How language misleads us, and we don’t even notice
Language begins as sound for almost all animals. Birds communicate through vocalizations, as do most mammals. Additional sounds can be employed, such as the slapping of a whale’s tail on the surface of the ocean or the chest-thumping of a male gorilla. We’re still in the very early stages of trying to understand communication in other species but already it’s clear that the range of concepts that can be expressed is significantly larger than most people imagined even a mere twenty years ago.
Our ability to use sounds to communicate is likely the most well-developed of all the species, though the caveat is we still understand almost nothing about the clicks of dolphins, the song of whales, and the astonishing colors-and-patterns signaling of cuttlefish.
Language gave us our earliest ability to cooperate among ourselves, and we see similar use of vocalizations among our closest primate cousins. We existed for tens of thousands of years on the basis of vocalization alone, passing on what meagre knowledge we’d amassed by means of simple tales handed down from adults to children (in addition, of course, to simply showing a technique so that others could learn by repetition).
Then, for reasons that are still unknown, we began to conceptualize abstractions. Early humans began painting scenes on the walls of caves. Doubtless they also drew pictures on the ground using pointed sticks, but naturally none of those scratchings has survived down the millennia. At some point we also began shaping clay to represent fertile females and priapic males.
Eventually we invented the first rudiments of persistent communication: shapes and symbols scrawled on stone or pressed into clay tablets. These innovations came very late in the day and seem to have arisen from the need to keep records in what were increasingly large and sophisticated societies. These societies had grown up in consequence of a change in the DNA of certain grass species that in turn enabled humans to begin practicing agriculture at scale. Clans turned into tribes and tribes turned into towns, which eventually grew into cities containing thousands of individuals. In such conditions, verbal records were woefully insufficient.
As best as we can tell from what remains, early records were a mixture of both speech-based and numerically-based information. It’s not difficult to understand why this should be so. In a city it was necessary to know that Joe had contracted with Sally to put a tiled roof over her dwelling, and it was also necessary to know that the grain depository was storing 201 jars of grain.
The first records were largely hieroglyphic and used pictograms to represent physical objects in a one-to-one correspondence. Thus a clay tablet showing 23 tiny bales of hay represented 23 actual bales of hay. Later, marks were developed to represent higher numbers, so one mark could count for 12 bales, thus saving time both writing and reading the inscription and minimizing the number of clay tablets required to keep track of things.
This is the stage before real math. All the clay tablet holder had to do was line up the bales and see if s/he could point to a real bale each time s/he saw a representation on the tablet. Real math appears the moment numbers become abstracted from physical items, because then we can begin to manipulate them. Two bales plus five bales; ten bales minus two bales. And then, later still, twenty-four bales divided equally onto two ox-carts; six ox-carts each with eight bales.
From this point onward, language and mathematics began to diverge. And we’ve failed to understand the enormous significance of this divergence.
Language is inherently imprecise. We say things like “Mary is bigger than Susan” and “James was bad yesterday” but these statements are in fact empty because there’s no definition or scale contained within them. They could mean almost anything.
Conversely, even rudimentary mathematics is essentially about precision (yes, even allowing for Gödel’s Incompleteness Theorem). We can say we have 20 bushels of corn and if we want to be even more specific we can state the weight of each bushel in whatever units of measurement happen to be current to our society.
Oddly, people failed to notice this fundamental difference between language and mathematics, and this failure to notice led to a great deal of confusion and wasted intellectual effort.
Poor old Plato, among others, was convinced that because words were abstract there must consequently be a perfect absolute “out there” for which our human words are merely imperfect shadows. Ferdinand de Saussure spent his entire career pointing out the obvious gap between signified and signifier and unfortunately gave birth to a whole field of structuralists and post-structuralists. And because he hadn’t bothered to learn about biology or evolution, Wittgenstein dreamed up his “private language” theory.
Worse yet, religionists of all kinds have persistently assumed that because we humans have in our lexicon simplistic undefined words like “good” and “evil” there must consequently be invisible magical creatures that embody these vague notions. And we all know the horrors that result from believing in nonsense: human sacrifice, persecution, the suppression of knowledge and learning, sexual repression, and near-infinite opportunities for the priestly caste to prey on the gullible.
Meanwhile mathematics continued to evolve so as to be able to deal with greater and greater complexity. Mathematicians don’t burn each other over conflicting definitions of “good” and “evil” but instead look to solve problems using an accepted structure that can generate results that in many cases can be confirmed empirically. Of course there are avenues of higher math that are seemingly dissociated from any practical application but these are surprisingly rare and it may simply be too early for us to assume there are no practical applications.
For the most part, math gives us a view into a world not accessible by language. Although we can use crude analogies to attempt to explain to the layperson how mass warps spacetime, the analogies are very imperfect whereas once we understand vectors and tensors we get a far more complete understanding of the phenomenon being described. Furthermore, we can calculate the effects whereas with analogies made of words we have no means whereby to predict precisely what we ought to see when we make empirical observations.
This capability has revealed to us at the grandest scale the workings of the universe and at the tiniest scale the behavior of sub-atomic components. And the behaviors of forces and particles at sub-atomic scales appears to be entirely contradictory to anything words would have permitted us to discover or describe.
Meanwhile math enables engineers to design heavier-than-air vehicles that reliably take off and fly long distances, to build bridges that don’t collapse when subjected to vehicles crossing them, and skyscrapers than don’t crumble under their own weight. None of this is possible using words.
Math, in essence, takes us beyond the quotidian. Language, in essence, is always rooted in the quotidian and always suffers from vagueness, imprecision, and potentially conflicting interpretations.
Yet for the most part we simply don’t see this obvious fact because we’re so used to using language as our primary mode of communication. This article is written in words, but it would be far more succinct if expressed mathematically. Unfortunately, were the concepts to be expressed symbolically, only a tiny percentage of Medium readers would be equipped to receive the message because few people really internalize math in the way that most internalize language.
It seems to me an astonishing fact that while we introduce children to the riches of language via stories and poems we present math most often in a dry, boring, abstract manner guaranteed to dull anyone’s enthusiasm for what is in reality a domain of extraordinary richness. Thus, because most people know little of math beyond the most elementary calculations, we remain unaware of how clumsy and limited we are when we’re confined to the world of words.
This is not to say that math is “better” than language, because mathematics is a language. But it is a language largely of symbols and precision whereas words form a language largely of vagueness and imprecision. It would be redundant to use math to express “Mary is hungry” or “Joe feels sad” but it is entirely possible; it is impossible however to use words to express (to take but one example) the Einstein field equations:
Language is inherently linear. When we read the poem by Catullus that commences, “Vivamus, mea Lesbia, atque amemus” we must follow it word by word to reach “nox est perpetua una dormienda.” But with an equation we can often see in both directions simultaneously, a phenomenon that we can illustrate with a simple algebraic equation:
In this tiny equation we have near-infinity in both directions: the infinitely large and the infinitely small. When N is very small, 1 approaches infinitely large and when N is very large 1 approaches infinitely small. Using words, I’ve had to write a lengthy sentence; in math I can merely glance at the equation and see it all instantly in my mind, stretching out into the vastness of space on either side.
If we more adequately understood the limitations of language we might be slightly less inclined to let empty words stir us up and make us fearful or angry or credulous. If we understood how words limit us and deceive us we could, perhaps, be slightly less foolish across the entire spectrum of human activity.
What does it really mean when we say “I love you?”
What does it really mean when we say, “I don’t like him?”
For each of the two sentences above there are likely thousands of possible definitions and interpretations. Yet each one of us will believe we know what is meant, and each one of us will be most likely be wrong. Yet if we write down 2+3 = 5 everyone who can perform simple addition in base-10 will know precisely what we mean, without any possible ambiguity.
So if math is the way to achieve greater understanding and precision, why don’t more of us feel comfortable with math?
Let’s imagine if we taught language in the same way we taught math: we’d hand small children a dictionary and a grammar book. We’d set them exercises in conjugating verbs and constructing relative clauses. What we wouldn’t do is interest them in the expressive power and beauty of language. We wouldn’t use language to tell stories. We’d simply bore them to tears with tedious linguistic chores that would ensure they’d want to stay away from reading and writing as much as possible.
So why do we teach math in such a dire manner? Why don’t we help children to discover the beauty and richness and expressiveness of math right from the very beginning? Why don’t we play with math rather than setting them arid sums? There are so many games in math, so many interesting twists and turns.
· How many people do we need in a room in order to have a 50:50 probability that two of those people will share the same birthday? (Answer: only 23 people) Why is this true? Are there two people in our class today who share the same birthday? If not, what does that tell us about probability?
· Should we stay or switch when confronted with the Monty Hall problem? (Answer: switching gives us a 66.6% probability of winning whereas staying gives us only a 33.3% chance). Why is this true? If there was ice-cream behind one of the doors how would we feel about our choice of staying or switching?
· Why should you not agree to pay someone one grain of rice on the first square of a chessboard, two grains on the second square, four on the third, eight on the fourth, and so on? Just imagine if we could trick someone into doing that for us!
When we begin to feel math in the same way we think we feel language, our minds begin to view reality in a more subtle and perceptive way.
Given where we are now, with the mindless follies of populism tearing our fragile civilization apart, I think we could all use a little more subtlety and perceptiveness in our fellow humans.