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.
- 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.
- 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.
- 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.
- 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.
- Both standard and nonstandard calculus generalize to other fields, such as topology, so that is not an issue.
- 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.
- 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.