>>12993365>yes but how do you know what 1^2 + 2^2... n^2 equals?
you realize that fancy formulas like the one about which the thread is about were found later on, rather than developed immediately as the starting point?
and where investigated earlier on.
the problem with math books is that they look at a finished process of developing and proving things. what I'm trying to say is that in most cases people who found a theorem started with a simple problem, then they noticed the general idea (a prototype of the final theorem) and they created lemmas as the steps to the theorem.
in math books we see a bunch of weird lemmas which are hard to understand why you would consider them without a context.
proofs should be like this
1. start writing the proof and the moment you run into a problem, start developing the lemma
2. repeat this each time so that each lemma is motivated by a real problem
3. finish the proof.
I hate books with the approach
1. introduce a bunch of seemingly random lemmas with proofs
2. reach the significant theorem and prove it by "from lemma 1 follows this, then lemma 2, lemma 3, and the theorem is proven".
this is unnatural