Numbers…How Do They Work!?


As usual, I was sitting here drinking coffee and thinking about things this morning…this morning oatmeal was also involved.  When I just sit and stare, I find all sorts of strange things go on in my mind, noise in the internal dialog.  These things take many forms, but they are usually predictable.  One thing that often happens is that I begin to tap out the drum cadence from when I was in marching band in high school; to any of my friends who read this that were also in the band, I assume you do as well.  Sometimes, I find myself staring at objects and internally “highlighting” geometric patterns, like finding Tetris blocks in floor tiles and such.  Other times, I find myself comparing background sounds or smells to completely unrelated things, like how this bird next door makes a two-pitch call that is the exact same sound that repeats over and over again in “Feel Good, Inc.” by The Gorillaz or how the eggs that my wife just cooked smell like a combination of anise seed and vinyl gloves.  It always strikes me as interesting what the brain does when you stop paying attention, the seeming randomness that “bubbles up”, as it were.

Today, I found myself counting.  Counting in French, no less.  This happens a lot, for some reason, like I’m unconsciously reviewing for a test.  I’ll be vacuuming the front room and then I just start counting internally…un, deux, trios, quatre, cinq, six, sept, huit, neuf, dix…no idea why.  It’s never in English, it’s always in French, or any other language I can think of numbers in…eins, zwei, drei, vier, fünf, sechs, sieben, acht, neun, zehn.

Any time the French counting ensues, and I become aware of it, I find myself going through a series of questions in my head.  First, why are the “numbers” in French different than in English?  Next, why is arithmetic apparently built into French and not into English?  Next, what does this say about how our brains work?  Finally, what does this say about how the universe works to make our brains function as such.  Now that I have a nifty blog, I decided to actually write this series of events down.

Why are the numbers different?

A lot can be said about different numbering systems and why we have them.  Of course, we now count in base 10, that is each digit in a number that we write down expresses some power of 10.  Why do we do this?  Probably because we have ten fingers and we use them to count.  A lot of languages have unique words for one through ten and then riff on those or combine them in some way to make the rest of the numbers, at least up to 100.  There are some interesting questions along the way.  For instance, in English, what makes eleven and twelve so cool that they get their own words, unlike 13 through 19?  In French, we have onze (11), douze (12), treize (13), quatorze (14), quinze (15), and seize (16) before we get to the “teens” like 17, 18 and 19…dix-sept, dix-huit, and dix-nuef.  Wat?

Base 10, of course, isn’t the only option for counting.  If we go with the idea that we use base 10 because we have ten fingers, why not use our toes as well, right?  Let’s count in groups of twenty instead of ten, something called the vigesimal system (rather than the decimal system).  Indeed, some cultures have developed that system, notably the Maya, the Aztec, and several African cultures.  The Mayan Calander is based on blocks of 20 and multiples thereof.

Another one that pops up is the sexagesimal system, base-60.  It originated way back in the days of ancient Sumeria and Babylonia, apparently motivated by economic trade.  The (perhaps apocryphal) story goes that Babylon chose 60 as the base unit of its currency because all of the surrounding city-states had currency that were in units that could be easily divided into 60 (which is divisible by 2, 3, 4, 5, 10, 12, 15, 20, 30, and 60).   The Egyptians used it a lot as well.  There’s not much left of that system today, except for the fact that the Babylonian calendar was situated on a circle (upon which the Sun travelled) that was divided up into 360 days, which is why there are 360° in a circle.  In addition, we still measure time in the sexagesimal system, with 60 seconds in a minute and 60 minutes in an hour.

Something that linguists have pointed out is that a lot of European languages have a vigesimal system above a certain point in counting.  On theory is that the Basque culture, which used a vigesimal system, imprinted their counting technique on Europe, which was then passed around by the Normans.  In French, for instance, the number 20 is vingt, the number 80 is quatre-vingts, literally “four twenties”.  In Danish, the vigesimal system is used for numbers between 50 and 99; tresindstyve is the Danish word for 60, which means “three times twenty”.  This brings me to my second point…

Why do some languages have arithmetic built in?

So, why “four twenties” of “three times twenty”?  Of course, the English “eighty” implies some sort of arithmetical difference from “eight”, but it is not explicit in the language.  In French, 21 is given by “vingt et un”, which is “twenty and one”.  In Danish, you have “enogtyve”, which is “one and twenty”.  In English, we say “twenty one”, but in French and Danish, the “and” is explicit.  Here, the word “and” is synonymous with “plus”.  In fact, the plus sign, +, is derived from a bastardization of the Latin “et”, which means “and” , probably from errors in transcriptions in the days before printing presses.  So, French and Danish have arithmetic built in to their numbers in a way that English does not.  Also, it’s interesting that English is Germanic, as is Danish, but the order of twenty and one that we use is the same as the French.  Those damn Normans and their invasion.

As I mentioned before, the French word for 80, “quatre-vingts”, implies the process of multiplication.  Danish is even more ridiculous than French in that their language also has fractions built in.  The Danish number for 50, halvtredsindstyve, means “one-half third times twenty”.  Here, the “3rd one-half” is 2½ (the “1st” is ½ and the “2nd” is 1½).  And, indeed, 2½ times 20 is 50.  How is that convenient?

So, what is the point of this so far?  Cardinality, the mental concept of number is universal; the “fiveness” of something is understood by everyone.  Numerality, the way we express cardinality using language, is most-definitely NOT universal and is, apparently, completely ridiculous.  Hell, even in English, a wise man once said “Four score and seven years ago…” instead of “eighty seven”.  Also, when do we hyphenate the number and when do we not?  Language is just stupid…

What can be learn about the brain from the way language expresses numbers?

So, we all count differently.  Who cares?  What I find interesting is what that difference implies about the way we learn things, in this case mathematics, and how we internalize data.

Language is a manifestation of the physiology of the brain and is restricted by the way the brain can interpret and understand input.  The psycologist, cognitive scientist, and linguist Steven Pinker, in a book called “The Stuff of Thought” (which is amazing and should be read by all), gives an interesting example in the way that children learn certain verbs.  He points out a certain class of verbs, known as object-locative and container-locative verbs, and how use then.

Say you have a process which involves putting an object into another object, such as water into a glass.  There are two ways you can structure this sentence and get the same point across.  You can say “I pour water into the glass”, in which case the word “pour” is object-locative because it acts on the object, water, which is being put into a container, the glass.  We can, however, also say “I fill the glass with water”, where “fill” is container-locative, as it acts on “the glass”, into which “water” is being put.

However, take a similar process, loading a truck with boxes.  One could say “I load the truck with boxes”, but one can also say “I load boxes into the truck”.  In this case, the word “load” can be used both ways.  It is both container and object-locative.  However, one would never say “I fill water into the glass” or “I pour the glass with water”.  It seems that “pour” and “fill” are one-way.

What’s more interesting, as Pinker points out, is that a child learning language will never make that mistake.  A toddler will never say, “Daddy, pour my glass with water!”  They screw up words, for sure, but that structure is always preserved.  Why?

Well, the process of loading and unloading a truck is reversible, presumably due to the nature of the objects being loaded and unloaded: they are solid.  Pouring and filling, however, are usually reserved for operations involving a liquid.  There is a certain implied irreversibility in this process that is not present in the load/unload sense.  The fact that I can even say load/unload implies that the word is different than pour and fill; what sense can we make of fill/unfill and pour/unpour.  Whatever pouring and filling is, you can’t “un” them, you have to do something else.  You fill/pour and then empty.

In addition, this idea applies to any language.  In French, “I load books into the truck” is “Je charge des livres dans le camion”, but “I load the truck with books” is “Je charge le camion avec des livres”.  Indeed, “to load” is “charger” and to unload is “décharger”.  There is not a word for unfill or unpour.

This all implies that the words pour, fill, and load are tied to physical processes in the world that we describe with language that is restricted by the way the process works and the way we gather the information.  Neat.

Now, when we consider the way we use language to describe number, what does that say about the way we work?  Is there something universal in the world about the idea of counting?  Number itself is universal, but is counting?  It doesn’t look like it for the ridiculous variations in the way we count.  But, does the way we count say something about our brains?  Are some people predisposed to, say, mathematics, simply because they have a language that more deeply ingrains the concept than others?  If I had learned French as my native tongue instead of English, would I be better at arithmetic now because I would have basically had to have learned arithmetic to count?  Who knows.  I’ve looked around and read some interesting papers from linguistics, mathematics, and neuroscience, but I can’t really find anything substantial.

Humans seem to have ease with counting to about 4.  Once you get past that, however, we have difficulty remembering individual pieces of information.  We start to do a process referred as “chunking”, we assemble the pieces of information into bigger pieces and then remember those.  Since our number systems have roots in practical application, perhaps some of the crazy features come from collecting things together to make it easier to remember?  Maybe we say “four-twenties” in French simply because we don’t have to remember another word for 80?  In Swiss French, however, the word “huitante” is used for 80, so who knows.

Like I said, language is stupid…

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