6 ms·
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I re
by chronolitus 6y ago
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift.
On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide like this, or 3-Blue-1-Brown videos would have made my day, then.
- ghaff 6y agoWikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.
- cs702 6y agoProbably true! That said, Wikipedia's mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the "reference of least resistance."
- ghaff 6y agoI'm at least ambivalent about a general purpose encyclopedia having whole categories of articles that are mostly only accessible to specialists. In an ideal world, there would probably be a companion "Mathepedia" or something along those lines with a different target audience. But, of course, Wikipedia doesn't really have the concept of a specific reader persona like a conventionally edited reference does.
- cs702 6y agoI'm ambivalent too, i.e., I agree with your original comment but also, personally, frequently find Wikipedia's "general purpose encyclopedia for specialists" useful. FWIW, there's Simple Wikipedia (https://simple.wikipedia.org https://simple.wikipedia.org), but it is, quite frankly, terrible for learning about anything of a mathematical nature beyond high-school algebra. Perhaps Wikipedia should add a "Learn About This" section to math, physics, engineering, and hard-science pages?
- C4stor 6y agoWell, it's not Wikipedia's fault if people producing intuitive content are not contributing to it. It's not like editors go out of their way to remove nice explanations.
- uoaei 6y agoThere are many anecdotal reports of tyrannical Wikipedia editors doing exactly this. They claim dominion over a subset of Wikipedia articles and then manicure them exactly to their own personal tastes. I'm sure the situation you describe happens often in various corners of the website.
- soVeryTired 6y agoI nearly based my Master's dissertation on a topic I discovered on a wikipedia math page when I was looking for something to cover. Turns out that subsection was maintained by the 'inventor' of topic and essentially served as a vanity page. The topic itself had no recognition in the community and if I had forged ahead on it, I would have failed my dissertation pretty hard.
- QuesnayJr 6y agoIf this is a math topic, you should bring it to the attention of the WikiProject Mathematics talk page: https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mat.... They might do something about it.
- soVeryTired 6y agoOh, I got rid of it years ago. Thanks though. Eventually a big-name mathematician (who some years later won a Fields medal) waded into the discussion and sided with me. I used my real name in that discussion so I won't give more detail than that :)
- C4stor 6y agoWell, I've personally contributed to small parts, and never encountered the problem. I don't doubt there are jerks on Wikipedia, but I would also bet that neither the author of this nice github repo or the praised youtuber mentioned in the comments even tried to commit anything into it. Not that they should for some reason, but too often editing wikipedia is presented/thought of as something reserved to a happy few with deep social consequences, when it's in fact the simplest thing in the world.
- QuesnayJr 6y agoThe thing is that writing an intuitive explanation is hard, while writing down the actual definition is easy. Also, one person's intuition is frequently another person's gibberish (hence the "monads are like a burrito" phenomenon).
- devalgo 6y agoTrue generally in Physics as well. What's the point of "understanding" Quantum Mechanics intuitively when you can do computations blindly without it!
- kgwgk 6y agoWhat's the point of "understanding” QM intuitively if it brings confusion rather than understanding?
- devalgo 6y agoYou literally can't understand something if you are confused by it. Your question makes no sense.
- mellosouls 6y agoPerhaps the comment was intended to convey the very different potential for intuitive "understanding" between visualisable classical theories and non-visualisable quantum theory. At a fundamental conceptual level it does not map to anything in our normal experience - attempting to force analogies etc can end up confusing things further. I agree with your original observation btw, but it's important to recognise the limits we may be bound by in explanation and comparison.
- kgwgk 6y agoThere is a difference between understanding and “understanding” (intuitively, no less!).
- aeternum 6y agoQM can be understood intuitively, the problem is we are still teaching the bullshit 'wave particle duality' when we basically know that fields are what is real. There's also no such thing as 'observation'. Fields interact by a precise and well-tested mathematical function. That function does have the effect of mutating the state, but why should there be such a thing as pure observation? There's nothing magic about conscious observation.
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- ericol 6y agoThis, yes. To this day of the several dozen that I visited, I'm yet to find a math related article that serves the purpose of explaining the concepts laid in it.
- athinggoingon 6y agoI wish there was an "intuition" wiki where you can learn the intuition first and then a link at the bottom takes you back to the Wikipedia page where you can stare at the math until you really get it.
- mikez302 6y agoI am not aware of such a thing. But this is arguably the closest thing I could find: https://betterexplained.com/ https://betterexplained.com/
- lordgrenville 6y agoA good description I read of Wikipedia maths articles was that they're like the blackboard of a graduate-level class after everyone has left. All the information is there, but if you're not already familiar with the subject there's no way you'll grasp anything.
- machawinka 6y agoGreat analogy.
- SilasX 6y agolol agreed. Every Wikipedia article about math is: <math term> is a <math term2> that <math term3s> a <math term4>'s <math term5>. It is an example of a <math term6> that cannot <math term7> a <math term8>. Me: I learned nothing from that. (Yes, which is partly my fault, but it's really not helpful for an intro to any math topic unless you already have mastered 99% of the topic and just want to resolve the last few things. The encyclopedia model really goes against what you need for the general population here. Arbital tried to do a more helpful model here but is defunct; Khan Academy is a generally better for this but requires more investment.)
- chongli 6y agoThe problem is that math is inherently more abstract than any other topic. It’s so abstract that the vast majority of math terms have no simple, intuitive translation into everyday English. Everything is built upon a tower of abstractions that have developed over centuries. In this community, we’ve witnessed the difficultly first-hand with the much-maligned monad tutorials. A mathematician would have no trouble understanding monads —- they’re just a matter of reading the definition and the laws. That’s what you do every day when you’re studying math in university. Read some definitions and some theorems, play with things a bit, try to prove stuff, then move on to the next topic.
- JadeNB 6y agoI love explanations of concepts that appeal to people in new and different ways, and I thank anyone clever, patient, and gifted enough to produce them. The world is a better place when more people understand are not afraid of more things. But I think it's important to remember that mathematics is not some evil thing! There are people who train carefully so that the terse mathematical explanations are the intuition. There's nothing wrong with not being one of those people, but knocking the mathematical explanation is denying them their intuition just as their explanation denies you yours. In a word, it takes all kinds, and I think we should all be as patient with others' ways of learning and understanding as we hope that they will be with ours.
- rusk 6y agoThat was beautiful
- ABeeSea 6y agoMost mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.
- potiuper 6y agoIf the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.
- imranq 6y agoI guess there's also an aesthetic component to limit the extent of generalization
- martincmartin 6y agoThe book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-the-gradient-of / meters. And, or course, the vector you get from a cross product only has 3 degrees of freedom because we live in a 3D world. The cross product of two 2D vectors is a scalar, the cross product of two 4D vectors needs 6 numbers to describe it. Even in 3D, if you reflect the original vectors in a mirror, the cross product now points the opposite direction, i.e. it depends on the handednes of the coordinate system.
- billfruit 6y agoCan you mention the name of the authours..
- martincmartin 6y agoIt's by Gabriel Weinreich.
- zentiggr 6y agoI'm not personally knowledgeable enough to suggest how this would behave in geometric algebra, I'm just smart enough to think that this would behave much cleaner in that framework. Would really like to have that free time to delve and be able to suggest the alternative formulation myself.
- soVeryTired 6y agoThat sounds like the beginnings of intuition for exterior algebra. While the gradient is a vector (i.e. it lives in a vector space), it's better thought of as being different to a standard 'displacement' vector. Look at a topographical map of a landscape and note the contour lines. As you zoom into the contour lines (if they're detailed enough), they'll start to look more and more like parallel lines, densely spaced for a steep slope, or sparsely spaced for a mild slope. These parallel lines are the gradient. A gradient and a vector 'fit together' to give a real number. The more parallel lines the vector pierces, the bigger the number [0]. So a gradient 'eats' a vector, spitting out a real number, and vice-versa. Just like a row vector 'eats' a column vector and spits out a real number. I'm saying 'gradient' here, but really what I mean is 'one-form'. Language deliberately imprecise for all y'all mathematicians out there. [0] https://neutronstars.utk.edu/class/p616_s19/lect3/index.html#slide6 https://neutronstars.utk.edu/class/p616_s19/lect3/index.html...
- fsckboy 6y agojust a quibble, if your Calculus 101 covers div and curl, you are going to an intense STEM school and really could be expected to understand the math.
- uoaei 6y agoUnderstanding the math and understanding the importance of the mathematical concept are two different things. Personally, div and curl didn't quite click for me until I took fluid dynamics in my final year. I could do the homework but didn't really get why they were useful until it made sense why "div = 0 always for incompressible fluids".
- fsckboy 6y agook, but my point was that div and curl were taught in 102, not 101 (which you easily could have placed out of). therefore, the reasoning goes, if they were in 101, that must be a hyper 101. source: went to intense STEM school
- titanomachy 6y agoThis was one nice thing about studying physics, most of the math we learned was motivated by some application. I think my first exposure to div and curl was using the analogy of fluids flowing in the plane. Intuitively, divergence is how much fluid is being created (or destroyed) at a point, curl is how much a waterwheel would spin if placed at a point.