I alluded in one of my very first posts here to a calculus class that I was teaching using the ring . The class was a six-week, 5-day-per-week intensive course covering the usual material of a college Calculus I course. I’ve been promising Greg and Jim that I’d write up some of my experiences with the course. I think I have had enough time to process the experience — let’s talk non-nonstandard calculus.
Maybe a word about the name before we begin: I think that a lot of my philosophy on mathematics came from taking several analysis classes at Smith College from logician Jim Henle. I learned nonstandard analysis from him one summer, but also learned what he called “non-nonstandard analysis”. The idea was to get infinitesimals into calculus without the big ultrafilter-style logic machinery of standard nonstandard analysis. I’d like to say more about this version of non-nonstandard analysis sometime in the future, but not just this moment. But the main idea is this: we want infinitesimals, but we don’t want to bring in the heavy machinery if we can avoid it. The (serious) tradeoff is that we lose the transfer principal when we use non-nonstandard analysis.
Now, a bit more about the class I taught. Most students had not seen much beyond precalculus. Maybe they had been told how to formally take derivatives of polynomials, but not much more. So I had a fantastically clean slate to work with. This may have seriously affected the outcome of the class.
In the first serious lecture, I introduced as a positive number so small that . They thought this was a little funny, but came around to it after I drew an analogy to and : even if they had philosophical objections to thinking about or as “numbers”, their recent experience with complex numbers in high school put them in a good place for accepting as a useful formal symbol, if not an honest-to-goodness number.We could then quickly move on to computing the derivative of some elementary functions like :
I also pointed out that computers are using a number system which is closer to this than it is to : since floating point processors have finite precision there is a smallest positive denormal. It must square to zero. So if you are judging mathematical truth by “is relevant to things I know in the real world”, it seems that nilsquare infinitesimals aren’t such a strange idea.
On the second day, I gave them a few useful tidbits: as an axiom, I said . Then by looking at a right triangle of height , we decided that and . From here, it is easy to compute derivatives of relatively complex functions by hand. By the end of the third day, there were homework problems (generally solved correctly) like “use the definition of the derivative to find the derivative of ”:
so the derivative is .
By asking them to compute the derivative of things like or by hand, they rapidly re-learned (and remembered!) the angle-sum formulas for sine and cosine and the algebraic rules involving exponents. I never allowed formula sheets or calculators or anything like that in the class or on the tests — the kids were genuinely remembering the formulas and the derivations. Even the weaker students at the end of the course could all compute the derivative of as a routine computation:
but even better, they remembered or because these manipulations had become common to them. They used them every day in nearly every problem, and they remembered. No formula sheet hell involved.
When a number like appeared in the denominator, the students quickly figured out that they could multiply on the top and bottom by to “realify” the denominator. I was surprised that they saw this trick so quickly, until I realized that they had been doing the exact same thing with complex numbers for several years already. The derivative of , with no more than two steps left out:
On the final, I asked them to compute the derivatives of , , and from the definition. Almost uniformly, the students could create correct computations of the derivative. The next question on the final was “prove the quotient rule”. Nearly everybody in the class, even the C-students, did this derivation properly. They were not told beforehand that anything like this would be on the test.
Another thing which they were fairly good at was using differentials, since the differential was not some mysterious formal symbol to them but an actual infinitessimal value. From this perspective, it is easy to see how to use differentials to get good approximations. This also made it as easy to work with implicit differentiation as with explicit differentiation, which it turn makes computations of related rates much cleaner than usual. From my perspective, it also makes the usual geometric ways of demonstrating the product rule or the fact that more rigorous: the area added really is just some lengths times the width of my chalk. By treating infinitesimals on the same footing as finite numbers, the approximation schemes of calculus become more intuitive. I think every mathematician has discovered this on their own, in their own private language. Why not make the language commensurable with the computations that we do?
I’ll write plenty more about the other topics of the course in the future, but this should give you some idea of what the class was like and why I felt that nilsquare infinitesimals were a productive way to teach calculus. We really didn’t use limits at all: they were all replaced with infinitesimals and approximation schemes. It is true that limits, infinitesimals and approximation schemes are all equivalent ideas, but I believe the latter two more closely model our internal, geometric understanding of calculus. So: what do you readers think?