6
$\begingroup$

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense.

Can someone elaborate on this concept and whether there are any other things that are useful to know for a student in introductory calculus?

$\endgroup$
  • 10
    $\begingroup$ In my opinion Non-standard analysis introduces too many jargons and thus is totally unsuitable for anyone dealing with introductory calculus. At the same time such a student is better off if he remains totally unaware of the word "infinitesimal" and instead focuses his energies on the words like "infinite" or "infinity". $\endgroup$ – Paramanand Singh Jun 6 '15 at 6:08
  • 5
    $\begingroup$ No. ${}{}{}{}{}{}$ $\endgroup$ – copper.hat Jun 6 '15 at 6:23
  • $\begingroup$ @ParamanandSingh, was your teacher's first name Georg by any chance? $\endgroup$ – Mikhail Katz Mar 13 '16 at 14:37
  • $\begingroup$ @user72694: Well, I read most of mathematics after 18 years by myself through books and Internet. And I did not have any teachers in school time with name Georg (this kind of English name is very rare in India). Perhaps my views remind you of some teacher named Georg. $\endgroup$ – Paramanand Singh Mar 13 '16 at 15:45
  • 1
    $\begingroup$ I found the paper "Putting Differentials Back into Calculus (2009)" quite convincing in its argumentation. Personally I cannot but admire the ease and clarity of Silvanus P. Thompson's "Calculus made easy" (1910). I love the epilogue of that book. :) $\endgroup$ – gwr Sep 20 '16 at 14:05
2
$\begingroup$

One useful concept is the Leibnizian distinction between assignable and inassignable number (according to historian Eberhard Knobloch, the distinction originates with Cusanus; Galileo's distinction between quanta and non-quanta is also traceable to Cusanus). In Robinson's framework this is implemented in terms of a distinction between a standard and a nonstandard number. Thus, ordinary real numbers are standard, whereas infinitesimals and infinite numbers are nonstandard. The sum $\pi+\epsilon$ where $\epsilon$ is infinitesimal is also nonstandard. The two domains are related by the standard part function, also known as the shadow. This is defined for any finite hyperreal. The standard part rounds off each finite hyperreal to its nearest real number.

To illustrate how this is useful in calculus, note that the derivative of $y=f(x)$ can be computed as the shadow of $\frac{\Delta y}{\Delta x}$ where $\Delta x$ is an infinitesimal $x$-increment and $\Delta y$ the corresponding change in $y$.

$\endgroup$
  • 4
    $\begingroup$ How is any of the concepts useful for a student of introductory calculus? $\endgroup$ – Wojowu Feb 13 '16 at 21:30
  • 1
    $\begingroup$ @Wojowu, we just finished teaching introductory calculus to 130 students using infinitesimals and standard part. They found it very useful, as did our 120 students last year. I can send you an education study we wrote based on our teaching experiences last year if you are interested. $\endgroup$ – Mikhail Katz Feb 13 '16 at 21:32
  • $\begingroup$ I'd be really interested to have some information about this course! is there any information publicly available? $\endgroup$ – gniourf_gniourf Feb 16 '16 at 9:18
  • 1
    $\begingroup$ @gniourf_gniourf, you could consult my page for the course. $\endgroup$ – Mikhail Katz Feb 16 '16 at 9:25
  • $\begingroup$ @gniourf_gniourf, we just published an article describing our experiences and especially the students' experiences. $\endgroup$ – Mikhail Katz Jul 12 '17 at 13:21
3
$\begingroup$

Ed Nelson had demonstrated in his book Radically Elementary Probability Theory that a good part of probability can be made clear to freshmen without appealing to measure theory by using the Mises frequency approach. There are other gains for various courses. But above all nonstandard analysis allows students to understand better the triumphs and tragedies along the path of mathematics. Our ancestors Wallis, Gregory, Barrow, Newton, Leibniz, Euler, Cauchy, and many others were geniuses and to try and understand their ways of thinking is much better than to arrogantly accuse the great masters of being second rate thinkers in any style similar to that of Kline, for instance: "The net effect of the century’s efforts to rigorize the calculus, particularly those of giants such as Euler and Lagrange, was to confound and mislead their contemporaries and successors. They were, on the whole, so blatantly wrong that one could despair of mathematicians’ ever clarifying the logic involved." We learn math to fit life, and infinitesimals are part of the life of humankind.

$\endgroup$
2
$\begingroup$

Short opinion answer.

I think the fact that the use of infinitesimals can be made rigorous is the most important contribution of nonstandard analysis at the elementary calculus level. That should free students and instructors to work confidently with the intuition infinitesimals provide. I don't think it's useful or necessary to work with formal nonstandard analysis, any more than it's necessary to provide a definition of the reals sufficient to work rigorously with ordinary analysis.

$\endgroup$
  • $\begingroup$ Valid comparison indeed. Just as we don't provide the foundations of analysis (by constructing the real number system) in freshman calculus, we don't need to develop the construction of the hyperreals either. What is important is the procedures of infinitesimal calculus rather than set-theoretic foundations, at least at this level. $\endgroup$ – Mikhail Katz Feb 18 '16 at 14:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.