This is similar to the many proposals to replace the notation for numerals, or hexadecimal, or other math notation.
I understand and agree that the set of squiggles we use is completely arbitrary, but the sets of squiggles known as "latin alphabet", "greek alphabet", and "arabic numerals" are the set of squiggles I grew up with. And using them has some advantages, for example, my proficiency with those particular sets of squiggles allowed me to notice in short order that the Y combinator is not, in fact, the one reported in the linked article.
The expression given in the OP is, according to Wikipedia, sometimes cited as the y combinator itself (and is beta equivalent).
The arbitrariness in the notation here isn’t about whether to use Latin letters or some other alphabet, or to write application on the left or the right, but rather about whether to use expressions in a formal language or some other representation. In many fields, different notations or finding eg some graph structure, may reveal interesting aspects of the underlying structure of something (eg why do we write permutations with cycle notation not permutation notation? Eg Conway’s notation for knots).
Also there are issues with the lambda calculus notation using letters for variables: whenever you do a substitution you may need to rename some variables if their meaning would change. I think this can be confusing to people and is perhaps an annoying thing to have in your notation.
I'm not sure how well-known this is, but Conway notation can be thought of as taking a Tait graph of the knot (i.e., a checkerboard graph) then reducing all the series-parallel networks until you get some irreducible graph. He noticed that certain kinds of series-parallel networks can be represented by an extended rational number (includes infinity), which can dramatically simplify the analysis of small knots -- to the point he was able to classify all of them in an afternoon! So, Conway notation corresponds to an irreducible planar graph with an expansion of each edge by a series-parallel network, with each edge in the expansion labeled by a rational number. You can even work out a set of moves such that if two such diagrams correspond to the same knot then there is a sequence of such moves connecting them.
I do wonder if there's a similar analysis for lambda calculus represented as graphs... It might be that certain expressions play the role of rational tangles. (Though very likely not :-) )
> The expression given in the OP is, according to Wikipedia, sometimes cited as the y combinator itself (and is beta equivalent).
That's actually true, and very obviously so. I'm sorry.
> The arbitrariness...
I agree that clear notation is important; the pictorial style made me think of Feynman diagrams. My comment was a bit tongue-in-cheek, maybe unnecessarily so, but the point is that I fail to see how those pictures are clearer than plain text: you are not replacing complex stuff with a simple representation, you are just rewriting a simple formula in a different style.
> Also there are issues with the lambda calculus notation...
Are there? That's the same problem you face whenever you have dummy variables, e.g. the 'dx' in integrals.
Yeah I don’t see much from this notation (I also find it hard to tell the difference between the result of an abstraction and an application which feels like a somewhat important difference). I mostly wanted to suggest that diagrams or notation changes in general can be very useful.
It’s true that variable renaming comes up in other parts of mathematics but scope matters a lot less in most other mathematics, so much so that it is basically implicit and left to be inferred based on the mathematical context. You wouldn’t normally rename a variable to avoid a conflict so much as you would avoid giving it a conflicting name in the first place. The recursive nature of the lambda calculus means you can’t avoid the problem this way.
