[geokon]

This probably won't be a very popular take.. but to me Fourier Transforms are the perfect example of the opposite problem - where the educators are so hellbound on constructing a visual explanation that they end up brain damaging the students. I've seen this come up several times in my math education.

For the Fourier transform there are primarily two issue:

1. The "complex plane" - this tries to make complex numbers somehow less scary and more intuitive? But it's actually a useless crutch that gives people a false sense of understanding. The core issue is that you can multiply two complex numbers and get another complex number. If I give you two vectors or two points on a map and tell you to multiply them.. you can't - b/c that is not a defined operation. We have no intuition about how to multiple 2D numbers (you might think, oh dot products and cross products! but those are something unfortunately unrelated)

2. "Frequency Space" and an the constant suggestion that your signal is somehow being broken up into it's magical innate hidden frequencies. This is also very deceptive (or doesn't hold up in the discrete case). The basis is selected to be mathematically convenient (ie. orthogonal) and is based on your sampling window and properties of complex numbers. These may correspond to some underlying natural frequency that's occurring - or it may not. But it's not helpful to try to confound the two

If you instead approach the whole problem from a purely mathematical perspective of projection on an orthogonal basis and how to construct complex values that are convenient - then the whole setup is less "fun" but it actually becomes a lot more understandable. You can then move on from there and start asking yourself much more interesting questions about aliasing, ringing, fm/am, phase etc.

I feel it took be ~5 attempts of learning Fourier Analysis to unwire my brain and unlearn these bad visual intuitions that send you down the wrong path

[hpcjoe]

I gotta say I disagree. I got the space change bit quickly (piece-wise linear space into frequency space). Complex plane's aren't a crutch, really they are the firm theoretical basis in which to express the detailed discussion properly.

You could always just switch to the Fourier sine and cosine forms, and avoid all of the other theoretical basis baggage. Sort of like physics for poets (I'm a computational theoretical physicist by training), leaving out the more detailed derivations and background, for a more straightforward approach.

Moreover, the DFT is not the FT. In the limit as your sampled points get very large, it will approach FT. There's a great book covering lots of these things in depth[1], with a pragmatic as well as theoretical approach. I think I gave my daughter this one (math phd student) last year.

FTs aren't merely a change of basis, there is quite a bit more to them than that. For DFT you can look at the process as a sequence of operator applications, but in the FT case this becomes a continuous sequence. Hence the space bits.

[1] https://epubs.siam.org/doi/book/10.1137/1.9781611971514 I highly recommend this book.

[9erdelta]

Can't speak to Fourier specifically -- but heartily agree that a lot of times in math, teachers try to make it "easy" and provide visuals that are actually a hindrance. Instead of understanding _the material_ you understand the _crutch_.

[SkyBelow]

I think there is a special trick that happens many places in math. When you are given a visual, and then some integral or derivative is applied to it, you need to pause and make sure to truly include that in the mental model. Often I find math videos go far too fast through this section and I have to re-watch that specific spot and even pause and just ponder on how does the integral or derivative change what I was working with.

[lahvak]

Ad 1.: Multiplication of complex numbers is a very intuitive operation on complex plane, the moment you realize you need to use polar coordinates.

[marshray]

Link? (please :-)