Abstraction

Generalization and Specialization

Here, we can add logic as an external meta-abstraction that applies to ALL of these.
- With arrows pointing from Logic to each one.
Start with real-world, often physical, observations of objects,
Remove "special properties" (or "context") from the real world:
- If you are looking at a picture with three different lions, remove their differences,
- and then remove their "animalness", etc.
- this is called making it "context-free".
Look for:
- relations between them,
- operations that act on them,
- properties they might have.
- This is the most difficult part and cannot be taught completely.
Now see if the objects can be replaced with others.
- Do the properties / relations still hold?
Now see if the operations could be replaced with others:
- Do they still work?

Surface area in the naive, physical sense:
- how much water does it take to fill some flat surface?

Then generalize to:

squares: $A = s \times s$,
rectangles: $A = w \times h$,
triangles: $A = w \times h / 2$,
regular polygons: add up the triangles,
irregular polygons,
areas under arbitrary continuous curves of a single variable over an interval:
- add up areas of rectangles over the interval, take limit: Riemann integral
piecewise continuous curves over an interval,
"integrable" curves over an interval,
integrable curves over the partitioning of an interval by a function of bounded variation,
integrable curves over measurable sets: Lebesgue integral,
multi-variable version of this,
the $p$-adic numbers instead of $\mathbb{R}$,
an integral over a locally compact topological group with a left-invariant Haar measure:
- abstract harmonic analysis!