Continuing higher math as a hobby
September 3, 2020 11:55 AM Subscribe
If I find higher math fun but I'm not gonna do a degree in it, how else could I indulge in it as a hobby?
I really enjoyed the math courses I did as part of my computer science degree. I'd love to learn more math, and I'm intrigued by tales of people "doing" math. It's the kind of thing I'd want to indulge or dabble in - learn a new area of math (topography?), maybe try to create a proof of something already known (ie independently work out how to figure it out. I say already known because I think it's wildly j realistic I'd discover something new with my limited background and time...)
However, outside of doing a math degree - not relevant, and that includes auditing one - I don't know where/how to scratch the itch. Are there grad level math courses available online? Are there places people discuss and share proofs? Mathy reading materials I could follow?
(for the attempting proofs bit of the question, I guess I'm sort of imagining how in bioinformatics there's the project - name escapes me - where you get assigned progressively harder bioinformatics problems to try to solve. But I feel like in math it would help to be aware of other proven theorems I could use as building blocks...)
I really enjoyed the math courses I did as part of my computer science degree. I'd love to learn more math, and I'm intrigued by tales of people "doing" math. It's the kind of thing I'd want to indulge or dabble in - learn a new area of math (topography?), maybe try to create a proof of something already known (ie independently work out how to figure it out. I say already known because I think it's wildly j realistic I'd discover something new with my limited background and time...)
However, outside of doing a math degree - not relevant, and that includes auditing one - I don't know where/how to scratch the itch. Are there grad level math courses available online? Are there places people discuss and share proofs? Mathy reading materials I could follow?
(for the attempting proofs bit of the question, I guess I'm sort of imagining how in bioinformatics there's the project - name escapes me - where you get assigned progressively harder bioinformatics problems to try to solve. But I feel like in math it would help to be aware of other proven theorems I could use as building blocks...)
Not sure what your background is, I'm assuming you have some university level math? I would try working through a textbook. I know that in my experience in math and physics, textbooks were used more as references that supplement that course material. But many of them are designed to be worked through front-to-back. I did this with a physics textbook after finishing school and got a lot out of it.
posted by no regrets, coyote at 12:06 PM on September 3, 2020 [2 favorites]
posted by no regrets, coyote at 12:06 PM on September 3, 2020 [2 favorites]
The two math challenges that I know about are Project Euler and the Wolfram Challenges.
I've also enjoyed 3blue1brown on youtube.
posted by Phredward at 12:08 PM on September 3, 2020 [4 favorites]
I've also enjoyed 3blue1brown on youtube.
posted by Phredward at 12:08 PM on September 3, 2020 [4 favorites]
There's lots of great youtube content. Already suggested above are numberphile and 3blue1brown and I'll add mathologer.
posted by Obscure Reference at 12:41 PM on September 3, 2020 [3 favorites]
posted by Obscure Reference at 12:41 PM on September 3, 2020 [3 favorites]
There's this AMS A Page A Day calendar that you might enjoy -- keep your eyes peeled for the 2021 edition?
posted by batter_my_heart at 12:43 PM on September 3, 2020 [1 favorite]
posted by batter_my_heart at 12:43 PM on September 3, 2020 [1 favorite]
In case it hits the right sort of spot for you, I really enjoyed this Youtube series on measure theory - for me it hit the right level of detail and clarity and each part is fairly short.
If you want a book you can browse in, I highly recommend The Princeton Companion to Mathematics - it has great articles on all sorts of areas of maths and may give you more ideas about what parts of maths you'd find interesting to learn more about. The editor, Timothy Gowers is on Twitter - if you use Twitter there are other good follows in maths twitter.
posted by crocomancer at 1:29 PM on September 3, 2020 [1 favorite]
If you want a book you can browse in, I highly recommend The Princeton Companion to Mathematics - it has great articles on all sorts of areas of maths and may give you more ideas about what parts of maths you'd find interesting to learn more about. The editor, Timothy Gowers is on Twitter - if you use Twitter there are other good follows in maths twitter.
posted by crocomancer at 1:29 PM on September 3, 2020 [1 favorite]
Petra Bonafert-Taylor has an excellent course in complex analysis. And another vote for the Mathologer.
posted by phliar at 1:53 PM on September 3, 2020 [1 favorite]
posted by phliar at 1:53 PM on September 3, 2020 [1 favorite]
This is a topic I'm also interested in. One of the best things you can do is get yourself a textbook or two and start doing some problems! Here are some threads I bookmarked on Hacker News that have a lot of good suggestions:
Ask HN: How to Study Mathematics?
Ask HN: How to self-study mathematics from the undergrad through graduate level?
Ask HN: Are there books for mathematics like Feynman's lectures on physics?
Ask HN: Best resources to gain math intuition?
Bonus: Ask HN: How to self-study physics?
(You can also check out MIT OpenCourseWare here)
posted by stinkfoot at 1:57 PM on September 3, 2020 [1 favorite]
Ask HN: How to Study Mathematics?
Ask HN: How to self-study mathematics from the undergrad through graduate level?
Ask HN: Are there books for mathematics like Feynman's lectures on physics?
Ask HN: Best resources to gain math intuition?
Bonus: Ask HN: How to self-study physics?
(You can also check out MIT OpenCourseWare here)
posted by stinkfoot at 1:57 PM on September 3, 2020 [1 favorite]
With a programming background its definitely easier to scratch that itch! Project Euler as linked to above is a good one. Have you ever considered any data science? I love the stuff but I get bogged down in the theory, which may be what you're looking for? Some of the exercises at Kaggle might be fun for you!
posted by cgg at 1:59 PM on September 3, 2020
posted by cgg at 1:59 PM on September 3, 2020
Another great YouTube channel is Black Pen Red Pen.
posted by wittgenstein at 3:50 PM on September 3, 2020 [1 favorite]
posted by wittgenstein at 3:50 PM on September 3, 2020 [1 favorite]
Check out Talk Math With Your Friends and the associated YouTube channel. This is a weekly "virtual colloquium" on math and math education. The talks are pitched for a broad audience, and they allow much more time for questions than many lecture series.
posted by yarntheory at 3:54 PM on September 3, 2020
posted by yarntheory at 3:54 PM on September 3, 2020
For reading materials, one possible source is the Mathematical Association of America (MAA), which was founded to support mathematical explanations. The MAA gives a bunch of prizes for mathematical writing, and the associated pages include links to many of the papers:
posted by yarntheory at 4:00 PM on September 3, 2020
posted by yarntheory at 4:00 PM on September 3, 2020
The accepted answers to the questions on https://math.stackexchange.com/ may be interesting to a hobbyist. There's plenty to choose from and you can always find something that stretches your mind a little. ( Note that I'm not talking about answering the questions, although that probably also be educational - just following the logic of the answers. )
posted by metadave at 4:26 PM on September 3, 2020 [1 favorite]
posted by metadave at 4:26 PM on September 3, 2020 [1 favorite]
My father is a mathematician at a university. The math dept. is -- in non-covid times -- constantly having guests in from other universities to discuss their research in colloquia that are open to the public. Once you are familiar with a specific branch of math through self-study or however so that you could follow the lecture, I'd think attending such events where you ask questions and mingle with mathematicians would be fun.
Math at university level seems to be divided into branches that don't necessarily communicate much with each other. My father is in logic and set theory, for instance, and doesn't often cross paths with group theorists or topologists, say. Picking a particular branch and getting deep into it might make it more fun. And from what I can tell, proving theorems and knowing what theorems are out there needing proofs and how all the mess of proved and unproved theorems relate is the bread and butter of mathematics, so practicing and getting good at proving stuff is probably your first step. (There's even a branch of math called proof theory.)
posted by bertran at 6:00 PM on September 3, 2020 [2 favorites]
Math at university level seems to be divided into branches that don't necessarily communicate much with each other. My father is in logic and set theory, for instance, and doesn't often cross paths with group theorists or topologists, say. Picking a particular branch and getting deep into it might make it more fun. And from what I can tell, proving theorems and knowing what theorems are out there needing proofs and how all the mess of proved and unproved theorems relate is the bread and butter of mathematics, so practicing and getting good at proving stuff is probably your first step. (There's even a branch of math called proof theory.)
posted by bertran at 6:00 PM on September 3, 2020 [2 favorites]
I asked my mathematician father about your question. He thought coming from computer science you might get a lot out of studying a certain unsolved problem in computability known as the p=np problem. It's unsolved but also a very deep problem, so studying it would give you a lot of insight into mathematics more broadly, I gather.
posted by bertran at 7:05 PM on September 3, 2020
posted by bertran at 7:05 PM on September 3, 2020
Free Textbooks might find you something to read. Maybe somebody pointed out some good math books out of the lot.
The Universe of Discourse: category 'math', I started following mjd because he was Just Another Perl Hacker who wrote the fantastic Higher Order Perl book, but soon realized that a ton of his posts are math and he's always answering things on math.stackexchange.com that are WHOOSH! over my head. I still have a Group Theory problem that he left when signing my copy of HOP that I'll get around to someday.
Stand-up Maths is another of the standard math YouTubers, he has a second channel that's now doing hard puzzle like stuff. But like Nuberphile it's a bit along the lines of math engagement without going too deep into gory details.
Nth 3blue1brown and Mathologer for sorta getting into details in different ways.
posted by zengargoyle at 8:03 PM on September 3, 2020
The Universe of Discourse: category 'math', I started following mjd because he was Just Another Perl Hacker who wrote the fantastic Higher Order Perl book, but soon realized that a ton of his posts are math and he's always answering things on math.stackexchange.com that are WHOOSH! over my head. I still have a Group Theory problem that he left when signing my copy of HOP that I'll get around to someday.
Stand-up Maths is another of the standard math YouTubers, he has a second channel that's now doing hard puzzle like stuff. But like Nuberphile it's a bit along the lines of math engagement without going too deep into gory details.
Nth 3blue1brown and Mathologer for sorta getting into details in different ways.
posted by zengargoyle at 8:03 PM on September 3, 2020
Oh, and if you haven't yet, you should probably read Gödel, Escher, Bach. It rolls a bunch of things math, programing, unproveability, formal systems, into an interesting mess.
posted by zengargoyle at 8:09 PM on September 3, 2020 [1 favorite]
posted by zengargoyle at 8:09 PM on September 3, 2020 [1 favorite]
I recommend the book Measurement by Paul Lockhart. Don't be put off by the unassuming title. It is an unconventional development of math from simple geometry through calculus and differential equations, in a very readable narrative that is totally unlike the textbooks on these topics. Lots of hand drawn diagrams on almost every page.
posted by JonJacky at 11:23 PM on September 3, 2020 [4 favorites]
posted by JonJacky at 11:23 PM on September 3, 2020 [4 favorites]
It's definitely possible to be an amateur mathematician. I have a friend who is a playwright who has published a few papers in number theory, despite having little formal training. So go for it!
Depending on what you want to do, there are a few options.
1. Watching videos online (e.g. 3blue1brown) are very engaging and informative, but I do think that there is a limit as to how much you can learn from them. This leads to...
2. Working through a textbook of some kind is probably be best way to really "get" mathematics. You would be doing this anyhow if you were taking courses at a university, and you would have a community of classmates who would be there to struggle through them with you. This is one of the challenges of being self-taught.
However, I think that watching good videos for inspiration, plus working through good textbook problems, plus asking questions when you are stuck on math.stackexchange is likely a pretty good combination.
So what textbooks should you consider? That's a good question, and it strongly depends on what you find interesting. Here are a few that come to mind:
Principles of mathematical analysis: If you like calculus, this is an excellent and challenging book. Our year two honours calculus stream was based on this. It starts with defining what exactly real numbers are (depending on the edition, you get two very different definitions, but they are equivalent), and then works out what it means to do single and multivariable calculus. It's tough, but excellent.
Abstract algebra and solutions by radicals: I think this is the book that we used for a third-year course in Galois theory. If you haven't heard about it, Galois theory is one of the coolest ideas in all of mathematics. It's classical (stretches back to ancient times), one could argue that it motivated almost all of modern mathematics, it has a great tragic tale behind it (read about Évarist Galois), and it's just a beautiful set of ideas. If you ever looked at the quadratic equation and wondered where it comes from and what it means, and if you could extend it, then this is absolutely worth reading.
Category theory in context: As someone with a compsci background, category theory may also appeal to you. This is about relationships between different mathematical "objects". This is also an excellent book with a great collection of problems throughout, and it builds motivation for why category theory is A Thing. It may be a bit more advanced, at least in its expected familiarity with ideas from other parts of mathematics. But it's a great book.
On numbers and games: This is a strange one, and I have to admit that I have only started looking at it recently. It's by the late John Conway, and explains his construction of the Surreal Numbers. If you go through the first book above and find the actual definition of the Real number line to be weird, then these will blow your mind. But I think that other than at a challenging conceptual level, this is actually a pretty low-prerequisite book.
Anyhow, I'll leave it at that for now. Best of luck, and if you have any questions, feel free to memail me!
posted by vernondalhart at 11:25 PM on September 3, 2020 [4 favorites]
Depending on what you want to do, there are a few options.
1. Watching videos online (e.g. 3blue1brown) are very engaging and informative, but I do think that there is a limit as to how much you can learn from them. This leads to...
2. Working through a textbook of some kind is probably be best way to really "get" mathematics. You would be doing this anyhow if you were taking courses at a university, and you would have a community of classmates who would be there to struggle through them with you. This is one of the challenges of being self-taught.
However, I think that watching good videos for inspiration, plus working through good textbook problems, plus asking questions when you are stuck on math.stackexchange is likely a pretty good combination.
So what textbooks should you consider? That's a good question, and it strongly depends on what you find interesting. Here are a few that come to mind:
Principles of mathematical analysis: If you like calculus, this is an excellent and challenging book. Our year two honours calculus stream was based on this. It starts with defining what exactly real numbers are (depending on the edition, you get two very different definitions, but they are equivalent), and then works out what it means to do single and multivariable calculus. It's tough, but excellent.
Abstract algebra and solutions by radicals: I think this is the book that we used for a third-year course in Galois theory. If you haven't heard about it, Galois theory is one of the coolest ideas in all of mathematics. It's classical (stretches back to ancient times), one could argue that it motivated almost all of modern mathematics, it has a great tragic tale behind it (read about Évarist Galois), and it's just a beautiful set of ideas. If you ever looked at the quadratic equation and wondered where it comes from and what it means, and if you could extend it, then this is absolutely worth reading.
Category theory in context: As someone with a compsci background, category theory may also appeal to you. This is about relationships between different mathematical "objects". This is also an excellent book with a great collection of problems throughout, and it builds motivation for why category theory is A Thing. It may be a bit more advanced, at least in its expected familiarity with ideas from other parts of mathematics. But it's a great book.
On numbers and games: This is a strange one, and I have to admit that I have only started looking at it recently. It's by the late John Conway, and explains his construction of the Surreal Numbers. If you go through the first book above and find the actual definition of the Real number line to be weird, then these will blow your mind. But I think that other than at a challenging conceptual level, this is actually a pretty low-prerequisite book.
Anyhow, I'll leave it at that for now. Best of luck, and if you have any questions, feel free to memail me!
posted by vernondalhart at 11:25 PM on September 3, 2020 [4 favorites]
The FiveThirtyEight Riddler is a fun recreational-math column (weekly, Fridays) that often requires college-level math and interesting combinatorial techniques (as well as computational solutions from time to time). It'd probably be right up your alley.
posted by Johnny Assay at 4:00 AM on September 4, 2020
posted by Johnny Assay at 4:00 AM on September 4, 2020
Check out the math page on Khan Academy. Self pace, an unbelievable number of courses, from 1st and 2nd grade basics but then goes really deep, way into college courses. I think (not certain but I think so) I think that they even have volunteers to tutor you one on one in areas you might get stuck in.
A *lot* of great arithmetic and mathematics courses -- the school is weighted in that direction -- but also a bunch of physics, and some history done quite well; it's sortof a teaser, I'd love to follow where this guy leads on most any topic
All of these courses online are flattening the differences, students all over the world; I knew a bunch of computer programmers from India, and also some from China. Also electrical engineers. A lot of them had to share a textbook(s), sometimes they'd xerox the entire book and in that way going from 27 textbooks but now there are 43 more books. But now with everything online, well, everyone has a cell phone or tablet. The competition to get into great schools there is fierce, not like here.
I'm going on -- sorry. But check out Khan Academy.
posted by dancestoblue at 5:28 AM on September 4, 2020
A *lot* of great arithmetic and mathematics courses -- the school is weighted in that direction -- but also a bunch of physics, and some history done quite well; it's sortof a teaser, I'd love to follow where this guy leads on most any topic
All of these courses online are flattening the differences, students all over the world; I knew a bunch of computer programmers from India, and also some from China. Also electrical engineers. A lot of them had to share a textbook(s), sometimes they'd xerox the entire book and in that way going from 27 textbooks but now there are 43 more books. But now with everything online, well, everyone has a cell phone or tablet. The competition to get into great schools there is fierce, not like here.
I'm going on -- sorry. But check out Khan Academy.
posted by dancestoblue at 5:28 AM on September 4, 2020
If you completed a CS degree and took math along the way, I don't think Khan academy will be nearly enough for you -- it's doesn't go beyond linear algebra and calculus (single and multivariable) at the college level which are courses most engineering majors take (at my school, most people took all of that freshman year). I have a math degree and Khan is far too basic for me for reference.
Try MIT OCW -- lots of class videos and materials online in full. Other major universities also put courses online in full. I think it's best to choose a course that has solved HW/exam problems so you can check your work because as you get deeper into math, problems get very difficult to solve without a ton of time and effort.
posted by shaademaan at 5:43 AM on September 4, 2020 [2 favorites]
Try MIT OCW -- lots of class videos and materials online in full. Other major universities also put courses online in full. I think it's best to choose a course that has solved HW/exam problems so you can check your work because as you get deeper into math, problems get very difficult to solve without a ton of time and effort.
posted by shaademaan at 5:43 AM on September 4, 2020 [2 favorites]
A view observations: There are two math sites on StackExchange. MathOverflow is for research-level questions. It may be fun to look there, but it's not the place to ask the easy questions. Mathematics.StackExchange is open to homework questions on up. One way to do math as a hobbyist is to answer questions in areas you know about, or can brush up on as needed. I sometimes answer mostly math-based questions about Math.NET Numerics on their own board or on StackExchange.
Numberphile is interesting, but it spends most of its energy on simple Number Theory problems and math tricks. The explanations on Computerfile are more meaty, especially in the area of cryptography and data security. The videos on deep learning are also interesting, and illustrate a point. This area of research in not yet fully mature, i.e. there is still fairly rapid progress being made, and as such as left some unsolved problems behind. It produces helpful, but inexplicable, results. It might be interesting to explain results better starting with the simplest examples.
Back 15 years or so, my daughter spent a summer as an undergraduate doing research in topology. It was explained to me that topology, or some parts thereof, still have enough simple problems outstanding to make that possible.
Finally, I think it would be profitable to pick a possible area or two to focus on. No one knows everything. I also think it's unlikely to obsess on any of the great unsolved problems like p=np or the Reimann Hypothesis. They're just too hard. But they may point you to an area of math you can thrive in.
posted by SemiSalt at 6:52 AM on September 4, 2020
Numberphile is interesting, but it spends most of its energy on simple Number Theory problems and math tricks. The explanations on Computerfile are more meaty, especially in the area of cryptography and data security. The videos on deep learning are also interesting, and illustrate a point. This area of research in not yet fully mature, i.e. there is still fairly rapid progress being made, and as such as left some unsolved problems behind. It produces helpful, but inexplicable, results. It might be interesting to explain results better starting with the simplest examples.
Back 15 years or so, my daughter spent a summer as an undergraduate doing research in topology. It was explained to me that topology, or some parts thereof, still have enough simple problems outstanding to make that possible.
Finally, I think it would be profitable to pick a possible area or two to focus on. No one knows everything. I also think it's unlikely to obsess on any of the great unsolved problems like p=np or the Reimann Hypothesis. They're just too hard. But they may point you to an area of math you can thrive in.
posted by SemiSalt at 6:52 AM on September 4, 2020
In addition to MIT OCW, if you go to the university web pages of random math professors, there is a good chance some of them will have course websites for courses you are interested in, which sometimes include homework problems and solutions, and commonly very detailed typed lecture notes. So looking at university course catalogues for classes you might be interested in is a good place to start, as is just Googling the name of a common course ("measure theory") or the title of a popular textbook in that course plus "syllabus".
Here's one example of what I mean, on a random topic. A ton of courses have pages like this.
One thing I haven't been able to find is a good venue to discuss and ask questions about mathematical topics. I think most enrolled students are asking questions of each other on forums internal to their classes, so there's no place to have a lively discussion elsewhere. math.stackexchange.com is okay, but has frustrating Stack Overflow moderation + is overrun with people trying to get someone else to do their homework.
posted by vogon_poet at 7:55 AM on September 4, 2020 [1 favorite]
Here's one example of what I mean, on a random topic. A ton of courses have pages like this.
One thing I haven't been able to find is a good venue to discuss and ask questions about mathematical topics. I think most enrolled students are asking questions of each other on forums internal to their classes, so there's no place to have a lively discussion elsewhere. math.stackexchange.com is okay, but has frustrating Stack Overflow moderation + is overrun with people trying to get someone else to do their homework.
posted by vogon_poet at 7:55 AM on September 4, 2020 [1 favorite]
Response by poster: Wow, thank you so much everyone.
posted by Cozybee at 10:09 AM on September 5, 2020
posted by Cozybee at 10:09 AM on September 5, 2020
This thread is closed to new comments.
posted by jozxyqk at 12:05 PM on September 3, 2020 [2 favorites]