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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?

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    $\begingroup$ Professor Tokieda is definitely on to something. And to be absolutely clear: in the excerpt, he is suggesting that for grasping epsilon-delta, insufficient mastery of one's native language is a stumbling block; he is making no claim whatsoever that studying linguistics is any prerequisite. $\endgroup$
    – ryang
    Commented Oct 15, 2023 at 4:41
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    $\begingroup$ The question (rightly) asks for research to support an assessment of a psychology-of-learning issue raised in a talk. None of the current answers, including the accepted one (!?!), provide that. [This site, to avoid opinion-only answers, used to emphasize research-based Q&A — in fact, overemphasize, imo. Perhaps it has swung too far the other way.] $\endgroup$
    – user1815
    Commented Oct 15, 2023 at 15:40
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    $\begingroup$ Maybe it's a lack of mastery of quantifier logic. $\endgroup$ Commented Oct 15, 2023 at 15:43
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    $\begingroup$ @MikhailKatz Note also that Tokieda claims to have taught himself mathematics as an adult. It's a different matter when, as a young person, you have to put together the parts of mathematics coming down the assembly line, as run by the teacher. I don't mean that the teacher is bad, just that it's different when you're in control of what, whether, and how fast you learn. $\endgroup$
    – user1815
    Commented Oct 15, 2023 at 15:48
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    $\begingroup$ Some persons here have edited the question so that it is now a completely different question from the one originally posed. $\endgroup$
    – user103496
    Commented Oct 16, 2023 at 7:20

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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.
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    $\begingroup$ "I'm confident that my years of mathematical training would allow me to parse Thucydides much faster than the average college freshman." Please elaborate? How? I have had years of math training too, but I can't parse Thucydides AT ALL! $\endgroup$
    – user95017
    Commented Oct 15, 2023 at 1:12
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    $\begingroup$ @user95017 but you would be able to learn doing it more easily than somebody who hasn't had that math training. $\endgroup$ Commented Oct 15, 2023 at 12:18
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    $\begingroup$ @leftaroundabout Truly? Kindly elaborate why please? How's parsing Thucydides related to math training at all??? $\endgroup$
    – user95017
    Commented Oct 15, 2023 at 22:40
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    $\begingroup$ @user95017 It's what I said in my answer: the ability to parse complicated multi-part statements and extract precise meaning from them. This can be learned in many contexts (CS, philosophy, math, linguistics, and others) and is transferable between contexts, to a large extent. $\endgroup$
    – user22788
    Commented Oct 15, 2023 at 23:39
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    $\begingroup$ Unfortunately the question has been totally changed by some persons here. So, while this answer might answer the original question, it no longer answers the current version of the question. $\endgroup$
    – user103496
    Commented Oct 16, 2023 at 7:18
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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.

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    $\begingroup$ Part of the cognitive challenge of your example sentence might be that the assertion is true: we tend to find it hard to give the conditions for disproving something that we know to be true, because the logical meaning of the challenge diverges from the intuitive meaning. $\endgroup$
    – wizzwizz4
    Commented Oct 15, 2023 at 20:05
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    $\begingroup$ No, I mean the elephant thing. There are always non-rainy days, therefore for every elephant just pick one of those days. To have a chance at proving your friend incorrect, we need an eternal downpour, which isn't how the weather works. $\endgroup$
    – wizzwizz4
    Commented Oct 15, 2023 at 20:50
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    $\begingroup$ See, this is exactly what I'm talking about. Your bizarre contrivance only makes sense if everyone's all on the same “this is an analogy” page (and additionally understands what it's an analogy for). Where, to the new student, does “The existence of non-rainy days is an assumption that cannot be made.” come from – especially since the question includes “if the day is rainy”? This example is actively harder to understand than the epsilon-delta definition of the limit. $\endgroup$
    – wizzwizz4
    Commented Oct 15, 2023 at 21:20
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    $\begingroup$ How about: "For every $\varepsilon$lephant, there is a $\delta$ay, such that for every subsequent day, the elephant bathes." $\endgroup$
    – Stef
    Commented Oct 16, 2023 at 14:29
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    $\begingroup$ To @wizzwizz4 's point, you misunderstood your own example. You wrote ‘Our special elephant will be one that always bathes whenever the day is rainy’, but finding such an elephant is not enough; I also need to prove that every day is rainy. It's not just that I can't assume that non-rainy days exist; I must prove that they don't exist to prove my friend wrong. $\endgroup$ Commented Oct 17, 2023 at 21:47
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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.

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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.

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    $\begingroup$ Tokieda would perhaps say that clear language and clear thought are inseparable, in the sense that sloppy use of language is a sign of unclear thought, and insufficient command of language is an obstacle to clear thinking. $\endgroup$ Commented Oct 15, 2023 at 13:18
  • $\begingroup$ That is somewhat funny because math is supposed to be orthogonal to language (and sometimes is, as talent is concerned). But most proof in traditional ME is presented in language where the most minuscule parsing error will invert the meaning, or render the text meaningless. Textbooks are resembling what we call "Textaufgaben" in German ("one worker produces 5 items in 30 minutes" etc.) which is proven to put non-native speakers at a disadvantage. By contrast, pure "formula discussions" on a blackboard should, similar to chess, provide a common language removed from natural languages. $\endgroup$ Commented Oct 15, 2023 at 13:22
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    $\begingroup$ A related issue is that the language people actually use to communicate is not literal and precise. "I could care less" means the same thing as "I couldn't care less," just to take one example. A person could be a very effective communicator via natural language but still struggle when forced to communicate with extreme precision because that isn't how humans usually get ideas across to each other. $\endgroup$
    – user22788
    Commented Oct 15, 2023 at 23:45
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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).

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    $\begingroup$ I have found that university students majoring in humanities tend to score lower in tests in my math classes, but tend to understand the ideas (such as epsilon and delta definitions) better than students majoring in science. So I don't find it at all silly to seek to explain the difficulty of epsilon-delta in terms of insufficient command of natural language. $\endgroup$
    – Simon
    Commented Oct 16, 2023 at 8:29
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    $\begingroup$ @MikhailKatz, you may be right, but I'm not sure that the opposite view is necessarily "silly". Meanwhile, I think that skilled users of their natural language would be greatly helped by increased rigour in the example definitions you gave. For instance, writing "f is continuous if and only if" rather than "f is continuous if". I note also that you seem to have added in an edit, the quantifier that was missing from h. I think that is an improvement. $\endgroup$
    – Simon
    Commented Oct 16, 2023 at 13:03
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    $\begingroup$ @MikhailKatz thank you for your reply to my latest comment on what you rightly point out is your answer (not a comment, like mine). I am sure that your papers are very impressive, and I won't presume to correct your style there. You are of course quite right that definitions frequently use that contraction. I have always found that annoying, and I suspect that a skilled humanities student, capable of parsing Shakespeare or Thucydides, might find it a needless stumbling block, when attempting to learn mathematics. I myself found it illogical, when I first encountered it. $\endgroup$
    – Simon
    Commented Oct 16, 2023 at 22:30
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    $\begingroup$ I think Tokieda would counter that English majors are precisely those least likely to "speak really carefully and making sure that you understand absolutely everything that you are saying and every word and every phrase counts" and most likely to "blah blah blah, just talk away". $\endgroup$
    – user103496
    Commented Oct 17, 2023 at 14:06
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    $\begingroup$ I don't think he's referring to being "talkative" or having "better command of English grammar", but rather being able to express themselves clearly and precisely and "making sure that you understand absolutely everything that you are saying and every word and every phrase counts" (which is perhaps the opposite of "talkative"). $\endgroup$
    – user103496
    Commented Oct 17, 2023 at 14:13

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