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That old topic. Ironically, I find people who work with math (chosen to work with math, that is) to probably be quite unqualified to truly understand the proble
by nothis 11y ago
That old topic. Ironically, I find people who work with math (chosen to work with math, that is) to probably be quite unqualified to truly understand the problem. He opens by saying that learning avoiding mistakes before any bigger concepts is a major root of this but then again, people do make mistakes in math (even math professors) and it's a field where any mistake can completely invalidate the work. Also most people learn math for practical uses rather than as a science or field of research which also gives a different perspective (for most people, it's more important in life to not drop a zero when calculating a bill than it is to understand, say, a Homomorphism).
Math simply is hard. It goes against all our good-enough, subconscious brain functions we use every day to make basic "bigger or smaller" decisions. The problems mathematicians call "interesting" are counter-intuitive pretty much by definition and now imagine someone thinking like that writing a math book for children. The language used is vast and was built over literally millennia of mathematicians appending their own little modifiers above, below, and all around other mathematical symbols, resulting in monstrosities like the function for the Fourier Transformation which can actually be described quite intuitively geometrically.
Mathematicians want "simplicity" and "beauty" but their definition of either is absurdly different from any that would take into account basic human intuition. That's the problem. And there barely are any mathematicians that would admit that this could ever be a bad thing.
- anon4 11y agoI'm not sure how you would define the Fourier Transform geometrically without needing 4 spatial dimensions, at which point intuitive geometric reasoning is impossible.
- lordCarbonFiber 11y agoFar from impossible, it's even reasonably intutitive. Wikipedia has an excellent graphic https://en.wikipedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_%28small%29.gif https://en.wikipedia.org/wiki/File:Fourier_transform_time_an.... Then for people for which that's too abstract, I find this graphic helps a lot in explaining. http://static.nautil.us/1635_42a3964579017f3cb42b26605b9ae8ef.png http://static.nautil.us/1635_42a3964579017f3cb42b26605b9ae8e... You picked a really bad example for the unintutiveness of mathematics.
- occamrazor 11y agoI'd suggest most of homological algebra as an example. It's called "abstract nonsense" for a reason. Personally, my intuition gives up at spectral sequences.
- BurritoAlPastor 11y agoSpeaking in my capacity as a person who doesn't understand Fourier transforms, I'm afraid I have to say that your "intuitive" graphics are anything but. I can't even make sense of your second graphic. Just a data point for you.
- lordCarbonFiber 11y agoFair enough, it works a lot better as a gif. Out of curiosity, does this clear things up, or just as confusing? https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_series_square_wave_circles_animation.gif https://upload.wikimedia.org/wikipedia/commons/1/1a/Fourier_...
- BurritoAlPastor 11y agoOooh. Yes, actually; I think I have an idea about what's going on (and why they're so damn useful) now. Thank you!
- echo_oddly 11y agoDoing a Fourier transform is simply applying a set of inner products from the function, f, onto a set of orthonormal basis functions. Put a post in the ground, measure the length of the shadow cast by the sun. That's how to do an inner product. So you can at least describe the process using few dimensions. Then you can scale it up to higher dimensions from there.
- deleted 11y ago[deleted]
- Jtsummers 11y ago> opens by saying that learning avoiding mistakes before any bigger concepts is a major root of this but then again, people do make mistakes in math (even math professors) and it's a field where any mistake can completely invalidate the work. The problem is that an emphasis on avoiding mistakes leads to paralysis. Though he phrases it differently. Students don't want to be wrong so they just don't instead. As he points out later in the piece, learning from our mistakes is key. If I never learned to walk because I fell the first time and failed walking class, I'd be crawling even today. An overly negative assessment of the students' works, an overemphasis on right and wrong, rather than correction and expansion to different methods when some method has failed a student, ends up producing many students who can't do math.