Hyperreal Numbers

The folks at Uncommon Descent gave a post about hyperreal numbers, as an alternate way to do calculus. It’s easier to use, more natural, and so can open up more math to more people: a good thing!

I find the hyperreals interesting for a number of reasons. First of all, I think that infinities and infinitesimals are somewhat of the equivalent of Intelligent Design for mathematics. Infinitesimals were essentially banned from mathematics in the 1800s because it was said that they were inconsistent and non-rigorous (this is why calculus switched from infinitesimals to limits). This move was largely philosophically motivated, with Hilbert and others trying to naturalize mathematics.

However, in the 1960s, infinitesimals were proved to be equally mathematically rigorous as other standard mathematical entities.

In any case, from a practical side, infinitesimals make calculus, limits, and other sorts of mathematical ideas a LOT simpler to work with. In fact, using these types of numbers, Calculus becomes pretty much identical with Algebra.

The Hyperreal Number System

(Extra info here.)

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