Mathematics is hard for mathematicians to understand too

The views quoted are just as cryptic as modern mathematics. Did mathematicians lose the ability to convey stuff tin plain simple ways?

Probably they are trying to romanticize something that may not sound good if told plainly.

Face it. Mathematics is one of fields strongly affected by AI, just like programming. You need to be more straight forward about it rather than beating around the bush.

To simply put, it appears to be a struggle for redefining new road map, survival and adoption in AI era.

Usually you'd have to write that as \sum_{v \in \{a, b, c\}} v; one of the ways I think conventional math notation could in fact be improved would be by separating the aggregate function of summation from the generation of the items, allowing you to write \sum \{a, b, c\}, at the minor cost of having to write \sum_{i = 1}^N i^2 as something like \sum |_{i=1}^N i^2.

It's not conventional to write commutative-but-not-associative functions as infix operators, but I don't think that's due to some principled reason, but just because they're not very common; non-associative operators such as subtraction and function application are almost universally written with infix operators, even the empty-string operator in the case of function application. The most common one is probably the Sheffer stroke for NAND (although Sheffer himself used it to mean NOR in his 01913 paper: https://www.ams.org/journals/tran/1913-014-04/S0002-9947-191...).

You can go a bit further in the direction of logical manipulability, as George Spencer Brown did with "Laws of Form" (LoF): his logical connective, the "cross", is an N-ary negation function whose arguments are written under the operation symbol without separators between them, and he denotes one of the elementary boolean values as the empty string (let's call it false, making the cross NOR). ASCII isn't good at reproducing his "cross" notation, but if we use brackets instead, we can represent his two axioms as:

    [][] = []  (not false or not false is not false)
    [[]] =     (not not false is false)

In this way Spencer Brown harnesses the free monoid on his symbols: the empty string is the identity element of the free monoid, so appending it to the arguments of a cross doesn't change them and thus can't change the cross's value. Homomorphically, false is the identity element of disjunction, which is a bounded semilattice, and thus a monoid.

This allows not only the associative axiom but also the identity axiom to be simple string identity, which seems like a real notational advantage. (Too bad there isn't any equivalent for the commutative axiom.) It allows Spencer Brown to derive all of Boolean logic from those two simple axioms.

However, so far, I haven't found that the LoF notation is an actual improvement over conventional algebraic notation. Things like normalization to disjunctive normal form seem much more confusing:

    a(b + c)  → ab + ac          (conventional notation, rewrite rule towards DNF)
    [[a][bc]] → [[a][b]][[a][c]] (LoF notation)

It's a little less noisy in Spencer Brown's original two-dimensional representation (note that the vertical breaks between the U+2502 BOX DRAWINGS LIGHT VERTICAL characters are not supposed to be there; possibly if you paste this into a text editor or terminal it will look better)

    ┌─────    ┌────┌────
    │┌─┌──  → │┌─┌─│┌─┌─
    ││a│bc    ││a│b││a│c

but not, to my eye, any less confusing.

Well said.

I'd say the ability to take complicated definitions and to not have to through a rigorous definition every time the ideas are referenced are, in a sense a form of abstraction, and a necessary requirement to be able to do advanced Math in the first place.

> What they are missing is that the notation is the easy part.

This is so wrong it can only come from a place of inexperience and ignorance.

Mathematics is flush with inconsistent, abbreviated, and overloaded notation.

Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.

> No, the argument is even dumber than that. The person who writes a poem hasn't created any literature.

Sure they have, by virtue of writing it down. It becomes literature when it hits the paper (or computer screen, as it were).

(Unless you mean to imply that formulating an original poem in your mind counts as "writing", in which case I guess we illustrate the overarching point of value in shared symbols and language and the waste of time in stating our original definitions for every statement we want to make)