Grossones and non-standard analysis

This paper is fantastic. I always wondered where more concrete examples of the use of non-standard analysis were. For anyone not familiar with NSA it is a fantastic conceptual explanation for a lot of the paradoxes of trying to intimately understand calculus, limits, and integrals as they exist in reality. (Velocity as a derivative is based on finite quantized lengths and times and hence has structure that is much different from a limit that approaches an arbitrarily small number as a proto-example for understanding the motivations of NSA.)
http://arxiv.org/pdf/1203.3150v1.pdf

http://en.wikipedia.org/wiki/Non-standard_analysis

I was trying to figure out how the grossone from the paper connects to the hyperreal / shadow discussion of NSA. http://en.wikipedia.org/wiki/Hyperreal_number

From google "grossone nonstandard analysis'

http://www.math.nsc.ru/LBRT/g2/english/ssk/ems_2011.html

Sergeyev confronts his ideas with the nonstandard analysis of Abraham Robinson, defining his grossone as “the number of elements of the set of natural numbers.” In fact, the role of this would-be mysterious entity can happily be performed by the factorial of an arbitrary infinite number which are galore in nonstandard analysis.


The only other 'concrete' problem I remember NSA being used for was Navier Stokes and Monads. Some examples:

https://www.google.com/search?client=ubuntu&channel=fs&q=navier+stokes+monad+nonstandard+anyalsis&ie=utf-8&oe=utf-8#hl=en&client=ubuntu&hs=Fr8&channel=fs&sa=X&psj=1&ei=XAdpT8KJGKqLsQLqgKWtCQ&ved=0CBsQvwUoAQ&q=navier+stokes+monad+nonstandard+analysis&spell=1&fp=1&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&cad=b

This is really cool though. I always remember reading about those limit area fractal things trending to zero and thinking 'uhh, that can't be..' I remember there was always some trick to work around it basically hand-waving past the obvious problems and this is just so much more satisfying as an answer.

Also same field of study applies to all that fractal compression stuff.

Also like that lens that was designed by an evolutionary algorithm, there were also some minor math proofs that were done by similar processes. An evolutionary algorithm is basically by definition a fractal since it's an iterated function system that acts on itself as input recursively, except the iteration isn't the same each time, it's a random mutation + a fitness selection. So this branch of analysis would apply to such problems.

http://forums.xkcd.com/viewtopic.php?t=57691&p=2050100

There is some mostly critical commentary of grossones. The guy has a PhD though and NSA has always been criticized as being too strange/controversial in it's benefits. It still seems like the foundations of the paper are pretty solid though with their connection to the hyperreals by a factorial. That and the problem he's solving is already solved so a more elegant proof is not exactly all that controversial to begin with. Some of the other stuff on his page might be more speculative though.

Also:

You might be interested (or not) to know that John H. Conway has developed a similar numeric system called the "surreal numbers", which are defined from games, not sets, and which is completely coherent, though its applications are in a completely different area. It does have the equation infinity+1>infinity, though.

Game of numbers, hah.