Quote:
Originally Posted by tdvance
There is one situation in which you can divide, not by zero, but by "infinitely small numbers"--the field of surreal numbers (it's on Wikipedia) of John H. Conway (the Game of Life guy). It's pretty...surreal in that there is no set of all surreal numbers, but it includes all real numbers, every cardinal number including the infinite ones, something much like the ordinals (but using different rules of arithmetic), and lots more different kinds of infinity, and their reciprocals which are infinitesimals. Still, division by zero in the surreal field is forbidden.
A modification of this (some guy name Robinson I think) leads to Freshman calculus, with all the tricks of dividing by infinitesimals to get derivatives--tricks invented by Newton--for which he was criticized because, though the methods just happened to produce the right answers, were not very rigorous. Weierstrass invented the modern definition of the limit to make it rigorous, and Robinson figured out how to make Newton's manipulation of infinitesimals rigorous, showing why it worked (i.e. it didn't just accidentally work, there was something to it).
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The guy that you are thinking of is Abraham Robinson who wrote the book
Non-standard Analysis (Princeton University Press)
. He uses ultrafilters to construct a system in which there are infinitesimals, and can develop calculus using that methodology. this is a recognized branch of analysis, but there are relatively few specialists. We had one in the department when I was in graduate school.
Paul Halmos was known for opposing non-standard analysis, but as I recall he ran into trouble when a theorem relating to invariant subspaces of operators on Hilbert spaces was proved using non-standard methods and he was unable to construct a standard proof.