Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

Question

Nonstandard calculus is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, Calculus originally used infinitesimals. The problem was that it did not have a rigorous foundation, which is why mathematicians started using the epsilon-delta definitions instead, although they still used infinitesimals informally. Much later, nonstandard analysis came along, which did put infinitesimals on rigorous foundations, but by then it was not necessary since epsilon-delta definitions were available. Nonstandard calculus is calculus except based on nonstandard analysis instead of analysis. There are still people who think that nonstandard analysis should replace those definitions, however. I will put an informal introduction to how it works at the end of this post.

Would teaching students calculus using nonstandard calculus make it easier to learn? Answers should also take into account future calculus learning, not just learning in the course itself.

I would prefer to focus on "normal" students. I think for bright students, teaching a little bit of both would be beneficial, since I think comparing the approaches teaches the idea that there can be radically different ways to study math and arrive at the same results. However, for most students, I think mainly focusing one or the other would be better.

Here are some of my observations. They are kind of long, so feel free to skip/skin them. The most important is probably points 4 and 6, since they are about how standard and nonstandard calculus are different. The other points are about how they are the same.

1. Of course, the mathematical foundations of nonstandard analysis would be much too complicated to cover in such a course. The same is true of standard analysis though, so this is not a concern.
2. Nonstandard calculus is compatible with calculus. By that I mean that anything you can prove in analysis can be proven in nonstandard analysis, and anything you can disproof is analysis can be proven in standard analysis, so there will never be a contradictory result. In terms of provable statements, the only difference is that nonstandard analysis proves some things about infinite and infinitesimal numbers that standard analysis is not concerned with (since it does not use such number), and even those can be translated into equivalent statements about real numbers in standard analysis. The main difference is how they got about proving them.
3. Definitions in nonstandard calculus involve much fewer quantifiers than those in standard analysis. Limits, continuity, and uniform continuity all have a quantifier complexity of $1$, whereas in standard calculus they have 3 or 4. This however is balanced by the fact that it is much more difficult to do arithmetic with hyperreal numbers than the real numbers. You can not even build a hyperreal number calculator. All in all, proofs in both are about equally as long.
4. Although the proofs are equally as long, they are much more intuitive in nonstandard calculus than in standard calculus. With standard calculus, you usually motivate theorems with ideas like infinity, or being infinitely close, or dividing infinitesimal numbers. Then you translate those intuitions into a series of epsilon delta definitions, with many inter-dependencies, never including your intuitions in your proof, just their translation. In nonstandard calculus, most of the intuitions are included literally in the proof. Infinity is no longer just an idea, its a number. They are also superior to informal proofs using infinitesimals (like scientists sometimes use) since it never proves an contradictions. You still have to do the work, but the work is not separate from the intuition.
5. Both standard and nonstandard calculus generalize to other fields, such as topology, so that is not an issue.
6. The main issue I see is when they take Calculus in the future. If the students never take another formal calculus course, this is not a concern, but if they do, all the proof methods they learned in nonstandard calculus will no longer be taught in light of new material. The grader may not know how to grade a proof using nonstandard calculus, as well (which is reasonable, since it invokes a whole host of concepts not used in standard calculus). One thing that helps though is that proofs from standard calculus still make sense in nonstandard calculus. For example, even though limits no longer have an epsilon-delta definition, they do have an epsilon-delta theorem. In fact, most limit proofs using the definition can be mechanically translated into a proof using the theorem. Since the epsilon-delta theorems are true by, well, definition in standard calculus, they would be compatible with future calculus courses. Additionally, but having the students carefully study the epsilon-delta theorems, they will likely be able to see the connection between the material in future calculus courses could be translated into nonstandard calculus. Unfortunately, this results an "overhead" cost of using nonstandard calculus over standard calculus. However, in my experience, I find that writing a proof first in nonstandard calculus and then translating it to standard calculus is easier mentally (unless I immediately see the standard calculus proof). This may be unique to myself however. However, for some courses (especially less proof based ones), learning nonstandard calculus will actually make the course easier. That's because many courses less concerned with rigor will just use infinitesimals anyways, ignoring foundational issues. Students who learned nonstandard calculus with have an advantage, since they know how to manipulate infinitesimals rigorously.
7. Another potential issue is materials. For any introductory calculus course, this is just potential, since there is textbook that teaches introductory calculus, using nonstandard calculus proofs, that the author distributes for free. There are also some other ones of varying costs.

Answer 1

This has certainly been tried before. See for example,

H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover.

Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-375.

Answer 2

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ability to manipulate limits, and an awareness that derivatives and antiderivatives were defined as limits and could be found as limits if necessary.

You're not going to understand Maxwell's Equations or the Schrödinger Wave Equation or Fourier transforms without understanding integrals and derivatives, but you can perfectly well understand them without knowing the rigorous definition of a limit.

It seems to me that an introductory calculus course isn't going to be an introduction to proving that calculus works, any more than an introductory algebra course is going to start with a rigorous definition of the real numbers—it's going to be an introduction to the concepts of differentiation and integration and how to use them.

Obviously, proving rigorously that calculus works does require rigorous definitions, and at that point something new has to be introduced. Maybe a strange new definition involving $ε$ and $δ$, and maybe a strange new number system in which there are different sizes of zero which are defined not to equal each other or zero.

I think awareness at that stage is likely to be:

• $\frac{dy}{dx}$ is defined as a limit, and so is $\int y dx$.
• but we sometimes treat $dy$ and $dx$ like algebraic quantities
• this is a bit dodgy even though it works, since we're effectively either dividing $0$ by $0$ or multiplying $0$ by $\infty$
• the way round this involves something to do with limits.

If $ε–δ$ is explained clearly at that stage—ie in terms of making $ε$ arbitrarily small by making $δ$ small enough or $N$ large enough, and not as a dense expression full of quantifiers—then once someone has grasped it, they'll be convinced it's rigorous. But if they're introduced to hyperreal numbers instead, they'll have lingering doubts about whether arguments using them are actually valid. It means another layer of understanding is needed, namely the background theory of the new number system.

So I think it's more useful to focus on understanding the principle of $ε–δ$ definitions and techniques clearly than to introduce an unfamiliar number system. Apart from anything else, this will help them to understand other textbooks in the future, which must likely won't use hyperreals.

Answer 3

My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual."

I have some experience teaching some NSA-based material to first-semester calculus students at a community college. Here is the book I wrote for that purpose. My approach is "bilingual." The philosophy with which I approached this was not that students should learn NSA in excruciating detail -- that would be kind of silly IMO, and would detract from the main thrust of the course, which is basically to be able to do rule-based differentiation.

However, scientists and engineers still use Leibniz notation and manipulate infinitesimals using algebra. They've been doing it for 300 years, and they never stopped doing it just because there was a short gap between the invention of the limit and the invention of NSA. These students will see such manipulations in their physics courses, and they will see and be expected to use them in their careers. So we should give them some systematic idea of what techniques are appropriate when performing these manipulations. If this was 1850, we would teach them a body of techniques that we knew from experience gave correct results, e.g., throwing away higher orders of $dx$. Today we have more secure knowledge that these procedures can be put on a sound logical footing, but that actually has little effect in reality on what body of techniques we use.

The goal should not be to eliminate the $\epsilon-\delta$ definition of limits. This is not possible because there is a clear consensus among mathematicians that $\epsilon-\delta$ is something students should have already learned by the time they go on to their second-semester course, and we need to prepare these students properly. In fact, $\epsilon-\delta$-like ideas are fundamental to many of the common modes of reasoning about calculus. E.g., we can do a numerical simulation of the motion of the moon with a small time step, and say, "If I make the step size small enough, I should be able to make the error as small as desired." Newton and Leibniz would have understood that idea, and so would every physicist of succeeding centuries, long before the definition of the limit.

When you get right down to it, using NSA doesn't really turn out to produce any incredible simplification of freshman calculus. For example, it would be nice if we could just prove the chain rule and L’Hôpital’s rule by manipulating infinitesimals using algebra, but the nuts and bolts don't actually work out quite that trivially. In the case of the chain rule, you have the issue that the derivative is not the quotient of infinitesimals but the standard part of that quotient. In the case of L’Hôpital’s rule you also have all the various forms of the rule (repeated application, limits at infinity, $\infty/\infty$, etc.). These are complicated to prove (and many books don't prove them all), and the complication isn't really reduced very much by using NSA.

Keep in mind also that at this level, our students never really get a full-blown introduction to the real number system. No commercial textbook I've ever seen systematically introduces and applies anything beyond the first-order properties of the reals. (They may state the completeness property, but they never use it to prove things like the intermediate value theorem.) This is material that belongs in an upper-division analysis class. Since the hyperreals have the same first-order properties as the reals, there is actually very little that we can meaningfully say about the hyperreals to students at this level.

I do explicitly name the hyperreals and describe how their properties differ from those of the reals (this is mainly in section 2.9 of my book). However, I don't think it's a good idea to go into the kind of depth that Keisler does.

Answer 4

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "x+h". The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit of series, and small section on ODEs.

You are swinging at the wrong opponent with this obsession on the definition of a derivative (but not surprising given the theory bent of many math majors).

If you want to make things easier, cut partial fractions, integration by parts, etc. Do nurses and business students really need that? (Of course this does mean tracking because science and engineering do.)

P.s. I found your question hard to parse given the notation (upside down A). This is ironic given you are planning to make things easier for knuckleheads worse than I. Not showing it here...makes me leery of your proposals. Can't you discuss pedagogy and coverage without such a segue into exploring the math theory like that?