Patterns, Play, & Processes:
Sounding Topological Graph Theory
Harvard College ‘22
Undergraduate Thesis in Mathematics and Music
By Jessica Shand
Advised by Peter Kronheimer (Mathematics) & Christopher Hasty (Music)
The following consists of written excerpts and recordings from the introduction and appendix to the undergraduate thesis Patterns, Play, & Processes: Sounding Topological Graph Theory (2022), an exposition on topological graph theory featuring five original musical works. The full text is available online.
Introduction
It is at the strange and unforeseeable intersections and confluences of people, of ideas, of memories and dreams and visions—more often than not, in spaces far removed from the halls of academia—that innovation, disruption and, above all, togetherness and community may blossom. In these moments of in-between, that which has always been taken to be true and inevitable may for once be thrown, even if temporarily, into doubt. If the reader-listener were to take just one message away from the present work, we hope this might be it: that at the interstices, in the gaps between seemingly disparate languages and domains, we may find the space to imagine, and then bring into being, entirely new modalities of thought, expression, and experience.
The teaching of any subject relies upon assumptions, made implicitly or explicitly, regarding the nature of that subject. As a discipline of study, mathematics is no exception. In the words of René Thom, the French topologist best known as the founder of catastrophe theory in mathematical dynamics, “all mathematical pedagogy [...] rests on a philosophy of mathematics,” whether that be a personal philosophy or a codified, institutionalized one (Thom 1973). This has important consequences not only for the transmission of mathematical knowledge in and out of the classroom (Hersh 1979), but also for the generation of new insights in the field, and therefore for the development of mathematics as a whole. From a Foucauldian point of view, the mechanisms of hierarchical observation, normalizing judgment, and examination amount to nothing less than the regulation of one’s thoughts and behaviors (Foucault 1995). In the case of mathematics, these kinds of mechanisms—filtered through the lens of a given philosophy on the subject—in turn produce and reproduce inextricable links between power and knowledge (known in sociology as le savoir-pouvior, or “power-knowledge”) in the field and, to a degree, in society more broadly (Foucault 2008).
Falling back on long-standing lineages and traditions in the field, the vast majority of contemporary researchers and professors of mathematics share an absolutist view of the field. As a philosophy, absolutism has, of course, taken on a variety of meanings across many different intellectual contexts. In political philosophy, absolutism is understood in terms of the practice of unlimited centralized authority, as observed in royal monarchies and dictatorships; in Newtonian physics, absolutism regards space and time as independent aspects of objective reality; and in moral philosophy, absolutism refers to the existence of absolute standards against which all moral questions can be judged. The common thread across each of these framings, of course, is the presumption of the existence and certainty of the absolute. Accordingly, in a mathematical context, absolutism asserts that mathematical knowledge is in some sense universal, objective, and atemporal.
The earliest known presentations of mathematical knowledge consist of logical deductions of theorems from postulates and axioms, as in Euclid’s Elements, the classical Greek treatise on geometry and elementary number theory. Considering that the axioms of such texts were not regarded as temporarily adopted assumptions, but were rather taken to be basic truths, requiring no justification beyond their own self-evidence (Ernest 1998), we ought to locate the origins of absolutism as a philosophy of mathematics at least as far back as antiquity. Much later, in the seventeenth century, René Descartes renewed the epistemological quest to find absolute foundations for mathematical knowledge as a central pillar of all human knowledge. It is Gottlob Frege, however, who is credited with setting the tone and agenda for modern philosophies of mathematics. Indeed, following Frege, all three of the major schools of thought in the philosophy of mathematics since the early twentieth century—logicism, formalism and, to a degree, intuitionism—fall under the umbrella of absolutism as it has been described here.
In the 1910s, the absolutist idea of mathematics as a source of infallible knowledge motivated Alfred North Whitehead and Bertrand Russell to attempt to provide a rigorous foundation for all mathematical knowledge in the three-volume Principia Mathematica (PM), whose title comes directly from Newton’s own Philosophiæ Naturalis Principia Mathematica. One would expect that Kurt Gödel’s famous theorem that no finite system can be used to derive all of mathematics—the proof of which arrived only a couple decades after the publication of PM—would have deemed the logicist program of Whitehead and Russell all but a fool’s errand. Nonetheless, absolutism has persisted in mathematics well into the twenty-first century as the dominant underlying philosophy of mathematical pedagogy, research, and discourse.
For an example, consider the following exchange between the author and a former faculty member of Harvard University’s mathematics department, which took place at an undergraduate advising fair several years prior to writing:
Author (A): What does research in pure mathematics look like?
Professor (B): What do you mean?
A: I guess what I mean to say is that I’m considering pursuing pure mathematics as an undergrad, but I hesitate because it’s not clear to me what the research process looks like. Especially compared to applied mathematics, where you have these more empirical problems that you’re working on, with very practical applications. Like, do you just come up with theorems and stuff?
B: (Chuckles). Ah. Not quite. (Looks away). It’s not very easy to describe. But if you’d like to try it sometime, what I would say is that...I have a large, plain, white wall in my office. If you wanted to do research, I’d pull up a chair for you, and we might spend three or four hours staring at one particular spot on that wall. Then we might choose another spot to fixate on, mostly in silence, perhaps muttering here and there. Then we’d call it a day. (Looks back at A).
A: Okay, thanks! (Applied math it is?)
We do not include this dialogue to suggest that one person’s attitude completely characterizes mathematics pedagogy or research at one of its leading institutions, let alone in the much wider world surrounding the field. However, we invite the reader to recognize that this is far from the only anecdote of its kind, and as such, we hope that it adds color to the specter of absolutism in mathematics, which we conceive of not as an abstract idea or theory, but as an observable phenomenon with real-world consequences. As Paul Ernest points out, for one, it cannot be considered a coincidence that the absolutist position corroborates public images of mathematics as “difficult, cold, abstract, theoretical, [and] ultra-rational, but important and largely masculine” (Ernest 1996; emphasis added).* Ultimately, what this means is that the ways we learn, share, and talk about mathematics have profound implications for our shared perceptions of what mathematics is and, relatedly, what it means to be a mathematician.
*It is worth emphasizing here that as constructions of gender difference are contingent and unfixed, the last of these characterizations—the idea that mathematics is somehow masculine—relies on a specifically Western framework of hegemonic masculinity, as theorized and expanded upon in gender studies since the 1980s.
Sounding Mathematics
In this joint thesis in mathematics and music, we focus our efforts both directly and indirectly on addressing only one facet of absolutism in mathematics, one that has at some point touched every domain of academia: subject-object duality. Rejecting this ideology of binary opposition between subject and object, which is steeped in Cartesian dualism, we assert that mathematics—as well as music, for that matter—is, in fact, objectless. We maintain that we can never fully remove ourselves from the mathematical structures, models, and abstractions we have created, just as we cannot separate a sound recording or a score from its moment of origin in time and space.
In music theory and musicology, the literature has only recently begun to adopt the language of intersubjectivity, which posits that reality is created by the interaction of subjective and embodied minds. Taking this framework a step further, actor-network theory (ANT) assigns agency in the construction of reality to all actors within natural and social networks of relations—including, for example, non-human animals and machines (Latour 2005). Our encounters with these kinds of frameworks in music scholarship are precisely what has led us to seek out alternative philosophies of mathematics that do away with subject-object dualism, instead calling upon interdisciplinary “maverick” paradigms (Aspray and Kitcher 1988). The viewpoint we have landed on at least for the time being, known as fallibilism, asserts that mathematical knowledge is, as the name suggests, corrigible, dynamic, and an ever-evolving work-in-progress that is dependent upon deeply human social and cultural practices.** And while fallibilism does not constitute any of what are considered to be the primary schools of thought in philosophies of mathematics, it was the view taken and defended, under various guises, by the likes of Wittgenstein and Lakatos (Ernest 1998).
This thesis is a creative one in that we introduce the reader to the subfield of mathematics known as topological graph theory by combining formal mathematical exposition with original songs and compositions, the latter of which are intended to elucidate a fallibilist take on the field. We emphasize and embrace the fact that in writing music “about” mathematics, we are engaging in a fundamentally non-neutral endeavor. In particular, the music of this thesis—and the discussion around it—offers three distinct, but neither mutually exclusive nor completely exhaustive, lenses into mathematical practice: storytelling (as a kind of pattern-seeking), games and puzzles (as a kind of play), and the interplay of rhetoric and revision (as a kind of process). It is our hope that these offerings may key the listener into a vision of mathematics and music as understood and experienced by only one member of the two fields’ vast ecosystems of participants and practitioners.
**In its foregrounding of human social and cultural practices, our view might be called humanist. However, following the precedent set in philosophy and the humanities, it may serve future research to go beyond even a humanist lens toward a post-humanist vision of mathematics, whatever that might mean. We speculate further on this in the conclusion to this work.
Why Topological Graph Theory?
Not unlike the name of one of the Harvard University Dining Services (HUDS)’s delectable lunchtime offerings, the grilled pizza sandwich, it may seem to the non-mathematician that every word in topological graph theory is completely unpredictable from the last. As we will show in this thesis, however, much of the field in fact arises naturally from several quotidian, easy-to-understand, and even frivolous problems and puzzles—problems and puzzles that turn out to have incredibly rich and complex solutions.
At its core, graph theory is the study of relationships in a network. The structures it deals with are familiar to our contemporary society; namely, by graphs, we mean structures of the sort we have all seen in family trees, in webs of strings secured to bulletin boards by push-pins, or in simple games of connect-the-dots. On the other hand, topology relies on notions of continuity, studying the properties of mathematical structures that are independent of transformations like stretching, squeezing, and twisting. The intersection of topological graph theory, then, conceives of the discrete structures of graph theory in terms of the continuous ones of topology. It uses topology to study graphs; occasionally, it also uses graphs to study topology, providing intuitively appealing combinatorial interpretations of subtle topological concepts (Archdeacon 1996).
Given its ties to manifestly culturally embedded activities, topological graph theory offers fertile grounds on which to explore some basic ideas of modern mathematics as the outcomes of social processes, rather than as absolute, intangible objects. We emphasize, however, that while graph theory in particular has long maintained a close relationship with practical motivations and applications, what we conceive of as social, cultural, and historical contingencies of mathematical knowledge in this thesis are not specific to this subfield. To the contrary, such contingencies can be observed even in the most unexpected realms of mathematics, in the areas that may seem the furthest removed from our day-to-day lived experiences.
How to Read—and Listen—To This Work
This thesis consists of a written document and five recorded audio tracks, not bound by any one genre or style, but connected together by the underlying question: what does it mean to practice mathematics? There is certainly no one correct way for the reader-listener to engage with the materials, though the following information may be of use in navigating them:
All of the tracks incorporate stereo panning, to differing degrees. Therefore, listening with over-the-ear headphones when possible may produce the most effective stereo image.
The “sound on” icon indicates the inclusion of an audio track in a given section. Track titles appear together with the section whose content or context most directly relates to the music, with two tracks bookending each of Chapters 1 and 2, and a final track concluding Chapter 3.
In the digital version of this work, each track can be accessed individually by clicking on the track title. To avoid navigating between many tabs or windows, all of the tracks may also be accessed from a single shared folder, which is available here.
It may be of interest to the reader to listen to a given track alongside or in conjunction with the section in which it is presented, but this need not be the case.
Information on the background for each track, the personnel involved in each recording (e.g., session musicians), and lyrics is aggregated in Appendix A. While informal notes on some of the ideas and inspirations for the tracks are included for completeness, the listening experience may also be a highly personal one, and the listener is encouraged to engage in a way that is most personally fruitful.
In general, we contextualize each chapter by introducing material relevant to the philosophical discussion in the main body of the thesis; Appendix A then offers a fuller extension and elaboration upon that discussion, both showcasing the music and grappling with the relevant philosophical questions. Keeping this in mind, the reader may wish to note that neither the main body of the thesis nor the material in Appendix A can be considered purely mathematical or purely musical in nature. Both components incorporate elements and pursue inquiries within both fields, and thus either or both may be of interest to readers of various inclinations.
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Appendix A: On the Music / Liner Notes
It is widely recognized that the advent of modern warfare has relied on the extensive cooperation of scientists and mathematicians with government intelligence agencies and militaries, which is exemplified by the operations of both the Allied and Axis forces during WWII. Prominent researchers and academics were recruited on both sides of the war for varying projects, according to varying ideologies, and with varying mixtures of reluctance and willingness. On the Allied side, one of the foremost graph theorists of the 20th century, W.T. Tutte, made breakthrough contributions to the cryptanalysis of the Lorenz cipher, enabling the British to decode Nazi German communications. And then there was the Manhattan Project, undertaken in the United States and assisted crucially by such mathematicians as Stanislaw Marcin Ulam, Richard Hamming, and child prodigies J. Ernest Wilkins Jr. and John von Neumann. In addition to designing the explosive lenses used in the Trinity and Fat Man bombs, von Neumann, a Hungarian Jew, infamously helped choose Hiroshima and Nagasaki as the first targets of the atomic bomb, overseeing computations to project death tolls and to determine “the distance above the ground at which the bombs should be detonated for maximum effect.”
Considered in light of the influence of mathematics and mathematicians on the war—from code-breaking to the creation of weapons of mass destruction and everything in between—Königsberg emerges paradoxically as a site of both creation and of destruction. It situates the blossoming of an entirely new vein of mathematics and mathematicians, a tradition and a lineage that, in some indirect yet non-negligible way, helped bring about the same military technologies that would physically destroy Königsberg—and many other cities, and ultimately some 3% of the world population over the entire duration of WWII—centuries later. Of course, one cannot even approach an understanding of WWII without carefully examining the full political, social, and historical context of the Holocaust, the racism and fascism of the Nazi regime, and the palpable, existential fear experienced by those who lived through its horrors. Nonetheless, the fact that rapid mathematical and scientific development took on a central role in wartime strategy to an extent that it hadn’t ever before, perhaps in all of human history, cannot be denied.
In grappling with these revelations in my own retelling of the story of the seven bridges, I thought back to a conversation I’d had with colleagues at the Artistic Freedom Initiative, the organization I’d been working at for the previous summer. We’d been talking about extreme performance art: self-mutilation, near-suicidal high-wire stunts and, yes, setting fire to public buildings. “There’s got to be a difference,” I’d said, “between art and...arson.” But now, having spent so many days and nights pondering the tragic fate of Königsberg and its residents, I’d begun to question what I had said that summer. To what extent have we drawn a line between artistic craft and “artfully” veiled forms of violence, to what extent can we?
We live in a society that glorifies brutality. Many among us idolize those whose craft it is to exploit human labor and natural resources for profit. And too often, we disregard the costs of scientific (and pseudo-scientific) research, an art of its own, in the uncritical pursuit of an ideal of progress. Ultimately, the same class in society whose definitions of violence have long dominated social, political, and legal discourse (as argued, for example, in Arblaster 1975) has also historically had disproportionate influence on what is and is not championed as representative of our culture, as the best of us, as art. Being part of a generation that is increasingly challenging this status quo, I feel an impulse larger than any one person to turn to increasingly extreme modes of performance—be it onstage, online, or in our everyday lives—simply to be noticed, to be heard. If anything, it seems, art and arson go hand in hand.
Lyrics:
young in our old town, we heard a tale about
tracing figure eights on seven bridges
legend had it, no one’d ever found a way around
without doubling back on the ground
we circled the river
but never worked it out
‘til they burned the bridges down
that golden summer as the army marched in
we gathered ‘round the fire, our faces aglow
and in unison we sang, war’s a wicked game
but someone, someone’s gotta win
abandon your guitar
load the ammunition
surrender to the rhythm
oh, aren’t we all done
waving our arms in the air?
I know you’d rather wield ‘em
but if I’m being honest, I’m afraid
it’s too late
to get ourselves back
to the garden
you’ll hope it’s over when the smoke’s all gone
and cobblestone has made way for concrete
but we’ll know it’s not done, that’s art and arson
chasing one another ‘til the dawn
so remember this song
and play on
play on
The late Israeli journalist Amos Elon gave a singularly chilling account of Kaliningrad upon visiting in 1993, barely two years after the city reopened following the collapse of the Soviet Union. “One cannot escape the uncanny feeling of the existence of the old Königsberg,” Elon writes, “like the negative of a damaged photograph, lying ten to twenty feet underneath the city’s surface, covered with rubble from the war and Stalin’s bulldozers” (Elon 1993). No matter the utter destruction and transformation that the landscape had undergone, and certainly irrespective of its renaming after the Soviet functionary Mikhail Kalinin, visitors remarked consistently on the ghostly presence of the past that gave Königsberg/Kaliningrad the feel of a “parallax city” (Amundsen 2020).
Following reopening, former residents and the children of expellees and refugees who had fled during the war also returned to the city in large numbers. Their journeys came to be known as Heimwehtourismus, or “homesick tourism,” a phenomenon compelled by nostalgia and (be)longing while also often entangled with trauma and loss, and distinguished by the assumption of the tourist gaze (Marschall 2015). In other contexts, assuming the tourist gaze means looking with interest and curiosity at that which differs from the familiar, from home. It is the perception of the other (Urry 1990). What does it mean to be a tourist in one’s own place of origin, taking notice not of that which is different, but that which is—unexpectedly, unsettlingly, unimaginably—the same? To come to terms with the seemingly inevitable repetitions of history, the circularity of time that can be apparent even within a single lifetime?
This song is a meditation on that sensation, reflecting on my own experience of leaving one home in Colorado to return to another at Harvard during COVID-19. An eternity and yet a whirlwind, nearly a year and a half after my classmates and I parted our ways amid the chaos and uncertainty of a global pandemic. Much of the human interaction we’d had for quite some time had been entirely technologically mediated; to accommodate social distancing guidelines, we’d done everything from celebrating birthdays and socializing with friends to taking classes and attending doctor’s appointments through teleconferencing platforms, usually Zoom. Upon finally getting back to Cambridge (after working from home for the year), I now knew only a small proportion of the student body. Most of my friends had chosen to finish up school online, and had since moved on to graduate school, full-time jobs, or any other of a number of possible real-world occupations and endeavors.
During the fall semester, I was stricken to see boys strolling along the streets in suits well after midnight, vying desperately to get into one of Harvard’s all-male final clubs—just as they had in that distant pre-pandemic time. I’d find myself walking into the same lecture halls full of twenty-somethings, squandering those sparing minutes between classes with noses buried in smartphone screens, tablets, laptops. Different people doing all the same things, in all the same places. A close friend of mine, and a talented journalist in her own right, put it best: it felt as though we’d borne witness to the next installation of a rotating cast of characters, come in to replace the last. I had to recognize that if such a cast existed, then we, too, were part of the play.
Lyrics:
remember those winter nights
the snowfall, a blanket of stars
we’d be warm inside (weren’t we warm inside?)
seventeen and superstitious, dreaming up a storm to keep us home you were home
the faces change
but it’s all the same
it’s all the same
now I walk these busy streets, never felt so cold
the cartoons we cozied up to, I’ve outgrown
still, I’ve got one foot in the past, rushes back every time I’m in a crowd ’cause twenty-two’s enough to spot the same old characters come back around someday I’ll come around
the faces change
but it’s all the same
it’s all the...
disembodied voices on your stereo
(now I’m the) disembodied voice on your stereo
The wish to regain contact with a playful, childlike state of being—a state of wonder and enlightenment devoid of attachments and prejudices—appears throughout much of classic Eastern philosophy and art. In Chapter 28 of the Tao Te Ching, the an- cient Chinese philosopher Laozi writes: “Being the stream of the universe / Ever true and unswerving / Become as a little child once more” (Laozi 1989). The idea of un-learning or deconstructing learned biases and behaviors as a necessary step to liberating and rewriting the self also features heavily in Indigenous thought and Black radical aesthetics. In the song “Daybed,” the creative polymath FKA twigs sings: “Jaded is my father / Childlike is my answer.”
This song takes as its inspiration the tension between this view of childhood as a state of unadulterated imagination and freedom on one hand and, on the other, the commodification of the child prodigy in Western society, which can traced back to the invention of the child celebrity performer in 18th- and 19th-century Europe (Graus 2021). The connections between and among music, chess, and mathematics as a “holy trinity” of prodigious talent have led to the shared belief, in some circles, that these “are [the] only three major areas of human endeavor which produce prodigies” (Zographos 2017). Lying at the intersection of two of those three areas, I’ve long felt troubled by the fetishization of the child prodigy, and feel passionately that it is past time for a reckoning with the role of institutions—and people—in confusing the lines between play, excellence, and the looming specter of the Faustian idea of genius.
Lyrics:
another day, another piece to play
the lights go down, curtains come up
someone announces your name
the crowd cheers right on cue
they’re all here to see you
one in ten million, you take to the stage
a flick of your wrist and it’s off to the races
the boy with the bunny-eared laces
ooh, ooh ooh, ooh ooh, ooh...
not a flaw in your allegretto
octaves in your left hand, twinkling in your right
you move on to the adagio
furrowing your brow
don’t they see it now?
can’t help but think that they’d all be mistaken
if there were really a lonely heart aching
in your sweet, solemn melody
ooh, ooh ooh, ooh ooh, ooh...
ooh, ooh ooh, ooh ooh, ooh...
take a bow, give an encore, they’ll say
at the end of the show, will they even know
you’re more than an entertainer
at the end of the show
I hope you’ll know
I hope you’ll know
In Mikhail Bakhtin’s theory of the carnivalesque, the transgressive, farcical, re- bellious rites and revelries that characterize medieval carnival enact a kind of play on the social and political order—the rigid dogmatism and mysticism of the high church, the greed of the nobility (Ehrenreich 2006). To some perhaps more frivolous extent, this is what the solution we gave to the three utilities problem feels like to me: an incongruity between reality and what we have constructed, imagining alternative realities in which we can live on a donut-shaped planet, or a non-orientable one.
In the vein of this idea of reimagining present realities, I wanted this song to emerge as a synthesis of organic and electronic sounds. All of the percussive effects come from the inside of a piano (applying varying levels of processing) or from field recordings taken in my home state of Colorado. The whimsical sound world draws from a lazy afternoon in the mountains near Breckenridge, a vibrant ski town by the Tenmile Range of the Rockies. A group of close friends and I had been hiking in the summer heat when we stumbled upon a hidden reservoir. As the sun brushed the sky with streaks of pink and orange, we waded in the cool water, skipping stones and marveling at the concentric circles radiating across the surface. We were the only ones there, in that idyllic opening in the woods. Instinctively, I pulled out the handheld recorder I’d been toting around for several months and set it down just above the shoreline.
I didn’t listen back to the recordings until months later, wishing to hold onto those hours as I remembered them. When I did, however, it felt as though I’d opened up a portal into a past life, re-discovering the sound of waves and water. In line with the theme of subverting hierarchies—but now, in the face of laws of physics that no one, not even the most powerful, can transcend—I wondered, what would our lives be like if time were a spatial dimension, or another valley to be crossed, or a material thing to be touched, tossed, skipped around, played with?
Lyrics:
strange and beautiful
as nature is alive...
liquid, like time
a corner to peer behind
a hill to roll down
the tick, tick
melting
moldable
what would it be like?
Recorded amid the March 2022 Russian invasion of Ukraine, which has triggered the mass exodus of nearly two million people at the time of writing, “cat’s cradle” eerily comes full circle back to the discussion around the first track in the collection. Perhaps, as one computer science professor suggested in a morning lecture, we are only now witnessing the unraveling of the systems we put into place to ensure that we would never repeat the horrors of World War II. Perhaps it was only a matter of time. In particular, considering the renewed criticisms levied against American imperialism in light of Russia’s unprovoked advance, it seems equally plausible that these systems—which, to be sure, rely on policies of mutually assured destruction—were never really meant to be secure. Not for the long term, that is. And certainly not for everyone.
In stark contrast with the first few tracks of this collection, which foreground the vocalist as narrator, here I swing toward a vision of music in which nonsense phrases and pre-verbal vocalizations come together with psychedelic, heavily processed guitars to create a wash of sound. In this vein, I draw aesthetic influence from Cocteau Twins, the Scottish band best remembered as an early pioneer of dream pop in the 1980s and ’90s.
Lyrics:
take me away the raven calls
embracing letters and lamp shades roll the dice
everything returns and renews itself
take me away the raven calls (blue ay mo)
embracing letters and lamp shades roll the dice (blue ay mo)