About Math

• understanding complexity

• complexity (entity, problem, idea, concept, object)

• divide and conquer strategy

  • analysis and synthesis

  • decomposition and composition

• abstraction

• generalization

• mathematical concept

Analysis and synthesis

When faced with a sufficiently complex problem, (save a few exceptions) we don't rely on intuitive understanding ("a flash of clarity") that will allow us to comprehend it in an all-encompassing manner, but on our trademark strategy, that made us famous across the chordata phylum, nicknamed "divide and conquer". Equipped with the latest evolutionary upgrade, the mind with abstract thought support, we can decompose (smash it into bits until it can be smashed no more) a single large problem into its constituent parts, that are simple bits, far more convenient to reason about. We analyse the pieces, gradually gathering information into knowledge. Once we completely understand all the pieces, we can start composing them into the higher-level structures, slowly abstracting away the lower-level details while we work our way toward the totality.

The process always starts with analysis/decomposition (the "divide" part) and it is followed by synthesis/composition (the "conquer" part). That is, we first decompose the big problem into a bunch of tiny problems which we can solve easily, then we compose these tiny solutions into an overall solution.

Abstraction is the process in which the well-understood details (of the studied object or concept) are going through the deliberate simplification as the process of understanding moves from the lower to the higher levels, toward the totality.

Generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences. The process of verification is necessary to determine whether a generalization holds true for any given situation.

In philosophy, essence is the property that makes an entity what it fundamentally is, which it has by necessity, without which it loses its identity. Essence is contrasted with accident - a property that the entity has contingently, without which it still retains its identity.