Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s ...
In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point ...
Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...
This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY. George Berkeley wrote in 1734 with reference to the early calculus that ...
Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...
I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2). In William Dunham's book 'Journey through Genius' ( ...
Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the ...
This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is: I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...
Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...