Wikipedia 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.
A 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.
lol 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.)
The 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.
I 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.
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."
I'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.
I'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), 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?
Well, 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.
There 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.
I 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.
Oh, 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 :)
Well, 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.
The problem is more due to a general culture that prefer rigour over clarity. Mathematics by definition is made of layers upon layers of intuitive conclusions. Then why do universities prescribe books with long dry derivations over clear explanations? Rigour is important to ensure correctness - and that should never be compromised. But that isn't the reason why humans learn mathematics. People learn it because it extends their imagination. Clarity and intuition are important for that and those who can't see it becomes disillusioned.
Rigour is better left to computers these days. They are better at following rules. We can have better proofs with software like metamath. Surprisingly, you can even gain deep insights by writing automated proofs (compared to manual proofs). Ideally, practitioners should consider simple and clear explanations as their primary goal and equations and proofs as necessary supplements. Think of this as literate programming for mathematics.
I think it is wikipedia's fault, but that it's not far from how it should be. Wikipedia is a secondary source, designed have be a reference taken from primary sources, rather than be a primary source teaching information. It's a weird blurry distinction, but seems to work well. It'd be great to have a pedagogy section that either explicitly linked out to pedagogical primary sources, or a section explicitly designed to teach.
This, 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.
The 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).
This statement is misguided. A lot of physical phenomena doesn't lend itself to intuitive understanding. Quantum Mechanics is a great example of this. You cannot expect subatomic particles to behave the same way that macroscopic objects do (and macroscopic behavior is what humans find intuitive because we experience it on a day to day basis).
Its the same misguided thinking that leads to the popular PopSci explanation of electron behavior being both wave and particle like. Why should we try to categorize the behavior a subatomic particle (an entity so distant from our experiential reality) as being analogous to one of two macroscopic entities?
I totally get the desire for intuitive understanding, and it should be encouraged, but sometimes you just have to put intuition aside and come to conclusions with pure mathematical reasoning.
Isn't this just a misunderstanding of what we mean by intuitive? It's possible to have an intuitive understanding of a thing that's not "intuitive" to humans.
Exactly. Just imagine the quantum world as a bunch of liquids floating around carrying their associated particles, and you have an intuitive grasp of (the debroglie bohm interpretation of) quantum physics.
I'd even be more ok with that, if mathematicians used more rigor in their notation. Math developed in a time where it was tedious to write this out by hand and a lot was hidden in abbreviations and assumptions. One thing that programming gets very right is that it is way more explicit (although obviously not perfectly) the behavior behind any notation.
It's one thing that Structure and Interpretation of Classical Mechanics (which I haven't read) got really right in concept. It's a shame, that approach hasn't been adopted more widely.
Take for example page 7 of the pdf with the heat equation. Even the description of the Laplacian is wrong. The value isn't the average of the points surrounding it. But if this simple function were written in python as a "update" function over a multi-dim array of temperature values it'd be clear exactly what's happening.
Another example of simple inconsistency in notation: superscript, does that mean squared or not? When writing two terms adjacent is it multiplication or an operator being applied? Where is the behavior of the operator defined that a student and "read the code"?
English is an extremely verbose and imprecise language. Mathematicians replaced a verbose imprecise language with a terse imprecise language. It's about time for rigorous fields to take the last step and introduce precise concepts using a precisely defined language.
> The second part describes the long-run behavior of the differential equation
> The temperature value that this point takes is the average temperature of the points surrounding it
And this is not correct. Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren't describing and evolution if its inaccurate for all points except a final state.
On top of that, even during the evolution the temperature isn't what's taking on the "average" of surrounding points, it the change in temperature wrt time that's being change by the average of points. And yet again, it's not an average of the surrounding temperatures is and average of the surrounding differences in temperature.
And this interaction we've had highlights exactly my point. English is an impressive language, and Mathematics is full of hand wavy explanations that come with simplicity context that isn't explicity and precisely defined. Mathematicians are used to it, and of course it's learnable like anything. My point is it's not necessary, it evolved in a different time with different constraints. Its a similar example of "the medium is the message" - math evolved when the only way to write was by hand, and duplication of definitions was manually expensive. We don't need these handwavy shortcuts now - we can make it easier for students to learn by being precise and providing definitions for them to see exactly what's happening - not expect them to learn from trying to recall every imagined context. And the fall back of "I had to learn it this way, so they should too", is a horrible excuse.
> Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren't describing and evolution if its inaccurate for all points except a final state.
The average of the surrounding points is the attractor temperature for the system. It is an asymptote which the temperature of the point is moving towards. It's like saying an oscillator (such as a spring) wants to be at neutral, even though it never comes to rest at neutral.
I'm not engaging with your larger point, I'm just quibbling with you saying page 7 is wrong. I think that "takes" in the english description says that the temperature approaches the value over time, whereas you interpreted "takes" as referring to the temperature at every point in time.
> I'm not engaging with your larger point, I'm just quibbling with you saying page 7 is wrong. I think that "takes" in the english description says that the temperature approaches the value over time, whereas you interpreted "takes" as referring to the temperature at every point in time.
I hear you, and I think we've probably approaching the end of the productive part of our conversation.
I do want to mention that the quote above, and interpretation of "takes" is exactly my larger point though. These definitions are all sloppy and prone to interpretation. Precise definitions would eliminate the need for all of this.
And since I can't help myself, thinking about this a little more, even your interpretation above is either faulty, or has to change the definition of "surrounding". If I have a point of average temperature, a doughnut or sphere of warmer points a small distance immediately around it and then the majority of all other points around that being colder, then the asymptote is actually towards the average of the of outer colder points, not the surrounding warmer points.
QM 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.
Perhaps 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.
