(idea) by m_turner Wed Oct 11 2000 at 5:32:24
We do have math capable of handling the concept of the smallest number after zero. Its a particular application of calculus called infinitesimal calculus. This method was invented by Leibniz in the mid 1670s, and published in 1684 - nine years before the earliest account of Newton's method.

Leibniz developed his calculus in order to find methods by which discrete infinitesimal quantities could be summed to calculate the area of a larger whole. This probably came from his metaphysical work on monads. Newton was working with infinitesimal changes of force and motion with respect to time.

In order to work with the smallest possible number, one must work with infinitesimals, around which the model of calculus has been developed.

To work with infinitesimals, it necessary to use the hyperreal number system. This is much closer to the math that Newton and Leibniz did. In 1960, Abraham Robinson established the framework for non-standard analysis.

First off, some definitions.

  • Internal - objects that exist within classical mathematics
  • Standard - limited objects within the internal set

There are three axioms added to ZFC theory (the basis for today's mathematics): Idealization, Transfer, Standardization. These lead to the three principles of non-standard analysis:

  • 1st principle: If E is an internal object which is defined from standard objects, then E is standard.
  • 2nd principle: All elements of an internal set are standard iff the set is finite.
  • Transfer principle: Let P(x) be an internal expression relative to x. Then P(x) is true for all x, iff P(x) is true for all standard x.
For any x (standard or not)
  • x is limited iff there is a limited integer greater than x.
  • x is unlimited iff it is greater than any limited integer.
  • x is infinitesimal iff its absolute value is less than 1/n for any limited integer n
  • x is perceptible iff x is not unlimited or infinitesimal
  • x is infinitely close to y iff x - y is infinitesimal

A new function is added called the Standard Part that operates on the hyperreal number set and maps it back into the real numbers. For 'e' designating Epsilon (an infinitesimal):
SP(1 + e) = 1
SP(e) = 0
It is not possible to take the standard part of an infinite number, but it is not difficult to take the standard part of the reciprocal.

Yes, this is a bit hazy in my memory. I happily defer to anyone who can explain non-standard analysis better than I. The point being, we do have a formal system for dealing with the smallest number greater 0, and it has been around for awhile.

A text book on infinitesimal calculus has been made available under a Creative Commons License when the book went out of print and reverted back to the professor (incidently, the one I took 2nd semester calculus under) who wrote the book. The book is available via PDF at http://www.math.wisc.edu/~keisler/calc.html







ENOUGH


The answer is 1 unless you're a freak.




No more writeups, please.


An editor who is also a mathematician writes:

I have just killed thirteen (13) write-ups in the above node, because none of them demonstrated sufficient knowledge of non-standard analysis and infinitesimals to make a useful contribution to it. Trust us, we mathematicians do know what we're doing. All your radical ideas have already occurred to us.

I like it! 4 C!s