tell you how many times I pronounce words incorrectly because I have only read it and never heard someone talk about it. Also embarrassing.

I have actually read that same thing about your [2] foot note and I wanna say I saw it here on HN but cannot remember when. It was pretty interesting and I can totally see how not learning math the proper way can cause a lot of issues related to research. In my case doing physics for simulations, its not as pronounced, because its a small user base but in a larger scale, I would be terrified of publishing my work for this exact reason.

And I by no means intend on convincing others to learn this on their own. I would actually suggest doing it the standard way because it was much more difficult and time consuming trying to learn this stuff by your own. Especially since I had no real person to talk to about it. I kinda wish I could have gone back and changed majors.

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

#72

Post by heimatau »

mmarx wrote: post #59
heimatau wrote: post #49

Earlier quoted context omitted.

I agree with this sentiment. I'm currently pursuing a Bachelor's in Pure Mathematics (or called 'Theoretical Math', eventually I'd like to go far as a PhD in it). I think the ideas of Math could be taught in a condensed way. Maybe it's already done but...education needs to be disrupted in order to do this. My current idea is that Math could be taught as a language and taught as a critical thinking class. A condensed…

That might suffice for solving real-world problems, but not for doing mathematics itself: Intuition will get you a long way, but for working out some of the finer details, you'll have to resort to rigour. Furthermore, without having gone through the rigorous training, you might not even know when your intuition doesn't reach far enough. Terence Tao[0] put it this way: »The point of rigour is not to destroy all intuit…

I think it's pretty clear, in my reply, I'm talking about the masses need for Mathematics. Which is for 'solving real-world problems'.

I agree with Terence Tao's sentiments.

Math, for the masses, is a great way to abstractly teach the masses how to critically think about things. Math, for the masses, shouldn't get bogged down in the rigour. But if one were to go on to Adv Math, then yes, rigour is needed and demanded of the mathematician.

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

#73

Post by yodsanklai »

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…

> 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 to make sense of it, you need to study maths for a few years.

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

#74

Post by visarga »

mungoid wrote: post #47
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…

I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…

The web is full of videos and PDFs with learning materials. But what is needed for learning to actually work is to have exercises to practice on. What I mean is fine gradation of difficulty and tracking prerequisites (notions needed in order to tackle a problem) so as to give students problems that are not too easy or too difficult, but just at the right level. I seldom find such problems/examples tuned to slightly above my level of understanding.

Same problem in programming and machine learning - people need a little hand holding in the form of a sequence of problems to solve that would never be either too difficult or too easy. Examples usually jump from Todo MVC to full apps, in one step, or in ML, from a simple MNIST example (or even the minuscule Iris dataset) to double LSTM with memory and attention. Where are the intermediary nice problems to learn on?

When I was learning math in school and high school there were loads gradual problems to solve, but at university suddenly there was just theory and almost no useful problems to practice on.

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

#75

Post by ianai »

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…

I would argue it's difficult to unlikely. People who attempt it though are sure to better themselves and probably their work output.

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

#76

Post by gjulianm »

mungoid wrote: post #47
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…

I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…

As a mathematician, and seeing what's on the article (and with no intention to downplay your achievements, which are impressive), that's not what I think of when I hear "advanced mathematics". Vectorial calculus and differential equations (ordinary, not partial) are basic courses in math degrees. For the things that the article explains, such as topology, group/ring theory, measure theory, functional analysis, etc (which are still nothing fancy that doesn't get reviewed in a degree, so not yet "advanced"), I think that self-learning is almost impossible unless you're near to a Terence Tao-level genius.

Here I talk from experience. I remember reading books on some of these subjects and understanding few things, without really getting a grasp of what they're talking about. A lot of times, the problem is that you don't know what is missing in your knowledge. You need a clear roadmap, you need relationships, you need to solve a lot of questions, you need to do exams and, most importantly, you need to test your knowledge. I cannot even count how many time I thought I understood some theorem only to do some exercise and see that I had absolutely no idea. Sometimes you notice yourself, sometimes you do it so bad that you don't even notice it is incorrect.

And, for these subjects, the material on the Internet starts to diminish and be less accessible (more oriented to professional mathematicians than to learners). Khan Academy does not have advanced courses, the definitions on Wolfram or Wikipedia are only useful if you have already a grasp of the subject (see for example https://en.wikipedia.org/wiki/Measure_(mathematics)#Definiti... - What is important? What are the critical aspects? Which are the subtle parts of the definition that you must read carefully?) and in Youtube you may find lectures, but usually they're like the books: you will be lucky if it's not a succession of theorems and definitions, and you still lack the possibility of checking and testing your knowledge.

So, while some parts of math can be learned independently, I don't think that advanced mathematics can be done. Myself, only after 5 years of mathematics I'm somehow comfortable to study subjects by myself, and it's still hard.

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

#77

Post by sixo »

mungoid wrote: post #47
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…

I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…

I don't mean to be rude by saying this, but the truly-difficult advanced math - the stuff that's really hard to build an understanding of by yourself, because it's fairly distantly separated from any obvious applications or anything you'd readily have experience with, and heavily obfuscated (to newcomers) by the notation and pedantic proof-focused thoroughness (appropriate for academic math, less so for applications) - starts a few courses after Diff eq.

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

#78

Post by thearn4 »

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…

Another thing that I'll toss in there (as a math PhD) is that the more advanced a topic is, the harder it is to truly grok it on your own without at least SOME connection to a subject matter expert. A mentor can really help you to understand something from several different perspectives, which is really critical to gaining your own expertise (versus a cursory understanding).

Now not all professors are great at this, but I would say that a great many would love nothing more than to talk about the things that they know very well.

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

#79

Post by mungoid »

aluhut wrote: post #67
mungoid wrote: post #47

Earlier quoted context omitted.

I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…

I wish I could make the jump again. When I was a kid, I loved Math. I even got one of this badges that were so popular in my east block country, for being the best kid in Math for my whole year group. Then we moved to Germany. Math level was far below mine, I got bored, started to do other stuff and lost it when they overtook me. Growing up and work did the rest. I lost it. When I had/have to do some math I'm doing w…

Yeah it becomes increasingly difficult as you get older. I really wish I would have gotten into it sooner. DIY math is great, but it taught me to be more of a loner than I'd like.

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

#80

Post by yodsanklai »

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…

> Imagine Dirichlet's box principle, a thing that a 5-year old should understand; now look at how is it taught in discrete mathematics.

The theorem is "there's no injective function whose codomain is smaller than its domain". It's not stated this way because mathematicians are snobs or to impress students! abstraction is the very nature of mathematics.

From https://en.wikipedia.org/wiki/Abstraction_(mathematics)

"Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena."

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