I think you're giving me an uncharitable reading, and in doing so support my stated concern:
> I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue.
On the topic of "heterogeneity" it would seem to follow from my saying "If you can follow along with Terry" that I recognise some people are capable to find his explanation clear.
My concern is providing that link as educational stead insightful.
On the heterogeneity of pure mathematicians it is very frustrating to see someone try to claim, in a public forum where people unfamiliar with the subject can read, that a Mathematician uses anything other than "numerical reasoning while trying to reason about a subject", and I feel it causes great harm to people outside of mathematics.
You say, "some use graphical reasoning", well where did their graphical intuitions come from? Are they born with it? Or perhaps they were taught graphical reasoning by way of numerical reasoning and now are able to just use the graphical reasoning on new problems.
This is part of what I was trying to express by referencing Gowers. Using identities to reason about a problem is fine, and arguably necessary, but introducing identities when teaching without explanation is befuddling and I argue deters learning.
The op said "I'd like to learn more about symmetry", you used the word "introduction" and Tao used the word "simple".
If the op said, "I went through a physics program in undergrad and grad school and I feel like I have all the mathematical pieces but I'm still struggling to put them all together... can anyone recommend a resource that avoids getting bogged down in prerequisite explanations and just focuses on the relations."
Then I would have scrolled past thinking, yeah that Tao article is perfect.
If you would have said, "After years of higher education this article made it click for me in ways multiple professors failed to but you'll need an equivalent amount of the numerical examples from my education of the various topics used to understand this."
Then I would have scrolled past thinking, yeah Tao does a great job of illuminating concepts.
If Tao would have used any other word than "simple".
Then I would have left out the "novella" jab. Though now I'm tempted to ask what about the short textbook was more difficult, because in my similar library experiences the short books were the ones that just had equations and identities and left out any of the numerical reasoning. Is that what you discovered when you Haha'd at that little textbook?
I worry for Mathematics when someone says, "Hey I think I might be interested in that thing..." and the response they recieve is a bludgeon of latin characters, and the dubious unqualified follow up "well other people get it".
It is now doubly hard for that person to explore their interest because they need to both find their own resources that actually teach the material, as well as decide whether its even worth it if this is the community they are working to become part of.