Nonstandard Analysis: Calculus Underground!
Submitted by on Fri, 10/30/2009 at 11:40am.
Calculus is almost exclusively taught geometrically (the way Newton originally formulated calculus). Some mathematicians do not prefer to use questionable spatial definitions but prefer a more rigorous approach. These are the constructivists, and if you're one of them you know how to sweat. It's not easy to prove things with a Constructivist approach.
There are several Constructivist approaches, in fact. Robinson used hyperreal numbers in his analysis. Others, such as George Lakoff and Nunez would disagree with this approach; more commonly, infinitesimals are used. It appears in fact that infinitesimal numbers satisfy all axioms for the definition of a real number. The whole infinitesimal approach is not something new though, as this was Leibniz's approach of thinking of derivatives on a curve vs. Newton's more geometrical descriptions.
The reason why infinitesimal usage was nearly wiped out comes from what Karl Weierstrass showed in the 1800's: he gave a pure nongeometric arithmetization of Newtonian calculus. The infinitesimals struck back underground in the 1940's.
Now the infinitesimals form a whole new number system called the granular numbers. There are also many different "layers" of infinitesimals. Granular arithmetic is not too hard luckily. One can have real numbers joined with strings of granular numbers similar to how real numbers and imaginaries join to make complex numbers.
Perhaps one great achievement of the infinitesimals is that once we use granular arithmetic, all calculus can be done without any concept of a limit! Calculus becomes arithmetic. For every calc proof with limits, one can find a corresponding infinitesimal proof.
The one main reason why Constructivists like nonstandard analysis is because it is far easier to understand the nature of infinite sums. Also, infinitesimals are said to have a well-known conceptual advantage too.
Keisler is one of the leading dudes in nonstandard analysis. His homepage has a link to a free pdf that he wrote:
http://www.math.wisc.edu/~keisler/calc.html
His home page:
http://www.math.wisc.edu/~keisler/