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> I have no idea how nautil manages to so consistently talk about mathematical concepts in a way that is so hard to make heads or tails of

Because they're writing for laymen. I have a degree in math and work in a subfield of CS that relies on math a lot, so I have a lot of practice explaining concepts to some of the smartest people I know without a math background. The tradeoff for your concise expression is that it uses jargon that adds cognitive overhead for people unfamiliar with college-level or higher math, even if they've got a high quantitative intelligence. A non-exhaustive list that would trip up, say, my friend with a PhD in pathology:

1) associative and commutative are not commonly-know terms

2) "A group is a collection of elements 'with' multiplication" is underdetermined. What does it mean to be "with"? The notion that multiplication isn't commutative is unintuitive without understanding that the definition of multiplication is flexible here, defined based on the type of the element in the group.

3) Using o as a composition operator is something that a layman is guaranteed never to have come across

4) What does invertible mean? It's not trivially obvious to a layman that some matrices A have no matrix B such that AB = I.

Etc etc etc. Note that each of these gaps and confusing points (for laymen) compound; I'll occasionally read a paper in a field I'm not an expert in (like economics), and the being bogged down every other sentence by confusion is a severe impediment to understanding how these pieces are composed into something novel. Hell, even when I understand each piece, sometimes the most foreign concepts haven't marinated long enough for me to use them as building blocks yet.

Whats your complaint here? "I have a math degree, how dare nautilus target anyone but me with their articles?"

This may be very dependent on your education system, but commutativity and associativity are terms I was taught around fifth grade, while function composition, groups and matrix inverses were high school maths.

I've lost contact with many of these after finishing my university, but they are still pretty common terms around me.

Where and when I went to school, commutativity and associativity were taught in elementary school, and again in high school because we had all forgotten it. Matrix math was something I learned in university.

However that was a long time ago and standards have changed. http://www.corestandards.org/Math/Content/HSN/VM/C/9/ indicates that matrix math is now considered high school material across the USA.

So those should still be terms that are familiar to most people in programming today, not some arcane post-grad math stuff like category theory.

In theory, yes.

But you know about theory vs practice.

Sure, I'm pretty sure I learned the terms in middle school too. But along with matrix inverses, it's not the kind of thing that people retain, especially since it was detached from any real framework with which to orient it; many students tend to learn it by rote as a descriptor detached from any usefulness or meaning. Eg, you don't need to do any reasoning about the concept of commutativity to know that subtraction isn't commutative; much younger children are capable of subtraction without grasping commutativity as an abstract concept. Associativity as an out-of-place concept is similarly useless for lower-level math when you can just use the much more concrete PEMDAS.

Function composition is a pretty trivial concept, but I referred specifically to using ∘ to represent it; this symbol isn't IME as widely-understood to the layman as simply using f(g(x)) (by contrast with college-level and higher math, where it's very widely in use).

The gist of my point was that even the smartest quantitatively-inclined people who went to my private school in a tony part of California aren't likely to have retained familiarity with these concepts unless their college education and/or career involves theoretical math. This is leaving aside people not as smart, or less quantitatively-inclined; my friends who fall into this bucket probably didn't remember what commutativity and associativity were by the time they _entered_ college, let alone later in life. Arguably this latter group wouldn't be part of the target audience interested in reading an article about theoretical math at all, but my friends in the former group certainly would read publications like Nautilus on general mathematical topics.

I don't know anything about representation theory, but the way you describe it I thought "how's this different from a homomorphism." All elements of a group (e.g. any x, y in your example), are invertible by the definition of a group.

The wikipedia sentence definition makes more sense to me: "a "representation" means a homomorphism from the group to the automorphism group of an object"

This seems natural, since if you take _any_ algebraic structure (e.g. a ring), then its automorphism group is a group. Which is a reason why group theory is worth studying - since _any_ algebraic structure can be studied using group theory by considering its automorphism group. (Also, an automorphism group is just the set of symmetries, so group theory can be described as the study of symmetry).

This is really concise, thanks! Removes a lot of the mysticism that is all too present in mathematics writing for laypeople (this explanation is just as understandable).

Weird, I felt the exact opposite. In the article I can use the pictures to understand exactly what all of these mathematical terms mean.

No one who graduated college with a degree in English is going to know what "multiplication is associative, there is an identity, and every element has an inverse. Multiplication is generally NOT commutative." means, and that's the first sentence...

What phrasing would have been better in your opinion?

> For example, reflecting the triangle and then rotating it 120 degrees reorders the vertices the same way as if you’d merely performed a different reflection.

With pictures, from the article.

You have to have fully internalized those concepts already for that explanation to be useful. So it's good for people who already have a somewhat solid mathematical foundation and working knowledge.

For it to be useful to a layperson, you'll need to provide concrete examples and more of an explanation of each concept. Or, if you want to help people who are visual learners, something like a triangle.

Not nobody. I know high school kids who would understand that. Furthermore I submit that someone who understands the phrases "linear algebra", "matrices", and "the trace of a matrix" is likely to understand that sentence. Anyone who knows what a "Lie group" is, DEFINITELY understands that sentence. (All of those phrases occur in the article without real explanation.)

As for what that sentence means, it means that given any two elements we can "multiply" them. Multiplication is associative, meaning that x * (y * z) is always the same as (x * y) * z. There is an identity simply means that an element we can call 1 exists such that 1 * x is always x. There is a multiplicative inverse means that for every x there is some element we can call x^(-1) such that x * x^(-1) = 1. By not commutative I mean that x * y doesn't have to be the same as y * x. (An example of non-commutative is that if you swap the first two items in a list then the first and third you get a different result than if you swap the first and third then swap the first two items.)

But still, you have answered my actual question.

What that site attempts to do is, at length, build some intuition about a topic, so that it can talk about it. By contrast my explanation just reminds someone who understands the fundamentals of the topic of the concepts so that we can start talking. So someone like you has a chance to understand something. Someone like me has to work to keep track of where they are, only to say, "You're just talking about group homeomorphisms."