In my last post, we took a look at how our choice of unit has both mathematical and linguistic consequences when we try to talk about *one* of something, particularly in a few weird cases. One of the themes (here unit = "theme") that came up in the course of the discussion is the notion that there are certain objects that lend themselves to **counting**, and others that lend themselves to **measuring**. Moreover, the words we use in our reckoning of different objects/substances are informed by our mathematical interpretation of their underlying structure.

**Checkout Lines and Carnival Rides**

I grew up with two extremely precise parents: a teacher mother who routinely marks public signs with a Sharpie to fix grammatical and spelling errors, and a teacher father who routinely soliloquizes over dubious scientific claims in the media. Perhaps it's no accident that I both teach (and love) math and write (and love) this blog. One of the things I apparently learned/inherited from my mom is a visceral, knuckle-whitening cringe induced by express checkout aisles labeled "10 Items or Less" instead of "10 Items or Fewer." It's a reaction that has lodged itself firmly into the parts of my brain normally reserved for images of poisonous snakes and lion silhouettes. We experience this discomfort because there is a dissonance between the referent noun *items* (a countable substance) and the comparative adjective *less* (a measure word). It makes no more sense to speak of a number of items *less than* ten than it does to speak of a paint color *taller than* red. Height is not an attribute of paint color; measure it not an attribute of item count.

Of course no one is actually confused by that what the sign means: "If the cardinality of the set of items in your basket exceeds ten, please find another line." Still, the entire point of a grammar is to avoid ambiguity whenever possible. Even though there is no appreciable ambiguity in the express aisle, that we're even talking about this highlights an important aspect of our language: we *do* in fact draw a line in the linguistic sand with the measurable on one side and the countable on the other. We have different words that signal counting and measuring in the same way that, e.g., German has different words that signal *you* (singular) and *you* (plural). As good citizens, we try to use the right signalling words, and we're perhaps slightly irritated when others don't.

Once we've figured out whether we're measuring or counting, though, the grammatical questions are decided for us, so the important---and sometimes difficult---part lies in gauging which side of line we're on. In the express aisle, for instance, we have to determine what structure, exactly, underlies the nature of an "item." Well, we understand that the point of the express aisle bound is to get people through the line as quickly as possible, and the quickness with which one negotiates the line is a function of the number of scans that take place. In effect, we can measure checkout time in terms of boops. So, in this case, 1 boop = 1 item. And, since boops are atomic (it makes no sense to think about what half-a-boop might mean), we model them with cardinal/ordinal numbers. Your six-pack of Diet Coke? One boop, not six. Your ten yogurts, which are conspicuously *not* linked or priced together? Sorry, 10 boops = 10 separate items. There is certainly no room for *less than *here.

At the other end of the spectrum, my mom wouldn't look twice at a Tilt-A-Whirl line with a "No Riders Less Than 48 Inches Tall" sign. Presumably the point of the height restriction is to prevent too-small human beings from slipping out of the safety restraints mid-tilt or -whirl, and thus any height at all below the 48-inch cutoff is potentially hazardous. We want to exclude someone who is 45 inches tall, or 45.019 inches, or 42.31 inches. Since it seems as though we're modeling height with real numbers, we know that we're in measuring country. Now we don't have room for *fewer*.

**Time Keeps On Slipping/Discretely Clicking Into the Future**

Of course it's not always so simple. Depending on the context---and thus the *unit*---in question, things might be either countable **or** measurable. Consider time. On the one hand, time seems like the prototypical infinitely-divisible thing, hence calculus gives us exceedingly good predictions about the behaviors of physical bodies moving about the universe. On the other hand, we are sometimes only concerned with time meted out into discrete chunks, hence survey-of-history courses.

Let's pretend we're about to run a 100m sprint together. I might say to you, "If you can run this in **less than** 9.58 seconds, you'll have beat the world record." We're **measuring** seconds, so this makes perfect sense. If, however, we were talking about my teaching career, I might say something like, "Since I've been teaching for **fewer than **three years, I have to be observed by my principal." Now we're **counting** years. I don't think anyone would freak out if I said "less than three years," but it's extremely considerate for me to use "fewer" in this case, because it signals to my conversational partner that "years" is the relevant unit of account. In fact, almost any conversation about my teaching experience is likely to be couched in counting terms, because almost all of the important distinctions (e.g., pay) are binned into one-year increments, the fractional parts of which are totally irrelevant. In other words, we're modeling with cardinal/ordinal numbers, which require counting words.

The amphibious nature of concepts like time can lead to some interesting and confusing consequences. For instance, the ratio of the human lifespan to the rate at which the Earth orbits the sun makes it often convenient to group time into fairly large chunks. Large chunks generally demand counting language. On the other hand, we like to be pretty precise with our reckoning of time, and precision often demands measuring language. Which interpretation wins out might depend on cultural norms.

Consider the traditional Chinese method of determining age. You are born into some year. This is your first year. When the lunar year rolls over, even if that happens tomorrow, you have suddenly been extant during two different years, which makes you two years old. We're starting at one and modeling with cardinal/ordinal numbers, i.e., counting. In the West, we think this is nuts. The moment you sneak out of the birth canal (or, I suppose, the abdominal cavity), the clock starts running. Your birth is the zero marker, and everything is measured as a distance relative to that point. That is, your lifespan is *measured* from birth to death. To say that someone is 18 means two entirely different things in the two cultures. In China it means that he has taken at least one breath on 18 distinct calendar pages; in the West, it means his age measurement lies in the interval [18,19). Neither one is *a priori* more correct; it just means that someone who is 18 under the Eastern system might be barely old enough to start driving in most of the U.S.

But even in the West we're not entirely consistent in our choices. We are by and large measurers of time, but we still count it under certain circumstances. For instance, the year 72 CE belongs to the **First** Century of the Common Era, even though 72 has a zero in the hundreds place. That's a telltale sign of counting. But then we also might talk about the 1900's instead of the 20th Century, and "1900's" is based on a measurement starting at the zero point of Jesus's possibly apocryphal, possibly entirely literally true birth (well, not really the *zero* point...we jumped from 1 BCE to 1 CE without any Year Zero...even though we've been measuring for a long time, now, we got off to a lousy counting start). And a century seems to be about the inflection point: any chunk larger than that and we almost exclusively count (we might talk about the 2nd Millennium, but nobody talks about the 1000's), and anything smaller we almost exclusively measure (ever hear anybody refer to the 90's as "the 200th Decade?").

**What We Talk About When We Talk About Math**

Like so many things in mathematics, this counting/measuring business really comes down to deciding what domain we're in so that we can choose an appropriate mathematical model. Notice I've been saying throughout the piece that we "model" counting or measuring with real or cardinal numbers. And, like all choices of model, it boils down to convenience. The continuity of real numbers makes them very nice to work with sometimes; it's comforting to think that I can measure out any arbitrary amount of water or time that I might need, even though that's not strictly true. Nothing in the physical universe *really *has anything to do with real numbers. We don't know (almost) any empirical measurements beyond about 7 or 8 decimal places. And, even if we did, it seems as though the universe itself, being quantized, turns out to be metaphysically countable. In other words, we could get away with counting language alone, you know, if you didn't mind measuring carnival riders in Planck Lengths.

Of course we don't want to do that, so we've agreed to trade some verisimilitude for the pleasantries of measurement language, even if it slightly increases our grammatical efforts in the process, and even if that increased effort leads to the occasional error in signage. I'll try to keep that in mind next time I find my heart rate climbing in the grocery store.

*Aside: Many of the ideas about the business of measuring and/or counting time, as well as the seeds of my fierce loyalty to a singular "data," can be traced back to John Derbyshire's excellent book, *Prime Obsession.

Fawn NguyenMost common one I hear at school is "less students" -- I'm bothered by this because I thought you could count the number of students. But the language arts teachers are saying this, and English being my second language, I must not know what I'm talking about.

But then "10 items or less" never bothered me until you point it out! 🙂 I am, however, poised and ready to load up my shopping cart of mangos the next time I see them priced at ".50 cents" per. Thanks!

Chris LustoTake solace that you're right about "less students" being wrong. English Arts teachers screw up grammar in everyday speech at least as often as I screw up math in my everyday life, I imagine. I'm generally forgiving.

And yeah, if you see that 1/2 penny for mangos deal, buy me a few and I'll cut you a check.

EmmanuelNice post!

There actually is a way to "measure" so that at the same time you are also "counting". It is an interesting phenomenon, which you explore when you ponder measuring seconds and then counting years and at the end of the post. and you can read about it here: The Counting Measure.

Euclid's approach is really interesting since measurement and number are so intertwined. The greeks didn't employ measuring (with a marked ruler) in their theoretical geometry. Every number had to be constructible and the definitions of unit, number, part and prime are really interesting.