What in mathematics has the property of beauty?

Question

I might or might not be at an impasse in my writing... I have around 200 pages of notes, and I finally sat down and tried to compile some of the material, but I feel like the presentation is off somehow. There are so many details to go over, half the time it's like, "Where do I even start?" or even worse, "Am I forgetting some subargument that I need to go over before going over what I actually started with?"

At any rate, I'm assuming that implementations are the semiotic analogue of models and proofs re: semantics and syntax. So I want to codify what it is about them that might or might not at least partially/incompletely "justify" them, seeing as they do obscurely affect the space of theorems in a set theory.

I was also considering alternative formulations of intuitionism. I actually have assimilated Cantor to a technical alternative intuitionism, namely a theological one, where God, Who is all-powerful, certainly has the power to cause mathematicians to have intuitive knowledge of various things, here the mathematical realm. (To be sure/fair, I don't know whether Cantor thought that God had intuitively or discursively revealed the alephs and the omegas to him. But at any rate, God's own knowledge being a matter of intellectual intuition (as per Kant), God's knowledge of the mathematical realm is intuitive, so we still have an intuitionism in possible play, here.)

However, there is also the question of aesthetic judgment in mathematics. In my eyes, the property of beauty, such as it is (it might be a relation, IDK, I haven't read/thought enough about it to be honest), is intuitively given. Even if there is some arcane/convoluted process involving the mental gears of our Kantian faculty psychology, that is meant by the notion of beauty, yet if I remember correctly, since it is the power of judgment that is implicated in the process, and judgment is the vehicle of subsuming particular intuitions under general concepts, so this ends up sounding like 'knowledge' of beauty is interchangeable with an intuition of something as beautiful.

Now I suppose that's all unnecessary to the actual intended argument on this score, because my intent is to advert to aesthetic considerations as justifying conditions modulo semiotics, and it doesn't matter too much whether aesthetic consciousness is principally intuitive or discursive, or somehow both, or whatever. Although semiosis is a more intuitive process, on the surface, than syntactic configuration (which seems almost the sine qua non of discursion), it is irrelevant to the immediate argument whether it is also strongly discursive a process for some reason.

So my question is: 'where' is beauty in the mathematical realm?

I read through a Wikipedia article that covered this topic, and the article seemed to indicate that different things on different levels count as mathematically beautiful: a simple proof, a clever proof, etc. However, I want to focus on a paradigmatic example in order to illustrate what might or might not be at stake with respect to Benacerraf's implementation problems. This example is Euler's identity, normally stated as:

e^iπ + 1 = 0

IIRC, the Wikipedia article cited an opinion that part of/the reason for this being such a beautiful equation is that it showcases a deep relationship between various numbers that are fundamentally important to the development of the theory of arithmetic and other such functions on numbers. However, this makes the fact that the inscription refers to into the thing that is beautiful: it is beautiful that e^iπ + 1 = 0.

So consider the following list of alternative statements of the very same fact:

e^πi + 1 = 0

1 + e^πi = 0

0 = 1 + e^πi

2.71,,,^{3.14...(√-1)} + 1/1 = 0

The natural logarithm, when raised to the power of the imaginary unit (which itself is multiplied first by the ratio of a circle's circumference to its diameter), and then added to 1, equals zero.

I dór ceri-, ir na i rod plural rodyn -o i pennas er bui i in -o a rind's os- na i thar, a a er, na- lost. {This is an extremely frustrating Sindarin translation of the preceding. Don't try translating it back: to find words that would go through on the program, I had to rephrase the preceding by more or less 'literally' butchering it.}

(e ↑³ (i ↑² π)) ↑¹ 1 = 0

(e ↑³ ((-1 ↓³ 2) ↑² π)) ↑¹ 1 = 0

There is a supposition, on my part, that issues of glyphic efficiency are implicated in the display of glyphic elegance. The simplest efficient digitalization of the natural numbers is in binary format (i.e. having two symbols, whose concatenations 'stand in for' 2, 3, 4, ...), and for all the good it would do (and it might do some good!), we could then trade out the zeroes and 1's for F's and T's (for False and True), so then even ⊥ and ⊤, maybe. Certainly Cantor made an eternally wise choice by using the aleph glyph, and indeed subitizing the set of all natural numbers is most efficiently accomplished by having a unit symbol for infinity, then (though note that, if we are not to have a different glyph for every transfinite cardinal, we have to index our unit symbol to differentiate the cardinals to at least some extent (note that even inaccessible cardinals and their initial ordinals can be styled fixed points of the aleph/omega functions, after all)).

Kant said something somewhere about clarity of understanding being a matter of appreciating the distinctions between things: a term is clear when clearly differentiated from other terms. I assume there are possible glyphsets that satisfy this criterion to the minimally satisfactory extent, that is they have the minimum number of elements (glyphs) required to clearly differentiate the terms of the system. But I don't think this has to result in a more beautiful display of mathematics than if we used more glyphs, or rather I don't see why this simple efficiency would be the only criterion of mathematical beauty.

"Bonus points": and don't get me started on Kant's theory of the sublime as a counterpoint to beauty, what with his talk (if I recall) of the ratio of finitude to infinity. I would say that the finitist thinks we 'merely' lack justification for asserting an actual infinity as an individual 'number,' whereas the ultrafinitist goes further and says that those infinitarian assertions are antijustified, but then neither of these analysts would be able to believe in infinitely beautiful equations/theorems/what-have-you. And ultrafinitism, by conceptually 'destroying' an infinite amount of possible beauty (so to say), convicts itself of its own antijustification, however: in fact, ultrafinitism turns out to be infinitely grotesque.