I think this is something like what you're asking for
How to Become a Pure Mathematician (or Statistician) http://hbpms.blogspot.com/
Ask HN: How to self-study mathematics from the undergrad through graduate level?
101–110 of 231 posts
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#102Hello, I have actually done this. I learned Algebra up to a good amount of Vector Calculus over the course of four years mostly through self-study. I would leave for work an hour early and either sit in my car or go into a Starbucks and do math. Doing time before work is important. That's when you are at your best. Then after work I would sit in my car and do math for an hour. Then on the weekends, in the morning, I…
Bet you weren't married at the time. I'm trying to re-learn calculus myself, and I have to hide it from her because she gets mad at me when I try to do calculus problems: "why are you doing this? Are you doing this for work? You don't have to do this. There's no reason for you to be doing this."
"Why are you doing this?"
I'm trying to figure out where I went wrong with my life by retracing my steps. Starting with when we met in freshman calculus.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#103I started doing this, and the first stumbling block I hit was simply assessing at what level my math education falls apart and figuring out what courses I needed to refresh with, what materials I needed etc.
There's a market opportunity for building a math placement test product, or even better a fully automated math education platform with examinations at each stage, that takes you from K-12 refreshers all the way up to grad school. Something a bit more advanced than the existing online learning programs that are still very much based on the college lecture model.
There's a lot of products out there that do this for software development, but not for other subjects. I have noticed there are bits and pieces I can download as mobile apps, but they are designed to be supplements to certain college curriculum rather than an end-to-end education.
Obviously math is a subject where software is an optimal instructional tool, so much more could be done here
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#104As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards. There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning: At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which…
I've long wished for a series of interactive ebooks or websites based on that type of approach. Each would be devoted to one great theorem or problem.
Each would start with a presentation of the theorem and proof, presented how it would be presented today if it were a newly discovered research result being presented by professionals to professionals in the field.
At each step of the presentation, there would be two expansion options. One is to ask for filling in the details. A detail expansion keeps the presentation at about the same level of required knowledge, but takes smaller steps. You use a detail expansion when you understand where a step starts and end, but you just don't quite see how it made the connection.
The other expansion option is to ask for background or prerequisites. A background expansion is for when you don't have the background to even understand the start and end points of a step. It opens up material to teach you the background necessary to understand what is going on.
A key aspect is that this would all be recursive. You could do a background expansion on a background expansion, and so on, all the way back to common high school math.
The background expansions would just teach enough of their subject to support the step above. So, for example, if you used one of these interactive books to learn an analytic proof of the prime number theorem, and you started knowing nothing beyond high school algebra, you would end up learning all the calculus and complex analysis needed to prove PNT, but only such calculus and complex analysis as are needed.
What I wonder is if you could pick a set of theorems and problems for such books such that (1) someone could go through them all in about the same time as a conventional math degree takes, and (2) combined, the background expansions would have covered as much as a conventional degree.
If so, that could be an interesting way to keep motivation high because everything you are learning has a direct, visible, connection to advancing the proof of the interesting theorem at the top.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#105SOME ADVICE BEFORE YOU DO A DEEP DIVE INTO WHATEVER YOU END UP STUDYING MATH-WISE:
My Background: Current Undergraduate in CS and I recently added Mathematics
The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on mathematical thinking and proof techniques before/during/after reading any other dense math book.
Like you, I realize the value of having a mathematical mindset and want to have a deep understanding. When I added math as a major, I had a very hard time jumping from computational courses (typical math courses, geared towards any major) to theoretical and conceptual courses (proof-based courses that use all the fun and interesting math books everyone has linked here). These books helped:
https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat...
https://www.amazon.com/How-Think-Like-Mathematician-Undergra...
https://press.princeton.edu/titles/669.html
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#106There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices. Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically comput…
Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.
Neither did most historians, or most apple farmers. What's your point?
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#107First I think you need to think about a direction. "Learn Math" is a very broad and vague statement. You will never learn all of math. So you really need to think about why you're wanting to learn it, and pick a direction that suits that.
If you're wanting to learn math just because you like the subject, then it's wise to get a sampling of a wide array of different areas. Then you have a better chance of finding an area that you like best.
If you're goal is to learn it so you can apply it in some other area, you should focus mostly on learning what math applies in that area. Then find the prerequisites and start studying.
Studying alone in your spare time will be nothing at all like attending a university. The full immersion you get to focusing primarily on one thing, and the ability to engage others at will also focusing on the same thing is a great help in learning. So set your expectations right. You won't be able to reach the level of understanding a 60 year old math professor has. Since he/she has been fully immersed in the topic for probably 40-50 years. So set your expectations accordingly.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#108There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices. Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically comput…
Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.
From what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it.
Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#109I'm actually building an app specifically for this use case: https://getpolarized.io/ Polar is basically a personal knowledge repository. The idea is that you use Polar for all of your education. Either official education or continued learning. Every textbook or technical paper should go into Polar. You can then annotate the textbooks directly including text highlights, comments, etc. It supports spaced repetition sy…
Learning math isn't really about memorizing things.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#110Earlier quoted context omitted.
I always got more from working problems on my own with an occasional consult from peers or a teacher. A danger is that you think your group sessions substitute for working problems.
group sessions keep you honest and help you work through high level stumbling blocks. You're expected/obligated to do a fair amount of solo pick-and-shovel work in between.
Well in my experience they are a waste of time, but that's just me.