111–120 of 171 posts
This reminds me of an old joke in the Haskell community, where people who struggled to understand Monads would finally get it after a while, and would assume that whatever the last sentence they heard was the only necessary one for the explanation.
That color coding is nice. Not just as a reference to the parts of the math formula, but also within the sentence itself.
Earlier quoted context omitted.
It took me an embarrassingly long time to realize that it was a joke when people said "It's always the last place you look." Like well into my teens. But ever since I figured it out, I always look at least one more place after finding something.
I think that there are many people who don't recognize it as a joke and pass it on as great wisdom. I was also pretty old when I realized the tautology.
Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…
Does "blog post learning" ever really work? I have taught these kind of undergraduate subjects and, in the context of a course , the Fourier transform never struck me as something very complicated. First, it feels completely natural to write down a Fourier decomposition for periodic functions; the inverse transform to determine the coefficient is just (real or complex) calculus; and all that is left is to convince th…
On a similar note, I personally believe there are many concepts that have been explained much better by random YouTubers than academic textbooks. Textbooks are more comprehensive almost always, but having an intuitive understanding of something is obviously more important in many cases than simply remembering a long list of bullet points about it. You can always use reference material later. In real life, memorizing facts on their own is rarely useful.
The best answer here is that you should use both, or imo, "all three," of blog posts, video content, and academic textbooks/articles/courses. Taking a variety of approaches helps me get to a mental "click" quicker, personally.
Earlier quoted context omitted.
Disagree. If you truly understand something, you must have prooven it to yourself, everything is just a mental help ("donkey bridge" in german) that allows you to remember the statement better - weather it's right or not, you can only know once you've prooven it; and before you did that, it can often happen that your intuition on what's right is actually wrong.
I think a lot of people can understand simple statements in math such as Fermat’s Last Theorem or the Collatz Conjecture but proving them is an entirely different matter. While it may be the case that proving some statements is sufficient to understanding them, I would say it’s never necessary.
But usually true understanding and formal proofs go hand in hand.
P.S: Though as a disclaimer, I don't know maths beyond the undergraduate level, and I'm sure there must be very complicated concepts that mathematicians don't understand, but can reason formally about, as well as concepts that mathematicians feel that they intuitively understand but can't quite prove.
Additionaly, there's von Neuman saying "Young man, in mathematics you don't understand things. You just get used to them"
Earlier quoted context omitted.
>I have taught these kind of undergraduate subjects and, in the context of a course, the Fourier transform never struck me as something very complicated. It's great that it was "never complicated" for you. In contrast, the author (David Smith) of this blog post admits that he initially struggled with the Fourier Transform -- and wants to share some insights he gained after he understood it. He has already graduated w…
> It's great that it was "never complicated" for you. I don't think they meant "not complicated to learn", but "never complicated to teach, once its time had arrived in the natural progression of the course".
Which is, of course, true for give or take all subjects out there
Earlier quoted context omitted.
I think you need to study the Euler equation to understand the relationship between goniometric functions and exponential functions when calculating with complex numbers. One easy to remember formula that connects those functions.
Indeed. The title of that 3Blue1Brown video is, “What is Euler’s Formula actually saying?” That “easy to remember” formula had been presented to me countless times. It only made sense once I stopped thinking about it in terms of repeated multiplication.
Earlier quoted context omitted.
> It's great that it was "never complicated" for you. I don't think they meant "not complicated to learn", but "never complicated to teach, once its time had arrived in the natural progression of the course".
>but "never complicated to teach, once its time had arrived in the natural progression of the course" Which is, of course, true for give or take all subjects out there
I don’t think it’s at loggerheads with just-in-time learning or the real opportunity cost of more extensive study. I think it’s just true that hard things are hard whether you like it or not.
I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions. Looking at its formula, we can see it looks like we are projecting a function into a se…