1.0 Introduction
A couple of years ago I saw somebody in my local library who was obviously tutoring students in mathematics. I cannot recall how or why, but I started a question. He assured me that advanced high school seniors were taught calculus here. But the approach they teach nowadays does not require kids to learn epsilon-delta definitions of limits and continuity. This surprised me. I understand limits are difficult to wrap one's mind around. For one thing, one needs to not think in terms of dynamics, in some sense. And epsilonic definitions are rarely seen as natural to the beginning student.
I have since had similar conversations with a few youngsters. And they did not recall epsilon-delta definitions either. I realize that teaching and student recollection varies. Furthermore, the use of epsilon to represent a small distance in the space of the range of the function is a notational convention. Perhaps, some other symbol was used in their classes (although I doubt it). Furthermore, to engineers and practical-oriented students, they might be more interested in getting to problems with derivatives and integrals. (When I asked C. how his calculus class was, he said, "We're still on limits", which I thought expressed an impatience.)
I wonder about this. I have a theory how some might have justified a change to teaching in calculus since my day, although I can imagine other justifications that do not contradict my ideas below. Anyways, I only intend to raise questions in this post.
2.0 A Potted History of Calculus after Newton
When Newton and Liebniz invented the differential calculus, they had a problem with certain quotients. The slope of secants, drawn for two points on a "smooth" function, might be a well-defined ratio. But what does it mean to take a limit? Sometimes Newton seems to treat a denominator as simultaneously zero and non-zero. And this problem with infinitesimals (or fluxions) is compounded when one starts thinking about second derivatives and even higher orders.
Berkeley quickly pointed out these difficulties. I gather he was concerned to argue against the deism - to him, atheism - that often seemed to accompany Newtonian physics and cosmology. Why criticize the mote in your neighbor's eye without first casting out the beam in your own? Anyways, mathematicians recognized Berkeley had a point about calculus. But the mathematics worked in practice and seemed to be extraordinary useful for physics.
So mathematicians struggled for centuries, building an immense structure on what they recognized to be an unsound foundation. They also tried to rebuild the foundations. Cauchy, for example, made some improvements. As far as real numbers and limits are concerned, the decisive work came in the second half of the nineteenth century, with Weierstrass' epsilon-delta definitions and Dedekind's construction of the reals out of sets of rational numbers, known as cuts. Whether this was the answer, or whether this just moved the problems deeper down to questions about sets and logic, was not immediately clear. The work of Cantor, Frege, and Russell are of some importance here. The twentieth century saw intensive exploration of such foundational questions. Anyways, nobody seems to have ever found a contradiction in Zermelo-Fraenkel set theory, even if the absence of such contradictions cannot be proven. ZF set theory, with the axiom of choice in many applications, seems to provide a sufficient foundation for the working mathematician.
I guess that that is how the picture stood around, say, 1960. Newton's own approach to calculus was non-rigorous, but epsilon-delta definitions provide all the rigor introductory students of calculus need. Also, Alfred Tarski had invented something called model theory. Along came Abraham Robinson, who used model theory to develop non-standard analysis. Somehow, nonstandard analysis provides a rigorous justification of infinitesimals. (I wouldn't mind understanding the Löwenheim-Skolem theorem either.)
So maybe it does make sense to teach calculus, without the rigor of epsilon-delta definitions. Keisler wrote a textbook to illustrate the teaching of calculus on the foundations of infinitesimals, maybe easier for the student to understand and justified by the rigor of the advanced abstractions of non-standard analysis. Has this approach, revolutionizing centuries of understanding, won out in introductory calculus classes?
3.0 Other Special Cases in Introductory Teaching
I can think of a couple of other cases where what was in my textbooks in calculus and analysis was superseded, in some sense, in more advanced mathematics. I gather mathematical analysis is often informally defined as what the differential and integral calculus would be if taught rigorously. And Rudin (1976) is a standard introduction to analysis.
Rudin provides an epsilon-delta definition of limits. This definition is more general than you might see in (old?) calculus courses. In such less abstract courses, you might see two definition of limits. One would be for sequences, that is, for functions mapping the natural numbers into the reals. And another would be for functions mapping the real numbers into the real numbers. But Rudin's definition is for functions mapping an arbitrary metric space into (possibly another) arbitrary metric space. One might get the impression that some notion of distance between points is needed to define a limit. But, as was pointed out in the class I took with Rudin as the textbook, a limit of a function is a topological notion.
A common intuition for integration is as of the area under a curve. This notion can be formalized with the Riemann integral, and, for me, this is the first definition I learned. But another definition, Lebesque integration, is taught in classes on measure theory. Lebesque integrals are more general. Some functions have a Lebesque integral, but not a Riemann integral. But, if a function has a Riemann integral, it has the same value for the Lebesque integral.
I offer a suggestion in the spirit of a devil's advocate. Why teach the special case at all in these instances? Why not start with the more general case? Do those who concern themselves with the pedagogy of mathematics selectively advocate the teaching of the more abstract, general case? Is so, how do they choose when this is appropriate?
4.0 Conclusion
Is it now quite common - maybe, in the United States - to teach introductory calculus without providing an epsilon-delta definition of a limit? If so, does common justification of this practice draw on a non-standard analysis approach to calculus? Why should this extremely abstract idea influence introductory teaching, but not other abstractions?
Appendix: Two Definitions of a Limit of a Function and a Theorem
These are from memory, since I do not want to bother looking them up. The proof of the theorem, probably stated more rigorously, was a test question in a course I took decades ago.
Definition (Metric Space): Let f be a function mapping a metric space X into a metric space Y. L is a limit of f as x approaches x0 if and only if, for all ε > 0, there exists a δ > 0 such that, whenever the distance between x and x0 is less than δ, the distance between f(x) and L is less than ε.
Definition (Topological): Let f be a function mapping a topological space X into a topological space Y. L is a limit of f as x approaches x0 if and only if for all open sets B in Y containing L, the preimage of B, f-1(B), contains x0.
Theorem: Let f map a metric space X into a metric space Y. Then L is a limit of f as x approaches x0, in the metric space definition, if and only if L is also the limit of f, in the topological space definition, in the topologies for X and Y induced by the respective metrics for these spaces.
References
- George Berkeley. (1734). The Analyst: A Discourse Address to an Infidel Mathematician... [I never finished this.]
- H. Jerome Keisler (1976). Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt. [I barely started this.]
- Morris Kline (1980). Mathematics: The Loss of Certainty, Oxford University Press.
- Walter Rudin (1976). Principles of Mathematical Analysis, 3rd edition, McGraw-Hill.
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