I would like to get a better/broader understanding of mathematics. I got a M.Sc in Engineering and while math was never my strongest subject, I have studied, and can apply, fields like linear algebra, calculus, complex analysis, differential equations, boolean logic etc. My issue is that math, for math’s sake, never really interested me. I have always approached math as a tool to be applied to “real” problems.
I have decided to give math another try but i’m not quite sure where to start. My final “destination” i expect, will be many of the subjects i already known how to use and apply, but hopefully with a much better understanding of why things are like they are. I think i would like to start from the basics like set theory and number theory and work my way up from there. Maybe in a historical/chronological context where it makes sense. Is this the way to go or would i be better off with another approach?
Is this even possible as a self study in my free time? Where do i start? Any recommendations on books or websites that will help me?
What springs to my mind as a great place for you to start is William Dunham’s book Journey through Genius: The Great Theorems of Mathematics. It’s a wonderful, well-written, popular-level book that’ll give you some historical context and some good examples of beautiful mathematical reasoning. After you get your feet wet, you could go on from there to more systematic study if you’d like.
I found Linear Algebra to be pretty interesting when it came to the theory part of it. Fun, but probably not terribly difficult to understand the theories and proofs if you are half decent at math/logic.
However. I took it twice. The first time it was terrible and my grade wasn’t so hot. Then, a few years later after school I decided to take it again for fun in evening after work classes. Now it was easy and fun. So, I went back and looked at my old textbook (that was small, very expensive, and unsellable afterwards). No wonder it was hard and unfun. That textbook was horrible compared to the new one.
I don’t know how much of your stuff from college you still have, but use that if you can. It should ring a bell. I actually have all my textbooks and notes from the math and science classes I took (which was pretty much all the classes I took in college).
I still have all my textbooks and do use them as a reference from time to time. It’s not so much that i have forgotten what i already learned (though details are certainly fuzzy,) it’s more the way i have learned in the past that is unsatisfactory to me today.
In linear algebra, I know how to multiply matrices to transform vectors, I know how to find an eigenvector and how it can be used to solve a system of linear equations e.t.c. Linear algebra i can use as a tool, but i have only a vague understanding of why the tool works. The same goes for many other subjects in math.
I’m missing something that makes math more than an exercise in manipulating symbols. I think a better understanding of the basics might be it.
After I recommended Journey Through Genius above, as a place to “get your feet wet,” it occurred to me that, if you’d rather jump into the deep end, you might want to look at Courant and Robbins’ What Is Mathematics? It’s a broad overview of math for math’s sake, focusing on the ideas and concepts and not “math as a tool to be applied to ‘real’ problems.” There’s a lot there (though, oddly enough, nothing about linear algebra) to show you what math looks like to a mathematician (or at least, to someone who majored in math, as opposed to something like engineering).
For that, Linear Algebra Done Right (link) is the book you need. It does exactly what it claims to, and without steamrolling every little detail with determinants, so you can actually answer the ‘why’.
MIT provides a lot of courses online for free. Very handy if you want to learn the material. You do have to buy the textbooks, but you can watch the lectures for free.
I would take one of those subjects you mentioned, perhaps your favorite among them, and start peeling back the onion. Start with the textbook you already have. Understand all the definitions and key theorems. Your text may be too “applied” and lacks the proofs, at which point check the references in the text or find another book that does have the proofs. Keep going back in the subject until you understand as much as you’re willing to. Since you’re doing this for fun there’s no need to go all the way back to basics, but you can. Regardless of how deeply you go you’ll definitely understand more when you’re through and will have achieved your stated goal. Then move on to another math topic. Another way is to start from the basics and move forward (as they do in a math curriculum), but you may lose patience after a while. Starting with what you know and going backwards and stopping when you’re tired may work better. Just a suggestion.
Oh man. I want to relearn my math as well, and then keep going. I got as far as basic differential equations in college, and had some university exposure, but I never went beyond that. And I have need of it. I want to be able to describe the changing fall of sunloght on the windows of a house, among other things.
I’m just looking over the MIT site. I was just at the beginning!
You might also be interested in new fields of math.
Wikipedia and MathWorld are invaluable jumping-off points when it comes to figuring out what abstract algebra, for example, is and where it fits into the mathematical universe.
Here’s a little list of subjects that are what I would consider foundational, in some sense, to pure math, or math for math’s sake. You may want to look them up on MathWorld as well, or just google them:
As a start, i’m going to order a copy of “Journey Through Genius”, hoping that it will provide some inspiration and get me into in the right mindset.
After that i’ll probably do some reading on set/number theory while refreshing the more advanced topics. For this, I’ll probably start with calc 1 and actually work through the proofs this time. My old textbooks should be an OK place to start as i can always dig deeper into the references when needed (the actual books were written by my professor and are pretty bad).
There are some great websites out there i never knew existed!
