Ask HN: Independent Math Study
39 comments
To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does.
Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them.
My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.d... ).
Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them.
My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.d... ).
Saying the Putnam is a next step from AMC12 problems is like saying the NBA is a next step from pickup basketball with friends in middle school! There are people who can do Putnam problems for fun, but those people generally know who they are already.
Solving problems 1-4 on each day of the Putnam with an "unlimited" amount of time is not a ridiculous expectation. Putnam's difficulty is partly due to its time format.
Putnam problems are not considerably harder than AIME problems, and AIME is definitely the next step from AMC12. Anybody who can solve a few AIME problems can certainly solve A1 on the Putnam of almost every year.
> I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless
Useless? I dear to say that number theory is currently the most lucrative field of mathematics. Without number theory, modern day cryptography would not exist and thus everything that depends on secure communication of information would not exist. So forget about commerce over the Internet, bank wire transfers, credit cards, administrating computers remotely and, most importantly, hiding your huge porn collection from your wife.
And, combinatorics is useful for the study of algorithms. It is pretty much the foundation of computer science.
Useless? I dear to say that number theory is currently the most lucrative field of mathematics. Without number theory, modern day cryptography would not exist and thus everything that depends on secure communication of information would not exist. So forget about commerce over the Internet, bank wire transfers, credit cards, administrating computers remotely and, most importantly, hiding your huge porn collection from your wife.
And, combinatorics is useful for the study of algorithms. It is pretty much the foundation of computer science.
I'm aware of their applications; by "practically", I meant "almost". There are certainly compelling uses for number theory and combinatorics (though I'm not convinced the study of algorithms is one of them), but they're nothing compared to calculus.
> calculus isn't that important for mathematicians
That's a joke, right?
That's a joke, right?
No. Many pure math classes require no (or very little) calculus. Abstract algebra, number theory, combinatorics, and graph theory certainly fall into this category. Topology does, too, depending on which area you study and what you consider calculus. Sure, there are obviously fields that do rely heavily on calculus, as well as certain branches in the above fields, but my point was that it's nowhere near universally needed. I'm a graduate student at UCSD, and I can't remember the last time I used calculus in my research.
This is terrible, wrongheaded advice. It's like Pablo Picasso, in the middle of his Blue Period, trying to convince younger painters that red isn't a useful color for serious artists.
If you want to study graph theory or combinatorics [1], then calculus will be pretty much useless to you, and you'll naturally go years without using it.
Calculus is also useless in some situations in abstract algebra (which are said to have combinatorial character). There are other parts of abstract algebra, e.g. Differential Galois Theory [2], in which calculus is pretty important.
Topology is similar. Elementary topology is part of the foundation supporting calculus, while algebraic topology is one of the tools that's useful when we try to do calculus (or solve differential equations) in non-Euclidean spaces.
Fields making heavy use of calculus include differential geometry, differential equations (ordinary or partial), dynamical systems or control theory. That subsumes most of physics. Fields underpinning (and largely inspired by) calculus include real and complex analysis, measure and integration theory (aka axiomatic probability theory). Also functional analysis, which is a generalization of linear algebra, which is the bookkeeping methodology of calculus in higher dimensions.
[1] The first sentence here says it all: http://en.wikipedia.org/wiki/Combinatorics
[2] http://en.wikipedia.org/wiki/Differential_Galois_theory
If you want to study graph theory or combinatorics [1], then calculus will be pretty much useless to you, and you'll naturally go years without using it.
Calculus is also useless in some situations in abstract algebra (which are said to have combinatorial character). There are other parts of abstract algebra, e.g. Differential Galois Theory [2], in which calculus is pretty important.
Topology is similar. Elementary topology is part of the foundation supporting calculus, while algebraic topology is one of the tools that's useful when we try to do calculus (or solve differential equations) in non-Euclidean spaces.
Fields making heavy use of calculus include differential geometry, differential equations (ordinary or partial), dynamical systems or control theory. That subsumes most of physics. Fields underpinning (and largely inspired by) calculus include real and complex analysis, measure and integration theory (aka axiomatic probability theory). Also functional analysis, which is a generalization of linear algebra, which is the bookkeeping methodology of calculus in higher dimensions.
[1] The first sentence here says it all: http://en.wikipedia.org/wiki/Combinatorics
[2] http://en.wikipedia.org/wiki/Differential_Galois_theory
That post does not contain any advice. It contains facts, none of which were contradicted by your post (a typical property of facts). I never told him not to study calculus. Given our current education system, that would be impossible anyway. I guided him more toward real analysis and suggested some other areas of math that might interest him. Since his only background is high school math, I felt it would be best to introduce him to something where proofs play a central role. If he can't stand that, then he probably shouldn't go into math.
> calculus isn't that important for mathematicians
> Many pure math classes require no (or very little) calculus.
These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)
> Many pure math classes require no (or very little) calculus.
These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)
Nope. I'm well aware that people still study analysis. I just meant that one doesn't necessarily need to learn calculus before taking the plunge into serious mathematics. Even in analysis, there's quite a bit you can do without knowing the stuff from a standard calculus class (though it certainly helps).
I would claim that Calculus isn't that important for engineers / scientists / programmers either. Real Analysis is important if one needs to understand thing deeper. In the real world, problems can't be solved analytically... and many of the tools one learns in Calculus are kind of useless. I think Linear Algebra is much, much more important than Calculus. Linear Algebra is the arithmetic of higher mathematics, like Bellman said.
>In the real world, problems can't be solved analytically...
Your other suggestions notwithstanding, you and I live in a very different "real world" my fried.
Your other suggestions notwithstanding, you and I live in a very different "real world" my fried.
real world
Tell that to physicists “renormalizing”† it over and over all days long…
† I almost forgot not everyone on HN may know what that is, http://en.wikipedia.org/wiki/Renormalization
Tell that to physicists “renormalizing”† it over and over all days long…
† I almost forgot not everyone on HN may know what that is, http://en.wikipedia.org/wiki/Renormalization
If you can solve your problems analytically, then you're either working in a blessed field, or you're working with simplistic models...
It's been ages since my last university math class (I was a I math major), so I can't point you to any reference material, but I can say the following.
If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 is greater than 0, you'll learn to look at things differently.
In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.
If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 is greater than 0, you'll learn to look at things differently.
In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.
What's a good real analysis text these days? Is Spivak's calculus book still a common favorite?
I wouldn't know as it has been over 10 years since I looked at a math textbook. The thing about math is it doesn't change. Well at least the basics so it's safe to say any text that you find in the library would be a good source.
Where different text books may deviate from one another is how they prove a theorem. Like programming, you can usually get the same results by going down different paths. Some paths are more efficient than others, but that is predicated by what you know.
If you are just learning, the best thing to do is find textbooks with answer keys to assignments. Also with the advent of google and such, I would have to imagine you can probably find answers to a lot of the questions that would be posed in these text books so answer keys may not be all that important now.
Where different text books may deviate from one another is how they prove a theorem. Like programming, you can usually get the same results by going down different paths. Some paths are more efficient than others, but that is predicated by what you know.
If you are just learning, the best thing to do is find textbooks with answer keys to assignments. Also with the advent of google and such, I would have to imagine you can probably find answers to a lot of the questions that would be posed in these text books so answer keys may not be all that important now.
I quite liked Stewart's Calculus. http://www.stewartcalculus.com/
And it is really helpful to use some CAS (like Maple, Octave or Maxima) to visualize problems.
I've learned a lot of math that's beyond the scope of what I had in college. I usually find that it works best when it's on the way to something that I'm trying to do or understand. I never really tried to learn math for the sake of math -- I wanted to understand quantum computing algorithms, recommender systems and graph clustering -- and had to fill in the gaps so that the papers in the fields made sense.
I self-studied math for 2 years. I just attended lectures without officially enrolling to the university. I also did about half of the homework problems given in these courses (My math-student friend was envious of me: I could choose the interesting questions out of the homework paper, and ditch the boring ones!)
Some of my studies I also did with books, video lectures, and articles I found on the internet.
Some of my studies I also did with books, video lectures, and articles I found on the internet.
Always use more than 1 text, always do the problems, & always keep up a steady pace. I haven't found anything else to be really important.
This is really good advice. Also, remember that it means nothing beyond what it says, all you really have to work from are the definitions and the theorems. :)
If you wish to move beyond the level of learning methods to solve a very specific class of problems (like Calculus I/II/III teaches, no offence/looking down one's nose is intended), you'll need to eventually learn to write proofs. A good book to get you over the initial hurdles is Daniel Velleman's How to Prove It.
And in addition -- it greatly enhances ones ability of abstract thinking. At least in my case it was true :-)
Checkout this thread, http://news.ycombinator.com/item?id=108723
I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html
Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
Fancy that! I'm at the UW right now. I have been considering a few different routes. Philosophy, Classics, Music, Computer Science, or Math. I'd really like to go for a Math undergrad with a minor in one of the other subjects, but I'm going to need a supplement during the summer to catch up with some of the other math guys at the UW. Lots of competition.
Book recommendation: Princeton Companion to Mathematics
It's a good way to skim a lot of different mathematical topics for further exploration.
http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...
It's a good way to skim a lot of different mathematical topics for further exploration.
http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...
You could do a purely applied approach, look at some Data Mining books, like Witten/Franke and the Weka java framwork (there's quite a few good books, check amazon reviews, ) and the assortment of methods that are applied from basic logit/probits, through clustering, SVM, neural, evolutionary programming, .
First make sure you have a solid operational foundation on the basics. Advanced topics will feel so much easier.
For that I can recommend Discrete Mathematics and its Applications by Kenneth H. Rosen.
Optionally supplemented by Student's Solutions Guide for more elaborate answers to exercises.
Do as many exercises as possible.
For that I can recommend Discrete Mathematics and its Applications by Kenneth H. Rosen.
Optionally supplemented by Student's Solutions Guide for more elaborate answers to exercises.
Do as many exercises as possible.
I second Discrete Mathematics and its Applications. This book was the book I used in the first class that required a substantial amount of proof writing. A majority of it was easily tackled within a six week course.
You should take a look at a book called The Road to Reality by Roger Penrose. While it's geared more towards physics, this book has proven to me to be the most enlightening mathematics text I've ever read. Admittedly I'm only about 10 chapters in - it's a very dense book, and you'd do well to go through it slowly. But, if you're interested in math, this book will blow your mind.
http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...
http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...
I really recommend http://www.mathxl.com . I've used it for calc 3 and linear algebra in place of physically being in those classes. It will walk you through examples and keeps track of your weakness to review later. It seems to be powered by Mathematica, from what I can tell.
My advice to you is to find a really good book and go with the book program. I passed many college classes just by studying hard with a good book.
You could try MIT's free courses!
My question to HN is: How does one go about doing self-study in Math? It seems, of all the sciences, to be especially difficult to tackle without the built-in support of the classroom. I assume that like most things, it just takes a lot of hard work and study, but I'm curious if anyone out there has a rough plan for tackling a reasonably rich understanding of mathematics on their own. Sites, materials, etc. are appreciated.
Thanks!