Math About Math is Like Art About Art

by Asa Boxer

Achilles and the Tortoise or Zeno’s Paradoxes. Artist: Martin Grandjean.

One of the reasons descriptive arithmetic and mathematics are so compelling in terms of their truthiness is that they participate in archetypal structures. As I’ve pointed out elsewhere, archetypes persist despite changing circumstances. Indeed, archetypes easily find new skins; they put on the costumes of the age. They take to cultural shifts so naturally, it’s as though they took pleasure in the exercise. They find renewal and rejuvenation by adopting the language and attitudes of each generation. Consider modern adaptations of Shakespeare. They work because the archetypal structures are solid. Times change, but archetypes persist. Same with math. . . There are, however, some differences worthy of our attention. Let’s take a look.

The underlying notion of an equation, the part that recommends it is how it represents unchanging relationships. Typically, mathematics uses letters that stand in for whatever variable quantities might apply. This standing-in-for quality is similar to simile and metaphor, where an image stands for a concept or emotion. Referring back to “How Dark Physics Fails,” I’ll quote poet Robbie Burns’s (1759-1796) famous, “My love is like a red red rose.” In this example, the red red rose stands in for love. If we were to look at this equation mathematically, the important parts would be the syntax. A simile would be expressed as y ≤ x ≤ y. Meanwhile, a metaphor would read x = y.

These formulae can be more complex where, say, x = y₁ + y₂. For example, here’s a line from Leonard Cohen’s “The Flowers that I Left in the Ground”: “With your beauty I am as uninvolved / as with horses’ manes and waterfalls”; x = beauty, y₁ = horses’ manes, and y₂ = waterfalls. No doubt, using this formula, one may generate countless variations. Indeed, plenty of writers play at this method to generate poems that appear original. Those familiar with the original, however, will judge such writing “derivative” unless the borrower does something remarkable. My poetry mentor used to call this sort of writing “borrowed magic.” Among artists there’s an expectation that one discover new and surprising formulations. That said, there is value to applying this sort of reductive notation to structural elements, as it allows practitioners to apply similar techniques.

As you can imagine, all manner of logic can be reduced to mathematical formulations borrowed from set theory, functions, calculus, and so forth. The thing is. . .  math proves unproductive when applied to artistic content. An argument can be made that plastic arts involve math for perspective, the golden mean, colour mixing, the stabilisation of sculptures, and so on. But these elements are the purview of the maker and are ultimately beside the point because audiences don’t value art by those measures. Meanwhile, the reduction of poetic statements to formulae, far from providing insight, entirely denudes them of the qualities that make poems work; it strips away their beauty, and leaves us with a whole lot of form and too little content.

If anyone remembers Chemistry 101, they’ll recall P₁V₁/T₁ = P₂V₂/T₂, derived from Boyle and Gay-Lussac’s so-called “laws” regarding the behaviour of gasses. The reason we call them “laws” is because they represent relationships that do not change: they therefore represent an underlying order. What the above equation expresses is that the relationship between pressure, volume, and temperature remain constant—a change to one of these factors induces a change to the others in a proportionate manner.

(I have elsewhere suggested that we would be wise to take a step back from the rigidity implicit to the term “law” because scientific laws represent artificially framed conditions. In other words, the formulae are not reality; instead, they describe manipulations within limited frames. Similarly, literary archetypes draw stick-figures and exaggerated types rather than persons we find in the real world or even in literary works. I therefore think “guides” would be more accurate.)

The rules (or laws) of algebra introduce further excitement because they allow us to “solve for” any given variable. This solving property of equations is owing to basic relationships inherent to mathematics. Consequently, the archetypal power of math functions at two levels—(a) the description of the phenomena (the formulae), and (b) internally to the math (how formulae are solved). Recalling “How Dark Physics Fails,” the reliability of math is therefore predicated on both its correspondence and its coherence. The beauty here too is that it doesn’t matter which heuristics one uses to measure pressure, volume, and temperature, so long as you’re consistent with the measures. The relationship encapsulated by a given equation transcends such arbitrary devices.

Fountain 1917 Marcel Duchamp (1887-1968) Photo Credit: Alfred Stieglitz. Fountainis a piece that might be credited with anticipating Conceptual Art. Marcel Duchamp here draws attention to the artistic value in something as mundane as a common urinal. Awakening audiences to what might be called "found beauty" originally sparked the ideas that led to Postmodern Art, where the idea or concept motivating the work supersedes and eventually supplants preoccupations with aesthetic value. Note how the viewer is expected to imbue the work with value and meaning. Here the artist steps away and asks you to half-make the art. This emphasis on subjectivity is a hallmark of the Postmodern.

In previous articles—one on probability and stats and another on Einstein and Feynman—I commented on math about math that perhaps has internal coherence, but at best only hypothetical correspondence, at worst, none whatsoever. This sort of trouble affects literature and the arts more generally as well when we get art about art in place of art about life. I have in mind artefacts like Marcel Duchamp’s urinal put on display in 1917 as Fountain, and Erased de Kooning Drawing by Robert Rauschenberg from 1958, both of which I wrote about here. When we put ourselves at two removes from reality, we likely wind up with runaway fictions. I encourage these fictions, mind you; what I’m saying is that, when it comes to science, things like dark stuff, black holes, and various quantum speculations ought to be treated as fun science fictions rather than as doctrines.

Erased de Kooning Drawing 1958 Robert Rauschenberg (1925-2008). Rauschenberg's Erased de Kooning Drawing typifies Conceptual Art proper wherein meta-theatre lies at the heart of the work, for it is not only the tale of the two months and countless spent erasers that it took to undo de Kooning's art that lends the piece meaning, but also the tale of de Kooning's cooperation with Rauschenberg, generously providing a canvas depicting "something" he'd "miss"--a drawing he was proud of. With this work, Postmodern Art turns its back entirely on the lay audience, courting instead academic theoreticians, and effectively establishing an exclusive bourgeois in-group of art connoisseurs.

There’s more. You may have noticed that equations function across this symbol: = . . . hence, “equation.” When we use the = device, we’re making a statement of equivalence, essentially an analogy. As with linguistic analogies, especially with more developed literary conceits, analogies can be tricky. Part of the problem is that arithmetic itself isn’t necessarily true. To take a surprising example, biologically and reproductively speaking, 1 + 1 = not just 2 but {3, 4, 5. . . ~15}. In other words, the schema of arithmetic can be too abstract to apply universally. These are issues one ought to keep in mind when considering math as somehow constant and inviolable and as existing in a transcendent state—as something discovered rather than invented. For math to work, it must remain rigorously descriptive of a phenomenon.

Perhaps the most perplexing characteristic of our measures is that since our metrics are arbitrary, any quantity may be reduced to 1. I mean we can decide to make 3 cm or 70¹⁰⁰ km = 1 unit. As I’ve noted before, our quantising instruments have measured the outcomes of our experiments and established the objective reality of a quantised universe. I mean, we’re obsessed with measurements and hence with particles. The methods directly inform the observations.

This trouble with infinite and arbitrary divisibility shows up a side of math that we would do well to take seriously. When Zeno of Elea (c. 490–430 BC) presented the Achilles paradox, he put his finger on the way math can dream up non-existent realities. For those unfamiliar, the proposition is as follows: in a race between Achilles and a tortoise, Achilles gives the tortoise a head start. When the tortoise arrives at some given distance, Achilles enters the race. The problem is that in order for Achilles to reach the tortoise, he must first reach the head-start mark, and by then, the tortoise will have advanced. And by the time Achilles reaches that new point, the tortoise will have advanced again, and so on ad infinitum. As a result, Achilles can never catch up with the tortoise. Of course, this isn’t true of reality, but you might imagine it were true or at least plausible if let’s say you were blind or had been kept immobile in a room your whole life. Thus, math can generate the sort of hallucinations we’ve been seeing in AI.

If you recall, I addressed the many-worlds hypothesis in “The Flaws of Probability,” pointing out how the idea of there being as many universes as there are choices is a mirage of the probability math, and a very unlikely reality. This is another example of ideas based on math instead of on the world before us.

Getting back to our archetypes. . . It’s not only mathematical equations that are like archetypes but also mathematical constants. I like pointing out how incommensurable numbers (AKA irrational numbers) like π, φ, and ψ indicate a problem with our notation system, since our math language can only ever approximate them. Meanwhile, they represent fundamental geometrical properties, and are thus in many ways more real than our abstract number system. π, φ, and ψ represent archetypal ratios (relationships or underlying forces) that direct the unfolding of most natural phenomena: circles, spheres, distributions, spirals, cones, triangles, pentagrams, magnetic flux, electric flux, and so forth.

The obvious difference between the literary-psychological archetype and the mathematical one, is that they each refer to a different world: one to the inner world and the other to the outer. Let’s recall that we’re living in an age that believes we can measure happiness. . . worse, that we can obviate our inner worlds altogether, since it’s all just an accident of neurological chemistry. As we saw, however, math isn’t especially productive when it comes to understanding figurative language and art. It is equally inept at providing useful data regarding the psychological world. Consequently, those who believe that all phenomena must be explainable in terms of energy and matter (naturalists or, if you like, naive materialists) would prefer to explain away those phenomena that inconvenience their model.

Archetypes are the language of wisdom. I say “wisdom” rather than “intelligence” or “truth” because we tend to think of wisdom as ways of seeing the world that remain relevant amidst the variables. They are the features of our universe that hold firm, while everything else heaves and shifts and changes appearance. As the pre-Socratic philosopher Heraclitus (c. 540-480 BCE) said, “All is flux.” And philosopher of science Henri Bergson (1859-1941) echoed a couple of millennia later, “Everything flows.” Indeed. But there are also moorings, constants, archetypes, elements of order: proportions, relationships, emotions, and compulsions. There are patterns to how things flow. Both the inner and outer worlds have equally hard-wired elements shaping and driving them. And we’d be wise to respect their differences and learn to differentiate between them.

As poet Ted Hughes (1930-1998) put it:

Sharpness, clarity and scope of the mental eye are all-important in our dealings with the outer world, and that is plenty. And if we were machines it would be enough. But the outer world is only one of the worlds we live in. For better or worse we have another, and that is the inner world of our bodies and everything pertaining. It is closer than the outer world, more decisive, and utterly different. So here are two worlds, which we have to live in simultaneously. And because they are intricately interdependent at every moment, we can’t ignore one and concentrate on the other without accidents. Probably fatal accidents.

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Asa Boxer’s poetry has garnered several prizes and is included in various anthologies around the world. His books are The Mechanical Bird (Signal, 2007), Skullduggery (Signal, 2011), Friar Biard’s Primer to the New World (Frog Hollow Press, 2013), Etymologies (Anstruther Press, 2016), Field Notes from the Undead (Interludes Press, 2018) and The Narrow Cabinet: A Zombie Chronicle (Guernica, 2022). Boxer is also the founder and editor of analogy magazine.