Calculus with Infinitesimals
From the talk by TJ Johnson given at the Amatyc Conference, Boston, November 2010.
(Previously titled “Calculus Unlimited”)
(Previously titled “Calculus Unlimited”)
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Intro
- Calculus with Infinitesimals works with a new number system that offers a more intuitive way of looking at calculus.
- Our goal is to see if exposing students to “Nonstandard Calculus” (“Infinitesimal Calculus”) enables them to work better with standard Calculus.
- Who I am: Math, Music, Philosophy.
- Got interested in this topic starting with Lakoff/Nunez “Where Math Comes From”
- Where Mathematics Comes From, Lakoff and Nunez, 2000 Basic Books
- Studied Nonstandard Analysis in depth with Goldblatt and Keisler texts.
- Lectures on the Hyperreals; an Introduction to Nonstandard Analysis, Robert Goldblatt, 1998, Springer Verlag
- Elementary Calculus, an Infinitesimal Approach, H. Jerome Keisler, 1986
- Augmented textbook approach. Envision the title Calculus with Infinitesimals, (similar to Stuart’s title Calculus with Early Transcendentals)
- Survey of textbooks, exact definition of limit is segregated in all texts, non-existent in some. Calculus with Infinitesimals will keep it segregated too. We’ve modeled it using Stuart 6th edition.
- Augment the textbook in such a way that the instructor can use the infinitesimal material or not.
- Include one whole chapter section
- Section 2.4I The Infinitesimal Definition of a Limit
- It follows (Stuart’s) Section 2.4 called The Exact Definition of a Limit
- Introduces the subject and gives the grounding for subsequent material
- Insert “I-sections” into text, anywhere from half a page to a couple pages
- Mark with an “I” logo of some kind in the margin. (“I” for Infinitesimals)
- Appendices will including Equivalence of Standard and Nonstandard Definitions, The Hyperfinite Partition, …
- I’ve presented this material in class up through Chapter 2. Nothing overall conclusive yet. Be careful showing too many hyperreal limit calculations. Here are copies of student exercises taken from Keisler exercises sections 1.5 and 1.6.
- Mention previous studies, esp Sullivan (1976). My main criticism of studies and texts written so far is that they fail to develop an intuition that relates the hyperreal with the real. They have tried to replace standard analysis instead of working with it.
- Would like to accomplish two things in this presentation.
- Show you the Hyperreals
- Discuss the implications for teaching Calculus
- So, we’ll go though the textbook material that’s been developed so far and then have a discussion.
- Is it an overall benefit? As it is now, many of us teach Calculus without even covering “epsilon-delta”.
- The extra work put into infinitesimals has to be worth it in increased student understanding, motivation, and performance.
- One advantage could be proving theorems that we don’t bother proving now. Is showing the infinitesimal proofs worth the effort? We get by fine now without any proofs at all. Or do we?
- How much extra class time does it take? How do the students do with the material? Are there exercises?
- Does it really make the material easier to understand, or does it just get in the way?
