What are the questions one should ask of oneself in trying to understand definitions and lemmas? [closed]

Question

Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then we define the finite sum $\sum_{i=m}^{n} a_i$ by the recursive formula $\sum_{i=m}^{n} a_i = 0 $ whenever $n < m$ and $\sum_{i=m}^{n+1} a_i = \sum_{i=m}^{n} a_i + a_{n+1}$ whenever $ n \geq m-1$ .

Lemma Let $m \leq n < p$ be integers, and let $a_i$ be a real number assigned to each integer $m \leq i \leq p$. Then we have $\sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i =\sum_{i=m}^{p} a_i$.

MY QUESTIONS

1. What are the typical questions one would ask of himself in trying to understand such a typical sequence of definition and lemma?
2. Why did the author write $m \leq i \leq p$ and not $(n+1) \leq i \leq p$? Is the author trying to generalize $i$? If yes what is the benefit of this generalization?

Answer 1

1. Questions to ask yourself: "Can I create a concrete example myself? Why are the hypothesis sufficient? Are they necessary? Can I find a counter-example if I change them?"

2. The index of summation ($i$) must be between the bounds of summation ($m$ and $p$). The integer $n$ is just where the summation is being split, so it will be an upper-bound for $i$ in the first summation, and a lower-bound for $i$ in the second summation.

Answer 2

Mathematics, in contrast to the wasy it is presented in many textbooks, is not a collection of sacred rules of conduct, nor a sacred dictionary of meaning-of-words, but a certain form of an account of what mathematics is about, and what mathematicians do.

It is true that there has been "optimization" of these accounts over the last 200 years, ... but that in itself is a problem, in some cases! That is, the back-story is tricky enough so that without knowing it (which would be daunting already) the "calm" sequel is incomprehensible.

And then there's the stylistic conceit about definitions/lemmas/proofs. This can easily degenerate into a sort of acolyte-seeking-to-propitiate-whimsical-gods scenario.

While we're here, although "the Moore method" has its fans, I am not among them. In fact, I would claim that the implicit hypothesis in this is that mathematics is (conceptually) "shallow", in the specific technical sense (not necessarily insulting, though it sounds like it!) that not much prior experience or prior knowledge (the big two "priors") are relevant. (Another discussion: necessity versus relevance.)

The most-most-immediate answer to such questions, I'd claim, is to ask what "real question" this "stylized/formal" question is hiding.

Usually, the answer is that there's no real issue. More like whether you're sitting down... much less having fastened the seat-belt... before driving.

It is certainly true that other philosophical approaches to mathematics dictate or suggest other views, but, I'd recommend not thinking in terms of those things, but in terms of the "what are we trying to do?" criterion. I know, yes, these are not the usual things in "school", ...