For example with the element symbol you can do

   let x \in R a real number

   let U \subset R, U \ni 0 a neighborhood around 0 

1. identity: "He is John"

2. membership: "He is engineer"

3. inclusion: "Wolves are carnivores"

For the non-symmetric membership and inclusion relationships, in natural languages the order of the words does not really matter, because a speaker will recognize which of the 2 words connected by "to be" corresponds to a bigger set, of which the other word may be a member or a subset, so "he is engineer" and "engineer is he" will be understood to mean the same, even when one alternative sounds weird (i.e. Yoda speech).

This is why, unlike for the agent and patient of a transitive verb, which need special markers, e.g. the nominative and accusative case markers, in the languages that do not have a fixed word order, for the subject and the nominal predicate that are connected by "to be" no distinct markers are required, they can use the same case (e.g. nominative), because they can always be recognized regardless of their order.

"To be" can also express other relationships, like position in space or time, qualities or quantities and so on, all of which are also distinguished by the kinds of words that are connected by "to be".

In an unambiguous language, like in formal mathematics or in programming, each kind of relationship should use a different notation.

In English, it's words like "is", "was", "are" or the verb "be". It follows the rules of linguistics, not mathematics, so there shouldn't be pressing need to stress its non-commutativity.

I have no skin in this game, but I wouldn't be surprised that someone with a brilliant mind would also have, to put it bluntly, "a bunch of weird shit."

Then there's the oral history someone posted on Something Awful some years ago [2] -- from someone who may have accidentally indirectly derailed the entire movement.

[1] https://aristasia.guide [2] https://web.archive.org/web/20230817170434/https://forums.so...