In order for a proposition to be completely stated, how much detail needs to go into describing it?

Question

During my study of mathematics so far, I had come to realize that how a proposition is stated can be just as important as the proof itself.

For example, when I was working on various propositions on the relationships of limits, one I had worked on was proving that the sum of two limits is equal to the limit of the sum.

In Hardy, this theorem was stated as follows:

Theorem 1: If $\phi(x)$ and $\psi(x)$ tend to limits $a,b$, then $\phi(x)+\psi(x)$ tends to limit $a+b$

Though he builds context in the book, from the theorem itself he leaves it unclear as to what each of the variables are.

Another way I had seen someone state this particular theorem is:

Theorem 2: Suppose that we have two functions defined by the real numbers, $\phi(x), \psi(x)$. If $\lim\phi(x)=a$ and $\lim\psi(x)=b$ then $\lim[\phi(x)+\psi(x)]=a+b$

However, some small room for ambiguity still seems to remain. So I wrote the theorem this way

Theorem 3: Suppose there exists two functions $\phi(x)$ and $\psi(x)$ that both have the real numbers as their field of definition. If $\lim \phi(x)=a$ and $\lim\psi(x)=b$ where $a,b\in\mathbb{R}$, then $\lim[\phi(x)+\psi(x)]=a+b$

The above three theorems seem to demonstrate a trade-off between a concise theorem versus an unambiguious theorem.

My question is - what is the optimal amount of information one should contain in a theorem in order to balance the factors: conciseness and unambiguity, and what factors should one should keep in mind when determining this at a particular case?

Answer 1

It really does depend on context, and on what has immediately preceded the statement of any given theorem. For example, a corollary is sometimes introduced or stated immediately following a given theorem, and the premises of the theorem are sometimes assumed in the statement of the Corollary.

Personally, I prefer that theorems be "self-contained", so I prefer your statement in that you explicitly state $a, b \in \mathbb R$. However, I would simply introduce $\phi(x)$ and $\psi(x)$ as functions defined on the real numbers.

How explicit one needs to be depends, in part, on the audience to which a text/paper is addressed, the intent of the author, etc. To a large degree, it is a matter of style, with some authors opting for "more concise" and others for "more precise" with respect to minimizing any possibility of ambiguity.

But concise and precise need not always be at odds with each other.

Answer 2

"Optimal" depends on the expositor's style, the target audience, and other factors. If you're concerned with writing mathematics yourself, your style is something you'll come into naturally, so I think the target audience in the most important consideration.

For instance, if I were a beginning student attempting to read an "intro to proofs" sort of book, I would very much prefer and need to have very precisely stated theorems, where I am completely aware of what each variable stands for and where it comes from. Correspondingly, if I were to write such a book, I would refrain from using any of the shortcuts I use for higher-level work.

An example of what I mean by shortcuts would be when I write up analysis. If I'm dealing with a sequence, I'll often index it by some letter like $i,j,k,n,m$ and I won't really bother with saying that these are natural numbers unless the context also involves two-sided sequences. Sometimes instead of $$\sum_{n=1}^\infty$$ I may write $$\sum_{1}^\infty,$$ or when I mean $\limsup_{n\to\infty}$I may write $\limsup_n$. For the audience I'm writing for, these shortcuts are natural and unambiguous, since they are more used to this sort of thing, and I sometimes get a feeling that to explain them in too many words would be an insult to the reader's intelligence.

I lean toward precision myself, but I appreciate an economy of words and letters. I like works as a whole to be self-contained, so I expect sections and chapters of a paper or book to fully introduce recurring symbols at least once. But I'm more forgiving with individual theorems in a book - if you've already told me that $\Omega$ is an open region in the plane, then I don't mind if you don't redefine it each time. As a rule, the parts that stand alone should be introduced properly.