# Generalization and Abstraction

[cm] Generalization vs. Abstraction

Posina Venkata Rayudu posina at salk.edu

Dear All,

In the following I’ll make an attempt to answer Prof. Tim Porter‘s question: ‘What is Abstraction?’

Let an object O have a collection of properties P = {p1, p2, …, pi, …}, each of which could be any of the familiar (or unfamiliar) properties like length, weight, etc. From this we can go on to say that each one of the properties pi in the collection of properties P takes a value vi, i.e. pi = vi, which is basically like saying the weight of the object O is 10 kg or something like that. Now a generalization of the object O along a property pi would be an object Og, whose property pi is not constrained to take the value vi i.e. not restricted to pi = vi.

Abstraction, on the other hand, of the object O along the property pi would be an object Oa, whose collection of properties Pa doesn’t include pi. More explicitly, the collection of properties of an abstraction Oa is given by Pa = {p1, p2, …}.

In the attached (Number2Function) I tried to illustrate Abstraction in terms of an example: abstracting the notion of FUNCTION from the concept of NUMBER.

I’d appreciate very much any corrections and clarifications that you can, your time permitting, provide.

Thanking you,

Yours sincerely,

posina

P.S. Hope to see you all bright and early tomorrow morning!

P. P.S. I hope the above along with the attached note is anything but an abuse of some concept[s]. In Conceptual Mathematics there’s a warning against abusing in ‘Two Abuses of Isomorphisms’ on p. 89.

NumberToFunction.pdf

Subject: Re: [cm] Generalization vs. Abstraction

Date: Mon, 1 Aug 2011 21:30:43 -0500

From: Charles Wells

To: Posina Venkata Rayudu

You were right about generalization: Allowing a fixed property to vary, or simply be dropped. Abstraction is more radical than what you described. An abstraction of an idea defined by certain requirements (axioms)

r1, r2,…rn

consists of finding a *different* list of properties, often *completely* different, such that the objects satisfying the original axioms all have those new properties, but there are also objects satisfying the new properties that don’t satisfy the original ones.

For example, a continuous function on a metric space can be defined by requiring that it satisfies the usual epsilon-delta definition. But you can define it in more general topological spaces by requiring that the inverse of open sets must be open. This way of saying it doesn’t mention the distance between points and so works for spaces than are not metric spaces. But it is equivalent to the original definition for metric spaces.

My article at

http://www.abstractmath.org/MM/MMOtherAspectsUnderstanding.htm#abstraction

talks about this and gives pointers to some examples.

Charles Wells

Shape Theory: Categorical Methods of Approximation (Dover Books on Mathematics)