Do undergraduates struggle with δ-ε definitions because they lack a habit of careful use of their native language?

Question

I transcribed this excerpt starting at the 22-minute mark, of Okinawa Institute of Science and Technology’s May 19 2020 podcast with Professor Tadashi Tokieda:

For example, this is a bit too technical, but there's a definition.... a very very precise way of thinking about the limits and continuity and so on, which is goes on the name of Epsilon and Delta. So for every epsilon there exists a delta such that and blah blah blah. And this is a stumbling block for almost everyone. But when I came into mathematics as an adult already (you know I taught myself mathematics), and when I came to Epsilon-Delta, I felt no difficulty whatsoever. In fact I didn't even notice that it was supposed to be difficult. That's because I had been very rigorously trained in the use of languages as a linguist. And so, the idea that, you know, if you change the order quantifiers, of course, the meaning changes completely. It's comp. It was trivial, of course, I mean, compared with the task a difficult task of taking apart the syntax of, say something somebody, like Thucydides you know whose sentence can continue for a page, with subordinate clause upon subordinate clause. By the way, he writes really clearly, but in a complicated syntax. Well compared with that kind of thing, language in mathematics was very very easy. I mean there's nothing to it. I think the fact of the matter is, most people don't have a sufficient mastery of their native language. They never had the experience, they don't have had enough, shall I say a bit more gently, enough practice of careful use of their own native language. You know, do you speak really carefully and making sure that you understand absolutely everything that you are saying and every word and every phrase counts? The answer is no. The people just blah blah blah, just talk away. So, if you have a really careful habit of careful use of language, it's my personal belief that most of the difficulties in mathematics will go away. And it's just that mathematics is an unforgiving subject, where any misunderstanding any lack of understanding shows immediately, whereas in the rest of human endeavors, you can keep going by faking for quite a long time. So in that way, yes the language frames how you understand mathematics, but in that very very practical way. I think the best way to improve your chance of your future advancing mathematics is to practice and improve your native language.

Does research on students’ difficulties with the δ-ε definition of limits and continuity substantiate Tokieda’s asseveration that a primary cause is insufficient mastery—or at least lack of a habit of careful use—of their native languages? Does empty-headedness of linguistic syntax bedevil undergraduates on, and thwart them from, δ-ε definitions?

Answer 1

I wouldn't say so. By studying linguistics on a deep level, this person learned to parse complicated multi-part statements and extract precise meaning from them. This skill--which people in general are not born with--is transferable to math, and it's no surprise that they didn't struggle with epsilon-delta the way most people do.

But that doesn't mean linguistics should be considered a prerequisite to math. It could just as easily go in the other direction; I'm confident that my years of mathematical training would allow me to parse Thucydides much faster than the average college first-year. It's often said that programming and studying pure math help each other, etc. The point is that many of these skills are transferable.

To use a sports analogy, if someone is a professional tennis player, they could pick up squash much more quickly than the average person. But we wouldn't say that the only reason people struggle to learn to play squash is because they haven't sufficiently mastered tennis.

In my opinion, students struggle with epsilon-delta for two reasons:

1. It's just hard.
2. It is (unfortunately) often inflicted on students who are still getting used to the idea of rigorous mathematical proof, and they are still working to make sense of a number of aspects of that, while also grappling with the inherent difficulty of epsilon-delta arguments.

Answer 2

The issue you raise is similar to those raised here:

How to make students comfortable with the use of axiom of choice in analysis

[Note: the question at the link has "axiom of choice" in the title but it really has nothing to do with the axiom of choice!]

We can appreciate the linguistic complexity of the definition by listening to people struggle as they try to state the negation. Here is "fun" assignment that illustrates this:

Your friend asserts that for every $\epsilon$lephant, there is a $\delta$ay such that the day is rainy and the elephant forgets to bathe. How would you prove your friend incorrect?

Note the logical structure of the elephant sentence is very similar to the definition of a limit. Students give a variety of incorrect solutions. It presents a cognitive challenge.

Added: We can formulate the scenario like this:

$$\forall\epsilon\exists\delta\left[\delta(R)\wedge\neg\epsilon(B)\right]$$

The negation is

$$\exists\epsilon\forall\delta\left[\delta(R)\rightarrow\epsilon(B) \right]$$

To prove our friend wrong, we need to produce a counterexample. We need to find an elephant with a certain property. Our special elephant will be one that always bathes whenever the day is rainy, i.e., one that never forgets to bathe on rainy days.

I always find it tricky to formulate this negation, and whenever I need to prove a function is not continuous using the limit definition, I know I am going to have sort through this all over again!

But it is interesting to note that large language models (GPT, Bard, LLaMa,etc.) are extremely proficient with the elephant problem! Use the boldface font above as a prompt, and see for yourself. On the other hand, the language model will not be so helpful when trying to prove that a specific function is not continuous by using the definition, because of the algebra and other mathematical details. This suggests to me that Tadashi Tokieda is onto something. AI in the form of large language models is very proficient with natural language processing, and tends to be much better at the underlaying elephant problem than are humans like me. The difficult logic of the elephant problem is the same as that of showing a function is not continuous using the definition. This is why we like to have lots of necessary conditions to check continuity--using them can often help us efficiently detect discontinuity without needing to grapple with the elephant.

Answer 3

I think the linguist is looking at it backwards: to study the formal grammar of a language, you have to apply some complex mathematical (or logical) ideas.

Natural languages are imprecise, contextual, and intuitive. The millions of people who speak English fluently, and communicate effectively in it, are not "faking it", they are using an incredibly sophisticated set of unconscious skills.

Mathematicians have devised artificial languages, firstly to remove the ambiguities of natural language, and secondly as short-hand to refer to complex ideas. The rules of these artificial languages are rigid and plainly logical.

Linguists attempt, among other things, to detect underlying rules in how native speakers construct and understand language. They propose formal models of how the language works, and test them against examples from the real world. Those models treat natural languages as, at some deep level, as rigid as the mathematicians' artificial languages.

That level of "mastery of your own language" is like having a "mastery of your own car" that would allow you to strip it down to individual nuts and bolts and build it back up again - admirable, maybe, but for most people a sufficient mastery would be fixing a flat tyre.

Answer 4

I think there are two (main) reasons why "people just blah, blah, blah, just talk away." One is linguistic: They have clear thoughts but cannot accurately express them in words. The other is cognitive: They don't think clearly. Tokieda emphasizes the former because he had been "rigorously trained in the use of languages as a linguist". He overlooks the cognitive aspect because he probably didn't need rigorous training in clear thought; he's probably one of those (fortunate) people to whom clear thinking just comes naturally.

Either linguistic or cognitive difficulties can cause "blah, blah, blah" and can obstruct understanding of epsilon-delta definitions. But I'd seriously doubt any claim that a particular cause (or finite list of causes) accounts for all the problems students have with epsilon-delta definitions. Some students fail to understand such definitions (and mathematically precise reasoning in general) for quite unpredictable reasons.

Answer 5

The is a built-in irreducible logical complexity in epsilon-delta definitions that has to do with the language of logic, rather than natural language. This has to do with alternations of quantifiers. To illustrate the complexity, compare the following two definitions of continuity of a function $f$:

1. $f$ is continuous if for every $x$ and for every positive $\epsilon$, there exists a positive $\delta$ such that, for all $h$, if $|h|<\delta$, one has $|f(x+h)-f(x)|<\epsilon$.

2. $f$ is continuous if every infinitesimal increment $\alpha$ leads to an infinitesimal change $f(x+\alpha)-f(x)$ (this happens to be Cauchy's definition of continuity).

It does not require great analytic skills to notice the difficulty of definition 1, due to a pair of alternations of quantifiers.

It would therefore be silly to explain the difficulty of epsilon-delta in terms of insufficient command of natural language. If this were the case, calculus would be a breeze for English majors! (this is not known to be the case as far as I know; if anything, the opposite).