This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.
Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling.
Such is Mr Orlin’s conclusion, as I understand it. I hope it is the right conclusion. Another possible reason for students’ not studying is mere lack of interest. Thinking about mathematics is less fun than hanging out at the local shopping mall. But if a student avoids studying because it causes her self-doubt, this is a more hopeful situation. It means the student might actually appreciate the feeling of being smart. The student may learn to achieve this feeling through actually coming to understand some piece of mathematics.
“Not understanding Topology”, writes Mr Orlin, “doesn’t make me stupid. It makes me bad at Topology.” This is imprecise. Not understanding topology means he is bad, right now, at what is being presented as topology. This momentary badness may change with time, and the presentation of the subject may be changed for the better. Here is where a teacher might help.
We all vary. When, in a graduate course, I first encountered the axioms for a topological space, I was so fascinated by them that I wrote them down in a letter to a non-mathematical friend from college. He was not amused.
But like Mr Orlin, I have had difficulties in mathematics. The hardest time of my own mathematical life was in a calculus course, when I had to understand limits and write epsilon-delta proofs. I copy the following from my high school notes.
The entire structure of the calculus rests upon the foundation known as the Theory of Limits…
The rigorous definition of limit is a simple translation of the following statement. The function f approaches the limit L near a if given any preassigned tolerance ε > 0 we can find a control δ > 0, so that when x is within δ of a, and unequal to a, f(x) is within ε of L. The following definition took 2500 years and is attributed to Cauchy and Weierstraß. It is the definition on which all of the calculus rests.
It is fine to point out the importance of limits. Their difficulty might also be acknowledged. Indeed, the definition is anything but simple, at least when one has to use it. Epsilon-delta proofs are a common stumbling-block. A friend of mine transferred to another calculus class, thought to be easier, but apparently still rigorous; he later reported that, after the switch, he finally understood epsilon-delta proofs. When I teach calculus now, I recall to the students my own difficulties with limits. Students should be aware that the concept is hard for just about everybody. [See comments.]
There are schemes for making the learning of calculus easier. Donald Knuth’s proposal to use the big-O notation is reprinted in Alexandre Borovik’s blog, Mathematics Under the Microscope. Another possibility is to use the so-called non-standard approach of Abraham Robinson.
Non-standard analysis is a wonderful subject, and everybody who teaches calculus ought to know something about it. I have thrice taught a week-long course of non-standard analysis for undergraduates at a summer math camp. (In the course, I go back to the origins of calculus in Archimedes.) The non-standard approach to calculus makes limits easier in retrospect; but this is the retrospect of somebody who has already struggled with the epsilon-delta definition.
The definition is difficult, because it involves two alternations of logical quantifiers:
limx→af(x) = L means ∀ε ∃δ ∀x (ε > 0 ⇒ δ > 0 & (0 < |x – a| < δ ⇒ |f(x) – L| < ε)).
This is the “simple translation” referred to in the notes quoted above. The non-standard definition involves no alternations of quantifiers:
limx→af(x) = L means ∀x (x ≈ a & x ≠ a ⇒ f(x) ≈ L).
But the reduction in quantifiers is only an illusion. The quantifiers are hidden in the new symbol ≈. The formula x ≈ a means the difference |x – a| is infinitesimal, so that
x ≈ a & x ≠ a means ∀δ (δ ∈ ℝ & δ > 0 ⇒ 0 < |x – a| < δ).
Here δ must be restricted to the field ℝ of real numbers, because otherwise the formula would have no solution. As it is, if a ∈ ℝ, then the formula has no solution in ℝ. It has solutions in a larger field, *ℝ, consisting of hyper-real numbers.
One can just declare, by fiat, that *ℝ exists as desired. It is a proper elementary extension of ℝ, when the latter is considered as a structure in a perfectly enormous signature. One wants this signature to contain a symbol for every subset of every finite power ℝn. Ideally, one also has a sort for each power set ℘(ℝn), so that one can quantify over elements of this (as for example when defining the Riemann integral). One still does calculus in ℝ as usual; the non-standard aspect is that one can use the help of elements of *ℝ, as in the second definition of limits above, where x ranges over *ℝ, even though a and L are in ℝ. This all needs explicit discussion of symbolic logic.
If one knows “abstract” algebra, and in particular rings, then one can let *ℝ be a quotient ℝω⁄p, where p is a non-principal maximal ideal of the Cartesian power ℝω. One might also write this power as the product ∏ωℝ. A proper ideal of the power is non-principal if and only if the ideal includes the ideal ∑ωℝ. The field ℝ embeds in ℝω⁄p under the diagonal map x → (x, x, x, …) + p; the embedding is proper because the ideal p is non-principal. Actually choosing such an ideal does indeed require a special case of the Axiom of Choice. So one does not and cannot make the choice explicitly; one just assumes it has been done.
I mention all of these details, just to make the point that understanding limits rigorously is bound to be hard, no matter how you go about it. The non-standard approach adds its own difficulties, and there is good reason why this approach has not caught on. There are calculus textbooks that take the non-standard approach. I am aware of the examples by Keisler and by Henle & Kleinberg. The latter authors write in their preface,
A most natural place for Robinson’s insight is as a next (and possibly final) point in the evolution of the teaching of calculus. We can now develop calculus using infinitesimals and enjoy all of their simplicity and intuitive power, yet at the same time work in a mathematically precise and rigorous atmosphere. This approach, although quite new, has been used at a number of universities with remarkable success.
This success has not been so great that Robinson’s non-standard approach has become standard. Perhaps in time it will become standard. It is however foolish to suggest that any approach represents the ultimate stage of evolution. It is dangerous to suggest to students that anything in mathematics is simple. If the mathematics really is simple, then we need not waste any time telling this to the students; we need only show them.
I am familiar with a younger contemporary of mine who struggled with mathematics. In a college course, she asked an instructor to explain the manipulations that he had performed on the board. He told her, “It’s easy!” Perhaps he also repeated the manipulations, by way of showing how easy they were.
No Sir, the mathematics is not easy; this is why you are being asked to explain it. Instead of explaining, you cause your student to be ashamed of her own confusion. Obviously she must be really stupid, if she cannot see how easy your mathematics is. This is what you are telling her.
I suppose it is just possible that the shame of feeling stupid may cause a student to work harder. But I think it is our job as teachers to find a better power of motivation than this. If our subject is not intrinsically interesting, beautiful, captivating, fascinating, then why are we teaching it?
Moreover, if mathematics is easy, why need we bother to teach it? Χαλεπὰ τὰ καλά as the saying goes (Plato’s Republic 435c): Fine things are difficult. We cannot make the pain of learning go away. To deny this only makes the pain worse.
- Henle & Kleinberg, Infinitesimal Calculus, MIT Press, 1979; republished by Dover, 2003.
- H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach, second edition, Prindle, Weber & Schmidt, 1986; available from the author’s website.