Angie Hugueth (*), Fernando Zalamea(*)
Abstract. We study some global connections between Moby-Dick themes and Grothendieck’s mathematics, and we trace all known local references to Moby-Dick in Grothendieck’s writings. We explore Hugueth’s Moby-Dick drawings and watercolors, and we relate them to some Grothendieck mathematical, methodological, and philosophical concepts.
Keywords. Grothendieck, Melville, Moby-Dick, Mathematics, Literature.
1. Infinitude, depth, abysses, and contraries in Moby-Dick
Between 1849 and 1850, at thirty years of age, Herman Melville (1819-1891) produces Moby-Dick, or The Whale at an unbridled pace, a disproportionate literary work where genres are crossed and voices multiplied at the scale of the cosmos, in a tremendously complex attempt to reflect the world’s own multiplicity with loyalty in one multi-referential literary structure. Moby-Dick unites two of Romanticism’s great strands: the search for the infinite, the absolute, and the transcendent with the polyvalent, almost pantheist, fusion of the Self with the World. Ahab and Ishmael’s voices predominantly incarnate these two strands, but never bluntly and always combining resonances of one with the other. Many contrasts recur in the work –revolutionary mix of novel, tragedy, comedy, practical excursus, and philosophical treatise– but its great wealth is found in the extensive relational plot of intermediate echoes and reflections delineated on an underlying dual warp’s support – good and evil, light and shadow, salvation and condemnation, hope and pain, communion and solitude, transcendence and contingence. Ahab and Ishmael are often confounded in the same voice out of the depths, close to the heart of the universe’s mystery.
As he starts his chapter on “Cetology,” Melville indicates to us that, “Already we are boldly launched upon the deep; but soon we shall be lost in its unshored, harborless immensities”[1]. Infinitude and the abyss progressively come to be symbolized in the great white whale through an extraordinary procedure of accumulation that takes us from the real to myth. In fact, the progressive mythical vision that the reader obtains from Moby-Dick results from the superabundance of realist details that, overall, gradually cease to be distinguished as particulars and come to be generically fused with one Whole, conceivable only as a ‘transcendent’ dimensionality. In line with the Romanticist vision on how art and philosophy embody the Absolute in the particular and the general, the realist summa of particulars that Melville undertakes allows him to fuse literature and metaphysics, and to access the genericity of myth. This is powerfully reflect in some of Frank Stella’s sculptures in his Moby-Dick series (1986-1997)[2] (see Figure 1).
Figure 1
The Dying Whale
(Frank Stella, 1990, 101 x 110 x 88 in., aluminum on iron base)
Departing from the etymology of ‘whale’ (from the Danish hval, hvalt, rolling: prelude to the plunge in the abysses) and from the first eleven pages of epigraphs about the whale, until the last considerations about its survival, going through extensive notes on cetology, entire chapters on anatomy (dismemberment, head, forehead, nut, tail, skeleton), on metaphysical symbolism (whiteness, monstrous representations), or on variegated and contrasting data (history, gastronomy, fishing chronicles)[3] the myth as unfathomable summa of the real, as unlimited depth of reality, is built (see Figure 2).
Figure 2
The Lee Shore (with A Bossom Friend)
(Frank Stella, 1994-1995, 14 x 15 x 10 feet, stainless steel)
The patinas of myth are built through an accumulation of diverse pigments of reality, just as white can be seen as a coalescence of the other colors. In “The Whiteness of the Whale”(ch. 42) this coalescing white is a symbol of joy, but at the same time of something profoundly elusive close to panic, of marvel and repugnance. White, like all myth, incorporates an enormous range of contraries, contradictions that, at heart, are what safeguard its permanent re-interpretative wealth. Melville’s writing accumulates multiple strata of reality, interpretation and imagination, and combines multiple voices in a polyvalent plot of novelistic, documental, theatrical, and epic fragments whose net result is the construction of the great myth of the white whale. The thick patina of myth acquires a notable profundity thanks to a full interrelation of deep theological and philosophical symbols with a complex literary architecture.
The maelstrom, the inscrutable tides, the vortices through which the infinite sneaks in, constantly underlie Moby-Dick’s physical and metaphysical background. The difficult dialectics between Nothingness and the Absolute, the tendency to identify opposites in the supra-sensible, are particularly well represented in images linked to the abyss (see Figure 3). The mad and hazardous map of the real, contrasted with Being’s decanted imaginary perspectives, the bleeding and violent world of the whale cut to pieces, contrasted with its high and clean symbolic whiteness, are only fully perceived from the shores of the abyss that separates them.
Figure 3
In-Ferno and In-Depth: Moby and the Maelstrom
(Angie Hugueth-Vásquez, 2023, 21 x 29 cm, paper rough, watercolor, pencil, and ink)
2. Melvillean Themes in Grothendieck: global approach
On a broad, global perspective, one can identify some central Melvillean themes in Grothendieck’s mathematical and philosophical thought: (i) a deep dialectic between types and archetypes, between particularity and generality, corresponding to the fusing of material differentials into the integral whole of the Whale, (ii) a counterpoint between calculation and structure, between casual waves and concealed currents, corresponding to the fight between the surfacing and finally sunk Pequod –with its types (Ishmael, see Figure 4) and archetypes (Ahab)– and the unfathomable Moby-Dick, (iii) a back-and-forth between injections and projections, between foldings and unfoldings, corresponding to the immersions and explosions of the Whale, (iv) a longing for beauty in the plumbing for deepness, a search for an invisible soul beyond blind materiality, corresponding to the blunt metaphysical searches in Moby-Dick, (v) a construction of a polyvalent and multivalent style, a constant crossing between areas of mathematics (geometry, topology, algebra, number theory), corresponding to Melville’s unceasing blurring of the passages between novel, essay, nautical treatise, philosophical derive. All these themes acquire in Grothendieck a particularly powerful incarnation, through extremely detailed technical inventions (Grothendieck was proud to have introduced more than a thousand definitions in mathematics), in the same vein as Melville achieves the poignant lyrical tone of the novel through extremely detailed descriptions.
Figure 4
Llamadme Fernando (counterpoint imagery to “Call Me Ishmael”)
(Angie Hugueth-Vásquez, 2017, 29 x 21 cm, match box, paper, watercolor, pencil, and ink)
Our contemporary world has exacerbated a World Wide Web,[4] which was already clearly imagined in Moby-Dick. Grothendieck’s mathematical creativity responds in turn to an extremely sophisticated mathematical wide web. His famous methodological metaphor, immerging a nut in a tide in order to soften its hard shell and access softly its fruit, convokes large perspectives and large times, and situates mathematics in Braudel’s long duration, beyond circumstances and events. This is precisely the same mood present in Moby-Dick, where all discrete human follies get outweighed by the continuous presence of the Whale, Nature, and God. When Grothendieck traverses the oceans of locally compact topological vector spaces and nuclear spaces (Ph.D. Thesis, 1953), when he navigates around Banach and Hilbert spaces relating them through Bourbaki’s three mother structures, topology, algebra, and order (Résumé, 1953), when he plumbs the intimate structure of abelian categories and introduces infinitary axioms to insure the existence of enough injectives (Tôhoku, 1957), or when he discovers the invisible sheaf structure underlying Riemann-Roch’s theorem (K-Theory, 1957), just to mention his four outstanding works of the first decade (1948-1957),[5] we see that Grothendieck probes mathematics in an acute Melvillean style.
A search for deep archetypes, for Universal Relatives[6] which govern a multitude of illusions, either in Literature or in Category Theory, is a central Melvillean and Grothendieckian aim. In Melville, this is obtained through myth and metaphysics, formed by an accumulation of concrete descriptions. In Grothendieck, the same goal is attained through functorializations and sheaf constructions, elevated over a multitude of concrete examples, in number theory, algebra, complex variables, topology, homology, and homotopy. Sinked layers of invisibility govern material layers of visibility (our “Blind Eyes”, in Tarkovsky’s adage). On another hand, in an inverse movement (which recalls Florensky’s Inverted Perspective),[7] Melville finds in specific local details a reflection of a global warp, in the same vein as Grothendieck captures in precise local definitions a reflection of global dialectics. A back-and-forth between concrete particularity and abstract generality is in fact a common cognitive strategy for both Melville and Grothendieck.
3. Grothendieck and Moby-Dick: local approach
In a letter to Yuichi Tsuji, the Japanese translator of Récoltes et semailles (1983-1986), Grothendieck shows his enthusiasm for Moby-Dick:
The “White Wale” is a symbolic image, taken from the well-known book Moby-Dick (also the name of the whale) by Melville – the book is the story of the chase of the “White Whale” by captain Ahab, and it ends with the complete shipwreck of the vessel, while the whale runs forever… It is the symbol of the elusive ideal, always out of reach, even when one feels close to capture it, and that fascinates some men thrown into its search, that only live for it, and in the impossible hope of reaching it. I recommend you the book (one of the most beautiful that I know), and the author, Melville, certainly one of the rare truly original spirits of the last century, a great man…[8]
Other explicit references to Moby-Dick and Melville appear in Grothendieck writings: “The spirit, thrown in the chase of the elusive flesh of things, rides as Ahab behind the White Whale”[9]; “Thoreau forms, with Whitman and Melville the «Great Trinity» of American literature in the 19th century”[10]. Also, from a remembrance of the mathematician John Tate, we know that Grothendieck had available Moby-Dick at his night table, and that it was his favorite novel.[11] From another record of Karin Tate (1958), we are told that Grothendieck offered his wife Mireille a copy of Moby-Dick, to supposedly improve her frail domain of English (a typically naive move by Grothendieck, for whom everything was easy)[12].
As Ahab, Grothendieck looks for an “elusive ideal, always out of reach” and lives only for it, “in the impossible hope of reaching it”. The abstract ideal (the absolute Galois group, its combinatorial Grothendieck-Teichmüller version, motives, or derivators, for example) transcends all its concrete approximation strata (Teichmüller tower, standard conjectures, localizers). The amazingly precise anatomy of the Whale according to Melville, reminds us of Grothendieck’s extremely detailed anatomy of schemes and toposes. The sinking of the Pequod in the abysses is reflected in the never-ending plunging in La Clef des Songes (1987). An access to a universal Great River in La Clef is obtained through an infinite sum of little things of our everyday life, in the same way as the elusive Moby-Dick is perceived through a meticulous, material description of the whaling life. We are in presence of forms of accessing the invisible through a sort of depletion of the visible, where deep Romantic forces permeate both Melville and Grothendieck. Ahab’s despair –”Science! Curse thee, thou vain toy”[13]– resonates in the anathemas of the first epoch of Survivre (before trying to “survive and live”: Survivre et vivre) and in the prophetic curses over the Great Mutation in La Clef des Songes. The Self must be annihilated in order to become World, thanks to an intense back-and-forth between sensibility and suprasensibility, between heart and spirit, proper of the great visionaries, Melville and Grothendieck.
Yoneda’s Lemma (in fact, found simultaneously by Grothendieck in his Tôhoku) reveals a philosophy naturally in tune with Melville. On one hand, an autoreference, or reflexive iteration (through objects reflected in their representable functors), is the one which offers a deep understanding of a given situation. A myriad of morphisms related to an object, captured by its representable functor, is in fact the only full and faithful way to understand the object. In the same way, only the polyvalent (and sometimes overwhelming) descriptivism of Melville, through whaling techniques or philosophical disquisitions, is the one which can finally offer a full and faithful image of the Whale. On another hand, Yoneda’s Lemma proves that our restricted vision (objects in a category, and/or their representable functors) comes to be very little in comparison with the invisible presheaves which complete the category
, through the immersion
. A complex range of phantasmata (non-representable presheaves) offer new perspectives for Tarkovsky’s “Blind Eyes”, as both Melville and Grothendieck predicted.
In Récoltes et semailles, Grothendieck explores the complexities of mathematical creativity, exerting in particular a sustained analysis of his own achievements. Discovery is obtained in moments of “solitude”, where “kindness (…) love (…) contact (…) yin” are needed to be able to “listen to the voice of things”, and where, surrounded by “free air”, the “soul” can immerge itself in the profound structures of mathematics.[14] It is a plumbing of the depths, similar to what happens in Pip’s second fall from the whaling boat and his immersion in the inferior strata of the ocean in Moby-Dick:
The sea had jeeringly kept his finite body up, but drowned the infinite of his soul. Not drowned entirely, though. Rather carried down alive to wondrous depths, where strange shapes of the unwarped primal world glided to and fro before his passive eyes; and the miser-merman, Wisdom, revealed his hoarded heaps […].[15]
Wisdom lies submerged in the profound, in the infinite abysses of consciousness. Through successive foldings and unfoldings (close to the ubiquitous methodology of sheaves, sites, and toposes), the “soul lives the experience and reacts, is fixed or expanded, is folded or learns”.[16] A search for something “beyond” (either motives, toposes, universal groups, derivators, or the very White Whale) occurs through many particular residues and general signs.
Both in literature and mathematics, we are dealing with a new ars combinatoria, which solidly articulates the One and the Many, following precise local directions: (i) an stratification of knowledge (concrete mathematical regions, material approaches to the Whale); (ii) a ramification of interpretations (Yoneda’s philosophy, multitude of perspectives for the Whale); (iii) a recursive conceptual deconstruction – “dévissage” in Grothendieck’s terminology– (functors and natural transformations, iterated reflections between genres in Moby-Dick); (iv) a construction of relational webs between the deconstructed strata (sheaves and toposes, poetic accumulations and transferences in Melville). The result of non-dogmatic approaches, as Melville and Grothendieck offer, teaches us the possibility to cover a complex World with non-reductive actions, and with a wealth of metaphors and techniques which exponentiate the richness of reality through the multiplicity of the imaginative mind.
4. Hugueth-Vásquez’s Moby-Dick series
In this last section, we present a series of Hugueth-Vásquez’s watercolors inspired on Moby-Dick, and some short comments by Zalamea on these pictures, relating them to the common Melville-Grothendieck themes we have explored in our previous sections.
A main strategy in the work of Melville and Grothendieck consists in always contemplating opposites simultaneously, in weaving the recto and verso of a given situation (poetical, mathematical, philosophical). One can sense and wonder on these opposites between the visible and the invisible, in Serenity (Figure 5a) and Dual Serenity (Figure 5b), where Moby-Dick appears concealed below the surface, but a look at traverse light reveals the hidden Whale.
Figure 5a
Serenity
(Angie Hugueth-Vásquez, 2024, 53 x 38 cm, paper rough, watercolor, pencil, and ink)
Figure 5b
Dual Serenity
(Angie Hugueth-Vásquez, 2024, 53 x 38 cm, paper rough, watercolor, pencil, and ink)
The mariners of the Pequod[17], One and Many, represent both the singular person, lost in the immensity, or the whole of humanity, lost between panoplies of ghosts. Beyond tempestuous seas[18] which submerge our fragile surroundings, “invisible winged creatures (..) frolic all around us”[19]. Bond (Figure 6) offers a very powerful vision of the evasive connections between Mind and Nature, between Humanity and an Emancipated Whale (Figure 7), where our luring pride rests ignorant over forces which profoundly transcend it.
Figure 6
Bond
(Angie Hugueth-Vásquez, 2024, 46 x 31 cm, paper rough, watercolor, pencil, and ink)
Figure 7
Emancipation
(Angie Hugueth-Vásquez, 2024, 46 x 31 cm, paper rough, watercolor, pencil, and ink)
A gaze at the Self inserted in the Infinite is presented in Suspense (Figure 8). In Autoreference, air (thought) and sea (fierceness) cannot be distinguished, as the softness of a Blue floods everything, and gigantic leviathans transit everywhere (see Figure 9)[20]. A gate is opened to shadows and phantasmata, from which light is further derived, as a limit of penumbra. One can sense the presence of Poe’s Maelstrom beyond these approaches, that try to capture the essentials which evade our “Blind Eyes”. The gigantic whale turns out to be a small fish in the midst of general destruction.
Figure 8
Suspense
(Angie Hugueth-Vásquez, 2024, 46 x 31 cm, paper rough, watercolor, pencil, and ink)
Figure 9
Autoreference
(Angie Hugueth-Vásquez, 2024, 53 x 38 cm, paper rough, watercolor, pencil, and ink)
Hugueth-Vásquez’s watercolors incarnate forcefully some Melville-Grothendieck dialectics: (i) recto/verso, positive/negative, yang/yin, through Serenity and Dual Serenity (Figures 5, 5a), (ii) surface/bottom, lightness/deepness, visibility/invisibility, through Bond and Emancipation (Figures 6, 7), (iii) closeness/distance, present/future, variety/invariance, through Suspense and Autoreference (Figures 8, 9). The Self and the World, mediated by the eye of the Whale, in turn reminiscent of a maelstrom, appear in Suspense‘s prayer. An infinite iteration of images, generating an intricate stratification of knowledge, is exhibited in Autoreference; an escape is impossible, and there, the very tail of the Whale serves as a repository for the ocean, and as a funnel where everything falls in the abyss. In Bond, the Pequod rides supported over the Whale, without notice for our “Blind Eyes”, but the invisible back of the Whale knows better; the mariner becomes only definable through his unfathomable chase. Escaping from a thousand scars (discrete marks), Emancipation shows the soft swimming (continuous plasticity) of Moby-Dick, beyond the violent circumstances of whaling; the end of a dependence bond opens a way to transcendence. Serenity presents an environment far from being peaceful in the visible side of things (Pequod), which becomes contrasted in Dual Serenity with the calm underneath appearance of Moby-Dick. In this way, one can never be certain of a given appearance, finite or infinite, discrete or continuous, positive or negative. It is always in a mediation between opposites that knowledge, literature, and mathematics, exert their work.
5. Conclusion
Moby-Dick’s epic spirit encounters a natural counterpoint in Grothendieck’s mathematical works. Everything in Melville and Grothendieck points to a deep beyond, where our frail perception of materiality is augmented thanks to conceptual imagination. The wide range of the invisible is stridden through sophisticated literary and mathematical constructions. By a meticulous process of immersion, the writer and the mathematician plunge into the unknown, and are able to explore some universal archetypes which govern the differential multiplicity of types. Melville’s White Whale, or Grothendieck’s sheaves, sites, toposes, motives, derivators, offer multiple connections to understand better the complex fabric of the World. Diverse harmonic counterpoints between the Many and the One, between particulars and phantasmata, govern their holistic approaches to knowledge. From their commitment and hard work, emerges an aura of beauty which moves us intensely, and opens the way for a true understanding of life.
References
Artin, Michael, et. al., “Alexandre Grothendieck 1928-2014, Part 2”, Notices AMS 63 (4) (2016), pp. 401-413.
Cruz Morales, John Alexander, Ham, Lorena, Oostra, Arnold, Universales relativos. Festschrift Zalamea 2019, Bogotá: Editorial Nomos, 2019.
Florensky, Pavel, La perspective inversée suivi de L’Iconostase, Lausanne: L’Age d’Homme, 1992.
Jackson, Allyn, “Comme Appelé du Néant. As if Summoned from the Void: The Life of Alexandre Grothendieck”, Notices AMS 51 (4) (2004), pp. 1038-1056, 1196-1212.
Grothendieck, Alexander, Récoltes et semailles, manuscript, 1983-1986.
———————————, La Clef des Songes, manuscript, 1987.
Lowry, Malcolm, Under the Volcano (1940), A Critical Edition, Ottawa: University of Ottawa Press, 2015.
Melville, Hermann, Moby-Dick, or The Whale, Illustrations by Rockwell Kent, London: The Folio Society, 2009.
Wallace, Robert K., Frank Stella’s Moby-Dick. Words & Shapes, Ann Arbor: The University of Michigan Press, 2000.
Zalamea, Fernando, Grothendieck. Una guía a la obra matemática y filosófica, Bogotá: Universidad Nacional de Colombia, 2019.
Acknowledgements. We offer these watercolors and lines to the Memory of Frank Stella (1936-2024), whose Moby-Dick series (1986-1997) is perhaps the greatest artistic homage ever given to Melville and Moby-Dick.
(*) Center for Logic, Epistemology and History of Science (CLE), Universidade Estadual de Campinas (UNICAMP). Mail: aphuguethv@unal.edu.co; a212152@dac.unicamp.br, Instituto de Filosofia e Ciências Humanas. FAPESP Grant N. 2024/00087-3.
(**) Departamento de Matemáticas, Universidad Nacional (Bogotá, Colombia), and Centro di Studi Grothendieckiani, Istituto Grothendieck (Mondovì, Italia). Corresponding author. Mail: fernandozalamea@gmail.com. Web: https://unal.academia.edu/FernandoZalamea.
[1] Hermann Melville, Moby-Dick, or The Whale, Illustrations by Rockwell Kent, London: The Folio Society, 2009, p. 134.
[2] Robert K. Wallace, Frank Stella’s Moby-Dick. Words & Shapes, Ann Arbor: The University of Michigan Press, 2000.
[3] Ibid., XV, XVII-XXVIII; ch. 105, “Does the Whale Diminish?”; ch. 32, “Cetology”; ch. 67, “Cutting In”; ch. 74, “The Sperm Whale’s Head”; ch. 75, “The Right Whale’s Head”; ch. 76, “The Battering Ram”; ch. 80, “The Nut”; ch. 86, “The Tail”; ch. 103, “Measurements of the Whale’s Skeleton”; ch. 42, “The Whiteness of the Whale”; ch. 55, “Monstrous Pictures of Whales”; ch. 57, “Of Whales in Paint, in Teeth…”; ch. 45, “Affidavit”; ch. 83, “Jonah Historically Regarded”; ch. 65, “The Whale as a Dish”; ch. 82, “The Honor and Glory of Whaling”.
[4] Never ending warp and woof, Malcolm Lowry’s literary opus can also be understood as one of the major artistic mixtures of the 20th century. Trellis of echoes, repetitions, chants, conjures between life and work, interweavings of montages, collages, syncopated rhythms and counterpoints, superposition of times, places, flashbacks, turns, strides and reverses, sedimentation of complex strata and inversions, Lowry’s literary technique composes deep accumulations of signs. Under the Volcano invokes the endless multiplicity of the world through many webs. Cfr. Malcolm Lowry, Under the Volcano (1940), A Critical Edition, Ottawa: University of Ottawa Press, 2015.
[5] For a complete presentation of Grothendieck’s three major mathematical decades (1948-1957, 1958-1970, 1981-1991), see Fernando Zalamea, Grothendieck. Una guía a la obra matemática y filosófica, Bogotá: Universidad Nacional de Colombia, 2019, 618 pp.
[6] John Alexander Cruz Morales, Lorena Ham, Arnold Oostra, Universales relativos. Festschrift Zalamea 2019, Bogotá: Editorial Nomos, 2019.
[7] Pavel Florensky, La perspective inversée (1919)suivi de L’Iconostase, Lausanne: L’Age d’Homme, 1992.
[8] Letter signed in “Les Aumettes le 4.9.1986”. Cfr. Zalamea, Grothendieck, op. cit., p. 519.
[9] Alexander Grothendieck, Récoltes et semailles, PU66.
[10] Alexander Grothendieck, La Clef des Songes, N268.
[11] Cfr. Michael Artin, et. al., “Alexandre Grothendieck 1928-2014, Part 2”, Notices AMS 63 (4) (2016), pp. 401-413, citation p. 405.
[12] Cfr. Allyn Jackson, “Comme Appelé du Néant. As if Summoned from the Void: The Life of Alexandre Grothendieck”, Notices AMS 51 (4) (2004), pp. 1038-1056, 1196-1212, citation p. 1047.
[13] Melville, Moby-Dick, op. cit., p. 501.
[14] Grothendieck, Récoltes et semailles, op. cit., P5, P49-51, P27, P13, P16 (order of quotes, “P” stands for “Preface”).
[15] Melville, Moby-Dick, op. cit., p. 414.
[16] Grothendieck, Récoltes et semailles, op. cit., P9.
[17] See Rockwell Kent’s engravings “The Ship” (Chapter XVI.1), and “Biographical” (Chapter XII.2), in: Melville, Moby-Dick, op. cit., pp. 87, 72.
[18] Kent’s “The Candles” (Chapter CXIX.1), Melville, Moby-Dick, op.cit., p. 640.
[19] Kent’s “The Symphony” (Chapter CXXXII), Melville, Moby-Dick, op.cit., p. 692.
[20] Kent’s “Cetology” (Chapter XXXII.1), Melville, Moby-Dick, op.cit., p. 166.
