Monday, March 30, 2020

more math, more games: lumines, abstraction, operators

Lumines Remastered (via The Verge

I've had a lot of time on my hands lately.

During the first week of social distancing, I was hitting the calculus textbook pretty hard; a thirty-six-hour moratorium has been placed on the calculation of antiderivatives in my household. It wouldn't surprise me to learn the standard calculus course's "strategies for integration" unit is the pons asinorum of the college-level math student. It's been kicking my ass. So I'm gonna collect my wits, maybe develop a flowchart (for the sake of testing the elasticity of last session's "game is math" metaphor, we could liken this to drafting a map for an NES- or DOS-era RPG), and come back to it in a few days.

So when I haven't been editing the n-v-l, making myself go on bike rides, or informing myself into a low-key panic attack, I've been playing Lumines Remastered on Steam. I love the game as much as I did its PlayStation 2 iteration, Lumines Plus (in 2008–9, it was my go-to after coming back from smoking spliffs in the woods on hot summer nights), though even now I couldn't clear the board to save my life.

As perhaps you know, Lumines is a brainchild of Tetsuya Mizuguchi, and like his cult-hit rail shooter Rez, Lumines evinces  Mizuguchi's fascination with synaesthesic experience. Its exogenous marriage of block-puzzling and input-synched audiovisual sparklepop make Lumines a rare bird in an expansive aviary of Tetris clones.

Comparing Lumines to Tetris—as I heard myself doing when trying to convince my roommates to stop playing Animal Crossing and get the Switch version—is almost unavoidable. Tetris is the fountainhead of geometry-puzzle video games; without it, Lumines wouldn't exist. Dr. Mario, Puyo PuyoPuzzle Bobble, Super Puzzle Fighter II Turbo, and the rest can each be conveniently summed up as "Tetris, but with [blank]." In light of this ubiquity, it's tempting to attribute a universality to the Tetris experience. There's something almost primordial about its format and function: simple forms composed of four unit-squares in seven different spatial configurations, an evolving problem of optimization, and a time limit imposed by simulated gravity.

In truly excellent instances of minimalist design, the object imparts a sense of inevitability, as though it could not have taken any other form without violating an irrefragable principle of its conception. Tetris is so elegant, so elemental, that it might not be entirely crazy to conjecture a timeline without Alexey Pajitnov eventually producing something a lot like it.

Since its 1984 debut, Tetris has been playable on more platforms than I care to look up and list, many of which were quite basic by modern standards. If the endurance of the tune to "Korobeiniki" in the ears of American millennials is any indication, the lime-and-grey, blooping and crunching Game Boy version was for many years the title's most popular iteration. I first played Tetris on the NES, and an MS-DOS version snuck onto the family PC at one point. Every now and then, I used to see a coin-op version in a diner vestibule or a lonely corner in an arcade. My roommate is currently playing Tetris 99 on the Switch. But no matter what format, Tetris always recognizably performs as Tetris, and can be enjoyed as Tetris.

But it's difficult to imagine Mizuguchi signing off on a version of Lumines with four colors, four sound channels, no stage skins, and two songs on the soundtrack. The project was predicated on the availability of a platform that could interweave spatial reasoning imperatives with vivid, high-definition splashes of color and sound—the intent being, as Mizuguchi puts it, to braid left-brain logic with the emotional content that buzzes the right hemisphere. Tetris, on the other hand, was initially developed and ran on a Soviet-era IBM desktop computer, and most of the changes that came afterward were more aesthetic than functional.

Back in 2012, the Tetris Company sued the developers of Mino (a rank Tetris ripoff for the iPhone), claiming copyright infringement. The defense mounted a fascinating argument, as Ars Technica reported:
In defending Mino, lawyers for Xio Interactive didn't deny that they were heavily influenced by Tetris, copying almost all of the game's basic elements wholesale. The defense's main argument, instead, was that the things it copied—everything from the shape and color of the blocks and the way that they rotate to the dimensions of the playfield—were actually integral to the underlying rules of the game, and therefore not subject to copyright protection. The argument, basically, was that Tetris is so simple and abstract that every part of the game is a basic "rule" that can be legally copied.
Basically, the defense came before the judge and argued that the Tetris Company was behaving like the Joker in the "Laughing Fish" story. ("You can't copyright fish," a patent attorney stammers to the Clown Prince of Crime, "they're a natural resource!") The court ruled in favor of the plaintiffs, though the verdict stipulates that while Tetris is not so simple as to be universal, its basic elements (as the judge interprets them) are.

Interesting stuff.

Hmm. Since we're already talking about simplicity and abstraction, let's return to mathematics for a minute.

Tetris (via First Versions)

It's not unusual to hear or read somebody venturing to declare that mathematical principles represent the ultimate truth of things. The idea is not new: mathematics has been a gateway to metaphysics (and vice versa) since antiquity. The Pythagorean Brotherhood's belief that mathematical entities were the fundamental stuff of the cosmos became the basis of the Platonic doctrine deprecating material entities as vulgar approximation of the perfect, immutable abstractions that were truly real. "Number rules the universe," the Pythagoreans are said to have proclaimed; "God forever geometrizes," wrote Plato. The philosophers of the Islamic world followed Aristotle in ascribing divinity to circular motion and to spherical forms. The Judaic tradition of Kabbalah utilizes a recondite numerology, as do the divination practices of the I Ching. In the centuries after Galileo, Kepler, Newton, et al. demonstrated the coextensive generality of events in nature and of mathematical logic, the conviction that numbers and geometry are a window to the hidden workings of things is no longer as susceptible to the superfluities of symbolism and superstition. It does not strain the modern mind's credulity to entertain the idea that we occupy a kind of holographic existence effectuated by the monumental moment-by-moment computations of a grand cosmic operating system. In this view, our experiences are like the user-interface on a display monitor, while the imperceptible milling interactions of the ones and zeroes in the CPU constitute the real activity. Many typical specimens of this sort of discourse can be found by punching any combination of the words "god" "universe" "math" "operating system," etc. into your favorite search engine.

Even an epiphenomenalist and crypto-nominalist like myself struggles to reckon with the ontic status of mathematical entities.

But let's assume, for now, that mathematical "objects" are easy to account for without recourse to metaphysics, subjective experience, or even abstract concepts.

Some of the confusion surrounding "mathematical existents" and the "pure forms" of geometry must be the result of imprecise language, that immemorial vector for muddled thinking. One example Thomas Hobbes cites is the careless statement "A Living Creature Is Genus." No it isn't: a living creature is a living creature, and a genus is a category. Likewise, I'm sure that at some point in my life I've blithely uttered "the Earth's orbit around the sun is an ellipse." This is careless, and it is incorrect. A less problematic phrasing would be: "the Earth travels around the sun in an elliptical orbit."

Or consider: "a cantaloupe is a sphere." No, it isn't: a cantaloupe is a cantaloupe. A sphere is merely the property of sphericality reified through a grammatical distortion. ("A predilection for things," BF Skinner writes, "sometimes leads to absurd consequences in the search for defining properties.")  A cantaloupe exists. Spherical things exist, but good luck finding anything in the world possessing no attributes whatsoever beyond the precisely defined volume it occupies.

Along similar lines, we might echo David Hume (another damn'd latter-day nominalist) in maintaining that nobody can "conceive a triangle in general, which is neither Isosceles nor Scalenum, nor has any particular length or proportion of sides." Why not?

"Triangle" is the properties necessary and sufficient for us to respond to a real object as a triangular one, spoken of (erroneously) as a static entity. Triangularity is a packet of sense-information, a characteristic of any number of objects; insofar as these objects possess this characteristic, additional attributes may be predicated of each on the basis of their triangularity (but only ones pertaining to triangularity per se). But triangularity in and of itself has no existence beyond the objects in which we observe it. (Whether we're treating triangularity as a property of an enduring object in the world or of an evanescent stimulus-event is here a choice of semantics: we can't engage with one but through the other.) The triangle that comes before my mind's eye when I think "triangle" cannot be conceived without the baggage of experience—of actual triangular objects to which my response "triangle" has been conditioned. It will have to be of a certain color (otherwise I'd be visualizing nothing), and it will have to be either scalene, isosceles, or equilateral: if it's none of the three, I'm not picturing a triangle; if it's any one of them, the idealization lacks the generality to which must be ascribed a real universal or a general conception. (A "general conception" implies that the triangular form I'm envisioning will be identical to the triangular form you're envisioning when we're both asked to think of a triangle, or that when I communicate "triangle" to you, I've transferred to you my visualization of a triangular form.)

Possibly there is an undergraduate out there somewhere who's just taken a few theory courses and wants to tell us that triangularity (and any number of other things) is just a "social construct." Insufferable though he might be (prove me wrong), he's not mistaken. Whether or not there are people to perceive them, triangular objects have existed in the world and will continue to exist. But our responses to them—including the apprehension of them as triangular objects—are products of social conditioning.

We learn to discern and respond to qualities such as triangularity through processes which BF Skinner termed operant discrimination and differential reinforcement. We probably all have some recollection of being taught to identify shapes as young children through a regimen of conditioned general reinforcers (ie, praise and finger-wagging):

"What's this?"


"No, this is a triangle. Tri-an-gle. Say it."


"Now let's try again. What's this?"


"Very good!"

A stimulus pattern (in this case it is primarily visual, but also possibly tactile) thereby gains control over a part of our behavior. Our response to this pattern may be as basic as inconsequential as speaking or thinking the word "triangle;" but in most cases, such responses will be superseded by others unless the property is relevant to our activities.

In the course of our experience, the stimulus properties that gain control over the response "triangle" (and of the simple recognition of triangularity, and of any additional behavior dependent on that recognition) become circumscribed. A child might point to a fleur-de-lis and say "triangle;" an adult is less likely to make that identification because he or she has been corrected when they've used the term promiscuously. The same kind of refinement is observed in the child being taught to distinguish "oval" from "circle," and "cube" from "square."

Through analogous processes, we learn to perceive and respond to quantity. We're taught to count the fingers on our hand: "one, two, three, four, five." We're conditioned to respond in the same way to a discrete property observed in a pair of fingers and a pair of apples, independent of the apples and the fingers themselves. We're walked through the "if you have two apples and I give you two more apples..." exercise. "Four," the non-thumb fingers on our hand, and the apples we have when we begin with two and are given two more become a class of stimuli to which the same responses are arranged. The property common to the apples, the fingers, and the spoken "four" becomes paired with the visual stimulus of a printed "4," which we are rewarded for reproducing when presented with the textual prompt "2+2=__". We are similarly rewarded for responding "four" when asked "what's two plus two?" At length, we come to reinforce ourselves for performing these exercises correctly.

Numbers are, again, reifications of stimulus properties. There is no such thing as "two," though we may be holding more than one and fewer than three apples, pebbles, toenail keychains, pencils, LSD tablets, Prozac capsules, or what have you, and use "two" to call out a trait common to each of these pairs of things (and all other pairs of things).

So what's the confusion?

Isḥāq ibn Ḥunayn's translation of Euclid's Elements (1466 edition)

Against the real universals avouched by medieval scholasticism, the nominalist urged a praxis of reductionism. We don't need to posit a cosmos whose narrative is enacted by the participation of abstract categorical actors: it is enough to say we inhabit a world of absolutely particular substances, to any number of which we can ascribe similar qualities. We don't need a universal entity called "roughness" that confronts us when we run our fingers over sandpaper and stubble: it's enough to say that both substances arouse similar sensations when we touch them. Increasingly, mathematical reasoning was called upon to explain qualities as emergent aspects of quantity. The "gravity" and "levity" of the ancient Greeks' four elements was shown not to caused by the proclivities of material species, but by quantitative differences in density. The investigation of optics unmasked color as a valence, a function of periodicity. It was found that there is no "cold" and "hot," only lesser and greater magnitudes of thermal activity. Rectilinear and circular motion are not separate and incommensurable; an orbital curve is linear motion plus a radial deviation. And so on.

If these simplifications truly served to extirpate all perplexity from our understanding of the world, we probably wouldn't have a twentieth-century physicist and mathematician (Nobel Laureate Eugene Wigner) publishing a tremendous essay with the provocative (but pitch-perfect) title "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."

It begins with an an anecdote:
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle." Naturally, we are inclined to smile about the simplicity of the classmate's approach. 
Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense.
There's something fishy about mathematics: it's a universal busybody. Why should any one certain category of properties of things be so evidently integral to the workings of, well, everything? Ironically, the sloughing of medieval scholasticism's "universal natures" from the discourse of natural philosophy opened a vacancy for mathematical entities to colonize.

We'll let Alfred North Whitehead (emphatically not a nominalist) pick things up from where Wigner's joke leaves off (and provide us laymen with a hint as to why pi is germane to population statistics). From Science and the Modern World:
The science of trigonometry arose from that of the relations of the angles of a right-angled triangle, to the ratios between the sides and hypotenuse of that triangle. Then, under the influence of the newly discovered mathematical science of the analysis of functions, it broadened out into the study of the simple abstract periodic functions which these ratios exemplify. Thus trigonometry became completely abstract; and thus becoming abstract, it became useful. It illuminated the underlying analogy between sets of utterly diverse physical phenomena; and at the same time it supplied the weapons by which any one such set could have its various features analyzed and related to each other. 
Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. The history of the seventeenth century science reads as though it were some vivid dream of Plato or Pythagoras. In this characteristic the seventeenth century was only the forerunner of its successors. 
The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete facts.
This paradox vitiates the confidence of a gainsayer of universal or abstract entities. It's difficult to dismiss mathematical constructs as fallacious reifications when we cannot articulate the behavior of the universe without reference to them. We've said that there is no "sphere" template from which all the round objects of our experience were someways manufactured; but it is somewhat dissatisfying to leave it at that when the cosmos itself seems almost to manifest an interest in producing spherical things. We maintain there's nothing mystical or transcendental about mere quantity—but Euler's number finds us in every scientific discipline in which we care to look for it, atomic behavior is quantized (a fact which a panpsychist like Charles Hartshorne would undoubtedly adduce in claiming "atoms can count!"), and imaginary numbers (abstractions of abstractions!) have practical applications in engineering and physics. It's not just unintuitive: it's inexplicable.

I feel I've entered choppier waters than I can swim, but I've a last thought before I return to shore.

It stands to reason that the development of the mathematical and geometrical disciplines was subject to the exigencies of social need: people were trained to calculate and measure because these activities were useful. The subject matter must have evolved under similar conditions: practices relevant to the performance of the pedagogical agency as an effective social organ were preserved, and irrelevant or interfering ones were discarded. Calculation's immediate use might have been in the levying of taxes or the tracking of inventory stores, but improvements in its utilization depended on advancements in the discipline's theoretical and methodological aspects. The same could be said for pure geometry and its applications in engineering, architecture, and so on. The early use of mathematics in astronomy and the formulation of calendars, the tuning of instruments, cartography, and, yes, divination and theology, attests to the recognition that its material was uniquely adaptable to virtually any area of human endeavor or inquiry, and the pursuit of mathematical "truth" was hardly arcane frippery.

From its prehistorical foundations, mathematics' universe of discourse was derived from content which is ubiquitous to experience: extension, duration, valence, recurrence, singularity, variance, and so on. We are incapable of conceiving of any data which do not presuppose these elements, or stimuli which do not insinuate them. Humans exploring mathematics became incrementally more responsive to these elements, and their discipline matured to become our most effective means of handling these properties on a general basis, and independently of the continuous sensorial bombardment of a world of particular substances, through which we would otherwise only perceive them indirectly.

It is not so much, perhaps, that reality is mathematical—rather, mathematics is humanity's systematic response to the subtleties of reality. Given its intimate proximity to the ineffable, is it any wonder that mathematics should have become magnetized by it?

The pons asinorum ("bridge of asses") via Oliver Bryne, The First Six Books of
The Elements of Euclid in which Coloured Diagrams and Symbols are Used
Instead of Letters for the Greater Ease of Learners

Confession: that last section was intended to be a segue into what I originally sat down to write about, but it rather took on a life of its own. The amateur freeform blogging format may not be good for much these days, but makes a fantastic romping ground for delinquent ideas. Thank you for bearing with me.

I wanted to elaborate on a remark from last time: mathematics, there is no presentational curtain concealing the functions qua functions; no aesthetic barrier between the surface experience and the "real" game.
When we are, say, solving a problem in a calculus workbook, we might believe we are engaging with pure abstract reasoning without any interference from sensorial or narrative content. This is, we surmised, one of the ways in which math as a "game" differs from Lumines, Magic: The Gathering, Settlers of Catan, or what have you.


When I do math, I'm almost always in my bedroom, sitting at the foot of my mattress, propped up against a husband pillow. Most of the time, I'm wearing pajama pants. If I'm going to be pulling at my hair and gnashing my teeth for an hour or longer, I might as well arrange for the rest of myself to be comfortable.

Usually I'll put on music. For a long time I liked to listen to Bach's harpsichord concertos, but for the last year or so, Ravi Shankar's The Spirit of India has been my soundtrack for computation and dense reading. Psybient playlists on YouTube sometimes suffice. Lately I've had Mystery Science Theater 3000 on in the background, but I'm seldom paying much attention to it.

Coffee. Coffee is important. I'll always brew a mug of black coffee before getting started. (I used to have a bad habit of picking of cans of Starbucks doubleshots on my way home from work, but they're expensive, and the aluminum probably goes to an incinerator instead of a recycling plant). Most of the time I'll just take a couple of sips before getting started and then forget about it, and drink the rest cold when I wake up the next morning. But it's important that I have a cup of coffee on hand.

I like to write with Pilot G-2 pens (.07 mm). Blue ink. Always blue ink. Black is too blah, red is too loud. I need blue ink. I order gel refills by the dozen. And for scratch I used copier paper. At some point in the last few years I soured on ruled sheets; I can't use them anymore, not even for composition.

I write my ones with serifs; lately I've been crossing my sevens. I tend to write out "d/dx" instead of "dy/dx." I like the way an integral sign looks. I don't know when I stopped writing out multiplication signs. Sure, it's probably faster to write out "4 x 5" than "(4)(5)", but this is just another of my ways in which I've become set.

Hm. Apparently there is an aesthetics of doing math. An individualized aesthetic, of course, but one that nevertheless factors into the event.

For our present purposes, "reasoning" might be too loaded a term to describe what we're engaged in when we're concentrating on solving a problem in calculus. Let's look at it another way.

You open a textbook and read a problem: you might choose one at random, start at #1, or pick up where you left off last time. You copy that problem onto a sheet of paper, where it becomes the textual prompt for a chain of transcriptive responses, shaped up by a long history of conditioning dating back to your kindergarten arithmetic lessons. We are not acting with regard to quantitative entities and logical operators per se: we're responding to the print in the textbook, our own cognitive activity, and to the strokes of ink we leave on the paper. Our writing down du = 2x dx was precipitated by what we wrote down a moment before—u = 1 + x2—which was elicited from a previous textual stimulus, perhaps ∫ x/(sqrt(1 + x2 )) dx.

We can't separate from the act of "reasoning" the visual, tactile, and perhaps olfactory and aural stimuli which are the context of its occurrence. Probably the color of the ink, the texture of the paper, the lighting in the room, and the scent from the mug of coffee steaming on the nightstand influence what we're doing far less than the self-generated intraverbal activity set in motion by the visual reception of the symbols on the page, but they are never completely irrelevant.

In this view of things, the aesthetic pleasure of mathematical behavior is taken from subtle, simple properties of things: the patterns of colored ink standing out against a field of white, the friction of the pen nib against the paper's surface, the cadence of one's own voice murmuring "sine squared theta equals one minus cosine squared theta." In spite of mathematics' empyrean pretensions, the act of doing it is a fertile opportunity for discovering joy in the minute properties of quotidian objects. I've never appreciated gradations of my white walls or the shadows on my ceiling more than when looking up from a tough problem. (Well, excepting occasions when I haven't been sober.)

The remainder and bulk of the action remains internal: we are responding to our own responses. The integrand to be solved doesn't enter into our brains: that's a physical impossibility. Our grey matter doesn't "copy" the numerals and operator symbols. Rather, we enact a train of self-supplied stimuli which ends when we've produced a solution (if we haven't quit in frustration yet). The "functionality" we keep referring to does not reside in the printed signs, but in the conditioned behavior they activate in us.

If mathematics can fairly be called a game, it is in this regard an unusual one: instead of learning how to play it, we teach ourselves to be played by it.

1 comment:

  1. "let's return to mathematics for a minute", he says, as he wanders off, never to be seen again. But such is the nature of blogging.