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Comment by sergiosgc | Hacker News Reader

This must beg the question: is math formal language expressive enough? Too often, the ideas are really simple but the mathematical formulation is extraordinarily complex. The first time I realized this was with Fourier transforms. Analytically, it is hell to understand the fundamentals, while if you read the original description by Fourier, it becomes dead obvious (as most great ideas, after the fact)

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Math formal language is expressive enough to express everything most mathematicians want. The thing is that formal math language wasn't designed as a medium to transfer ideas, but rather as a notation to write them down concisely and make things easy to manipulate. That's why learning from books that are very formal is such a difficult experience, though it gets easier with experience. Unfortunately, to get to that level of experience, you pretty much have to be focused on pure mathematics for years, and if you're just trying to learn some math to get some other shit done, and there isn't any book written in more conversational style (which thankfully usually is the case), you're out of luck.

Have a link to the original description?

Apostol explained it with linear algebra, which made sense, but seems pretty magical in terms of how you'd notice sine waves make an orthogonal basis in the first place.

I would like to see his original description as well. I've never understood the Fourier transform intuitively, no matter how much my professors and textbooks attempt to explain it. At this point, I'm just applying algorithms without fully understanding why I'm doing it.

It was handed out as support material in a course. I can't for the life of me find it online. Anyhow, the transform hinges on two insights: you can use a point rotating in a circle as a basic building block and you can look at any signal as a sequence of spikes. This article looks like it follows the same approach: http://betterexplained.com/articles/an-interactive-guide-to-...

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