math & games: presentation & functionality

László Moholy-Nagy, Q 1 Suprematistic (1923)

In case anyone's keeping track: I acknowledge the very late update (and thank you for noticing). Revisions for the new n-v-l are on page 460 out of 700-ish, and proceeding slowly. I'm beginning to doubt it'll all be worth it in the end, and it's getting to where I'd almost rather be writing a series of essays on Dissidia Final Fantasy.

Almost.

This awful project's been cutting into my other hobbies and projects, too. It's been a while since I last read any meaty 700-page nineteenth century novels. My backlog of short story ideas is becoming less of a queue than a clogged sewer pipe. And it's been at least a few months since I last picked up the ol' calculus textbook.

Yes, yes—I'm still working through James Stewart's Single Variable Calculus: Concepts and Contexts after like eight years. In my defense, I have zero congenital talent for mathematics. I consistently scored D's in Algebra II, and I'd still blanch at showing an actual math student my scratch paper. What's more, I've been proceeding in fits and starts, and whenever I take a break for a while and pick it back up again, it takes at least a week to reinstall the mindset, to go back and make sure I actually learned the contents of a previous chapter, and reacquaint myself with all those damn formulas. (Come on, like you never forgot that cos x cos y = 1/2 [cos (x − y) + cos (x + y)] and couldn't be satisfied until you'd worked it out for yourself.)

I'd be more ashamed of advancing so slowly if I weren't pursuing this study (if you can call it that) on a purely recreational level. I'm not good at this stuff, but I have fun doing it. Somehow or other, it became a de facto substitute for video games. Instead of playing Metal Slug III, Einhänder, or Half-Life after dinner, I'll attempt to work through a few implicit differentiation problems or work out the volume of a three-dimensional form created by rotating a curve around a given axis. It can get pretty intense, let me tell you.

People in my life tend to find it odd that I do math for fun—possibly because most of the people in my life are arts and/or humanities people. Tonight I'd like to try explaining how and why I've been getting my kicks from calculus, as opposed to video games, in my latter years.

It's not that I don't like video games anymore. I love them, and I know I'm incapable of enjoying them in moderation. Once in a blue moon, I'll download something on Steam and let it take over my life for one to three weeks. Last year I began playing System Shock 2 (for the first time, and yes, it was incredible) on a Sunday. By Friday, I looked like a haggard vagrant. I was staying up so late that I skipped shaving in the morning so I could lie face-down in bed for an extra twenty minutes. I put off doing laundry, and went to work wearing the least-dirty shirts and trousers I could find in the hamper.

At least it wasn't a JRPG. And god help me if I ever get back into fighting games. My last full-blown bender on a 2D fighter was when BlazBlue: Calamity Trigger came out in 2009 (how the years glide past), and my memories of that time are saturated with nostalgia and embarrassment. Though I'm loath to recycle similes, there's no fitter way to talk about my experience with BlazBlue than to compare it to a summer-long cocaine binge. It was a goddamned blast in the moment, but at this point in my life it's probably for the best that I'm not playing ranked matches for four to eight hours a day and allowing my emotional state to be contingent on my performance.

Since I've been too focused on revisions to allot an hour or two a night to decompressing via solving for x, lately I've been getting home from work and watching FightCade matches on YouTube for maybe twenty or thirty minutes. Another tired metaphor: this is methadone for the fighting game junkie—a quick vicarious thrill in lieu of the ecstasy and agony of committing myself to contending.

Competent Street Fighter III: Third Strike players are fascinating to watch, but only when the matches aren't Ken vs. Chun vs. Yun vs. Ken. The CPS2 Marvel Vs. games are reliably entertaining, especially the frenetic, dissolute aggression of X-Men Vs. Street Fighter. But the high-level Vampire Savior matches have been the most interesting. The third (and, sadly, final) entry in the Darkstalkers series has always been one of my favorite games, and I'm realizing I never really knew how it should be played. Browsing Mizuumi to better understand what I've been seeing has been illuminating and rather humbling. If you ever want to be made to feel like an ignominious casual gamer and/or get blackout drunk, read the Mizuumi entry on Morrigan and take a shot every time you come across technical patois that doesn't make sense to you.

Mizuumi's FAQ does an excellent job of explaining what Vampire Savior is all about, and by way of contrast, it underscores the reason why I was always frustrated reading mainstream gaming publications' reviews of 2D fighters. Most of the time, the critic on the magazine's payroll had no idea what he was talking about. Maybe the situation has improved since the genre bounced back to life in such a big way, but I remember IGN or GameSpot's review for the PS2 release of King of Fighters XI (my favorite in the series) running something like: "um, well, there's a lot of characters, and that's good, but the graphics are recycled and that's not so good, and if you're the kind of person who likes King of Fighters games, this is definitely one of those games."

Fighting games are complicated. Running through arcade mode a few times and playing some matches with another novice doesn't qualify you to say anything about a title beyond glib remarks about its presentation and controls. It's like reviewing a sports car after driving it once around the dealership's parking lot.

Anyway, there was one line in the FAQ that caught my eye. Under the "Is this game for me?" section, the author specifically cites "a high appreciation for the characters and setting" as a requisite for enjoying Vampire Savior. This is probably true: after all, it was the game's vibe that drew me to it and held my fascination when it was rotated onto the big-screen cabinet at the Rockaway Mall arcade in 1997. True, one could probably say the same thing about any fighting game, but the contents of the rest of Mizuumi's pages on Vampire Savior bespeak the discrepancy between form and function here. Though the discordance isn't so extreme in in a fighting game as it is in, say, a JRPG (to use my favorite example), where the field exploration, menu-based combat and anime-flavored narrative often constitute an emulsion of elements instead of a stable solution, what's actually happening during a Vampire Savior match has very little to do with "characters and setting." All of that gorgeous, personality-infused pixel-art animation is actually just the chassis for a manifold of hitboxes. The only area of Majigen that's at all relevant to the serious Vampire Savior player is the Fetus of God stage, whose expanded size makes it verboten in tournaments (though it's only selectable for versus matches on the console versions anyway). In competitive play, you won't be seeing or hearing those victory portraits, flavorful winquotes, and character-specific BGMs for more than one second after a match before the HERE COMES A NEW CHALLENGER flourish ushers in the next game.

Vsav arcade flyer

Characters and setting are irrelevant when the rubber meets the road. And yet, it's impossible to dispense with them. Imagine if Capcom revisited Super Street Fighter II Turbo or Third Strike—arguably two of the most well-designed fighting games of all time—and made them even better by applying miraculous patches that almost perfectly balanced all the characters without making any player feel like their main has been nerfed. But: the patch also turns all the characters into featureless grey humanoids, converts all the backgrounds into white space, and replaces all the audio with sound effects from Arkanoid.

Would anyone touch it? Probably not.

Not too long ago, Capcom took a hard lesson from the reception to its ill-starred Marvel Vs. Capcom Infinite. Released during Marvel's de facto X-embargo, it was the first game in a franchise that began with X-Men: Children of the Atom to completely exclude X-people from its roster. When fans grumbled about not getting to play as longtime favorites like Magneto, a couple of Capcom reps told GameSpot that there was no reason to be upset, since a fighting game's characters (as distinct from the bundles of frame data they represent) don't matter:

If you were to actually think about it, these characters are just functions. They’re just doing things. Magneto, case and point [sic], is a favorite because he has eight-way dash and he’s really fast, right? So, our more technical players, all they want to do is triangle jump and that kind of stuff. Well, guess what, Nova can do the same thing, Captain Marvel can do the same thing. Ultron can do the same thing. Go ahead and try them out.

You can imagine how well that went over. Marvel Vs. Capcom Infinite got slaughtered by rival release Dragon Ball Fighter Z, not the least because ArcSys wasn't telling people that they didn't really want to play as Vegeta because, look, Frost is in the game (Toei wants us to promote Dragon Ball Super), and come on, one guy with a small hitbox and a beam attack is as good as any other. (Historical footnote: this was very probably one of the reasons that Street Fighter III reached the height of its popularity some years after its release. It took time for fans to get over all the absent Street Fighter II characters before they could appreciate all the things III did right.)

But the gap between aesthetic and instrumentality is real, and no video game can be free from it. This is no more a shortcoming or fault of the medium than the fixed location of a stage play. It's an essential property of electronic gaming. And most of the time, it probably isn't even noticeable. My sense is that it tends to only emerge at very high levels of skill and engagement. For instance: classic DOOM has been around for so long, its source code has been so thoroughly analyzed, and it has enjoyed such an active afterlife through fan maps and mods that the metagame has reached a level of wonkiness on par with those of the most eminent fighting games. You can see hints of id Software's growing awareness of DOOM's distilled essence in the progression of the official releases. Notice how the level design in the first three episodes of the original DOOM evinces the developers' concern for evoking a sense of place through the level designs. The research installation areas convincingly feel (by 1993 standards) like futuristic industrial facilities, and the hell levels' textures and layouts adhere to what seems more or less like the sensible architecture of a plutonian labyrinth or citadel. By the time id Software was stamping its seal of approval on fan WADs and selling them as Final DOOM, it's evident that the designers weren't worldbuilding so much as constructing machines, devices to challenge players in deliberately conceived ways. We can observe the same ethos at play in fan mods like Kaizo Super Mario World and Rockman 4 Minus Infinity. In such extreme crucibles of  design and play, aesthetic is beside the point: the colorful graphics and accreted proprietary lore burn away like slag shed from the pure metal of functionality.

Which brings us back to mathematics.

In retrospect, the transition from video games to math-for-fun was more natural than my computation-averse high-school self could have anticipated. It's not inaccurate to call mathematics a game, or at the very least a field for play. When you're solving for a variable to balance an equation, proving that one train of trigonometric functions is the equivalent of another, or ascending the sublime ladder of Euclid, what you're doing is acting to achieve a definite objective according to given rules. This language is reminiscent of, if not identical to, the fundamental terms in the "ludological" analysis of video games.

When speaking of games, the words "engine" and "rules" are often used interchangeably: the actions available to a player at any time are precisely those which the software provides for. The rules of Street Fighter II amount to what happens when the player hits the buttons and wiggles the joystick; the player's objective is to hit the buttons and wiggle the joystick in such a sequence that his opponent's health gauge exhausts before his. In a JRPG, the player's objective varies on a situational basis. When it isn't "find person/thing on the field map to trigger a cutscene and open the way forward," you're usually negotiating the short-term exigencies of enemy encounters, making informed guesses about the viability of fighting as opposed to fleeing (given the strength of enemy groups, the state of your party, and your available resources), or dealing with a boss, wielding your individual party members' number-changing executables to reduce your foe's hit points to zero while keeping your people alive. The rules are, as in all video games, immanent in the digital play-space. You may press/tilt the joypad/stick to move your in-game avatar around the areas of the in-game environment programmed to be traversable. You lose the game if each of your party members reaches 0 HP. To prevent this from happening, your are provided with tools which you may acquire, use, and exploit in whatever ways the game allows. You win when you see the "The End" screen after the credits. Granted, this isn't a very rigorous definition, but to give a parsimonious and comprehensive general description of how a given console RPG works would be too onerous and abstruse an undertaking for the purposes of this blog, and is frankly better left to the professionals. Either way, it's wonky stuff, and in both cases (fighting games and console RPGs) success depends on preparation, concentration, and ingenuity in the face of unforeseen problems. This might explain why mathematics has been scratching more of less the same itch for me.

I am definitely not qualified to synopsize the "rules" of mathematics, so I'm not even going to try. But logician and mathematician Alfred Tarski certainly is, and I happen to have his classic Introduction to Logic and to the Methodology of Deductive Sciences (1941) on my bookshelf.

[C]ertain principles concerning the construction of mathematical disciplines have emerged as follows. 

When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions of this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. The sentence which determines the meaning of a term in this way is called a DEFINITION, and the expressions themselves whose meanings have thereby been determined are accordingly known as DEFINED TERMS. 

We proceed similarly with respect to the asserted statements of the discipline under consideration. Some of these statements which to us have the appearance of evidence are chosen as the so-called PRIMITIVE STATEMENTS or AXIOMS (also often referred to as POSTULATES, but we shall not use the latter term in this technical meaning here); we accept them as true, without in any way establishing their validity. On the other hand, we agree to accept any other statement as true only if we have succeeded in establishing its validity, and to use, while doing so, nothing but axioms, definitions, and such statements of the discipline the validity of which has been established previously. As is well known, statements established in this way are called PROVED STATEMENTS of THEOREMS, and the process of establishing them is called a PROOF. More generally, if within logic or mathematics we establish one statement on the basis of others, we refer to this process as a DERIVATION or DEDUCTION, and the statement established in this way is said to be DERIVED or DEDUCED from the other statements or to be their CONSEQUENCE. 

To demonstrate what all this means and how it works, Tarski later on walks the reader through the construction of a fragmentary theory of arithmetic. Here are the axioms pertaining to addition:

Axiom 6. For any numbers y and z there exists a number x such that x = y + z; in other words: if y ∈ N and z ∈ N, then also y + z ∈ N 

Axiom 7. x + y = y + x 

Axiom 8. x + (y + z) = (x + y) +  z 

Axiom 9. For any numbers x and y there exists a number z such that x = y + z 

Axiom 10. If y < z, then x + y < x + z 

Axiom 11. If y > z, then x + y > x + y

And the introduction of subtraction:

Definition 2. We say that x = y - z if, and only if,         y = z + x

Obviously most of us rarely consider the logical edifice presupposed by 2 + 2 = 4. But if we waited until students are capable of understanding the rules of deductive inference before teaching them arithmetic, most probably wouldn't be learning their multiplication tables until about the age of eighteen. (By the same token, we might also say that the seven-year-old child hunched in front of a TV playing Super Mario Bros. in 1987 is equally indifferent to the sedulous design principles which make Level 1-1 a ludological masterpiece.)

The rules of mathematics consist of a set of agreed-upon terms, postulates, and relations, all designated by simple symbolic language, and of the additional relations that can be extrapolated from the a priori expressions. The objective is to wander the universe of discourse without producing any false or unfounded statements.

In other words: it's a sandbox game.

Sophie Taeuber-Arp, Point on Point (1931–1934)

For the amateur (or the unhappy student coerced into playing), math is like an MMORPG in which you begin with basic equipment and meager abilities, and by questing and grinding, you ascend to higher levels at which you become capable of exploring new territory and surmounting greater challenges. This is pretty much what I've been up to. And just as it's more satisfying to lay into a monstrous RPG boss with the characters you've spent weeks building into tough, multifunctional units capable of using their abilities in tandem than it is to mash the "Fight" command and watch everyone poke at a pack of goblins with the box cutters they start the game equipped with, learning to negotiate more and more difficult problems using previously mastered methods and operations is increasingly gratifying as you progress from algebra to analytic geometry to differential calculus to integral calculus (taking a few detours in between to get acquainted with trigonometry and logarithms, as one might switch a unit's job in Final Fantasy Tactics to improve its proficiency in a secondary action ability). At higher levels—which I don't suppose I'll ever reach—the game is less about computation and manipulating expressions than exploring a kind of pure mathematical postgame wherein one might attempt to solve challenges regarded as impossible, advance the metagame, or test (and possibly redefine) the limits of the game's rules.

But, again, those sorts of things are for dedicated pros. In the short term, I'll be satisfied if I can internalize the proof of the fundamental theorem of calculus. If I were much more ambitious (and less aware of my low level cap), I might set my sights on devising an alternative proof. Not that I ever can or will, but it's exciting to know that the feat is or may be possible in the same sense that it excites me to see a 1-cc run of a brutal shmup or the possibility that I might someday watch a recording of some mad soul beating Wolfenstein 3-D in six minutes while wearing a blindfold.

Am I waxing vague? Am I not really saying anything about why this is all so enjoyable to me? Probably so. And to simply state that I like playing at math because it holds my attention, tests me, and makes my neurotransmitters dance could be said of any hobby, irrespective of the activities of which it consists.

But imagine being asked to explain why you enjoy video games without being permitted to make reference to anything that has to do with narrative, presentation, or the mechanics of any one particular game. For that matter, what does the connoisseur of movies say about film when she's prevented from speaking about plot, characters, dialogue, set design, cinematography, or any specific event depicted in any certain movie? How does the museum visitor tell us why this or that painting is her favorite if we forbid her from mentioning colors, textures, composition, verisimilitude, or "what it means" in her description of it?

In mathematics, we are automatically at such a loss because we are dealing with pure abstract functionality. A symbol means precisely what it means and nothing else. An operation does precisely what it does. Every relation stands precisely as defined between the terms which instantiate it. When we dispense with the framing devices of "word problems," these terms don't pretend to reference anything beyond a universe of discourse that has divested itself of all reference to material objects and real events.

Sometimes, while playing games, during instants of intense concentration, I catch the sporadic, fleeting glimpse of the functions behind the forms, like Neo seeing the "world's" code at the climax of The Matrix. During these moments, the onscreen portrayals disclosed themselves as mere operations, and I understood fully what was happening and what I was doing. Street Fighter and King of Fighters matches where I seemed to view the frames progressing in slow motion, and could read their properties as through a HUD. Evading enemy fire in Einhänder and ESP Ra.De. and comprehending, for maybe up to two seconds at a time, the trajectory of every individual bullet on the screen, and how they could be dodged. Walking into an ambush in DOOM 2 or Half-Life and noticing myself perceiving the imps, grunts, arachnotrons, and turrets not as the graphics by which they were rendered or in terms of the in-game consequences they threatened, but as a multivariate equation to be systematically broken down.

It never lasts: the spell of the aesthetic lowers the clouds back over my eyes. The imps and arachnotrons become imps and arachnotrons again. The bullets are a confusion of dots flashing against a scrolling background. Alex and Ryu revert to animated martial artists grunting and kicking, when a moment earlier they were a manifold of 60 frames-per-second chess pieces inexorably advancing to the logical outcome of the board state. But in mathematics, there is no presentational curtain concealing the functions qua functions; no aesthetic barrier between the surface experience and the "real" game.

I may have more to add on that point later in the month.

There's a passage from Alfred North Whitehead's Science and the Modern World (1925) that I saw years ago, quoted out of context, which was one among innumerable nudges impelling me toward opening a math textbook for the first time since high school and making an earnest, good-faith effort to learn what it had to teach: "the pursuit of mathematics" (Whitehead wrote), "is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings." I sometimes forget that when I first opened up my secondhand copy of Single Variable Calculus and began reintroducing myself to quadratic formulas and linear equations on the first pages, I did so with an ulterior motive of eventually being able to parse the arcane formulas on a given Wikipedia entry pertaining to some concept in physics. I still think that would be nice, certainly; but since then I've fallen under the spell of math for its own sake. To say I'm in the throes of a divine madness would be to egregiously overrate my own intellectual faculties, but perhaps there must be a touch of it in me if I'm more interested in scratching out equations in my spare time than engaging with entertainments that are generally more—well, entertaining.

It could also be that amidst the relentless overstimulation of twenty-first century life, even a casual, dilettantistic pursuit of mathematics offers a refuge, as Whitehead says. If not a retreat, then a purge: an iconoclastic juice cleanse to flush some of the representational, hyperreal excess from one's system, and to disconnect from the anger, fear, vanity, pointless hype, misinformation, vulgar commercialism, stultifying spectacle, overproduced and exponentially multiplying lore, political fanaticism, and glib nihilism that radiates ceaselessly from every LCD screen in range of a network signal. To altogether recuse oneself from the overall situation would be irresponsible and immature; but the quiet contemplation of symbolic puzzles that imply, with a veritably mysticistic suggestiveness, a general order and truth to the real world of things, has worked for me, at least, as a form of self-care.