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Let's take an example in Wu-Ki Tung, Group theory in physics:

Theorem 3.4: Irreducible representations of any abelian group must be of dimension one.

Proof: Let $U(G)$ be an irreducible representation of the abelian group $G$. Denote by $p$ a fixed element of $G$. Due to the abelian nature of the group, we have $U(p)U(g) = U(g)U(p)$ for all $g ∈ G$. According to Schur’s Lemma, $U(p) = λ_p E$. This applies to all $p ∈ G$. Hence, the representation $U(G)$ is equivalent to the one dimensional representation $p → λ_p ∈ C$ for all $p ∈ G$.

I would rewrite it as:

Because in an abelian group every irrep commutes with another irrep, then according to the Schur’s Lemma, they are just multiples of the identity operator, forcing them to be one dimension only.

In my opinion, the rewrite is much better. There is no need to prove the theorem anymore, but they are blended into a flowery but self-contained text with no single symbol, and readily to connect to the next idea. 7 sentences in 6 lines reduce to 1 sentence in 3 lines, but the true intuition is still preserved, and I don’t see how this is vague or makes the formalism lost at all. In other words, we have the best of both worlds.

But in the discussion Making intuitive formalism and concrete flowery text, it is said to be awful or incomprehensible, even to those who understand the topic. Do you know why is that?

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    $\begingroup$ The first line of the rewrite is inaccurate. It should be about two matrices from a single irrep, not two irreps. $\endgroup$ – Andreas Blass Jul 28 '18 at 13:10
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To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some point, mathematical sentences where written without symbols; the introduction of letters to denote mathematical object improved quite a lot the depth and intricacy of reasoning available and communicable.

While it is sometimes possible to write mathematical statements and proofs with few to no mathematical symbols (and I do prefer statement using fewer math symbols), this is not always reasonably possible, and always requires to be very precise (notably with grammar). Your example is telling in this regard: you confound commutation of two representations (which does not make much sense in the context) and commutation of images of two elements under one given representation. This makes your proposition quite problematic.

Even written with great precision, a proof without math symbols can be difficult to decipher, because the reader also needs to be hyper precise in reading it. The slightest slip in the interpretation of which pronoun refers to which object, and the meaning is lost.

That said, the habit of using symbols can sometime impose itself too heavily, and it is often possible to improve a proof by making it lighter, changing a sentence, limiting the introduction of notation. But the question is not math symbols versus flowery language, it is about how to best convey the proof to your intended audience. This is a craft that can be improved without limit.

Also, note that separating the statement from its proof is very important, as they play very different roles. You want the statement to be prominent, because it will usually be referred to several times.

Last, let me give a try at the proof you give in example.

Proof (most natural version to me). Let $U$ be an irreducible representation of an Abelian group $G$ and consider any $p\in G$. Since $G$ is Abelian, for all $g\in G$ we have $$U(p)U(g) = U(pg) = U(gp) = U(g)U(p)$$ and the operator $U(p)$ commutes with $U$. By Schur's lemma, this implies that $U(p)$ is a scalar operator. Since this is true for all $p$, $U$ is a scalar representation and, being irreducible, it must have dimension $1$.

Proof (attempt without symbols). Given an irreducible representation of an Abelian group, the operator associated with any element of the group must commute with the representation. By Schur's lemma, this implies that this operator is scalar. Since this holds for all elements of the group, the representation is scalar, and being irreducible it must have dimension one.

I do not think much is gained in replacing $G$ by "the group" and $p$ by "element of the group", and in the first proof it was possible to give more details (only suitable for beginners, but possibly crucial for them). Expressing the displayed equations without symbols would be clumsy and certainly unclear. Moreover, this is a very easy proof; try to only state the Five Lemma without maths symbols!

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    $\begingroup$ I realize that only by reading other's version that I can see how hard to follow the proof without notations. Maybe this is best for note-taking for oneself, because they can follow the flow in their own mind? Usually when I take note like this I have to reconstruct the sentence structure as well, to nicely follow with the surroundings. I also usually combine and rewrite many theorems together to produce what I call "intuition" or "big picture" $\endgroup$ – Ooker Jul 29 '18 at 16:35
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Honestly, both seem pretty bad and don't prove the statement in an obvious way. For example, there is confusion betwen irreps and the representations of individual elements. You can make the $G$-equivariant maps in Schur's lemma into intertwiners because $G$ is abelian, but it really deserves to be said that this is what you are doing.

I agree that it is useful to reduce notation if possible, but your rewrite isn't even grammatical. i.e. It isn't "the Schur's lemma", but simply "Schur's lemma". That lets alone the question of what it means for an irrep to commute with another irrep. What you are using is the commutation on the individual elements. There is such a thing as a representation ring (https://en.wikipedia.org/wiki/Representation_ring) where you could talk about the irreps commuting or not, but this is logically further down the line from the basic result you are proving.

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    $\begingroup$ Would the idea of rewriting be still good if I fix the technical and grammatical wrongs? $\endgroup$ – Ooker Jul 28 '18 at 17:04

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