Giving Conics the Old College Try

After nearly three years of not writing in this blog, I was reminded just now of another essay I’d written in college, this one about math history, and was wondering whether to give it the “Are Video Games Ambient?” treatment and post a revision of it here. Well: after reading through “Apollonius and the Pervasiveness of Conics in Classical Greek Geometry” afresh, I think it’s safe to say it’s not going to make it to this blog any time soon. For that matter, at this point in time I found even “Are Video Games Ambient?” itself difficult to understand, the confusion, I believe, due mostly to the original draft of the essay, rather than my revision efforts (my 2020 postscript I found clear enough). But I think some of the ideas in my math history essay are, at least, worth preserving.

The idea that the conics, or conic sections — circle, ellipse, parabola, hyperbola — are “pervasive” in the geometry of the ancient Greeks, and my attempted investigation of why that might be the case, formed the backbone of my essay, and yet, as such, was met with a fair amount of skepticism from my professor. What does it mean, that these shapes are pervasive? Why is that interesting? But: the beliefs I didn’t state outright probably contained a strong clue as to what intrigued me most about this so-called pervasiveness of conic sections.

One of my grand ideas at the time, I believe, was that mathematics could be helped along in its progression, or at least the solution of mathematical problems could be aided, by metamathematics: that is, the mathematics underlying the structure of mathematics itself. In particular, I wanted to look at mathematics through the lens of the complexity of mathematical structures, where complexity was a particular characteristic that could be quantified somehow, perhaps in something like a computational-complexity sense. The idea was that the solution to a math problem should not have to be more complex than the problem itself, that it should flow, in some direct way, from the statement of the problem. The fact that this idea is clearly false (for example, in the case of the four-color map theorem) only made me want to restrict my attention to a small “toy domain” within mathematics, for which my idea would stand a better chance of holding up: as this domain, then, I picked Euclidean geometry.

I was, perhaps, aware that, despite the prevalence of Euclidean geometry in the highest levels of mathematical competitions around the world (an area in which I’d also had more than a passing interest), it’s on pretty much the opposite shore, within the dominion of math, of anything approaching the current or cutting-edge. This style of geometry has been so played out, it’s become a game. But: that was part of why it was so fascinating for me. In a world dominated by circles and straight lines, what were these shapes — ellipses, parabolas and hyperbolas — that kept popping up? They seemed inordinately complex. Well… how complex are they? Could there be some primal definition of conic sections — the simplest possible way of describing them — from which all the manifold ways of looking at these shapes would be natural consequences?

(I was also heavily into the idea that some consequences, some deductions, would be more natural — simpler, in a hopefully technical sense — than others. I related this to the divide made between “analysis”, the process through which a proof is created, and “synthesis”, the most direct way of stating the proof. It seemed to be a big problem that, in modern mathematical presentation as in the expositions of the ancient Greek mathematicians, synthesis would always seem to push analysis by the wayside. Happily, I have since run across some exceptions, such as this blog post from mathematician and Fields Medalist Timothy Gowers about the cubic formula.)

Suffice it to say, I failed to find such a primal definition. The part of my essay that impressed my professor the most was the section (and link to a webpage) about Dandelin spheres, which to me still do seem to inform the most natural argument I know of linking the conception of conics as cross-sections of a cone with their more familiar planar properties (foci, directrices). In the essay, I made the argument that, although the discovery of Dandelin spheres didn’t take place until the nineteenth century, it was not inaccessible to ancient Greek geometrical modes of thinking. Although that, at least, still seems reasonable to me, the fact that conic sections also appear in projective geometry, for example, deserved a better explanation than I gave.

Ironically, it wasn’t until later on that I learned about so-called quadric surfaces, any cross-section of any of which is a conic, which I believe finally can give some substance to my nebulous idea of conic sections’ naturalness and, thus, pervasiveness. I still have a sort of fantasy about teaching a course (now that I am, you know, a math teacher) about conics and the different geometrical contexts that give rise to them, and there are ideas in my essay that I continue to think are worth exploring. It’s going to take a lot of work, however — whether in this blog or outside of it — to unpack them!

Are Video Games Ambient?

After not posting in this blog for almost two years, I decided to return to it by posting what I thought was a worthwhile philosophical project from my past.  This is a revision of an essay I wrote in college, in 2006, for a class about video game culture.  At the end, I’ve included a current update of my opinions as a postscript.  Hope you enjoy!

The Ambient Narratives Of Video Games

Introduction

It should be little wonder that Brian Eno admires Will Wright.  Here is Everything Bad Is Good for You author Steven Johnson, in a New York Times Magazine article on Wright’s video game Spore (2008): “‘We’ve both been working on similar lines,’ Eno said recently… ‘and I suppose we’ve converged.  Instead of making fixed definitive things that we’ve put out into the world, I think we’ve both decided that it’s much more interesting to make things that even we can’t predict.’” (Johnson, 53)

Brian Eno is considered one of the musical pioneers of the late twentieth century: having developed and popularized the style known as ambient music, he has paved the way for many of the developments that music has undergone since.  Wright occupies a similar position in his own field.  In addition to Spore, previous Wright games – The Sims (2000), Sim City (1989) – have simultaneously expanded the domain of what a video game can be, and been immensely popular (Johnson, 53).  It is my goal in this essay to argue that, in fact, video games have much in common with ambient music, and that this resemblance can be seen most clearly in a feature often overlooked in discussions of video games: their narratives.

Ambient Music and Video Games

What is ambient music?  Eno biographer Eric Tamm claims that the sense of the phrase as it is used today was invented by Eno:

When Eno chose the term “ambient”… the word’s rich connotations must have been prominent in his mind.  It was music that could tint the atmosphere of the location it was played.  It was music that surrounded the listener with a sense of spaciousness and depth, encompassing one on all sides, instead of coming at the listener.  It blended with the sounds of the environment, and seemed to invite one to listen musically to the environment itself….  if not completely free of individual taste, memory, and psychology, as in [composer John] Cage’s ideal, it nevertheless lacked the bathos of self-importance and confessional displays of open psychic wounds. (Tamm, 131-132)

Each of these themes will recur in this discussion of video game narratives: atmosphere, spaciousness, function within an outside environment, and the decentralizing of psychology.  To these qualities I will add one more, stated by Eno himself in his liner notes to his album Ambient 4: On Land (1982): “I took advantage of the fact that music… has the option of creating its own psychoacoustic space.” (Eno)  Indeed, tracks on On Land are titled “Lantern Marsh”, “Lizard Point”, and “Dunwich Beach, Autumn, 1960”.  Can the same evocation of landscape be applied to video games?

In his article “Game Design as Narrative Architecture”, video game theorist Henry Jenkins argues that video game narratives should in fact be viewed spatially, as Eno’s On Land invites itself to be: modeled not by the linear progressions associated with the narrative of a film or novel, but in several dimensions at once.  He lists four ways to accomplish this: “[S]patial stories can evoke pre-existing narrative associations; they can provide a staging ground where narrative events are enacted; they may embed narrative information within their mise-en-scene; or they provide resources for emergent narratives.” (Jenkins, 123) We will investigate each of these methods, and match them up with Tamm’s features, listed above, of ambient music.

To begin, however, let us recall Jenkins’ motive: “to offer a middle-ground position between the ludologists and the narratologists” (Jenkins, 119) – that is, between two competing schools of interpretation of video games.  Ludologists argue that video games are valuable because of the gameplay involved, narratologists that they provide valuable contexts for narratives.  Neither Jenkins nor his critics, however, call attention to a second, related dichotomy: are “narratives” meant to be something apart from gameplay, appearing for instance as a bubble of text that interrupts an action sequence, or to be something inferred from the gameplay itself?  I call the former imposed narratives and the latter implied narratives; I will argue for analogies between ambient music and video games’ implied narratives.  In particular:

1. Tamm’s “tint[ing] the atmosphere” matches with Jenkins’ “pre-existing narrative associations”.

2. Tamm’s “encompassing one on all sides” matches with Jenkins’ “staging ground where narrative events are enacted”.

3. Tamm’s “lack[ing] the bathos of self-importance and confessional displays of open psychic wounds” matches with Jenkins’ “emergent narratives”.

4. Tamm’s “blend[ing] with the sounds of the environment” matches with Jenkins’ “embed[ding] narrative information within their mise-en-scene”.

1. Tinting the Atmosphere / Pre-Existing Narrative Associations

An implied narrative for a video game can take the form of a summary of events, similar, for example, to a capsule summary of a football game.  In this case, imposed narratives are absent: the summary here is drawn from the fabric of the game itself.  This distinction is missed by Jenkins when he states that “many of my own favorite games – Tetris, Blix, Snood – are simple graphic games that do not lend themselves very well to narrative exposition.” (Jenkins, 119) Although these games are abstract, they are no more so than football; the explanation of either may be filled with esoteric terminology.  It is imposed narratives that do the most to interrupt game play: “[A]fter the first time through [a cutscene], most players ‘press start to skip,’ cutting right to the action.” (Beck and Wade, 24) The implied narratives that can be drawn out of a game like Tetris, in contrast, point the way to a seamless divide between narrative and gameplay; it is this feature which I link to ambient music’s “tinting the atmosphere”.

One video game that features prominent implied narratives is Nintendo’s Super Mario Bros. 3 (1990) for NES.  Its surface-level narrative is quite simple: Mario must rescue the kings of the Mushroom World, who have been transformed into animals by the minions of Bowser the Koopa King.  This is an imposed narrative, however: this is what the game itself tells the player that its plot is.  To get a sense of the storylines developed over gameplay, on the other hand – the game’s implied narratives – we could imagine players describing their experiences playing through the game.  There is great differentiation in Super Mario Bros. 3 between the game’s eight Worlds, through which the player progresses sequentially.  An important element of this differentiation is atmosphere, which Jenkins describes thus: “If, for example, the attraction centers around pirates… ‘every texture you use, every sound you play, every turn in the road should reinforce the concept of pirates.’” (Jenkins, 123) The eight World themes of Super Mario Bros. 3 are, in order, Grasslands, Desert, Water, Giant, Sky, Snow, Pipes, and Darkness, and each theme is enhanced by a combination of gameplay and graphics to give a sense of place.  The Desert, for instance, features quicksand in which Mario can sink, while the Snow World features ice blocks on which he slips.  This effect equates to Jenkins’ “evoking pre-existing narrative associations”: those which the player would already associate with such concepts as deserts and snow, which are not inherent to Super Mario Bros. 3’s gameplay in the abstract, and yet enhance the player’s experience of it.  These level themes, in a manner many earlier games lacked, then, tint the experience of playing Super Mario Bros. 3, just as ambient music would “tint the atmosphere of the location it was played”.

2. Encompassing on All Sides / Staging Ground

The foregoing analysis of Super Mario Bros. 3’s implied narrative gives us an extra correspondence.  Jenkins states that “The organization of the plot becomes a matter of designing the geography of imaginary worlds” (Jenkins, 124), and uses this statement to argue for an idea of spatial narratives as a “staging ground”.  In such constructions, the role of the main character is diminished with respect to the surroundings, which themselves become the important story element.  The story could, indeed, be said to “encompass one on all sides, instead of coming at” the player.  One should note that this second characteristic of implied narratives in video games piggybacks on the first: a strong sense of place is important for a designer to attain before an attempt is made at achieving an immersive environment in a video game.

The encompassing nature of such stories, too, does not preclude them from taking their course as stories.  Tamm comments on Eno’s proto-ambient album Another Green World (1975), with a clear analogy possible to the sequence of levels in a video game: “One could characterize Another Green World as a ‘concept album’ in the sense that it presents a series of pieces clearly arranged with a view to the whole.” (Tamm, 103) Although there is no explicit “concept” that unites Another Green World’s music, neither is there one to Super Mario Bros. 3’s sequence of Worlds; nevertheless, the overall progression and coherence of the game’s story are clear.

3. Decentralizing of Psychology / Emergent Narratives

A sense of place also leads into Jenkins’ concept of emergent narratives: an idea similar to, though more restrictive than, my notion of an implied narrative.  An emergent narrative is specifically determined by the development of the player’s own course through the game.  Jenkins uses The Sims as an example; to show the ambient characteristics of emergent narratives I draw the focus to Capcom’s Mega Man 8 (1997) for Sony PlayStation.  This is a game with strong, atmospherically distinct stages – one, the Astro Man stage, I characterize as having a “hippie” theme – which can be selected in any order by the player.  Although the player can thus construct a personal, spatially significant implied narrative by traversing the game, the player character, Mega Man, is not present for these decisions.  Indeed, one of the characteristics of a narrative integrated suitably into a video game is often that the main character is simply a proxy, a tool for the player’s exploration of the world.

In order to match this characteristic of video games to Tamm’s “lacking the bathos of self-importance and confessional displays of open psychic wounds”, I point again to the distinction between imposed and implied narratives.  Here Tamm quotes Eno on the construction of his piece “2/1”, on Ambient 1: Music for Airports (1978): “’And then I started all the [tape] loops running, and let them configure in the way they chose to configure.  So sometimes you get dense clusters and fairly long silences, and then you get a sequence of notes that makes a kind of melody.’” (Tamm, 136)  Here is the beginning of Eno’s move away from “fixed definitive” ways of composing, which separation he readily compares to Wright’s games.  In fact, the “melodies” in “2/1” are themselves only implied, their chance construction resulting in a mode distinct from melodies made to impose a certain emotion on the listener.  Instead, works such as “2/1” and The Sims imply that the end user ought to have the freedom to superimpose their own emotions on the material provided.  The silence of such console role-playing game heroes as Crono, of Squaresoft’s Chrono Trigger (1995) for SNES, exists also to further their functioning as a proxy for the player: the narrative that emerges around Crono during a playthrough of Chrono Trigger has little independence from the player’s decisions.

4. Blending with the Environment / Narrative Information in the Mise-en-Scene

Finally, is it unfair to match the natural insertion of ambient music within a larger environment with Jenkins’ idea of a game embedding traces of its own narrative in its mise-en-scene?  It is true that his listed methods of making video game narratives spatial do not make particular mention of the world outside.  Yet at times, the mise-en-scene of a game is precisely that, as Jenkins informs us in his description of Electronic Arts’ Majestic (2001): “We follow links between web sites; we get information through webcasts, faxes, e-mails, and phone calls (Jenkins, 127).” The player acts as a detective, filtering the game’s narrative drive through the complications of a real-world environment.  It is surely part of Majestic’s draw that its playing field is so readily conflated with the outside world; likewise, a listener to ambient music will not necessarily be able to distinguish between the sounds that come as part of the music and those that enter from outside.

Conclusion

The conceptual shift that would lead to video games’ societal acceptance is one that has happened before, in music.  Although Eno’s ambient albums were initially only at the margins of the popular music world, they have nevertheless blazed the trail to such genres as ambient techno; in this way, the culture’s set of musical possibilities has expanded.  The correspondences I have pointed out in this essay between ambient music and video game narratives are signs pointing towards the future acceptance of video games, too, by society, unaccustomed narrative structures and all.  I hope that the set of analogies I have brought forward in this essay will ease this transition.

Bibliography

Beck, John C., and Mitchell Wade. Got Game: How the Gamer Generation is Reshaping Business Forever.  Boston: Harvard Business School Press, 2004.

Eno, Brian.  Liner notes, Ambient 4: On Land (album).  Virgin Music 66499, 2004 (originally released 1982).

Jenkins, Henry.  “Game Design as Narrative Architecture,” in Noah Wardrip-Fruin and Pat Harrigan, eds., First Person: New Media as Story, Performance and Game.  Cambridge, MA: MIT Press, 2004.

Johnson, Steven.  “The Long Zoom,” New York Times Magazine (Oct. 8, 2006), 50-55.

Tamm, Eric.  Brian Eno: His Music and the Vertical Color of Sound.  Cambridge, MA and New York: Da Capo Press, 1995.

Postscript (2020)

When I wrote the first draft of this essay, in 2006, I loved video games.  Now, fourteen years later, as the societal acceptance of video games I had hoped so much to see in 2006 has occurred, my own appreciation of video games has plummeted.  However, especially since my reading of David Sheppard’s 2008 Eno biography, On Some Faraway Beach, in 2017-18, my appreciation of ambient music has only risen.  Although I still believe I made some good points in this essay about the ambient qualities of video game narratives, I think it’s important to recognize that in many other ways, video games have to be considered decidedly un-ambient.  A distinction I failed to draw in 2006 was that of types of hierarchy between the real world and the world created by a work of art: a piece of music, or a video game.  I gave the example above of Majestic, which now is considered to be part of the broader “alternate-reality game” genre, as an analogue of ambient music as listened to as part of a continuum of background sound.  What did not occur to me was that, while playing Majestic — for example, when receiving an in-game phone call — Majestic has primacy, with the physical world serving as a background to it.  Ambient music, on the other hand, is music designed to itself be a background.  I’m actually listening to ambient music while writing this postscript: it generally hasn’t been a focus for me, but instead has simply enhanced — “tinted”, if you will — my experience of writing it.  The hierarchy is exactly backwards.  Although a narrative can surely be ambient within the context of a video game, the video game itself tends to grab fast onto the majority of one’s attention: something that a good piece of ambient music will not do.

In the last several years, the styles of art I’ve been drawn to have really changed quite a bit, and on the whole, their range has become narrower.  Mostly, I read and I listen to music, whereas I was still often watching movies and, of course, playing video games back when I first wrote this essay.  An advantage that reading has, from my perspective, is that it’s generally self-directed — I have a lot of flexibility regarding the way I read — and, at the same time, it’s not all-encompassing.  I’m still grounded in the world outside, more than, I would say, while playing a video game.  I’m more conscious of the fact that I’m reading, what it means physically to accomplish this act, the position of my body; after a while playing a video game, the act of pressing the buttons on the controller used to become automatic, unconscious.  I also find reading and listening to music to be, ultimately, a more creative form of cultural enjoyment than playing video games, even if video games in their creation are just as creative a cultural genre as books or music.  I can combine reading and listening, essentially, either with each other or with other activities, in many ways.  One thing that I appreciate ambient music in particular for these days, then, is exactly the space it leaves unused in the mind.

I suppose a lot else could be said, too, about this change in my views, not all of it flattering: of course I don’t want to cast my lot in with video games now that they’re culturally mainstream; I’ve instead become a crotchety oldster.  I want to hold space open, though, for something about the current mainstream in particular that I have issue with, though I have a hard time defining exactly what that is; I think, though, that it’s whatever terrified me, going off as a video-game-loving teenager to a tech school, about the culture I saw there.  It’s only as time’s gone on that those types of attitudes have seemed to me to be inextricably tied up with video games, attitudes that I would describe as set against the more aesthetic experience of the world around us that, I now believe (especially now that I’m home almost all the time!), ambient music provides such a powerful way to experience.

The Elusive Function of Poetry

I’ve been reading poet Gary Snyder’s book The Real Work: Interviews and Talks 1964-1979.  In the interviews and talks, Snyder puts up such a coherent worldview concerning aesthetics and, indeed, how to live, that I find it amazing, sometimes while reading, that poetry fits into his life at all.  Indeed, in one of the interviews, Snyder is asked, “As you see it, what is the function of poetry?” Snyder responds:

You ask me what is the function of poetry so I think, “What is the function of poetry since 40,000 years ago?’ In all cultures of the world — total planetary overview.  And in that sense the function of poetry is not only the intensification and clarification of the implicit potentials of the language, which means a sharpening, a bringing of more delight to the normal functions of language and maybe making language even work better since communication is what it’s about.  But on another level poetry is intimately linked to any culture’s fundamental worldview, body of lore, which is its myth base, its symbol base, and the source of much of its values — that myth-lore foundation that underlies any society.

One thing I do appreciate about this answer is that it’s descriptive rather than prescriptive — that, when asked what the function of poetry is, Snyder’s impulse is basically to say, “Well, let’s look at what its function has in fact been.” And, one can indeed imagine that for most of the history of poetry, poetry has fulfilled the functions that Snyder lists.

But, one question that Snyder seems to dodge with this answer is that of what the function of poetry might be today — a question about which I have long wracked my brains.  If the answer is still to be that it is “linked to any culture’s… myth base, its symbol base, and the source of much of its values”, I find that answer lacking with regard to a culture whose poets are, for the most part, individuals committed to individual expression, competing with each other, whether they like it or no, for people’s money, time and attention.  I’m all in favor of individual expression, mind — just not the competitive aspect that always seems to follow.  The idea that the symbol base of our society should depend on who wins out in this competition not only is rather abhorrent to my sensibility, but also fails to even answer the question of what the poetry that doesn’t win out, that doesn’t manage to link itself to this symbol base, is for.

I’ve realized recently, I think, reading over much of my own poetry at the same time as reading through The Real Work, that for much of my own history of writing poetry, I’ve found in it a way to create or intensify a community, believing that through poetry, the deepest and most interior voices of individuals come out and can be shared.  (That doesn’t even necessarily mean that my favorite poetry is “lyric”, confessional poetry; instead, I would say that each person’s own decision about what to put forth as poetry is itself a function of what they deeply care about, in itself a self-expression.) This has held true for the high school class in which I first got to love poetry, for the open-mic scene I was involved with for a few years, and for the current poetry reading group I’m a part of.  But, I think, there’s a problem with that too, which is that it doesn’t seem to imply anything about poetry’s being necessary.  In the world we live in, in which some of the most ambitious and far-reaching self-help programs around find their way into the business section of the bookstore, ways of creating community that have nothing to do with sharing interior voices are at least as valued, if not more; and, I would be a snob to say they were less real.

What I want to propose instead is Snyder’s first function of poetry, which he almost appears to skim over: “intensification and clarification of the implicit potentials of the language, which means a sharpening, a bringing of more delight to the normal functions of language and maybe making language even work better”.  This, I think, is a better argument for the continuance of poetry, as long as language forms the basis of our interactions.  In an earlier entry in this blog, I wrote that “poetry, by very virtue of being the original form of literature, should count as the default form of literature” — literature whose atoms are mere words, not the characters or plots whose internal consistencies are such an overarching worry in the construction of other forms.  The structure of a poem is allowed to unfold from the words themselves.  The enhancement of the normal use of language thereafter seems like a natural thing.

Sometimes, thinking about the best poems I heard on the open-mic scene that decade or more ago, I think of them in these terms: these poems created a weavery of language that could, at its most transcendent, momentarily take the place of the society that I found myself so confined in, with its ploddingly uniform value systems and no forward motion.  But, that weavery of language isn’t sustainable in anything close to the long run.  Like Joseph Cambpell’s Hero with a Thousand Faces, we need to seize the linguistic treasure of the poetic otherworld, and bring it back into this world.  It just might be that the cultivation of poetry as a skill into the future depends on that retrieval.

 

Poetry, Ambiguity and Puns

Recently — I think I was reading Laura Riding’s poetry — it occurred to me that one of the things that appealed to me the most about poetry was its ambiguity. But: there’s a question about what the word “ambiguity” itself means. Suppose I claim that a poetic passage is ambiguous: I could mean either that its meaning is so vague as to spark nothing more than a cloud of associations in my mind, or that there are two (or, in rare cases, more) definite referents, either of which could be meant, without much mixing between those meanings.

The latter case, I’d say, is typified by the pun, which — especially if it’s embedded in a larger piece of writing — is a type of play with language that I find very honorable. Far from being “the lowest form of wit”, the pun means to me a form of humor that easily pushes away the temptation to mock anything, and results in a laughter of, instead of cruelty, delight at the coincidence it exposes. I also find the use of puns very valuable to me in my own poetry (example: “Before I went off to Wall Street to study the ways of the Dow”). They’re great for packing as much meaning as possible into a short passage, and also for “exposing” connections between things, perhaps on a subconscious level — such connections being, I would say, one of poetry’s important hallmarks.

The pun is also clean in another way: the ambiguity it produces is itself unambiguous. There exist a definite, small number of references each pun can point to. Those afraid of becoming too unmoored from meaning can still relax at the sight of a pun. One could even say that what a pun means is its exact multiplicity of meanings. And, therefore, complete ambiguity cannot be achieved by a pun.

The type of ambiguity that is itself ambiguous — whose set of possible references is itself open to interpretation — may, I think, be often denied, with respect to either its poetic value or its credentials as passable writing at all. But, I think it’s of immense worth to poetry. Otherwise, for example, what do we make of something like Wallace Stevens’ “Thirteen Ways of Looking at a Blackbird”, one of my favorite poems:

VI.
Icicles filled the long window
With barbaric glass.
The shadow of the blackbird
Crossed it, to and fro.
The mood
Traced in the shadow
An indecipherable cause.

… Soon, too, after realizing that what one of the things that appealed to me most about poetry was its ambiguity, an inversion of the thought struck me with a kind of horror: maybe one of the things that appeals to me the most about ambiguity is its poetry. Horror because, if ambiguity is primarily a feature of poetry, well, then (following Laura Riding herself), poetry doesn’t really model the world. Riding, who gave up poetry as an art and a career when she was in her thirties, insisted that poetry’s aim towards aesthetic surfaces waived any claim it had to truth.

Yet, I still think that I believe the many poetry fans who must still be out there, who (sagely nodding) understand that there in fact is truth in poetry. Here are the stakes as I see them, then: does the ambiguity that I find so important to poetry itself form part of that truth? Or, to put it another way: is ambiguity a feature of the world?

Poetry, Up Against Philosophy

Last Sunday, at my poetry-reading group, I read an excerpt from G. K. Chesterton’s Ballad of the White Horse, his epic-style poem about the ninth-century English King Alfred’s repulsion of a Viking invasion.  I read what I found to be the most moving part in the entire poem: the song of the villain, the Viking King Guthrum.  It begins:

He took the great harp wearily,
Even Guthrum of the Danes,
With wide eyes bright as the one long day
On the long polar plains.

For he sang of a wheel returning,
And the mire trod back to mire,
And how red hells and golden heavens
Are castles in the fire.

“It is good to sit where the good tales go,
To sit as our fathers sat;
But the hour shall come after his youth,
When a man shall know not tales but truth,
And his heart fail thereat.

“When he shall read what is written
So plain in clouds and clods,
When he shall hunger without hope
Even for evil gods.

“For this is a heavy matter,
And the truth is cold to tell;
Do we not know, have we not heard,
The soul is like a lost bird,
The body a broken shell….”

It’s a stunning expression of a pessimistic philosophy.  Not that I would call myself a pessimist — and, part of the pleasure in reading this section was knowing that Alfred himself, who has come into the Viking camp anonymously as a minstrel, would soon rebut it with his own song — but, the actual rebuttal I found disappointing after imagining what a potential rebuttal might consist of; and Guthrum’s song remains my emotional high point of Chesterton’s Ballad.

It also reminds me a bit of W. H. Auden’s “As I Walked Out One Evening”:

But all the clocks in the city
Began to whirr and chime:
“O let not Time deceive you,
You cannot conquer Time.

“In the burrows of the Nightmare
Where Justice naked is,
Time watches from the shadow
And coughs when you would kiss.

“In headaches and in worry
Vaguely life leaks away,
And Time will have his fancy
To-morrow or to-day….”

Auden’s stanzas, and many of Chesterton’s, are in the classic form of a ballad: the so-called common meter, rhyming ABCB.  And there’s a reason, too, why “ballad” is a word applied to songs as well as to poems: this type of thing sticks in your head.

Here’s my point: Chesterton and Auden reach me with their pessimistic philosophy in a way that a mere exposition of such a philosophy wouldn’t.  Now, this reach doesn’t actually consist of making me think pessimistically about life: instead, it’s poignant.  But, the lines of their pessimistic philosophy are ones I already know by heart.  Guthrum and Auden’s clocks sing a well-known song, make it beautiful, and make it ring in your head.  Philosophy is the preparation: with these ballads, you live with what the philosophy is saying — encapsulated as an element in your life.

Is it too much to say the same for all poetry?  Even though it often wants to introduce philosophical ideas to its audience, I don’t think poetry is nearly as good at it as philosophy itself is.  In poetry, things are more relative: we have to deal not only with the meaning of the words themselves, but with the meaning of the fact that the poet is saying these particular words, which, I find, often offsets the literal meaning.  If you want to take a position and mean it, you’re better off with philosophy.  And, as someone who so often thinks the structure of one’s internal ideas determines one’s life, well, that’s pretty important.  What does poetry have to give, instead, up against philosophy?

And, let me argue exactly this: that it makes philosophy itself stick in your head.  It takes the elements of a familiar philosophy and translates them into an earworm, or, if not literally, at least a mind-worm.  And, it can make the dourest of philosophies beautiful.

Schneiderman’s Throwdown

Rob Schneiderman is the author of my favorite article in Princeton University Press’s The Best Writing on Mathematics 2012, edited by Mircea Pitici, which I’ve talked about before in this blog.  Schneiderman’s article, “Can One Hear the Sound of a Theorem?”, was the first piece of writing I ever read that finally provided me with some sort of answer to the question, which I’d been struggling with for a while, of what exactly was this vaunted link between music and mathematics I’d heard so much about.  Schneiderman’s answer: music and math are both self-contained systems of expression, ones that require no references to the outside world to do what they do.  Other comparisons are fluff and bluster.

And really, I’d say, Schneiderman (who apparently knows my great-uncle) is best in his fluff-and-bluster mode.  Here’s one choice quote:

The problem is that mathematical content comes in the form of proven statements about well-defined structures, and attempts at “explaining” musical phenomena usually involve structures that are not well defined, with conclusions justified by carefully chosen examples and multitudes of counterexamples ignored.  And any logical development of well-defined structure is inevitably based on dubious or pedantic musical principles, so that the resulting conclusions can say precious little about what is important in music.  (p.97)

Over and over again, Schneiderman demonstrates that many of the links often drawn between music and math, taken in their entireties as disciplines, are superficial, tenuous, or worthy of extreme skepticism: they fail to get at the heart of what’s so beautiful and gripping in at least one, most likely both, of music and mathematics.  Take this excerpt, too:

… after mentioning musical affinities of Galileo, Euclid, Euler, and Kepler, the author [of the book Emblems of Mind, Edward Rothstein] includes Schoenberg, Xenakis, and Cage among a short list of examples that seem to point back from music to mathematics.  Even most mathematicians with an affinity for these composers would… surely recognize that this juxtaposition is way out of balance.  This comparison leads to such contradictions as claiming the existence of “a systematic logic that guides musical systems” but then admitting later that the great musical compositions “create their own form of necessity, the binding coming not from logic but from the unfolding of ideas…” (p. 106)

The juxtaposition, of course, is way out of balance because the group of twelve-tone composers that includes Arnold Schoenberg, Iannis Xenakis and John Cage can in no way be said to stand in for all of music.  And this, indeed, is Schneiderman’s point: that perhaps mathematics can form a basis for certain pieces or indeed compositional styles, but that that basis is merely one choice to be made within the universe of music, and has no bearing on the nature of that universe itself.  (As for music forming the basis for mathematics, contrariwise — forget it.) Properly looked at, too, this point of view is liberating.  Why indeed must a piece of music unfold according to an internal logic as tight, dare I say foreordained, as mathematics?

Yet Schneiderman implicates even the august Princeton Companion to Mathematics as falling victim to such false comparisons (p. 107).  What’s going on here?  A desperation, even among mathematicians who should know better, to imbue mathematics with just a drop of the intuitive beauty we all know can arise from a fragment of music?  … Or something more sinister: a desire among the quantitative explainers of the world to own music as part of their own field, even if its internal richness is reduced thereby?  A notion of mathematics as the golden road to absolute knowledge, outside of which further paradigms are unnecessary…?

To such attitudes, Schneiderman offers a throwdown.  Music, as he sees it, exists in a realm apart, and that realm’s dialogue with math’s goes both ways.  It’s a good thing, too: my love of math and my love of music combine to result in that much more love.

Math Therapy

If mathematics is supposed to be the underlying structure and grain of the universe, but so many people have trouble resonating with it, then why not math therapy — to restore one’s personal harmony with math?  Sometimes I think that, at its best, the part of my job that’s actually teaching math should approach this ideal.  When you peel away the techniques of classroom management, when you peel away the signposts to important facts and theorems — what, ultimately, I mean, are you trying to pass on?

Even more so recently than ever before, I’ve come to feel that the search for interesting and rewarding mathematical topics leads in a plethora of unexpected directions.  I imagine that for many of these, even calculus might be mostly or entirely unnecessary.  Why shouldn’t it be one of the multitude of natural states available to a person to make, and then try to answer, interesting mathematical inquiries?  Why shouldn’t it be an integrated part, if one chooses, of personal, of social life?

Maybe I’ll write a story about a math therapist.

Is Music Philosophy?

I’ve been reading David Sheppard’s biography of professional music-maker Brian Eno, On Some Faraway Beach.  Eno is perhaps best known as the popularizer of the “ambient” aesthetic in music, but there was a time, the late 1970s and early 1980s, when he was entranced by African musical sensibilities as well.  Sheppard records Eno, in that period, remarking about his “Fourth World” collaboration, the album Possible Musics, with trumpeter Jon Hassell: “We talked about music as embodied philosophy, for every music implies a philosophical position even when its creators aren’t conscious of it.”  Eno would soon take his philosophy of Africanized rock to the band Talking Heads, for the creation of whose album Remain in Light he would, Sheppard implies, exert an undue amount of control as producer — perhaps swamping the contributions of some of the band members in the process.

Eno’s remark — music as philosophy — is reminiscent of anti-web-2.0 technologist Jaron Lanier’s proposition, in his book You Are Not a Gadget, that “We [technologists] make up extensions to your being….  These become the structures by which you connect to the world and other people.  These structures in turn can change how you conceive of yourself and the world.  We tinker with your philosophy by direct manipulation of your cognitive experience, not indirectly, through argument.” (pp. 5-6) It’s a persuasive idea when Lanier states it, and it’s key to his book: he infers that technologists really need to think about the philosophies that their creations would foster before releasing them on the world (and that, all too often and to deleterious effect, they don’t).

Is Eno’s version of the argument as persuasive?  Don’t get me wrong: I’d love for music to be, or even to imply, philosophy, thus forging a fierce link between two of my strongest interests; but, after listening yesterday to Eno’s collaborative album with German band Cluster, After the Heat, with this question in mind, I didn’t feel so sure.  Yes, Eno’s music is more philosophical, in the sense of “direct manipulation of your cognitive experience”, than most; few musicians seem as conscious of the different contexts, at least, in which their music can be listened to.  But, it’s notable that nowhere does Sheppard record exactly what the philosophies Eno links his music to actually consist of.  At times, actually, the picture Sheppard paints of Eno reminds me of Dean Moriarty, the Neal Cassady character in Jack Kerouac’s On the Road:

“And he said, ‘Yes, of course, I know exactly what you mean and in fact all those problems have occurred to me, but the thing I want is the realization of those factors that should one depend on Schopenhauer’s dichotomy for any inwardly realized…’ and so on in that way, things I understood not a bit and he himself didn’t.” (p. 6)

Don’t get me wrong: Eno’s music is exciting, relaxing, mysterious: he runs a great gamut of emotions.  And, there’s something to be said for the high Beat aesthetic of continuous conversation about whatever, or as Allen Ginsberg put it in “Howl”, “whole intellects disgorged in total recall for seven days and nights with brilliant eyes”, whether the conversation makes ultimate sense or no — something Eno has, apparently, been pretty much fantastic at for a long time.  But, so far — and, maybe it really is just because music has so much less to do with lifestyle in the here and now than in Eno’s heyday — I’m going to have to guess the pessimistic answer to this post’s title question: No, music is not philosophy; or, at least, it hasn’t proven that it is.  It’s okay, though: actual philosophers, anyway, can rest easy.

Mathematics, Interpreted Freely

I’ve been interested for a while in Princeton University Press’s annual Best Writing on Mathematics series, edited since 2010 by Mircea Pitici.  In his introduction to the 2015 edition, Pitici writes:

“Interpreting mathematics points toward protean qualities of mathematics not immediately obvious in doing mathematics per se.  An accepted mathematical result is merely the egalitarian premise from which each of us can part with the commonly shared view by interpreting it idiosyncratically, as we please or even as it suits us.  … Lack of reflection on the proper context of applied mathematical thinking perverted the humanities, the social sciences, and even the study and practice of law — to name just a few examples.” (p. xiv)

What a liberating view of mathematics and its relation to the world!  Pitici’s 2012 edition, too, includes an article by one Ian Hacking entitled, “Why Is There Philosophy of Mathematics at All?”.  Hacking argues for two main answers to his title question: one, that it is “astonishment that engenders philosophy of mathematics” (pp. 238-239) — astonishment, that is, that certain abstruse but awe-inspiring mathematical objects are even out there — combined with, two, an idea from “Mark Steiner… who asked more or less my title question, ‘Why is there philosophy of mathematics?’ He answered, in effect, application.” (p. 245)

And, for all those who’re worried that appeals to applied mathematics might sully the philosophical purity of math here — well, never fear, because Hacking distinguishes seven kinds of application, only a couple of which would normally be considered applied math (and the failure to distinguish between which, I imagine, could well have caused Pitici’s “perversions”).  They are: “Math Applied to Math” (about which Hacking states, “Why should there be so much ultimate connectedness behind so much apparent diversity?… This question needs a lot of philosophical work, right now” (p. 248)); “The Pythagorean Dream” (in which “mathematics of a deep and simple sort really is just the structure of reality” (p. 249)); “Mathematical Physics”; “Mission-Oriented Applied Math” (these last two being perhaps what most people think of as applied math); “Common or Garden”; “Unintended Social Uses” (i.e., to preserve an elitist hierarchy, which I find the least interesting inclusion on Hacking’s list); and finally, “Off the Wall”, for which Hacking quotes Wittgenstein: “Why should not the only application of the integral and differential calculus etc. not be for patterns on wallpaper?  Suppose they were invented just because people like a pattern of this kind?  This would be a perfectly good application.” (p. 251)

… The point that Pitici and Hacking are striving for, it seems to me, is that, yes, there is something unsettlingly alien about mathematics, something whose exact nature philosophers have disagreed about for eons, but that, in fact, this very alien quality is what creates (mostly unrecognized) freedoms in how we relate mathematics to the experiential world.  And, in my own experience teaching math with physical objects, certainly, I find a delicate tension between defaulting to instilling the conventional “what you see is an approximation of something mathematically ideal” notions, and something wilder.  The objects that we use to model mathematical ideas never seem to end up being as precise as we want them to be; in a mathematically governed universe, I mean, why would it take so much effort to create something in order to demonstrate mathematics?

de Gua’s Theorem

Last week I was first made aware of de Gua’s Theorem about triangle areas in 3-dimensional space.  It’s (I find) a supremely pretty theorem, easy to state, and not horrendous to prove either: I even attempted to demonstrate a proof I had come up with to some precalculus students this past week.  (Anyone who’s interested in a proof should let me know!) So why had I never heard of it before?  This is the kind of theorem for which I expect several gorgeous demonstrations….