This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me.
Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote:
from the items $p_{i}$, find a common pattern in the direction (direction of pattern) $Q$ and denote the pattern as $a$.
For example $a(Traffic \; light \; to \; stop, Blood, The \; Sun \; at \; dawn, Seeing)$ , where $Q = Seeing$ may denote the color red, because all object has the property red colour in the direction of vision.
And $g(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, S)$ denote
from the items $p_{i}$ [ each $p_{i}$ has a unique attribute $S(i)$ ] denote find a object $g$ which has all the property $S(i) \forall_{n: > i \in [1, n]} p_{i}$.
For example, $g(Knife, Screwdriver, Scissor, Common \; day \; usage)$ [Where each $p_{i}$ (Knife, Screwdriver, Scissor) has a unique attribute $S(i)$ (Common day usage)], may denote a Swiss Knife, because it has all the property $S(i) \forall_{n: i \in [1, n]} p_{i}$ (It has the common use of a knife, common use of a screwdriver, common use of a scissor)
Is the function $a$ and $g $ are similar to what mathematicians call abstraction and generalization respectively ? If not, then please explicitly state what it is with a lot of example. (An example) What can be treated as a generalization and (separately) abstraction of $1^2 + 2^2 + 3^2 + 4^4 + ... + 66^2 $ (The last term is quite arbitrary)?
Before reacting , please be clear what I want (with a cue from Polya):
I am not asking in English SE, so I ain't excepting what is commonly mean to the general people (they may decipher it wrong) by the term. Mathematicians usually do something when they "abstract" or "genaralize" something. I am asking what is usually done by them, when they do that (With good non-trivial and boundary cases examples).