Originally posted by DeepThought
As I get it it's just the real numbers, but with infinitesimals added between zero and the positive reals and some infinite numbers added on the right of the set of reals. I think it can be made consistent, which is the interest to mathematicians, but I think they are more a mathematical curiosity than a useful calculational tool. From a pure maths poi ...[text shortened]... sistent. The Wikipedia page is reasonably good.
I noticed this quote in the hyperlink in that link:
" ... the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, ..."
although many other philosophers would disagree with that.
I wonder who is right? I can't tell without a huge amount of proper research into infinitesimals which I haven't done but my intuition has always made me highly suspicions of the concept of infinitesimals (a number between zero and the reciprocal of any positive integer no matter how large? That seems like nonsense to me ) even though, strangely, I have heard of infinitesimals often been mathematically used with apparently correct results.
Anyone with an opinion on this? Are infinitesimals 'pseudoconcepts' ?