Editing Nonstandard calculus (section)

==Motivation==

To calculate the derivative <math>f '</math> of the function <math> y =f(x)=x^2</math> at ''x'', both approaches agree on the algebraic manipulations:

: <math> \frac{\Delta y}{\Delta x} = \frac{(x + \Delta  x)^2 - x^2}{\Delta  x} = \frac{2 x \Delta x + (\Delta x)^2}{\Delta x} = 2 x  + \Delta x \approx 2 x</math>

This becomes a computation of the derivatives using the [[hyperreals]] if <math>\Delta x</math> is interpreted as an infinitesimal and the symbol "<math>\approx</math>" is the relation "is infinitely close to".

In order to make ''f ''' a real-valued function, the final term <math>\Delta x</math> is dispensed with. In the standard approach using only real numbers, that is done by taking the limit as <math>\Delta x</math> tends to zero. In the [[hyperreal number|hyperreal]] approach, the quantity <math>\Delta x</math> is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that <math>\Delta y /\Delta x</math> is infinitely close to 2''x'', so the derivative of ''f'' at ''x'' is then 2''x''.

Discarding the "error term" is accomplished by an application of the [[standard part function]]. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably [[George Berkeley]].

Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level.  Thus, the [[epsilon, delta technique]]s that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught [[infinitesimal calculus]]", to quote a recent study.<ref>{{citation
 | last1 = Katz | first1 = Mikhail
 | author1-link = Mikhail Katz
 | last2 = Tall | first2 = David
 | author2-link = David Tall
 | arxiv = 1110.5747 
 | doi = 
 | issue = 
 | publisher = [[Bharath Sriraman]], Editor. Crossroads in the History of Mathematics and Mathematics Education. [[The Montana Mathematics Enthusiast]] Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC
 | pages = 
 | title = Tension between Intuitive Infinitesimals and Formal Mathematical Analysis
 | volume = 
 | year = 2011| bibcode =2011arXiv1110.5747K}}</ref> More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta.