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> 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

> 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 a student, your job is to study, that's what you're expected to do and normally your degree will give you access to a job (esp. if your university is reputable). Now suppose someone learns advanced maths on their own. There's a huge opportunity cost. Not only it takes a lot of time, and the few lucrative jobs that make use of maths are in finance. I suspect financial institutions are very conservative and rarely recruit someone without a proper academic background. An other thing when learning things alone, is that your job is twofold. You must be teacher and student at the same time. You need to find the material, impose yourself some pacing, decide when it's ok to move on etc... It may be ok when you want to learn a new technique in a field you already know, but something as broad as "learning advanced mathematics" seems impossible.

elorant 10y ago
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.
- sotojuan 10y ago
  This is a good point. Many of us learn a new language or tool as a hobby after work. Why not consider math like that?
- FabHK 10y ago
  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 https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
- icu 10y ago
  I agree wholeheartedly! Math is the queen of the sciences, or the silent language of the world around us. Learning math outside of high school has also helped me identify 'snake oil' statistics in my industry and challenge the validity of information I've been presented as fact.
Chris2048 10y ago
What I actually need is "learning math in a short time", With a few, minimal yet illustrative examples (ala katas), plenty diagrams/illustrations and other mental aids, an no rigour - not a single bit of set-theory! The proofs and rigor can come later... Incidentally, I'm a dev in finance, looking to move into quant dev. I have a math degree (completed 2007) and I'm doing a "CQF" to catch up with the relevant quant knowledge. I think cyptography, and stats/data analytics might also be a good area for mathy-dev.
- heimatau 10y ago
  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 class would like like 'this is an equation...here is what we can do with it (derivatives, areas/3D/4D, etc)...but...99.99% of you won't need to know it this way. You need to use math in a way that indirectly teaches you how to creatively look at problems in life.' I'm not sure why everyone is forced to learn math without knowing WHY they are forced to know it. Creative problem solving is one of the best takeaways, imho, for the masses. As for Adv. Math...I think it's not effective for most people's career paths and skillset they will require in the real world.
  - mmarx 10y ago
    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 intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.« [0] https://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there’s-more-to...
zamalek 10y ago
> I wonder if it's even possible. From a practical perspective it definitely is. I've picked up a fair amount of graph theory and with nothing but extreme persistence have grokked and used some fairly advanced stuff[1][2] (2nd-year dropout). It was, however, work-related. Just don't ask me to proof anything. > the few lucrative jobs that make use of maths are in finance There is also competency on the table here. Graph theory crops up day-to-day with the business software work I'm doing (three separate deliverables). Calculus is used to a point of absurdity in game development - e.g. the Fresnel term. Machine learning? Calculus, linear algebra, tensors. Profiling? A basic understanding of statistics. Compilers? Category theory, graph theory. Physics engines? ODE. It's extremely valuable to know this stuff. [1]: https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_... [2]: https://eprint.iacr.org/2012/352.pdf https://eprint.iacr.org/2012/352.pdf
- ryanmonroe 10y ago
  >Just don't ask me to proof anything I don't mean to offend, but being able to prove things is generally the main focus of advanced mathematics. If you can't prove what you know, or at least have a rough outline of a proof you could construct after referring to something, you haven't learned it in the same way those at a university have.
  - zamalek 10y ago
    No offense taken, you are completely correct. I can, however, still explain it and use it.
  - empath75 10y ago
    And? You can use advanced math without having a deep understanding of it.
    - ryanmonroe 10y ago
      I just wanted to point it out. The thread title is "Learn Advanced Mathematics Without Heading to University", which could be read with the implication "Learn advanced mathematics, at the university level, without going to university". Under this context, someone replied that this is possible because s/he learned it, but can't prove what s/he's learned. Of course not being able to prove the useful theorems you've learned doesn't render them useless (although perhaps less useful as you're less likely to know precisely when they can be applied and how to extend them to new situations), but in the context of the discussion it seems like an important distinction to make.
paulpauper 10y ago
One problem is understanding the notation, as sometimes steps are omitted. I remember one time spending 10 minutes trying to figure out what an author meant, only to learn later he was using something called a 'total derivative'. this means that variables like x,y are actually functions of time . Having a professional simply explain it instead of having to infer the meaning from the author would save a lot of time
- kpil 10y ago
  As someone that learned some math in university about 20 years ago, and probably have forgotten most of it, I have a hard time when reading something mathematical that interrests me today. Maybe the authors of the papers that I read aren't always that pedagogical, and I get totally lost when someone tosses in a variable only to half-heartedly define what it is a page later. I think it's mostly due to that I suck at math, and need to figure out obvious things on my own - but perhaps also due to my programmer-view of the world were you typically define things before you use them... But learning on your own is probably hard. I got irritated once when I needed some not totally trivial transformations for a GIS application. I spent some evenings repeating from my old books, but it was unfathomably boring, so I gave up as soon as I got my transformation working :-) c
mungoid 10y ago
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 being a math tutor. Almost none of the math I use today as a physics developer was learned from schools. I also never had that piece of mind you mentioned, because I was constantly juggling several things at once while going to school. My knowledge of advanced mathematics at the time of my graduation was pretty non existent. I think the most advanced math I had was Algebra 2 or something like that, and the Professors just basically read verbatim from the book. A few years after Uni, I started teaching myself Calc, Trig, Vector maths, Diff Eq and Physics strictly from what I have found on various sites, software and books. Because of that, I ended up getting a physics simulation developer position at a software company. Because in my companies view, being able to teach yourself all that math is much more impressive than being taught from a University. I hated math during High School and College, but since then, I have found that I absolutely love math, and I will never stop trying to learn or do new things. My degree was two small lines on my CV, while about 50% of what I had on my CV was all learned on my free time, by myself. So learning math without a College or University is totally possible, and in my situation, worked way better. Sites like Khan Academy, Wolfram, Youtube, etc. all give you the resources and leave it up to you to progress at your own pace, for free.
- yodsanklai 10y ago
  Congratulations, and I'm glad you proved me wrong :)
  - mungoid 10y ago
    Thank you! I will admit that it was by no means quick or easy and there were many, many times I wish I could have had an actual person with me to explain it. Not to mention the frequent of wanting to give up when something wasn't 'clicking' and i felt i couldn't do it.
- heimatau 10y ago
bitL 10y ago
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 look at how is it taught in discrete mathematics. I remember myself being really upset after 5-year study of theoretical backgrounds and some things finally clicked and I realized how simple they were and how much was just a ballast to reach them. Often mathematicians invent a theory in their teens and spend the rest of their lives to fight with unexpected monsters in boundary conditions they created. Similar to making a distributed middleware backbone and then debugging it with all unexpected network error/split brain stuff coming in.
- hgh 10y ago
  Could you explain the principle in terms a 5-year old would understand?
  - _w7am 10y ago
    If you have n boxes and n+1 things, then it is guaranteed that one box has more than one thing.
- deleted 10y ago
  [deleted]
- xamuel 10y ago
  It's like this. If you're on a C++ team where the whole team heavily uses C++'s features as well as Boost, then you should write your code accordingly. This'll make your code more concise, and clearer to the other members of the team. At the same time, it'll make it much more obfuscated to, say, a C programmer.
  - bitL 10y ago
    It's more like everybody is forced to use C++ for everything. Would you rather write a distributed transactional system in C++/CORBA or in Java? How much time are you going to lose by choosing C++ and in debugging it comparing to Java? Or even substitute Assembly language for C++. (Disclaimer: I love C++, this is just an example)
ianai 10y ago
I would argue it's difficult to unlikely. People who attempt it though are sure to better themselves and probably their work output.
thearn4 10y ago
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.
clarry 10y ago
I find that educational institutions are incredibly good at killing motivation. And I can't have peace of mind knowing that only the constantly accumulating debt helps me pay the bills during a five or six-year session that I might not be able to endure due to the said killing of motivation. And that even if I graduated with a degree, it wouldn't automatically secure a job I like or a happy life.

elorant 10y ago

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.

sotojuan 10y ago
This is a good point. Many of us learn a new language or tool as a hobby after work. Why not consider math like that?
FabHK 10y ago
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 https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
icu 10y ago
I agree wholeheartedly! Math is the queen of the sciences, or the silent language of the world around us. Learning math outside of high school has also helped me identify 'snake oil' statistics in my industry and challenge the validity of information I've been presented as fact.

Chris2048 10y ago

What I actually need is "learning math in a short time", With a few, minimal yet illustrative examples (ala katas), plenty diagrams/illustrations and other mental aids, an no rigour - not a single bit of set-theory! The proofs and rigor can come later... Incidentally, I'm a dev in finance, looking to move into quant dev. I have a math degree (completed 2007) and I'm doing a "CQF" to catch up with the relevant quant knowledge. I think cyptography, and stats/data analytics might also be a good area for mathy-dev.

heimatau 10y ago
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 class would like like 'this is an equation...here is what we can do with it (derivatives, areas/3D/4D, etc)...but...99.99% of you won't need to know it this way. You need to use math in a way that indirectly teaches you how to creatively look at problems in life.' I'm not sure why everyone is forced to learn math without knowing WHY they are forced to know it. Creative problem solving is one of the best takeaways, imho, for the masses. As for Adv. Math...I think it's not effective for most people's career paths and skillset they will require in the real world.
- mmarx 10y ago
  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 intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.« [0] https://terrytao.wordpress.com/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there’s-more-to...

zamalek 10y ago

> I wonder if it's even possible. From a practical perspective it definitely is. I've picked up a fair amount of graph theory and with nothing but extreme persistence have grokked and used some fairly advanced stuff[1][2] (2nd-year dropout). It was, however, work-related. Just don't ask me to proof anything. > the few lucrative jobs that make use of maths are in finance There is also competency on the table here. Graph theory crops up day-to-day with the business software work I'm doing (three separate deliverables). Calculus is used to a point of absurdity in game development - e.g. the Fresnel term. Machine learning? Calculus, linear algebra, tensors. Profiling? A basic understanding of statistics. Compilers? Category theory, graph theory. Physics engines? ODE. It's extremely valuable to know this stuff. [1]: https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_... [2]: https://eprint.iacr.org/2012/352.pdf https://eprint.iacr.org/2012/352.pdf

ryanmonroe 10y ago
>Just don't ask me to proof anything I don't mean to offend, but being able to prove things is generally the main focus of advanced mathematics. If you can't prove what you know, or at least have a rough outline of a proof you could construct after referring to something, you haven't learned it in the same way those at a university have.
- zamalek 10y ago
  No offense taken, you are completely correct. I can, however, still explain it and use it.
- empath75 10y ago
  And? You can use advanced math without having a deep understanding of it.
  - ryanmonroe 10y ago
    I just wanted to point it out. The thread title is "Learn Advanced Mathematics Without Heading to University", which could be read with the implication "Learn advanced mathematics, at the university level, without going to university". Under this context, someone replied that this is possible because s/he learned it, but can't prove what s/he's learned. Of course not being able to prove the useful theorems you've learned doesn't render them useless (although perhaps less useful as you're less likely to know precisely when they can be applied and how to extend them to new situations), but in the context of the discussion it seems like an important distinction to make.

paulpauper 10y ago

One problem is understanding the notation, as sometimes steps are omitted. I remember one time spending 10 minutes trying to figure out what an author meant, only to learn later he was using something called a 'total derivative'. this means that variables like x,y are actually functions of time . Having a professional simply explain it instead of having to infer the meaning from the author would save a lot of time

kpil 10y ago
As someone that learned some math in university about 20 years ago, and probably have forgotten most of it, I have a hard time when reading something mathematical that interrests me today. Maybe the authors of the papers that I read aren't always that pedagogical, and I get totally lost when someone tosses in a variable only to half-heartedly define what it is a page later. I think it's mostly due to that I suck at math, and need to figure out obvious things on my own - but perhaps also due to my programmer-view of the world were you typically define things before you use them... But learning on your own is probably hard. I got irritated once when I needed some not totally trivial transformations for a GIS application. I spent some evenings repeating from my old books, but it was unfathomably boring, so I gave up as soon as I got my transformation working :-) c

mungoid 10y ago

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 being a math tutor. Almost none of the math I use today as a physics developer was learned from schools. I also never had that piece of mind you mentioned, because I was constantly juggling several things at once while going to school. My knowledge of advanced mathematics at the time of my graduation was pretty non existent. I think the most advanced math I had was Algebra 2 or something like that, and the Professors just basically read verbatim from the book. A few years after Uni, I started teaching myself Calc, Trig, Vector maths, Diff Eq and Physics strictly from what I have found on various sites, software and books. Because of that, I ended up getting a physics simulation developer position at a software company. Because in my companies view, being able to teach yourself all that math is much more impressive than being taught from a University. I hated math during High School and College, but since then, I have found that I absolutely love math, and I will never stop trying to learn or do new things. My degree was two small lines on my CV, while about 50% of what I had on my CV was all learned on my free time, by myself. So learning math without a College or University is totally possible, and in my situation, worked way better. Sites like Khan Academy, Wolfram, Youtube, etc. all give you the resources and leave it up to you to progress at your own pace, for free.

yodsanklai 10y ago
Congratulations, and I'm glad you proved me wrong :)
- mungoid 10y ago
  Thank you! I will admit that it was by no means quick or easy and there were many, many times I wish I could have had an actual person with me to explain it. Not to mention the frequent of wanting to give up when something wasn't 'clicking' and i felt i couldn't do it.
heimatau 10y ago

bitL 10y ago

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 look at how is it taught in discrete mathematics. I remember myself being really upset after 5-year study of theoretical backgrounds and some things finally clicked and I realized how simple they were and how much was just a ballast to reach them. Often mathematicians invent a theory in their teens and spend the rest of their lives to fight with unexpected monsters in boundary conditions they created. Similar to making a distributed middleware backbone and then debugging it with all unexpected network error/split brain stuff coming in.

hgh 10y ago
Could you explain the principle in terms a 5-year old would understand?
- _w7am 10y ago
  If you have n boxes and n+1 things, then it is guaranteed that one box has more than one thing.
deleted 10y ago
[deleted]
xamuel 10y ago
It's like this. If you're on a C++ team where the whole team heavily uses C++'s features as well as Boost, then you should write your code accordingly. This'll make your code more concise, and clearer to the other members of the team. At the same time, it'll make it much more obfuscated to, say, a C programmer.
- bitL 10y ago
  It's more like everybody is forced to use C++ for everything. Would you rather write a distributed transactional system in C++/CORBA or in Java? How much time are you going to lose by choosing C++ and in debugging it comparing to Java? Or even substitute Assembly language for C++. (Disclaimer: I love C++, this is just an example)

ianai 10y ago

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

thearn4 10y ago

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.

clarry 10y ago

I find that educational institutions are incredibly good at killing motivation. And I can't have peace of mind knowing that only the constantly accumulating debt helps me pay the bills during a five or six-year session that I might not be able to endure due to the said killing of motivation. And that even if I graduated with a degree, it wouldn't automatically secure a job I like or a happy life.