Re: Math "From Scratch"

In <1125877967.959142.79900@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
09/04/2005
at 04:52 PM, rodney.elliott@xxxxxxxxx said:

>Whilst revising the basics of undergraduate calculus I got to
>thinking about all the math that I/we all take for granted and never
>question because its so fundamental. I have been wondering if it is
>possible to develop the rudiments of a workable number system "from
>scratch", justifying any assumptions made along the way.

Sure, but the issues that you discuss later relate to Etymology rather
than to Mathematics.

>For example, lets say you start with a blank piece of paper and your
>goal is to end up with R - the set of real numbers. What you are
>likely to do is start with the set N of natural numbers and then
>work your way up from there,

Only if the goal is to develop a model of R. Otherwise I'd just give
suitable axioms.

>1. What are the origins of the squiggles we make on paper to
>represent the natural numbers - 1, 2 etc?

That is an historical question and has nothing to do with the
Mathematics. Is there a sci.etymology news group? Similarly for your
second question.

>3. Is the concept of quantity and variations in quantity learned or
>innate?

That's Psychology rather than Mathematics.

>It just seems to me that there is nothing natural about N
>whatsoever.

Well, there's nothing irrational about sqrt(2) and nothing complex
about 1+i; the map is not the territory. Nor is there any reason to
start from N. IMHO it makes more sense to start from Group Theory,
using suitable manipulables. The Montessori people might be able to
help there.

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Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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