
Colorized Math Equations - ghosthamlet
https://betterexplained.com/articles/colorized-math-equations/
======
aqsalose
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.)

~~~
dandare
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...](http://www.ktorides.com/wp-
content/uploads/2015/01/colors-750x410.png)

~~~
jhanschoo
If I use colors with uniform luminosity will color-blind have difficulty
distinguishing them?

------
cm2187
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.

~~~
TrinTragula
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

~~~
cm2187
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.

~~~
aqsalose
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.

------
catnaroek
This approach eventually breaks down. Could you give a useful colorized
explanation for Urysohn's metrization theorem? Or the Vitali covering lemma?

Colors are useless. Just learn to think abstractly. This is what (supposedly)
distinguishes you from lesser animals.

~~~
gugagore
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.

~~~
catnaroek
> 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.

~~~
gugagore
> On their own, examples are pretty much useless.

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.

~~~
catnaroek
> 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.

~~~
gugagore
> 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.

