As I said when I spotted this on Twitter, I think highlighting multiple things at the same time is a bit confusing. Looking at the equations, I, for one, can't easily spot any connections between the colorized parts of the equation and the textual description; it becomes all the same rainbow porridge.
What I'd like instead: highlight only a single thing in a single equation, and accompany the highlight with a text that explains it. Then repeat the (image with highlight, textual explanation) pairs until you've explained everything. Benefits: 1. I believe for many people like me, attention and focus on a single "highlighted" item at a time is much easier. 2. You can use only a single "highlight" color that can be colorbind friendly (it's enough that it is noticeably different than black) 3. You don't need to explain the equation in a single text passage (unlike in the examples, where it looks like the author wants to explain all of the equation in a single sentence so that all different concepts in the equation are represented in different colors both in a single text and the equation itself.)
Also, in my experience, yellow/orange is terrible color for text / plots in print. (Peter Flach's machine learning textbook uses yellow extensively for similar "text highlights", and it made many sentences in that book frankly unreadable. I simply could not read light yellow text on white paper background.)
It is very individual, the "rainbow porridge" works really for me, while your model may be better for other people.
Finding distinct colors for multi-colored diagrams is a science on its own. They should have the same level of luminosity (so that they are readable on white background) and equal "distance" on the hue axis. The answer is the Munsell color scheme http://www.ktorides.com/wp-content/uploads/2015/01/colors-75...
Mathematicians love greek letters, and if there are a couple of them it's easy enough to read the equation, but you can easily end up with 5 or more in a single equation, then have to spend extra brain time mapping which symbol refers to what variable.
What I often do when I want to better understand a formula is to rewrite it by replacing greek letters with plain words. Less elegant but more readable.
They don't "love" greek letters, you simply run out of letters. And you can't use words since that could be easily mistaken for a product between single variables. Once you learn them, things start getting way, way easier
That would be the case if all the other letters in the alphabet were used in the equation. This is rarely the case. In fact more often than not, all letters of the greek alphabet are used.
By custom, Greek letters usually mean different things than letters of Latin alphabet. And also it's customary that certain letters in Latin alphabet are used for specific things.
For example, Latin letter 'x' usually means a variable of interest / data point. Latin letter 'a' could be instead a constant or variable that is thought to be fixed for the time being.
If you want more unknown variables in addition to 'x', you can use y, z, maybe w, but if you want more than that you are out of letters and are often better off with super/subscripting (if it makes sense in the context), because a, b, c, d would be interpreted as constants. Especially in statistics/ML x is reserved as symbol for the independent variable and y is reserved as symbol for the dependent variable; z, w are common for latent variables. f,g,h are common shorthands for functions.
i,j are double-booked as an "arbitrary index symbol" or imaginary units. k,m,n are also constants, and often reserved specifically as constants that denote the dimensionality or sequence lengths (m times n matrix, sums x_1 + ... + x_n, sequences (x_1, ..., x_n), etc). (k can be also be an index in addition to i and j.) o is easily confused with 0 or a circle that is usually a function composition sign (I usually see the letter o only in the CS small-o-notation context, rendered in fancy typeface). p,q,k are again constants or sometimes alternative indices in complicated sums where i and j are not enough, but from p to v you have relatively underused letters. However, they are not totally "free" either: s and t and r and l are again common symbols for scalar variables like length or time or rotation; in statistics, p often is reserved for probabilities (conditional probability of x given m is p(x|m) etc). In linear algebra u and v can be arbitrary vectors, or unit vectors, or in complex analysis they are real / imaginary parts of a complex valued function.
Upper case letters are reserved for matrices and operations or (especially consonants like K or C) sometimes again constants.
In other fields things might go differently, but the general gist is that I do often feel like the alphabet is not enough.
Most often Greek letters get quite much use when you need symbols for angles or rotations (for example, Fourier transformation pairs) or as (hyper)parameters of probability distributions.
> Mathematicians love greek letters, and if there are a couple of them it's easy enough to read the equation, but you can easily end up with 5 or more in a single equation,
Is there any fundamental reason why Greek letters are harder to read than Latin ones? Harder to write, I can accept, if Greek is not your first language.
> What I often do when I want to better understand a formula is to rewrite it by replacing greek letters with plain words.
Try actually writing proofs using verbose variable names, and you'll see as clearly as daylight why single-letter variable names are preferred.
I taught myself computer programming before I learned advanced algebra and calculus etc. in school later. I have always felt that math notation was just a type of obfuscated source code with very poor variable naming that you generally couldn't run on the computer -- you are supposed to compute it by hand. In other words, I believe that many aspects of math are quite primitive and shitily designed.
I hope most people care about understanding something more than they do distinguishing themselves in your view. For many people, concrete examples can help them understand an abstract idea. Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea.
I'm not familiar with your examples, but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract, because it would be even better if I had a way to conceptualize of it that made it simple and obvious.
Finally, most _things_ breakdown. Doesn't mean that they aren't useful in a limited context.
> For many people, concrete examples can help them understand an abstract idea.
Sure enough.
> Even if someone doesn't grasp the full abstract idea, at least they might have a concrete example that exemplifies the idea.
Examples are only useful inasmuch as they pave the road to understanding abstract definitions. On their own, examples are pretty much useless.
In any case, your comment has nothing to do with the original topic, namely, whether coloring different parts of mathematical definitions is particularly helpul. It seems to me that the real problem here is that a certain kind of people has great difficulty understanding the syntactic and semantic structure of mathematical definitions. Now:
(0) The syntactic structure of mathematical definitions is the usual syntactic structure of English, or whatever your first language is. If it hasn't been drilled into your head in elementary school, sue your elementary school.
(1) As for the semantic structure of mathematical definitions, well, I guess there is no way around thinking hard about it, annoying as though it may be.
> but it is true that complex nuggets of math have evolved to be simpler, more elegant and sometimes more powerful. I try not to pride myself to much for understanding something complicated and abstract
Simplicity usually comes from having better abstractions (e.g., how Stokes' theorem evolved to its modern version in terms of differential forms), not from rejecting abstractions altogether.
This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it. Furthermore, you don't really get to decide what other people find useful for their understanding.
> Simplicity usually comes from having better abstractions
Absolutely. And sometimes using better abstractions involves less abstract thinking. I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be.
Notice that in the course of your argument, you keep bringing concrete math examples to make your point. You are not rejecting abstraction and yet you are not purely speaking abstractly either.
I would say that school teaches the (negative) integers is an abstraction over "what they actually are", the construction of the integers from natural numbers. Yet, thinking about that construction, or even knowing what the word "equivalence class" means requires abstract thinking.
In any case, I think it's a pretty trivial idea to show which parts of a natural language phrase correspond to which parts of something written in math. It's a gloss. You can do it with colors, or gesturing, or whatever.
Finally, I believe you are thinking too mechanistically about humans, especially humans learning about math. Elsewhere you suggest that (ask about whether) recognizing Greek letters should be as easy as recognizing letters from the Roman alphabet you are familiar with... Yeah, I think if you haven't used Greek letters enough, it's going to be harder to visually parse something with Greek letters. So now some more of your cognition is being spent on something that is wholly unrelated to the task at hand. Apparently you can even measure that German people are a little slower at some arithmetic problems because their names for quantities don't always list the digits in the same order of significance. Your WORD for a number causes you to be faster or slower at arithmetic. So I think using unfamiliar symbols might also make a difference.
Here you are suggesting that the syntactic structure of math definitions is the same as the syntactic structure of English. Right! They are written in English! However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses. The notion of syntax is an abstraction, and from that abstract point of view, there is no difference in the expressiveness of English and Mathlish. But humans apparently don't process not even natural language syntax as mechanistically as you suggest they do. Even though shallow and deep trees can be generated by the same grammar, more complex parse trees are harder to understand. You have to be careful with an abstraction because sometimes the details (like how many words are in a sentence, or how many layers of quantifiers there are) matter.
> This sounds like being familiar with an example of an abstract idea is as ["pretty much" as] useless as not knowing anything about it
Right!
> I am not persuaded that thinking concretely is akin to rejecting abstractions, as you seem to be.
I'm not arguing against thinking concretely. I'm arguing against highlighting the syntax of English sentences and math formulas just because some people couldn't parse them, let alone semantically analyze them, otherwise.
> Notice that in the course of your argument, you keep bringing concrete math examples to make your point.
Of course. As I previously said, examples serve to illustrate a general concept.
> it's a pretty trivial idea to show which parts of a natural language phrase correspond to (...)
Indeed.
> You can do it with colors, or gesturing, or whatever.
You can do it with your brain.
> Your WORD for a number causes you to be faster or slower at arithmetic. So I think using unfamiliar symbols might also make a difference.
Um, wat. Operating on a number doesn't require thinking of the word you use for that number.
> However, even in the realm of natural language, humans will have a hard time understanding utterances with nested quantifiers, negation, and clauses.
And they systematically let people graduate from high school in this pathetic state? Maybe I got it wrong: Don't sue your elementary school. Sue your country's education ministry.
> And they systematically let people graduate from high school in this pathetic state?
You're being nasty. There's nothing pathetic about recognizing that almost everyone experiences limits in their cognition. If you don't, then that must be very nice.
Perhaps you misunderstood. People can understand a sentence with a few levels of negation, quantification, embedded clauses, but not too many. And sentences in math are often more complex than sentences in casual speech. You made a bold claim that math syntax is equivalent to natural language syntax, and I am demonstrating why that claim is wrong, or at best not relevant to coloring math expressions. I understand you are claiming they are formally be the same (who knows, since no one has completely formalized "The Math Language" or probably any natural language, but I understand your point in the context of baby first order logic syntax). And I am claiming that that doesn't matter.
> I'm arguing against highlighting the syntax of English sentences and math formulas just because some people...
No one is saying you should colorize every textbook or every anything, so I don't understand what you are arguing.
> You can do it with your brain.
Yeah, and you need to feed your brain some input. There are lots of ways you can encode the same information. If you suddenly no longer had vision, someone could read to you unambiguously every math expression you needed, but I bet you're going to suffer a hit in your math performance. Despite using your brain.
> Um, wat. Operating on a number doesn't require thinking of the word you use for that number.
Obviously. A computer operates on a number without having a "word" for a number. (Well...). And some humans are better at being computers than other humans. That's not responding to my point that shows that people must be thinking about words, because you can measure delays when the words aren't very good, like the number words in German or French. Yeah, I agree it's "Um, wat" in an interesting way, but I fear you're just dismissing it.
I'm not arguing against examples. I'm arguing against using colors to make up for the inability to understand abstract structures. In my experience, the people who argue for this are usually “math educators” who don't do much mathematics themselves.
What I'd like instead: highlight only a single thing in a single equation, and accompany the highlight with a text that explains it. Then repeat the (image with highlight, textual explanation) pairs until you've explained everything. Benefits: 1. I believe for many people like me, attention and focus on a single "highlighted" item at a time is much easier. 2. You can use only a single "highlight" color that can be colorbind friendly (it's enough that it is noticeably different than black) 3. You don't need to explain the equation in a single text passage (unlike in the examples, where it looks like the author wants to explain all of the equation in a single sentence so that all different concepts in the equation are represented in different colors both in a single text and the equation itself.)
Also, in my experience, yellow/orange is terrible color for text / plots in print. (Peter Flach's machine learning textbook uses yellow extensively for similar "text highlights", and it made many sentences in that book frankly unreadable. I simply could not read light yellow text on white paper background.)