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I suppose this is a question about mathematical convention. In a problem in Introduction to Probability by Bertsekas and Tsitsiklis, they ask the reader to prove an identity. But then their proof is just using words:

In mathematics, why is this allowed? Can you say that this is more correct a proof that is, "Oh, it's obvious!" or "Just keep distributing $A$ over and over ad nauseum and you get the term on the right"?

I'm not trolling. I'm genuinely curious as to how thorough one must be when using words as proof.

Exactly as thorough as you would have to be using any other kinds of symbols. It's just that vast messes of symbols are hellish for humans to read, but sentences aren't. Adding symbols to something doesn't make it more rigorous, less likely to be wrong, or really anything else. Symbols are useful for abbreviating in situations where this adds clarity, and making complex arguments easier to follow, but shouldn't be used where they do not help in this regard.

Yes they can and I'm of the opinion that symbolism and notation should be avoided unless it serves to simply the presentation of the material or to perform calculations. For example you want to cut a cube so that each face has a three by three grid of smaller cubes similar to the Rubix cube and with a little thought and experimentation once might conjecture that six is the minimal number of cuts. The best proof of this that I know of is simply "Consider the faces of the center cube." They require six cuts because there are six faces and it follows immediately. No symbols or calculation but still logical and mathematically sound.

Natural language for expressing mathematical statements can be indeed vague and ambigous. However, when you study mathematics, one thing you will usually learn at the beginning is how to use mathematical terminology in a rigid, unambigous way (at least for communication with other people trained in mathematical terminology). This process takes usually some time if you are not a genius (I guess it took me about two years at the university until I became reasonable fluent), so unfortunately I fear I cannot tell you a small set of rules which kind of language is "right" for mathematical proofs, and which is "wrong".

So IMHO the answer is clearly "yes, words are fine, when used correctly by a trained expert". Amazingly, one could say the same about proofs using formal symbols.

Note that historically, before the 18th century, proofs using natural language was the de facto standard in mathematics. The symbolics we often use today was developed in the 18th and 19th century.

Two points:

(i) Historically, all proofs were done in words—the use of standardised symbols is a surprisingly recent development. This is obscured a bit because a modern edition of, say, Euclid's Elements is likely to have had the words translated into modern notation.

(ii) Before symbols can be used they have to be defined, and ultimately that definition will be in words. It's easy to forget this, especially with ones that we use all the time and learnt in childhood. But, for example, we once had to learn that $2+3=5$ was short for "Two things together with three things is the same as five things".

Though a lot of us learnt instead that $2+3=5$ meant "Three things added to two things makes five things".

Now, these two definitions are different. One makes $2+3$ into an operation done to $2$, and treats $=$ as an instruction to carry it out; the other says that the number on the right has the same value as the expression on the left. The notation, though, doesn't make this distinction, and it's possible to spend years using the $=$ sign as though it meant "put the result of the operation on the left on the right".

So in this case we've got one string of symbols, $2+3=5$, a correct definition and a misleading definition. And how do we clarify which is the correct definition of the symbols? By choosing which verbal definition we mean.

Of course, more advanced symbols will most likely have some mathematical symbols in their definitions—but ultimately, we'll get back to words.