Re: Distinct linear orderings on Z

From: robert j. kolker (nowhere_at_nowhere.net)
Date: 03/22/05 

Date: Mon, 21 Mar 2005 22:09:17 -0500

Dave Seaman wrote:
>
> Exactly what result do you consider to be counter-intuitive? That it's
> possible for a bijection to exist between a set and a proper subset of
> itself? That bijections can be used to establish an equivalence relation
> (a relation that is reflexive, symmetric, and transitive) between sets?
> That this equivalence relation is given a name?

The issue of count-intuitivity is vexing. Why should the power of a
mathematical concept or system by limited by intuition when logic can
carry out the consequences of the axioms of a system? It is true that
intuition can sometimes help find a proof or solve a problem. but the
limits of intuition can equally inhibit understanding of the
consequences and blind the intuitive to other non-intuitive
possibilities. Topoloy is full of example of constructs and structures
that boggle the visual intuition. The solution is simple. Do not rely
totally on visualization. Use analogy or metaphor instead.

It seems to me that progress in mathematics and physics has been made
largely by abandoning intuition or common sense when it becomes an
impediment.

Bob Kolker

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