All right, I was never very good at math. But I do have a pretty good sense of timing and proportion. When I set my alarm clock at night, I tend to wake up one minute before it goes off in the morning. I also have a good sense of physical spacing and, when I was working with layouts and typography, I could spot a misalignment on the order of one point—about equal to a millimeter.1 In high school, I was good at geometry, which involves a mixture of spatial sense and deductive logic. But algebra and its equations were mostly a puzzle for me, and quadratic equations were my downfall.
Still, I can appreciate mathematics. And I understand that the difference between our lives today and the world and its technologies in ancient and medieval times has a lot to do with applications of trigonometry, analytical geometry, and calculus. But sometimes I think our popular notions of logical proof and certainty go overboard when it comes to mathematics.2
First, mathematics is a human language. But it is unlike other human languages, which can be innately fuzzy. Spoken language is always changing: evolving its pronunciations, such as the English vowel shift; contracting long words into shorter ones by dropping vowels, consonants, and whole syllables; trying to group word forms together to capture new ideas—the Germans are good at this—before chopping them down again; and tinkering with a word’s basic meaning and sense.3 By comparison, the unspoken and more academically based language of mathematics is precise and relatively unchanging.
But math is still just a language, useful for describing things and stating ideas. It helps to state an object’s velocity as so many miles per hour or feet per second, rather than just saying “pretty fast” or “very fast.” Mathematical equations are used to describe motions, transformations, and the actions of large groups, and so they are suited to express thoughts in physics and chemistry. More recently, mathematics has come to dominate certain biological studies such as genetics, cytology, and epidemiology. Most recently, mathematics in the form of numerical models—which are simply a series of equations joined together and set in motion over some arbitrary time scale—have come to describe and predict the action of complex processes such as macroeconomics and weather.
As a language, mathematics is merely the expression of an underlying human thought. The language of mathematics may accurately describe a situation that would be hard to express accurately and unequivocally in English or any other spoken language. It might be harder still to hold that thought immobile in any spoken language over a span of several generations, given the tendency of languages to evolve. Anyone who has grappled with the word choices and shades of meaning in Shakespeare or the King James Version knows this.
But expressing a thought or a relationship or a transformation in the language of mathematics does not prove the truth of the underlying idea. Languages—including mathematics—can express and describe, they can invoke and use logic. But by themselves description and logic are not a proof of anything.
Second, mathematics and its underlying logic may be misapplied. It is possible to make mathematical statements that may be arithmetically and perhaps even logically true, but that do not describe anything in the real world.
I can write a series of logical syllogisms or propositions about black swans and white swans that may be rational and lucid and have nothing to do with real birds. I can describe my trip to Los Angeles, not in so many miles, but in gas stops and bathroom breaks, or in the number of Starbucks® Frappucinos® consumed between them. I can then perform complex transformations with these coffee drinks, but the computations still won’t get you anywhere. Lewis Carroll had exquisite fun with these kinds of cogitations in Alice’s Adventures in Wonderland and Through the Looking-Glass.
A great deal of modern physics—from the calculations underlying fusion experiments to the mathematical models underlying much of quantum mechanics, string theory, and cosmology—smacks of this sort of dealing. Schrödinger can describe the state of his cat in a box with a wave function showing the cat to be both alive and dead until the box is opened and the wave resolved. It’s a perfect metaphor—a language equivalent—for the unresolved and unknown state of a particle traveling through space, which can only be resolved and known by physical observation. But for anything so large and complex as a cat in a box with bottle of cyanide, applying equations is ludicrous. The cat is either alive or dead—and in the former case, it knows itself to be so. The underlying idea of a cat actually suspended between life and death, although elegantly expressed in mathematics, is nonsense.
Similarly, in string theory, the unification of the four basic forces of physics requires that reality in its smallest measures—too small to be noticed in everyday life or detected directly by observation—be composed of multiple dimensions wrapped around themselves. It’s an elegant proposition and mathematically impeccable, but it’s also unprovable by physical experiment. The underlying idea, expressed in mathematics, leads nowhere.
Third, the application of a mathematical idea may not in itself be complete. A system of equations that tries to express very complex systems, like weather or currency movements or animal migrations, must make choices involving the number of variables to include and how to interpret and weight their actions and effects. The main choice is always between completeness—and so an increasing complexity that may approach an infinity of variables—and manageability and comprehensibility.
Here I am reminded of another type of modeling: the choice that a war game designer must make between playability and producing realistic feeling and results. Simple games with few exceptions and die-roll modifiers are the most playable, but they can become so abstract that one loses the feeling of battle and gets all-too-predictable results. Complex games that include many calculations for actual battle conditions—unit morale, lines of communications, visibility and cover, state of supply, weapons calibers and penetration values, windage, time of day, etc., etc.—are elegant and yet unplayable. A three-minute battle turn can take half an hour or more to play. But, in the other direction, consider chess, with its simple rule set, as a model of diplomacy and warfare: the play becomes so abstract that it no longer feels like either diplomatic maneuvering or war.
Fourth and finally, math itself is still a human creation. Some of its principles—the order of the integers, the proportions of the fractions, and oddities like pi and the golden ratio—would seem to be ordained by a non-human hand and descended from nature. But other principles, used in everyday equations, are the invention of human minds working against human-imposed limitations. The invention of imaginary numbers—for example, the square root of a negative number, where arithmetic defines multiplication of two negatives to always have a positive result—is an attempt to climb over this limitation. Or to identify a proportion that cannot be expressed as a ratio of two whole numbers, like pi or the square root of two, as irrational is just a species of fussy bookkeeping. The human mind wants the world to be flat and all fractions to express themselves simply—but it isn’t, and they don’t.
In the same way, the application of mathematics—especially when it comes to creating equations that map and study problems in economics and meteorology—is still a human activity. In that way, math is an expression of human ideas—not much different from creating a story or myth where all the parts come together neatly to satisfy the human psyche. But how many economic models have accurately predicted the state of the stock market or the national money supply at the end of the next quarter? How many weather models—absent satellite observations that can be measured and tracked on a map—can tell you exactly when the rain will start falling?
While mathematics may follow fixed, discoverable, and rational rules, its applications do not. Human beings can still fool themselves and tell half-truths and whole lies in any language.
“My equations show”—
be they ever so precise—
does not make it so.
1. And I’m a demon—practically OCD—about straightening pictures, rugs, placemats, and other things that can go crooked. I believe this comes of having an landscape architect for a mother and a mechanical engineer for a father. They ran a ruler-straight home with everything spaced just so.
3. By “sense” I mean the basic nature of the word. A word can go from ameliorative (tending to praise or find the good in something) to pejorative (tending to fault or demean) over time. An example is our English word “nice,” which used to mean “simple” and now means “pleasing” or “charming.” It can also mean “precise” or “perfect,” as in “That was a nice golf shot.” When my family moved to New England, I learned that “wicked,” rather than meaning “evil,” could be used to mean “extremely” or “cool,” as in “That was a wicked fast ball.” And so, of course, a wicked fast ball can also be a nice fast ball.