Math: It doesn’t add up

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by Richard Nilsen
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How many sides does a triangle have? Don’t be too quick; it’s a trick question.
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Usually, math is not thought of as something where you can have opinions over answers. It’s one of math’s most reassuring qualities.
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Too often, we take what we hear at face value. Facts turn out not to be facts. No one changed your family’s name at Ellis Island. Didn’t happen.
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These are not just myths, they are just things that sound like they could be true and so become embedded in our midden of common knowledge. No, Eskimos do not have 30 or 43, or 90 words for “snow.” Human beings do not use merely 10 percent of their brains. A triangle has three sides. This is all stuff for the Cliff Clavins of the world.
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Sometimes this stuff gets caught in our mental wheel spokes because we simply don’t look closely enough. Take the Fibonacci series. We are told that this interesting pattern of numbers governs much of what appears in nature, including the spiral patterns we see everywhere from whelk shells to spiral galaxies. The problem is, observation does not support this idea, at least not as it is usually presented.
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The series is created by starting with a zero and a 1 and adding them together, and continues by adding each new number with the previous, making the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. The series has many interesting properties, one of which is the generation of the so-called “Golden Section.”
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To the Greeks, the golden section was the ratio ”AB is to BC as BC is to AC.” It also generates the Fibonacci series and is said to define how nature makes spirals.
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Look at the end of a whelk shell, they say, or the longitudinal section of a nautilus shell, and you will see the Fibonacci series in action.
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Yet it is not actually true. When you look at whelks, you find spirals and the Fibonacci series creates a spiral, but the two spirals are quite different:
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The mathematical spiral opens up much more rapidly. The shellfish has a tighter coil. The whelk’s spiral makes roughly two turns for every turn the Fibonacci spiral makes. Math is precise, but nature is various.
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What I am most interested in here is not just the agon of conflicting beliefs, but rather the faith in mathematics, and the sense that math describes, or rather, underpins the organization of the world.
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I cannot help thinking, in contrast, that these patterns are something not so much inherent in Creation, as cast out from our brains like a fishing net over the many fish in the universe.
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Take any large string of events, items or tendencies, and the brain will organize them and throw a story around them, creating order even where none exists.
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Consider the night sky, for instance, a rattling jostle of burning pinpoints. We find in that chaos the images of bears and serpents, lions and bulls. Even those who no longer can find the shape of a great bear can spot the Big Dipper. The outline seems drawn in the sky with stars, yet the constellations have no actual existence outside the order-creating human mind.
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Our own lives — which are a complex tangle of events, conflicting emotions and motives — are too prodigal to fit into a single coherent narrative, even the size of a Russian novel. Yet we do so all the time, creating a sense of self as if we were writing autobiographies and giving our lives a narrative shape that makes them meaningful to us.
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We usually believe the narrative version of our lives actually exists. Yet all of us could write an entirely different story by stringing events together with a different emphasis.
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The question always arises: Are the patterns actually there in life and nature, or do we create them in our heads and cast them like a net over reality?
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The issue is central to a brilliant movie made in 1998 by filmmaker Darren Aronofsky called Pi. In the film, a misfit math genius is searching for the mathematical organizing principle of the cosmos.
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His working hypotheses are simple:
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“One: Mathematics is the language of nature.
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“Two: Everything around us can be represented and understood through numbers.
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“Three: If you graph the numbers of any system, patterns emerge.
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“Therefore: There are patterns everywhere in nature.”
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The movie’s protagonist nearly drives himself nuts with his search until he cannot bear his own obsession anymore.
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But the film also questions in a roundabout way whether the patterns exist or not.
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In the film, when different number series — each 216 digits long — seem to be important, an older colleague warns our hero that, once you begin looking for a pattern, it seems to be everywhere.
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It’s like when you buy a yellow Volkswagen and suddenly every other car on the road is immediately a yellow VW. Nothing has changed but your perception.
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Mathematicians find patterns in nature, yet math itself is purely self-referential. It can only describe itself.
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As mathematician/philosopher Bertrand Russell put it: “Mathematics may be defined as the subject in which we never know what we are talking about nor whether what we are saying is true.”
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In other words, “one plus one equals two” is no different from saying “a whale is not a fish.” You have only spoken within a closed system. “A whale is not a fish” tells us nothing about whales but a lot about our language.
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It is a description of linguistic categories, rather less an observational statement about existence. Biology can be organized as a system of knowledge to make the sentence false — indeed, at other times in history a whale was a fish.
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Before Carl Linne, who created the modern biological nomenclatural system, there were many ways of organizing biology. In his popular History of the Earth and Animated Nature, from 1774 and reprinted well into the 19th century, Oliver Goldsmith divided the fish into “spinous fishes,” “cartilaginous fishes,” “testaceous and crustaceous fishes” and “cetaceous fishes.” A mackerel, a sand dollar and Moby Dick were all kinds of fish.
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Let’s face it, although the Linnaean system is useful, it is kind of arbitrary to organize nature not by its shapes, or where it lives, but rather how it gives birth or breathes.
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“One plus one” likewise describes the system in which the equation is true. It is only a tautology. Real knowledge is metaphorical, hence, “artists’ math.”
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Artists have a different way of counting, of doing arithmetic and of contemplating geometry. It’s what makes them artists.
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For an artist, one plus one equals three. It is a very clear formula: There is the one thing, the other thing, and the two together — a knife, a fork and a place setting. Three things.
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And a triangle has five sides. There are the normal three, and then the front and back. You can turn any triangle over from its back and lay it on its belly. Cut a triangle from a piece of paper and hold it in your hand. Your thumb is on one side of the triangle and your index finger on the other. Add’em up: Five.
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Computer programmers talk about fuzzy logic as if they discovered it.
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It is artists who wake up each morning in a Gaussian blur, after all. It is artists who first understood that all numbers are irrational numbers.
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The primary difference between a mathematician’s logic and an artist’s is that the artist is unable to leave the world behind: The mathematician, the logician, the philosopher deal in abstractions; the artist deals in plums.
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The artist lives in a world of things. Real things: palpable, noisy, smelly, difficult and beautiful. He mistrusts any answers not rooted in them.
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The sentimental view of artists has them constructing “castles in the sky,” but the artist scratches his head over this, because to him, it is math and philosophy that are constructed out of thin air.
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“No ideas but in things,” wrote poet William Carlos Williams. Like the plums that were so cold and so delicious in his poem.
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Don’t get me wrong: One should not dismiss the practical world out of hand. It is good to know how to balance a checkbook, and artists’ math does not carry much clout with the bank.
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An artist is likely to use something called “gut mathematics.”
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The artist knows, as a banker usually doesn’t, that the shortest distance between two points is a leap of the imagination.
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She also knows that three is more interesting than four. It just is. Ask any artist.
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And when an artist talks of pie charts, she wants to know if it is cherry or lemon meringue.
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Like the old math gag: “Pi R square.” “No, pie R round, cake R square.”
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It’s fun to joke about artists’ idiosyncrasies, but there is a serious side to all this.
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When we see yammering faces on TV shouting each other down over ideology, the artist is the one who can remind us that the world isn’t made up of theory or system, but is made up of hubcaps and clamshells. Ideology means very little to an asparagus.
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The world falls into peril every time a system denies physical reality. It is abstractions, after all, that fueled the Cold War, abstractions that justify suicide bombing; it was theory that built Auschwitz.
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Artists remind us of flannel, of smoke, of mud. These are the things we share with our family and our friends. These are the things that ultimately count.
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No ideas but in things.
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It reminds me of a line written by the poet Tom Brown:
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Richard Nilsen inspired many ideas and memories at the salons he presented through the years when he was an arts critic and movie, travel, and features writer at The Arizona Republic.   A few years ago, Richard moved to North Carolina.   We want to continue our connection with Richard and have asked him to be a regular contributor to the Spirit of the Senses Journal.   We asked Richard to write short essays that were inspired by the salons.

1 comment
  1. Thank you for connection – a very engaging and interesting piece of writing – Kindest Wendy

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