In Praise of Pedantry

Abbott: Funny thing, Lou: there’s an infinite number of numbers.

Costello: How many?

Abbott: An infinite number.

Costello: What infinite number?

Abbott: Oh no, there is no infinite number.

Costello: But you just said there was!

Abbott: No I didn’t. I said there’s an infinite number of —

Costello: There! You said it again!

— Abbott and Costello, in a number of their movies (specifically, the number zero)

Earlier this year I dug a mathematico-linguistic rabbit hole on Twitter when I wrote:

Dozens of people posted on that thread with very different takes on my question. One of them, Fred Klingener, reported to me the following actual example of infinity in real life:

Another respondent, Akiva Weinberger, dug a secondary rabbit-hole of his own for his linguistics pals on Facebook, who have their own brand of nerdiness distinct from, but parallel to, the nerdiness of math folks.

After jumping down both holes and crawling through all the tunnels, I emerged blinking into the light of day with renewed appreciation of the way different people can use different words to talk about the same thing.

If you’re wondering why I found the phrase “infinite solutions” problematic, you might find it helpful to know that in some mathematical contexts an individual solution can be infinite.

For instance, consider the equation x³ = x in the extended real number system that consists of all the ordinary real numbers along with bonus numbers +∞ and –∞ which satisfy various properties such as (+∞)³ = +∞ and (–∞)³ = –∞. This number system doesn’t have a good press agent, but it’s an actual Math Thing, and in the extended real numbers the equation x³ = x has two infinite solutions (+∞ and –∞) and three finite solutions (1, 0, and –1). If I were to ask various people (after introducing them to the extended real numbers) “Does x³ = x have infinite solutions in the extended real numbers?” some might say “No, it has only five” and others might say “Yes, two of them.” Likewise, if I ask “Does the inequality x² < 1 have infinite solutions?” some might say “Yes, because there are infinite numbers between –1 and 1 and they’re all solutions” while others might say “No, because +∞ and –∞ are not solutions”. What’s going on here? If we don’t want to get caught up in an Abbott-and-Costello-ish escalation of confusion, we need to recognize that the word “infinite” could refer either to individual solutions themselves or to the set of all solutions. That is, the word could be either an adjective or a cardinal. This may seem like an overly nice distinction, but if you’ve got mathematical sentences that some people call true and other people call false even though they don’t really disagree about the facts, then you’ve got yourself a nicety that deserves notice.¹

Personally, I view the phrase “infinite solutions” as ambiguous, so I avoid it and instead say that in the extended real numbers, x³ = x has solutions at infinity, and that x² < 1 has infinitely many solutions. But the responses to my tweet made me aware of just how diverse different English speakers’ mathematical idiolects are, and how even a single person can use words differently in different settings (formal written English is not the same as informal spoken English). It also made me aware of how English differs from other languages when it comes to measuring mathematical collections and how, as we’ll see shortly, English is in some ways more prone to ambiguity than other languages — which I suppose is what you’d expect from a language that can’t decide whether the nickname of mathematics should be “math” or “maths”.

Here I should confess that the use of the word “infinite” as a cardinal (the germ of the twitter conversation that was in turn the germ of this article) has distracted and irritated me but never genuinely confused me. By way of contrast, consider pronouns. I often get confused by pronouns in daily speech, and find myself thinking: “Oh no! Which of the two men she’s talking about did she just refer to? Should I interrupt to ask? Or do I need to temporarily maintain two parallel interpretations of what she’s saying in my head, updating both, until something she says later removes the ambiguity and I can abort that interpretation?” But when it comes to infinity, I’ve nearly always found people’s meaning to be clear from context.

UN-INFINITY

Before we tackle infinity and the fog of language surrounding it, let’s consider an adjacent mathematico-linguistic issue involving not things that are infinite but things that are as un-infinite as can be: the set that is empty, and the number that is zero, and the difference between them. Zero plays a bigger role in students’ practice in precollege mathematics (up to and including calculus) than infinity, and it seems that some students have serious “zero issues”. Tony Mann wrote, in reply to my question (conceptual error vs. language difference):

(Is that exam an urban math-myth, or can someone provide details?) Katherine Seaton described a calculus student who went from “the function is not differentiable at a” (that is, there is no derivative at a) to “the derivative at a is zero”. In the other direction, James Francese wrote “A large number of my calculus students decided that lim f(x) = 0 means the limit does not exist.”

Perhaps some of the confusion can be laid at the doorstep of the band They Might Be Giants, who in their song Zeroes rightly distinguish between the digit 0 and the number 0 but then describe the latter as “the number that means nothing-at-all”. To combat confusion between zero and nothing, teacher Elizabeth Hentges likes to reach into her pocket, pull out the contents — keys and phone — and show the class that she has $0 in her pocket but not nothing.

(We might also blame John von Neumann, who in trying to improve Georg Cantor‘s theory of infinity defined the counting number n as the set of all nonnegative integers less than n. Thus 3 is defined as the set {0,1,2}, which sure enough has 3 elements and so can serve as a prototype of 3-ness; 2 is defined as the set {0,1} (yup, it has 2 elements); 1 is defined as the set {0} (with 1 element); and 0 is defined as the empty set (with 0 elements). That’s all very well when you’re shoring up Cantor’s theory of infinity and all that, but “Logicians define zero to be the empty set” is not a helpful thing to share with high schoolers trying to understand calculus. I suspect that most of the kids who are confusing zero with the empty set haven’t heard of von Neumann. Yet. Please don’t tell them.)

INFINITE POSSIBILITIES

There are not actually infinitely many ways to say “infinitely many”, but there sure are a lot of them. Here are over a dozen phrases one might hear to describe a situation where a solution set is infinite.

Possibility 1. “There are infinity solutions.” … This seems to be widely derided as “mathematical babytalk”; it tacitly endorses the notion that infinity is a number. Teachers discourage the notion for good reason: If you treat infinity as a number and do it wrong, you get paradoxes that can infect the sensible kind of math as well. Whereas if you treat infinity as a number and do it right, you learn that infinity corresponds to not just one “number” but many of them (a topic I briefly touch upon in Endnote 2).

Possibility 2. “There are infinite solutions.” … Many people use “infinite” as a cardinal because they aren’t imagining situations in which an individual solution could be infinite. Timothy Gowers pointed out the Veritaseum video “How An Infinite Hotel Ran Out Of Room”, which uses the phrase “infinite rooms” to mean “infinitely many rooms” and the phrase “infinite people” to mean “infinitely many people”. To me, “infinite people” distractingly suggests people who are infinitely tall or infinitely heavy or infinitely smart. As for “infinite rooms”: if the rooms are individually infinite, wouldn’t a single one of those rooms accommodate all the guests, assuming they don’t care about privacy?

Possibility 3. “There is/are an infinity of solutions.” … I’m fine with both of these.

Possibility 4. “There are infinitely many solutions.” … That’s the one I favor, but many people who prefer “infinite solutions” say “infinitely many solutions” sounds “overly pretentious” (as opposed to insufficiently pretentious or just-pretentious-enough, I guess). The linguistic divide seems to become entrenched at age twenty or earlier: “infinitely many” sounds correct to people who are math majors but sounds excessively formal to people who are not math majors.

Possibility 5. “There is/are an infinitude of solutions.” … The word “infinitude” sounds pretentious to me (which makes me sympathetic to those who say “infinitely many” sounds pretentious to them).

Possibility 6. “There’s an infinite number of solutions.” … I’m okay with this one too, though the Abbott and Costello parody at the top of the essay shows the problem with this way of talking.

Possibility 7. “There is an infinite amount of solutions.” … I don’t like this, because “amount” suggests something continuous that we weigh on a scale or measure with a ruler, whereas solutions are discrete things, to be counted. Then again, the solution-set of x² < 1 forms an interval of length 2 — just is the kind of thing we might measure with a ruler and call an “amount”.

Possibility 8. “The set of solutions is infinite.” … As you might guess, I like this one, but to non-mathematicians, it comes across as pretentious. I mean, can’t we just talk about the solutions directly? Why do we have to corral them into a set?

Then we get options that (jumping off from the hierarchy of infinities I mentioned in Possibility 1) are more specific about which infinity is meant; I’ve relegated these to the Endnotes.²

OTHER LANGUAGES

John Baez tweeted:

Akiva Weinberger (the one who dug the secondary rabbit hole over on Facebook) concurred with John’s use of the word “weird”, summarizing the linguists’ response by saying “it does seem that it’s a case of mathematicians talking weird compared to the rest of the world”.

But is it the whole rest of the world, or just the English-speakers? Based on tweets that I saw down my own rabbit hole (zoological metaphor mismatch noted), I can report that other European cultures have different ways of including mathematical nuance through the use of word endings and word order.

The German language uses endings: “unendlich Lösungen” means infinitely many solutions, but adding an “e” to the end of the “unendlich” gives us “unendliche Lösungen”, referring to solutions that are themselves infinite.

In Italian, Portuguese, and Spanish, it’s all about word order. If you write “infinite soluzioni”, “infinitas soluções”, or “infinitas soluciones”, you’re referring to infinitely many solutions; but exchange the two words and you’re referring to solutions that are individually infinite. Disambiguation via word order is a handy feature for a natural language to have, and you won’t be surprised to learn that it has nonmathematical applications as well. For instance, in Spanish, “un amigo viejo” is a friend who is old while “un viejo amigo” is a friend you have known for a long time; “un pobre hombre” is a poor (unfortunate) person while “un hombre pobre” is a poor (destitute) person.

If you know a language with a fun way to disambiguate “infinitely many solutions” and “solutions that are infinite”, please post to the Comments!

Also let me know if you know of any students in Europe who (for instance) say or write “infinite soluzioni” when they mean “soluzioni infinite” or vice versa.

IS THERE A PROBLEM HERE?

Okay, descriptive philology is fun (for some of us), but let’s get back to actual math. We’ve seen that the phrase “infinite solutions” has two plausible meanings in English, yet many people seem to be unaware that it’s ambiguous. Is this a problem? Opinions on the Twitter thread ranged over the spectrum.

On the other hand:

SPEAKING OF …

Another loose assertion involving infinity that I sometimes hear is “pi is infinite”, meaning that the decimal expansion of pi goes on forever. Do the same students who say that pi is infinite also say that 1/3 is infinite?

Yet another kind of confusion can arise if we use the word “infinite” in a situation where there are different notions of bigness. In geometry class, we might describe a line or a ray as infinite and a line segment as finite, which makes sense if we’re talking about length. Then again, a line segment has infinitely many points. So, tell me: Is that line segment finite or infinite? As Sydney Gibson wrote, “Speaking precisely about math is difficult, and a more distinct skill from understanding math than people often realize.”

Some people who responded to my tweet thought that use of the phrase “infinite solutions” was symptomatic of a misunderstanding of the whole set-theory set-up of modern mathematics. Specifically, they thought that a student’s failure to recognize that a property like being infinite could apply to the set of solutions or to some of the elements of that set, and their failure to use language in a way that distinguishes between these two meanings, could indicate shakiness with the whole concept of sets. Jim Wolper wrote: “This is what happens when you speak about a set without being explicit in saying that you are speaking about a set. It’s conflating the set with its elements.” That is, the thing that’s infinite is the set of solutions, but if you aren’t comfortable with the idea of sets, you’ll impute properties of the set to its elements instead. (Teachers: What are your favorite examples of properties that a set can have even though all the elements of the set lack it, and vice versa?)

THINKING TOGETHER

Wolper’s diagnosis ties in with why I cared about my question to begin with. A lot of math becomes gobbledegook if you blur the distinction between sets and elements. Consider for instance the assertion (from the branch of math called model theory) that the existence of arbitrarily large finite models of a theory implies the existence of an infinite model. Never mind what all the words here signify; “the existence of arbitrarily large finite models” means that for any finite model you can find, I can find one bigger — which implies that there are infinitely many finite models. On the other hand, “existence of an infinite model” is just asserting the existence of one model — one model that happens to be infinite. If one’s way of talking about model theory doesn’t distinguish between models that are infinite and infinite sets of models, one won’t get very far.

If that example is too arcane, here’s one that’s more down to earth: “Horse can be black. White horse cannot be black. So white horse is not horse.” The classic paradox exploits the ambiguity of the word “is” (does “A is B” mean “A is synonymous with B” or “A is an instance of B”?), and the original Chinese version exploits the ambiguity of nouns in a language missing articles (so that “horse” could mean “a horse” or “horsekind”). The paradox shouldn’t confound people with a background in set theory. It’s true that the set of white horses is different from the set of horses, but that doesn’t mean that an element of the set of white horses can’t be an element of the set of horses. If one’s way of talking about philosophy doesn’t distinguish between properties of individual horses and properties of the set of horses, one won’t get very far.

But my native language is English, not Classical Chinese, so let me switch back into my lane and bash the English language a bit. Observe that the superficial resemblance between the sentence “I have wonderful children” and the sentence “I have two children” belies a fundamental difference: the adjective “wonderful” refers to the two children individually (when my children are together they’re not so wonderful!), whereas the cardinal “two” refers to the children collectively (neither of them has been two for a long, long time). So it makes sense that English speakers would have a hard time distinguishing between individual properties and collective properties when our grammar doesn’t reinforce this distinction.³

Here I am subscribing to a kind of linguistic determinism. The strongest version of linguistic determinism holds that you can’t hold a concept in your mind if you don’t have words for it. The weakened form I subscribe to holds that it’s difficult to think straight about topics for which one lacks the vocabulary to talk straight. This is especially true when cognition happens in multiple collaborating minds, as is frequently the case for minds doing math. If one’s way of speaking isn’t clear and unambiguous, one won’t get very far, and two will just go around in circles like Abbott and Costello.

Etymologically speaking⁴ (and what opening for a paragraph could be more pedantic than that?), the word “pedantic” means “teacherly”, so why should I be ashamed of that? Ain’t I — excuse me, am I not — a teacher? So I’ll declare myself to be pedantic and proud of it. Even if meticulously unambiguous speech would drive us all nuts (including me!) if we had to engage in it round the clock as both speakers and listeners, it’s important to have nerd-ese as a fallback option for two people who start to suspect they’re not on the same page.⁵ I’ve always liked the saying “I’m not pompous, I’m pedantic. There is a difference.”⁶ That’s the line I try to walk.

It should go without saying (but, alas, doesn’t!) that a conscientious pedant needs to be an equal-opportunity nitpicker. Consider the case of mathematician Edward Frenkel, who was denied admission to Moscow State University because he was Jewish — or rather, because he failed to live up to the ridiculously high standards that Jewish applicants (not gentile applicants) were required to meet. One of his examiners asked him to define a circle, and when Frenkel said “A circle is a set of points on a plane, equidistant from a given point”, the examiner cheerfully said “Wrong! It’s the set of all points equidistant from a given point.” That was the end of Frenkel’s chance to attend Moscow State. Fortunately it was not the end of Frenkel’s mathematical career, but unfortunately there were dozens of other people whose names we don’t know who didn’t have Frenkel’s happy ending. The story has a moral for all of us who function as academic gatekeepers: it’s your duty to apply the same standards to everyone.

Actually, let me amend that. You should have a single standard for all the people who aren’t you; but that doesn’t mean you should hold all those people to the same standard that you hold yourself to. Putting it differently: I think it’s important for us pedants to have a double standard when it comes to listening versus speaking. When a student says something that sounds like nonsense because they don’t know how to phrase what they’re thinking, a skilled teacher infers what they’re thinking (if the teacher’s experience leads them to know what the student must be trying to say) or asks them to clarify what they mean (if the teacher isn’t sure). In fact, I like to interpret the opening sentence of the White Horse dialogue — “Is ‘white horse is not horse’ assertible?” — as meaning something more like “Is ‘white horse is not horse’ defensible?”, or “If a student said such a thing, what kernel of plausible truth might you extract from it?” Part of the art of teaching is hearing a wrong answer, finding what’s right about it, and pivoting on that fulcrum to get the student to a better answer. You can’t do that if you insist on your students expressing themselves as clearly as you do.

So by all means be Spock⁷ when you speak, but be loose when you listen!

PEDANTIC POSTSCRIPT

I can’t believe I wrote a piece about pedantry without actually looking up the word. So I just did (four days before my “press deadline”), and it appears that I’ve been using the word in an overly narrow way. Pedantry, it turns out, isn’t just about how precisely or loosely we use words; it’s about excessive concern with minor details of all kinds. And this opens up the whole topic of fussiness in mathematics. Why are we mathematicians so fastidious? Why are we so quick to look for loopholes, or shoot down ideas (our own ideas or other people’s) by looking at pesky extreme cases? Is fussiness something incidental to math, or something at the heart of it?

If there were a nightclub of accepted mathematical truths, the standard of rigorous proof would be the ultimate obnoxious bouncer. “Sorry, your tie is crooked, you can’t come in.” Or: “You’re wearing a tie? Don’t ever come back.” If I were a theorem, I wouldn’t want to belong to a club like that. But I’m a mathematician, not a theorem, and I believe that the high barrier to entry at the Theorem Club has payoffs.

So I ask you, Readers: How many breakthroughs in math arose from pedantry in the broader sense of meticulous concern with formality? Consider hyperbolic geometry, for instance. Its discovery followed attempts to derive an “obvious truth” (the parallel postulate) from more basic assumptions, and was spurred by the way that all such derivations, when examined with scrupulous attention to detail, were found to be inadequate. Scruples are inhibitory, and finding flaws in proofs is on its face a negative enterprise, but when a lot of negative evidence piles up, something positive can come from it. Maybe I should write an essay called “The Fertility of Fussiness”⁸ someday!

If you found this essay interesting, you might want to check out a couple of “meta-pedantic” essays by Ben Orlin: The Persistent Pedantry of the Mathematical Mind and Advice for Snobs.

Thanks to Dan Asimov, John Baez, David DeBaun, David desJardins, Paulo Cezar Filho, Sean Fitzpatrick, James Francese, Sydney Gibson, Timothy Gowers, Sandi Gubin, Elizabeth Hentges, J. Diego Suarez Hernandez, David Jacobi, Arseny Khakhalin, Fred Klingener, Matthew Leingang, Peter Littig, Laura Manenti, Tony Mann, Ben Orlin, Evan Romer, Katherine Seaton, Mattie Wechsler, Akiva Weinberger, Glen Whitney, and Jim Wolper.

ENDNOTES

#1: I think it’s fitting — dare I say nice? — that the word “nice” has different meanings that we need to draw distinctions between.

#2: Most of the infinite sets studied in ordinary mathematics are either the same size as the set of counting numbers (in Cantor’s sense of the term “same size”) or the same size as the set of real numbers. Possibilities 9 through 11 pertain to the first case; possibilities 12 through 15 pertain to the second case. I won’t explain here why some infinite sets are more infinite than others, though if this is news to you, you might want to check out my essay Cantor’s Paradise and Skolem’s Paradox (scroll halfway down the two-parter).

Possibility 9. “There are aleph-null solutions.” … Aleph-null (also called aleph-nought or aleph-zero) is the smallest of the infinities; it’s the size (or more technically the cardinality) of the set of counting numbers. It’s also the cardinality of some sets that intuitively seem larger than the set of counting numbers. I learned about aleph-null at a young age from a book by George Gamow called “One, Two, Three, Infinity”. 

Possibility 10. “There are countably/denumerably many solutions.” … A set is denumerable if it can be put into one-to-one correspondence with the set of counting numbers. A set is countable if it is either finite or denumerable. So technically someone who says “countably many solutions” isn’t excluding the case where there are only finitely many, but I think that’s a quibble too far.

Possibility 11. “There are a countable infinity of solutions.” … That’s fussier but clearer than “countably many”.

Moving on to bigger infinities:

Possibility 12. “There are continuum-many solutions.” … This is the usual way a mathematician would talk about the solutions to x² < 1. It makes geometrical sense, since the set of solutions is a continuum, an open interval of length 2. We could also write that the set of solutions has cardinality c. c is defined as the cardinality of the whole real line, and the arctangent function gives a one-to-one correspondence between the real line and the open interval from –1 to 1.

Possibility 13. “There are two-to-the-aleph-null solutions.” … This is okay too, since in Cantor’s theory it can be shown that the cardinality of the set of real numbers equals two to the power of the cardinality of the set of counting numbers.

Possibility 14. “There are beth-1 solutions.” … This is quite arcane but correct. Beth-1 is defined as two to the power of beth-0, and beth-0 equals aleph-0.

Possibility 15. “There are aleph-1 solutions.” … If we’re talking about the set of real numbers, or the set of real numbers satisfying x² < 1, then “aleph-1” is wrong (well actually its rightness is merely independent of the currently accepted axioms of set theory, but that’s wrong enough), and I lost $50 back in college because of Gamow not knowing his aleph-beth (mathematically speaking). In “One, Two, Three, Infinity”, Gamow wrote “There are aleph-1 points on a line” (instead of beth-1) and I believed him. Years later, in college, I learned that there was an obscure prize called the Robert Fletcher Rogers Prize that nobody seemed to know about, offering cash to the top two presenters of mathematical talks. I persuaded my classmate Dan Freed to enter with me, and we agreed not to tell the other math majors. As the only two contestants, how could we lose? Dan prepared a good talk; I gave a goofy off-the-cuff discussion of infinity that featured Gamow’s conflation of aleph-numbers with beth-numbers. The judges gave Dan first prize and gave me no prize at all. I don’t blame them; I blame George Gamow. If anyone from Gamow’s literary estate is reading this, please contact me so that we can come to a mutually agreeable settlement.

#3. Linguist Mattie Wechsler points out to me that English grammar does support the distinction between determiners (like cardinals) and adjectives, as we can for instance infer from latent rules about word-order (“I have two wonderful children” is idiomatic, “I have wonderful two children” isn’t). As Wechsler says, “There are sentences, of course, where the surface structure is ambiguous, but because surface structures are flat and individual sentences cannot reveal unused architecture, not because English’s overall grammatical structure is ambiguous or undiscerning.”

#4. As I write the words “Etymologically speaking, the word ‘pedantic’ means…”, a little voice in my head is asking, in a sarcastic singsong-y voice, “Who or what is doing the speaking? Is the word ‘pedantic’ speaking etymologically?” Shut up, little voice.

#5: One of my favorite examples of miscommunication involved a friend of mine who was a librarian in Berkeley, California in the 1980s. A patron came in and said she wanted to read a book about crystals. My friend led the patron to the mineralogy section; the woman got very upset. “But … these are books about rocks and gems. I’m interested in CRYSTALS!” Clearly a more nuanced vocabulary would have helped the patron get what she wanted. (Or maybe my librarian friend just needed to get better at detecting the auras of New Age types.)

#6: There’s a variant that’s funny in a different way. “I’m not pompous, I’m pedantic. There is a difference. Let me explain it to you.” By compounding pedantry with pomposity, the speaker undermines the very distinction they’re trying to assert.

#7: I would hope it’s clear from context that I’m referring not to pediatrician Benjamin Spock, or to other actual people named Spock, but to “Mister” Spock, the fictional CPO (Chief Pedantry Officer) of the starship Enterprise, in Star Trek: The Original Series. Actually, I don’t like the way the writers had Spock give numerical answers with spurious precision (e.g., estimating the time until some catastrophe occurred in tenths or even hundredths of a second, which is meaningless in the context of a sentence that takes many seconds to utter), but that’s a bone I’ll pick some other time.

#8: In the meantime, you can read my not-quite-one-year-old essay “The Positive Side of Impossible”, which develops a similar theme.