Advice on Learning Advanced Mathematics

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As a philosophy student, I never went beyond Calculus. Since completing my studies, I began trading equities and derivative products. I've made efforts to study the mathematical aspects of finance, but lack the preparation to really understand the subject. Some of the required mathematics, is simply way over my head.

What advice would you give to someone studying the mathematics of finance on their own? What books? Any websites? I don't plan on going back to school for additional training. Any help would be appreciated.
I'm majoring in both philosophy and maths, and I've always taken a philosophical approach to mathematics. Mathematics is a very creative subject, so having a strong intuitive understanding is important. That's why I think it's important to read a lot. Different teachers and textbooks will prove theorems and describe maths in different ways. Getting a multitude of perspectives helps you understand what is going on.

I've never studied mathematics by just going to class and then trying to do exercises and assignments purely by what the lecturer says. Often there are better ways to do things, and it's good to do some research and find out these other ways. I don't think mathematics is about having a problem and being very insular in the way you try to solve it.

I don't know much about the mathematics of finance, and I've just been rambling, but I hope that you find something worthwhile in what I've said.
With the caveat that I have not actually studied much financial math, I recommend that you start with some work in probability and combinatorics. If you are comfortable with calculus, you will pick it up easily enough, and I think it will come in handy as your studies focus on wherever your interests may take you.

That said, recommending a book is a much different story. The is a wide spectrum of such books, ranging from too simple to too much. I have an interesting looking book called 'Probability Through Problems', where you work out all of the important theorems and applications as excercises (solutions are given). I have not looked too deeply, but this book may be a good place to start if you intend to learn yourself.

Otherwise, just browse for a book which looks readable, and one in which the excercises look interesting (because you want to do as many of them as possible).

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