Totalities

``totalities'' is a -pedagogical- word, not a mathematical word

Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Functions are machines that take in -one- input and spit out -one-
output.
This is the ``pointwise'' understanding: for any input, you get an
output.

However, you should also understand functions -as first order
objects-, as -totalities-: it's a machine -- you can plug -it- into
stuff.

I think the notions of:
- functional
- operator
- family of functions
clarifies this (I swear!!):
functional:
  takes in a function, spits out a number
``evaluation at $17$'' is a functional
operator:
  takes in a function, spits out a function
  EG, $f \mapsto f^2$,
  or $f \mapsto f'$
family of functions:
  (esp.: sequence of functions, continuous family of functions:
   think of a continuous family of functions as a -path-
   in -function space-!)
  takes in a number, spits out a function:
  $n \mapsto x+n$, or $n \mapsto x^n$, or $n \mapsto nx$ or whatever

Sets %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Sets are things that have elements, but you can operate on sets
-themselves-.
EG, union, intersection...

Categorically %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
More abstractly, after studying various mathematical theories,
you can consider a -theory- as a basic object!
A universe/model of set theory (abstract set theory),
a category, etc.
