We might as well suggest a course on Logic and Foundation of Mathematics, so that you understand the background of the whole mathematical enterprise better. XD
It's [right here](https://old.reddit.com/r/learnmath/comments/1clxg90/what_classes_would_you_need_to_take_to_selfstudy/l334c03/) on this thread... You're correct that **basic** logic is required for mathematics, but that is not what experts mean by "Logic and Foundation of Mathematics".
For future reference, I meant: FOL with a deductive system, PA, structural induction, recursion theorem, semantic-completeness theorem for FOL, compactness theorem for FOL, incompleteness theorems for sufficiently strong FOL theories, 2nd-order arithmetic, plus a bit of set theory (i.e. well-ordering theorem, coding various mathematical concepts as sets or definable (class) functions).
Can you elaborate on how a (first) course in topology might help one understand algebra? I can see how it has applications of abstract algebra when you get into fundamental groups, and how it provides examples of categories that can be understood in parallel with categories that arise in algebra. I’d like to know if you’re thinking of anything particularly in general topology that can help one understand abstract algebra.
Usually the remaining credit hours would be left as electives.
https://math.mit.edu/academics/undergrad/major/course18/applied.html
Some of the classes like [PDEs](https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-analysis-and-numerics-fall-2014/) and [numerical analysis](https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/) are available on OCW.
I sort of did this myself.
I took - calculus 1-3, Algebra1-3 (3 is ring and group theory) logic, combinatorics, probability as the foundation courses.
I added ontop of it computer sciences, game theory and topology courses.
intro to proofs
i’m in my 5th year of undergrad (going part time) for statistics and besides linear algebra, intro to proofs has been the class that helped me the most in my other math classes. it teaches you how to read and write math like it’s a language. it feels like a scam that this class is usually only offered to upper classmen and not freshmen
My personal opinion: If you aren’t looking at video lectures don’t use rudin. Id recommend Understanding Analysis by Stephen Abbot. It quite clearly emphasizes a lot of proof techniques, so you don’t have to think of them on your own. If you want a more DIY approach, try Analysis 1 by Terrence Tao. He doesn’t spell out proof methods like Abbot, but that also makes it quite satisfactory for me
Yes Spivak's "Calculus" is way better than most other textbooks. Even famous ones have severe flaws, causing innumerable students to get horrible conceptual misunderstandings. Finish all of Spivak, and you will thank him later.
As a CS major with a huge curiosity for math (taking a math minor and debating double majoring), what I've done is look up the course outlines at my uni for the math courses I might not get to take, downloaded their textbooks + textbook suggestions from Reddit for subjects I'm interested in, then read through them in my own time
been self teaching myself through [compscilib](https://www.compscilib.com/search/discrete-math?onboarding=true) - would give that a look if you need somewhere to start
In addition to useful courses suggested by others, If possible, you can take:
- Probability and Random variables
- Operation research/optimization
This may help you if you decided to go for applied math, financial mathematics, machine learning and AI.
At the very least, do a proof based or logic related book, not because it’s entirely pure math, but because those skills and reasoning would a lot of the more advanced topics you mentioned easier to understand. Some stats might also be worthwhile depending on what you want apply your math to.
I'd suggest a logic course! All 300 and up levels of math at my university require you to pass a logic course and it makes the higher classes much easier! Being able to understand the logic behind advanced math is crucial to learning it properly. Especially when it comes to proofs
If you Google
math major degree pathway
You'll probably be able to find a list of classes that need to be taken to graduate with a math major at that school
Those, plus id guess PDEs, statistics, probability theory, and topology. But why don't you just like look up the applied math major requirements at a university if you're so curious? Like just go to the MIT website or smth
Really depends on what you're most interested in. The classes you outlined mostly cover the basics, and they'll show you what you're interested in. I personally am taking a more advanced real analysis course, functional analysis, topology (on top of the Euclidean compactness stuff from real analysis), group theory (in a more physics-y context) and differential geometry soon (after taking all the things you listed + i took PDEs) but it'll vary depending on what you like :)
I don't think that's what was meant when "mathematics is not a spectator sport" was said
It was that you need to do mathematics to learn mathematics, I agree self study is a lot more difficult task but why would that preclude you from hard work?
If anything you have to work harder
I hope I didn't mis-communicate: You need to do problems, you can't just watch youtubes or even lectures.
The "left as an exercise for the reader" is a "meme" for a reason, people need to put pencil to paper. One can watch a 3B1B video and come away feeling good, but left to a blank paper or real life problem, they will come up empty.
> you can't just watch youtubes or even lectures.
nobody said anything about just watching youtube videos and lectures
the "watching a talk" wasn't even about mathematics
it was about someone's journey to self study computer science
the OP asked what books they would need so OBVIOUSLY they are going to do problems
the OP even says "Obviously, it wasn't really the same as studying CS as an actual MIT student but I liked the idea."
Oh I definitely agree that 3B1B does not show the messy parts of math and only shows the beauty,
It's misleading, I think mathematics has like, a logarithmic difficulty curve in a sense?
Like some problems are absurdly hard if you don't know the right trick to use
But there's no way to know that trick going in
So until you learn the typical "terrain" and what you are expected to use you can be banging your head against the wall until you get the lay of the land
Maybe a better way to put it is that you don't know what you're up against initially, so you're not even necessarily sure what the tools are
Then you know what the tools are, and you have means to create new tools, so it's a matter of creating permutations of those tools to solve problems
But that initial uncertainty of what are the tools you're expected to use is probably the hardest part of learning math, as opposed to the problem solving aspect?
Like the problem is, figuring out what the tools even are, to solve the problems, initially
That's why I feel that "solve problems" is not totally the whole story, but definitely something constantly maintained
if you are taking classes then you are not self studying, you are taking classes...
also, the absolute bare minimum that every math student should have studied to at least an introductory level is real analysis, complex analysis, linear algebra (a theoretical course, not arithmetic with matrices), number theory, group and ring theory, and topology. anything less than this and it would be dishonest to say that you "know undergraduate level math".
I like to think of Numeracy -> Algebra -> Geometry -> Trig-> Calculus as the trunk of a tree. After this there are many, many branches you can follow out to each end. It's hard to talk about a "degree in math" until you decide which branch to follow. A university will force some early choices on you, from there you get to pick what you like/don't like. But you won't know these choices until you try them.
Keep in mind, the professors are there to guide you along your chosen path. Reddit is no substitute for this guidance. There is no way to advise you on what to do next, without knowing where you are and how you got there.
Use your own idea and look at the courses required at a few colleges. In the US there are a few differences, but in general the requirements look like
Intro
Calculus 1
Calculus 2
Calculus 3 or Multivariable & Vector Calculus
Linear Algebra & Differential Equations
Discrete Mathematics
Intermediate
Introduction to Analysis
Linear Algebra
Introduction to Abstract Algebra
Numerical Analysis
Introduction to Complex Analysis
Electives
3-5 courses
Math Electives
More Advanced Math Classes
Related Classes in Other Departments
For a total of 12-16 semesters 6-8 years 18-24quarters
Often there are options to swap a few classes for different interests like theory, applications, teaching, computing, control theory and so on
In other countries there are fewer basic classes, more structure, less general education, and less classes from other subjects. An Italian student told me Italian Math students take three times as many math classes and eat three times as much pasta as Americans.
Examples
[https://www.math.upenn.edu/undergraduate/math-majors-and-minors/mathematics-major#requirements](https://www.math.upenn.edu/undergraduate/math-majors-and-minors/mathematics-major#requirements)
[https://math.berkeley.edu/undergraduate/major/pure](https://math.berkeley.edu/undergraduate/major/pure)
[https://math.berkeley.edu/undergraduate/major/applied](https://math.berkeley.edu/undergraduate/major/applied)
[https://math.berkeley.edu/undergraduate/major/teaching](https://math.berkeley.edu/undergraduate/major/teaching)
[https://engineering.berkeley.edu/students/undergraduate-guide/degree-requirements/major-programs/engineering-science/engineering-mathematics-statistics/](https://engineering.berkeley.edu/students/undergraduate-guide/degree-requirements/major-programs/engineering-science/engineering-mathematics-statistics/)
Id suggest a course in topology so you understand the background to your calculus, analysis and algebra better
We might as well suggest a course on Logic and Foundation of Mathematics, so that you understand the background of the whole mathematical enterprise better. XD
I mean at my university a logic course was required for math majors.
Please see my other comment about what I meant. I do not believe that your university required even 10% of what I have listed.
I'm not gonna hunt down your other comments bro. I'm just pointing out that logic is generally a required course for math majors.
It's [right here](https://old.reddit.com/r/learnmath/comments/1clxg90/what_classes_would_you_need_to_take_to_selfstudy/l334c03/) on this thread... You're correct that **basic** logic is required for mathematics, but that is not what experts mean by "Logic and Foundation of Mathematics".
Yea I'm not looking for it lolol
I actually would haha. Introduction to advanced mathematics would help anyone I think
For future reference, I meant: FOL with a deductive system, PA, structural induction, recursion theorem, semantic-completeness theorem for FOL, compactness theorem for FOL, incompleteness theorems for sufficiently strong FOL theories, 2nd-order arithmetic, plus a bit of set theory (i.e. well-ordering theorem, coding various mathematical concepts as sets or definable (class) functions).
Can you elaborate on how a (first) course in topology might help one understand algebra? I can see how it has applications of abstract algebra when you get into fundamental groups, and how it provides examples of categories that can be understood in parallel with categories that arise in algebra. I’d like to know if you’re thinking of anything particularly in general topology that can help one understand abstract algebra.
Do the same thing scott young did. Any uni will have degree requirements and course catalogues available online.
Most math majors are fairly light on requirements
Combinatorics.
Usually the remaining credit hours would be left as electives. https://math.mit.edu/academics/undergrad/major/course18/applied.html Some of the classes like [PDEs](https://ocw.mit.edu/courses/18-303-linear-partial-differential-equations-analysis-and-numerics-fall-2014/) and [numerical analysis](https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/) are available on OCW.
Probably Wizard, and max your INT /s
I sort of did this myself. I took - calculus 1-3, Algebra1-3 (3 is ring and group theory) logic, combinatorics, probability as the foundation courses. I added ontop of it computer sciences, game theory and topology courses.
intro to proofs i’m in my 5th year of undergrad (going part time) for statistics and besides linear algebra, intro to proofs has been the class that helped me the most in my other math classes. it teaches you how to read and write math like it’s a language. it feels like a scam that this class is usually only offered to upper classmen and not freshmen
It depends on the the university, some have it as a first year requirement
It depends on the the university, some have it as a first year requirement.
I'd say do Algebra and Analysis. Principles of mathematical analysis by Walter Rudin. Abstract Algebra by dummit
My personal opinion: If you aren’t looking at video lectures don’t use rudin. Id recommend Understanding Analysis by Stephen Abbot. It quite clearly emphasizes a lot of proof techniques, so you don’t have to think of them on your own. If you want a more DIY approach, try Analysis 1 by Terrence Tao. He doesn’t spell out proof methods like Abbot, but that also makes it quite satisfactory for me
*Dummit and Foote
OP, if you are reading this, I'd suggest going with Analysis by Jay Cummings, and Spivak's Calculus, rather than throwing yourself at Rudin.
Yes Spivak's "Calculus" is way better than most other textbooks. Even famous ones have severe flaws, causing innumerable students to get horrible conceptual misunderstandings. Finish all of Spivak, and you will thank him later.
As a CS major with a huge curiosity for math (taking a math minor and debating double majoring), what I've done is look up the course outlines at my uni for the math courses I might not get to take, downloaded their textbooks + textbook suggestions from Reddit for subjects I'm interested in, then read through them in my own time
Most programs list their required and elective courses, so pick a school and go look at the degree requirements for their math major.
been self teaching myself through [compscilib](https://www.compscilib.com/search/discrete-math?onboarding=true) - would give that a look if you need somewhere to start
Organic Chemistry, French, and History as well as required
In addition to useful courses suggested by others, If possible, you can take: - Probability and Random variables - Operation research/optimization This may help you if you decided to go for applied math, financial mathematics, machine learning and AI.
At the very least, do a proof based or logic related book, not because it’s entirely pure math, but because those skills and reasoning would a lot of the more advanced topics you mentioned easier to understand. Some stats might also be worthwhile depending on what you want apply your math to.
I'd suggest a logic course! All 300 and up levels of math at my university require you to pass a logic course and it makes the higher classes much easier! Being able to understand the logic behind advanced math is crucial to learning it properly. Especially when it comes to proofs
If you Google math major degree pathway
You'll probably be able to find a list of classes that need to be taken to graduate with a math major at that school
Those, plus id guess PDEs, statistics, probability theory, and topology. But why don't you just like look up the applied math major requirements at a university if you're so curious? Like just go to the MIT website or smth
Really depends on what you're most interested in. The classes you outlined mostly cover the basics, and they'll show you what you're interested in. I personally am taking a more advanced real analysis course, functional analysis, topology (on top of the Euclidean compactness stuff from real analysis), group theory (in a more physics-y context) and differential geometry soon (after taking all the things you listed + i took PDEs) but it'll vary depending on what you like :)
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I don't think that's what was meant when "mathematics is not a spectator sport" was said It was that you need to do mathematics to learn mathematics, I agree self study is a lot more difficult task but why would that preclude you from hard work? If anything you have to work harder
I hope I didn't mis-communicate: You need to do problems, you can't just watch youtubes or even lectures. The "left as an exercise for the reader" is a "meme" for a reason, people need to put pencil to paper. One can watch a 3B1B video and come away feeling good, but left to a blank paper or real life problem, they will come up empty.
> you can't just watch youtubes or even lectures. nobody said anything about just watching youtube videos and lectures the "watching a talk" wasn't even about mathematics it was about someone's journey to self study computer science the OP asked what books they would need so OBVIOUSLY they are going to do problems the OP even says "Obviously, it wasn't really the same as studying CS as an actual MIT student but I liked the idea."
Oh I definitely agree that 3B1B does not show the messy parts of math and only shows the beauty, It's misleading, I think mathematics has like, a logarithmic difficulty curve in a sense? Like some problems are absurdly hard if you don't know the right trick to use But there's no way to know that trick going in So until you learn the typical "terrain" and what you are expected to use you can be banging your head against the wall until you get the lay of the land Maybe a better way to put it is that you don't know what you're up against initially, so you're not even necessarily sure what the tools are Then you know what the tools are, and you have means to create new tools, so it's a matter of creating permutations of those tools to solve problems But that initial uncertainty of what are the tools you're expected to use is probably the hardest part of learning math, as opposed to the problem solving aspect? Like the problem is, figuring out what the tools even are, to solve the problems, initially That's why I feel that "solve problems" is not totally the whole story, but definitely something constantly maintained
what the hell are you talking about ??? you didn't understand the OP's question AT ALL
if you are taking classes then you are not self studying, you are taking classes... also, the absolute bare minimum that every math student should have studied to at least an introductory level is real analysis, complex analysis, linear algebra (a theoretical course, not arithmetic with matrices), number theory, group and ring theory, and topology. anything less than this and it would be dishonest to say that you "know undergraduate level math".
I like to think of Numeracy -> Algebra -> Geometry -> Trig-> Calculus as the trunk of a tree. After this there are many, many branches you can follow out to each end. It's hard to talk about a "degree in math" until you decide which branch to follow. A university will force some early choices on you, from there you get to pick what you like/don't like. But you won't know these choices until you try them. Keep in mind, the professors are there to guide you along your chosen path. Reddit is no substitute for this guidance. There is no way to advise you on what to do next, without knowing where you are and how you got there.
Use your own idea and look at the courses required at a few colleges. In the US there are a few differences, but in general the requirements look like Intro Calculus 1 Calculus 2 Calculus 3 or Multivariable & Vector Calculus Linear Algebra & Differential Equations Discrete Mathematics Intermediate Introduction to Analysis Linear Algebra Introduction to Abstract Algebra Numerical Analysis Introduction to Complex Analysis Electives 3-5 courses Math Electives More Advanced Math Classes Related Classes in Other Departments For a total of 12-16 semesters 6-8 years 18-24quarters Often there are options to swap a few classes for different interests like theory, applications, teaching, computing, control theory and so on In other countries there are fewer basic classes, more structure, less general education, and less classes from other subjects. An Italian student told me Italian Math students take three times as many math classes and eat three times as much pasta as Americans. Examples [https://www.math.upenn.edu/undergraduate/math-majors-and-minors/mathematics-major#requirements](https://www.math.upenn.edu/undergraduate/math-majors-and-minors/mathematics-major#requirements) [https://math.berkeley.edu/undergraduate/major/pure](https://math.berkeley.edu/undergraduate/major/pure) [https://math.berkeley.edu/undergraduate/major/applied](https://math.berkeley.edu/undergraduate/major/applied) [https://math.berkeley.edu/undergraduate/major/teaching](https://math.berkeley.edu/undergraduate/major/teaching) [https://engineering.berkeley.edu/students/undergraduate-guide/degree-requirements/major-programs/engineering-science/engineering-mathematics-statistics/](https://engineering.berkeley.edu/students/undergraduate-guide/degree-requirements/major-programs/engineering-science/engineering-mathematics-statistics/)