Measure Your Blessings

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.

Human Composure

Frequent commenter and child logic expert Christopher Danielson recently contributed a very cool video, entitled One is one…or is it?, to the Ted-Ed project in which he explores  what it is we really mean when we say “one.”  The very notion requires an implicit or explicit reference to a unit, and sometimes things aren’t quite as simple as they seem.  In particular, we might have units that are either built up of (composed units) or divided into (partitioned units) smaller sub-units.  Or sub-sub-units.  For instance, the loaf of bread in your cupboard is partitioned into individual slices; the pinochle deck you stash in the coffee table is composed of individual playing cards; and Pop Tarts are sold in boxes, which contain packs, which contain individual pastries. One is relative to your choice of unit.

Of course this is all very interesting, but the first thoughts that popped into my head during the dancing apple slices number involved the two conspicuous cases that seem to defy composition and partitioning: (1) units whose sub-units are human, and (2) units whose sub-units are tiny.  It’s still possible to smoosh them together and rend them asunder, but not so neatly.

Part the First

For just a moment let’s consider the North Carolina State Wolfpack.  Now that is definitely an all-the-way, one-hundred percent composed unit.  It is a singular pack, composed of singular wolves.  Unambiguous.  Until you read some press releases, which are simply pregnant with phrases like, “the Wolfpack are…” Which is very strange indeed.

Consider two fictitious news stories.  In the first, NC State is relocated by the NCAA to South Carolina.  In the second, an actual, literal pack of wolves (Canis lupus) is spotted migrating from Raleigh to Charleston.  The first headline would read, “Wolfpack Head Across the Border,” while the second would read, “Wolfpack Heads Across the Border.”  These two scenarios are mathematically identical; the only difference is that, in one case, the sub-units are figurative wolves.  So why do we require a different verb conjugation?  It seems that people somehow resist being subsumed by composed units in a way that, e.g., playing cards do not.  Admittedly this is a psycho-linguistic curiosity more than a mathematical one, but still…units can be slippery.

It’s maybe more obvious that people resist being partitioned.  After all, if you make it through life without anybody partitioning you, let’s call that a nontrivial success.  But it shows up in the language, too.  We are very rarely wont to consider a disembodied finger directly.  When we partition people, we replace the corporeal whole with a possessive placeholder, a pointer (ha!) to the original unit.  We hold the source object in memory in a much more vivid and deliberate way than with other objects.  An apple slice is an apple slice, culinary nuances notwithstanding, but “John’s foot” and “your foot” and “the crazy lady in 3B’s foot” require modification.  Unless you live under fairly abnormal conditions, indefinite articles no longer suffice: “a foot” or “the foot” rarely come up.  We make grammatical concessions more readily and more often for human “units” (suppressing, here, many jokes) than nonhuman ones.

Part the Second

Consider the strangeness of the following utterance: “I think the rice is done.”  Why aren’t they done instead?  There are tons of those little buggers!  There certainly exists a plurality of foodstuffs.  But when we deal with tiny sub-units (especially if they’re homogeneous), we have a hard time unitizing them naturally.  We can go clumpy: “I think the pot/serving/microwave bag of rice is done.”  We can go grainy: “I think the grains of rice are done.”  But both of those solutions feel deeply unsatisfactory.  There are contortions involved.  We either have to create a new group name (think pod of dolphins or murder of crows), or a new unit name (think kernel of corn or drupelet of raspberry).  Awkward either way.  It’s as if, on a fundamental level, we ache for rice to be singular entity, but of no definite unit membership.

And this weirdness, I think, is no longer merely syntactic, but deeply mathematical.  When we unitize the world in language, we make an important distinction between that which is countable and that which is measurable.  There is an analogy to be made here between discrete and continuous objects, respectively.  It is natural to consider the composed unit of a dozen eggs, because eggs are easily countable, and because it’s easy to count to a dozen.  It is much more difficult to create a conventional composed unit out of water molecules, because water molecules are hard to count, and because it would be hard to count to a number of water molecules that would be useful in most situations.  Thus, we measure water and treat it as an un-composed unit.  Somewhere in between those two extremes, we have things like rice, much more countable than water, much less countable than eggs.  In fact, it’s closer to water than eggs in its countability, so we treat rice as measurable/continuous, even though it’s technically countable/discrete.  Sand.  Salt.  Data.*

*I will fight to the death, via torturously long diatribes on gerunds and loan words, that “data” should be treated as singular in English, even though it’s inflected as a plural in Latin, all based on the composed unit argument above.  If you’re going to be a total weeny and use it in the plural, at least be consistent.  I had better never hear you talk about “an agenda” (singular), because “agenda” is also inflected as plural; each item is technically an “agendum.”  I’m watching you.

So tiny things resist composition, and they resist partitioning even more vehemently.  For one, they’re already tiny.  It’s inconvenient to let these things get any smaller (and our unit choices, after all, have an awful lot to do with convenience), and there’s no natural starting point from which to partition things in the first place.  If I wanted to decompose “sand” into parts, how big is this parent sand?  We’ve stumbled into a kind of reverse paradox of the heap.  A loaf of bread readily admits slices.  A _________ of sand admits grains.  Tough to fill in the blank.

An Interesting Competition

What happens when these two notions are pitted against each other?  Which one wins out in our brains and on our tongues?  How fortunate for my blog that the Miami Heat are currently playing in the NBA Eastern Conference Finals, and that the Miami Heat are one of the few professional sports teams with a singular name (as an fun exercise, try to list all the others—major sports only, no AA curling or anything).  Did you hear what I just typed?  “The Miami Heat are one of…”  But wait, that’s nuts!  I’ve never in my life heard anybody complain that the humidity are unbearable, so why should the capital-H Heat be any different?  Heat is an abstract and amorphous thing.  In our everyday usage, it’s definitely a measurable substance—like water, a singular.  But it’s also a unit composed of people.  And when I tell you about the current state of the NBA, I tell you that the Heat are in the Eastern Conference finals.

Our bias against making conglomerations out of people is so strong that it can overcome our natural tendency to treat both composed units and measurable substances as singular.  We hold ourselves in such high regard that we’re willing to regularly construct borderline nonsensical phrases to maintain our artificially inflated position.

Go, Pack!

Greek to Me

I recently had a chance to do one of my favorite (and my students’ least favorite) things: talk about words in math class.  Math words.  I also had the opportunity to use one of my favorite math-teacher-type resources: a dictionary.  I don’t mean the glossary out of a math book, or a page from Wolfram MathWorld, or any one of the approximately 10.5 million web results (as of this writing) that the Google spits out when prompted with “math” + “dictionary.”  I’m talking about a good, old fashioned English dictionary, one of three left in my room by the previous English-teaching occupant: Webster’s Ninth New Collegiate, circa 1989.

noe.2.thynges

Thanks to the book Group Theory in the Bedroom, by Brian Hayes, I finally found someone to blame for my geometry students’ daily growing hatred of mathematical language.  His name is Robert Recorde.  For those of you who, like pre-this-week me, have never heard of Robert Recorde, don’t worry: you’ve seen his handiwork.  Recorde was a 16th-century Welsh doctor, mathematician, and author of Whetstone of Witte (1557), the book in which the modern equals sign first appears.