chapter 7

The Habit of Clinging to an Ultimate Ground1

Every scientific ‘fulfillment’ raises new ‘questions’; it asks to be ‘surpassed’ and outdated…. In principle, this progress goes on ad infinitum. And with this we come to inquire into the meaning of science…. Why does one engage in doing something that in reality never comes, and never can come, to an end?

—Weber, Science as a Vocation

The notions of real interest to mathematicians like myself are not on the printed page. They lurk behind the doors of conception. It is believed² that they will some day emerge and shed so much light on earlier concepts that the latter will disintegrate into marginalia. By their very nature, they elude precise definition, so that on the conventional account they are scarcely mathematical at all. Coming to grips with them is not to be compared with attempting to solve an intractable problem, an experience that drives most narratives of mathematical discovery. What I have in mind harbors a more fundamental obscurity. One cannot even formulate a problem, much less attempt to solve it; the items (notions, concepts) in terms of which the problem would be formulated have yet to be invented. How can we talk to one another, or to ourselves, about the mathematics we were born too soon to understand?

Maybe with the help of mind-altering drugs? That’s what you might conclude from the snippet of dialogue from Proof quoted in the last chapter, but you would be wrong. It seems Proof’s author got that idea by reading Paul Hoffman’s biography of Paul Erdős, the Hungarian mathematician famous for finding unexpectedly rich structures in apparently elementary mathematics; for spending nearly all his life wandering across the planet, crashing in the homes of mathematicians and inviting them to join him in solving problems, sometimes in exchange for small monetary rewards; and for saying “a mathematician is a machine for transforming coffee into theorems.”3 Heraclitus considered most people “oblivious of what they do when awake, just as they are forgetful of what they do asleep.”⁴ Hoffman depicts Erdős, in contrast, as the Dr. Gonzo of mathematics: “for the last twenty-five years of his life… [Erdős] put in nineteen-hour days … fortified with 10 to 20 milligrams of Benzedrine or Ritalin, strong espresso, and caffeine tablets,” with the result that he spent the last quarter of the twentieth century fully focused on the kind of mathematics at everyone’s disposal, seeing things there that perhaps he was the only one awake enough to see.⁵

Grothendieck, by his own account, spent much of his life dreaming his way across the same landscape, forcing a path into a conceptual future that ceaselessly receded, riding not a drug habit but rather a refined mathematical minimalism. “He seemed to have the knack,” wrote number theorist John Tate, “time after time, of stripping away just enough…. It’s streamlined; there is no baggage. It’s just right.”⁶ Grothendieck’s search for what his colleague Roger Godement once called “total purity” encompassed his physical existence as well: in São Paolo in 1953–1954, he subsisted on milk and bananas, and years after his withdrawal from mathematics, during a forty-five-day fast in 1990, he went so far as to refuse to drink liquids, bringing himself to the point of death in an apparent attempt to “compel God … to reveal himself.”⁷ And yet what he was really seeking, in life as in mathematics, continued to elude his grasp.

INCARNATION

Sometimes the curtain can be persuaded to part slightly and the elusive items can be approached with the help of punctuation—quotation marks, for example, which mathematicians use frequently, especially in expository writing, in at least four distinct (though overlapping) ways:

1.  For direct quotations (the way everyone uses them).

2.  For implicit quotations (of something that is often said in the field, as in the first two examples in item 3).

3.  To substitute for the cumbersome procedure of formal definition, in other words as a synonym for “so-called.” Thus Eric Zaslow writes:8

We call such a term a “correlation function.”

Such a case is known as an “anomaly.”

and

The quantum Hilbert space is then a (tensor) of lots of different “occupation number Hilbert spaces.”

The last example is more complex than the first two—visibly, Zaslow is making it clear to the reader that he has in mind a notion deserving a formal definition, but that it would be distracting to present it here and unnecessary as well, since the reader can probably figure out what’s intended.

But the really interesting use of quotation marks is

4.  When an analogy that is, strictly speaking, incorrect offers a better description of a notion—a better fit with intuition—than its formal definition. For example,

you can think of actions as “molecules” and transitive actions as the “atoms” into which they can be decomposed.⁹

Since the sentence has exactly the same meaning without the quotation marks, one has to assume that they have been inserted to stress that the invitation to (chemical) intuition is also explicitly an invitation to relax one’s critical sense. Some seminar speakers write such quotation marks on the board; others make the quotation-mark gesture, wiggling two fingers on each hand while uttering the problematic word; still others preface an informal explanation by something like Let me explain this in words,¹⁰ which is literally meaningless unless understood as a warning that formal syntactical rules are temporarily suspended, to be replaced by some other kind of mathematics.

More piquant examples can be found in situations where normal semantics do not suffice for communication among specialists. Since I remember that it was advisable, before the 1991 two-week workshop on Motives in Seattle, to use the term motive only between scare quotes—I did so myself—I am gratified to see that Grothendieck made repeated use of type 4 quotation marks when explaining why he introduced the notion, in his unpublished 1986 manuscript Récoltes et Sémailles (ReS):

One has the distinct impression (but in a sense that remains vague) that each of these [cohomological] theories “amount to the same thing,” that they “give the same results.” In order to express this intuition … I formulated the notion of “motive” associated to an algebraic variety. By this term, I want to suggest that it is the “common motive” (or “common reason”) behind this multitude of cohomological invariants attached to an algebraic variety, or indeed, behind all cohomological invariants that are a priori possible.11

Cohomological, or its noun form cohomology, is the technical name for a method, properly belonging to topology, that introduces an algebraic structure in an attempt to get at the “essence” (a word some philosophers write in scare quotes) of a shape. So if the topological essence of the infinity sign (see figure 7.1) is that it consists of two attached holes, cohomology is a way of doing arithmetic (addition, subtraction, multiplication) with these holes. For example

4(left holes) − 5(right holes)

is a legitimate formula in the cohomology of figure 7.1, which is also an (unorthodox) picture of the cubic equation y² = x² (x − 1) (see chapters 2 and γ), whose holes are, therefore, meaningful in number theory.

Structure, already encountered in the second paragraph, is a loaded term. Historians stress the Bourbaki group’s role in making structures central to their mathematical architecture, following the structural revolution in algebra undertaken in the 1920s and 1930s by Emmy Noether and her protégé Bartel Van der Waerden.¹² Grothendieck was an active Bourbakiste during the late 1950s, and the word structure occurs on 124 of the 929 pages of ReS. On page 48, for example: “among the thousand-and-one faces form chooses to reveal itself to us, the one that has fascinated me more than any other and continues to fascinate me is the structure hidden in mathematical things” (my translation; emphasis in the original). Weil was a founder of Bourbaki; his insight mentioned in chapter 2, the one that converted me to number theory, was that the geometric essence of a problem in number theory could be grasped by means of the algebraic and topological structure of cohomology. And the motive Grothendieck hoped would provide the proper setting for Weil’s conjectures is a structure that clenches this essence even more tightly. Grothendieck explains the choice of term by musical analogies:

Figure 7.1. John Wallis’s symbol for infinity.

Ces différentes théories cohomologiques seraient comme autant de développements thématiques différents, chacun dans le “tempo”, dans la “clef” et dans le “mode” (“majeur” ou “mineur”) qui lui est propre, d’un même “motif de base” (appelé “théorie cohomologique motivique”), lequel serait en même temps la plus fondamentale, ou la plus “fine”, de toutes ces “incarnations” thématiques différentes (c’est-à-dire, de toutes ces théories cohomologiques possibles).*

Note the word incarnation. A philosopher might understand this on the model of an example or instance that points to an “essence” whose place in Grothendieck’s text is occuped by the “common motive.” On the same page, and several more times over the course of his rambling, lyrical—and often irascible—manuscript, Grothendieck uses the term avatar in scare quotes, possibly for the same reason:

Inspired by certain ideas of Serre, and also by the wish to find a certain common “principle” or “motif” for the various purely algebraic “avatars” that were known, or expected, for the classical Betti cohomology of a complex algebraic variety, I had introduced towards the beginning of the 60s the notion of “motif”. [En m’inspirant de certaines idées de Serre, et du désir aussi de trouver un certain “principe” (ou “motif”) commun pour les divers “avatars” purement algébriques connus (ou pressentis) pour la co-homologie de Betti classique d’une variété algébrique complexe, j’avais introduit vers les débuts des années soixante la notion de “motif”.]

Motives still attract quotation marks of this type. In 2010 Drinfel’d could write that

[m]orally, K_mot(X, Q) should be the Grothendieck group of the “category of motivic Q-sheaves” on X.13

(Note the philosophically fraught word category, which has a precise meaning in mathematics; we’ll be seeing it again.) Drinfel’d adds parenthetically that “the words in quotation marks do not refer to any precise notion of motivic sheaf.”

Placing quotation marks around motif or motivic sheaf—or for that matter using the word morally, as mathematicians often do, as a functional equivalent of scare quotes—permits invocation of the object one does not know how to define in much the same way that the name-worshipping Moscow set theorists brought sets into being by giving them names. But readers of the French mathematics of a certain generation, especially that associated with the Bourbaki group, can’t help but notice that a taste for Indian (rather than, say, Russian Orthodox) metaphysics inflected their terminology. Weil referred to “those obscure analogies, those disturbing reflections of one theory on another”—an early avatar of Grothendieck’s avatars—writing that “[a]s the [Bhagavad]-Gita teaches, one achieves knowledge and indifference at the same time.” Pierre Deligne, whose work did much to embolden mathematicians to remove the scare quotes from “motive,” explains the use of the word yoga, using the visual metaphor of panorama:

In mathematics, there are not only theorems. There are, what we call, “philosophies” or “yogas,” which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement…. A philosophy creates a panorama where you can put things in place and understand that if you do something here, you can make progress somewhere else. That is how things begin to fit together.¹⁴

“The sad truth,” Grothendieck wrote in a letter to Serre in 1964, was that he didn’t yet know how to define the [abelian category of] motives he had introduced earlier in the same letter, but he was “beginning to have a rather precise yoga” about these nebulous objects of his imagination. One might argue that Grothendieck in fact made precise definitions and formulated equally precise conjectures that would retrospectively justify the “distinct … but … vague” impression of which he wrote, and that the use of the words yoga and avatar, like the quotation marks, is a theatrical effect that serves merely to make the impression more vivid. One might, with equal justification, ask why just these rhetorical techniques do, in fact, enhance vividness. Working with the various “cohomological theories” of which Grothendieck wrote—just as Deligne did, viewing them as avatars of a more fundamental theory of motives—may leave no logical trace but does represent a different intentional relation to the cohomological theories in question:

A mathematical concept is always a pair of two mutually dependent things: a formal definition on the one hand and an intention on the other hand. He or she who knows the intention of a concept has a kind of “nose” guiding the “right” use of the formal concept.15

The author of this quotation, who seems to be using the word intention in the way I intended,¹⁶ did well not to lift the veil of (type 4) quotation marks from that which is better left to the reader’s imagination. Attempts at greater precision lead straight to the threshold of the abyss of speculative philosophy, where one seeks to explain what it means to take phenomena (impressions, mental images) as symptomatic of something that remains concealed. This applies to intentions as well as to avatars—which is not to say that the (intentional) relation I have in mind is purely subjective. If anything is peculiar about the use of intentional in connection with avatars, it is that the concealed “underlying theory” of which the phenomena are supposed to be symptomatic does not yet exist, as if the mathematician’s role were to create the source of the shadows they have already seen on the wall of Plato’s cave.¹⁷ I limit my speculation to claiming that it matters to mathematicians what they think their work is about, whether or not it matters to the work, and that this ought to be a matter of concern for philosophers.

In chapter 3, I suggested that the goal of mathematics is to convert rigorous proofs to heuristics—not to solve a problem, in other words, but rather to reformulate it in a way that makes the solution obvious. As Grothendieck (ReS, p. 368) wrote:

They have completely forgotten what is a mathematical creation: a vision that decants little by little over months and years, bringing to light the “obvious” [évident] thing that no one had seen, taking form in an “obvious” assertion of which no one had dreamed … and that the first one to come along can then prove in five minutes, using techniques ready to hand [toutes cuites].

“Obvious” is the property Wittgenstein called übersichtlich, synoptic or perspicuous.18 This is where the avatars come in. In the situations I have in mind, one may well have a rigorous proof, but the obviousness is based on an understanding that fits only a pattern one cannot yet explain or even define rigorously. The available concepts are interpreted as the avatars of the inaccessible concepts we are striving to grasp. So, I agree with Grothendieck and disagree with Wittgenstein when the latter writes (in On Certainty):

the end is not certain propositions’ striking us immediately as true, i.e. it is not a kind of seeing on our part, it is our acting, which lies at the bottom of the language game.

In mathematics this separation of seeing and acting seems artificial; seeing and conveying what one has seen is as important as any other form of acting as a mathematician.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

The word avatar was used in English as early as 1784 and in French by 1800 as a form of the Sanskrit word avatara, denoting the successive incarnations of the Hindu god Vishnu.¹⁹ By 1815 the word was so familiar that Sir Walter Scott could refer to Napoleon’s possible return from Elba as the “third avatar of this singular emanation of the Evil Principle.” The meaning relevant to mathematics of “transformation, manifestation, alternative version” first appeared in French in 1822 and in English in 1850.

Que d’avatars dans la vie politique de cet homme! Cette institution va connaître un nouvel avatar (one reads in the Dictionnaire de l’Académie Française). Educated French speakers were comfortable using the word in this way in the 1950s, if not earlier. Deligne may have been the first to give it its modern mathematical sense, in a widely read paper from 1971, where he wrote (in French) “One should consider this as an avatar of the projective system (1.8.1).” Physicists Sidney Coleman, J. David Gross, and Roman Jackiw used the word slightly earlier, and no doubt independently, to roughly the same end.20 Today, Google Scholar has hundreds of quotations about avatars in geometry—an algebraic-geometric avatar of higher Teichmüller theory—or topology—this operadic cotangent complex will serve as our avatar through much of this work—or mathematical physics—topological avatar of the black-hole entropy—or any other branch of algebra or geometry. The incorporation of the word in the standard lexicon of mathematicians likely has little or no relation to the development of video games or virtual reality experiences like Second Life and can be traced rather to its use by Grothendieck in Récoltes et Sémailles, to identify cohomology groups as symptomatic incarnations of the objects of an as-yet inaccessible category of motives.

The next time I use the word category, I will begin to explain what it means, along with the word structure, with which it goes hand in hand, but which I have deliberately left undefined. For the moment, bearing in mind that a motive is a certain kind of algebraic structure, the expression “category of motives” should suggest that such a structure can be grasped only in relation to other structures of the same kind and that the category provides the formal unifying framework in which such relations are made manifest. And just as the introduction of Galois theory brought about a change of perspective, in which the search for solutions of polynomial equations was replaced by a focus on the new structure of the Galois group, you may anticipate that the design of a category of motives meeting Grothendieck’s specifications will likely usher in a new mathematical era, in which the motives themselves—not to mention the equations whose “essence” they capture—will no longer be central. Attention will turn instead to the structure of the category to which the motives belong,²¹ along with other structures to which it can be compared. This is nothing as brutal as a paradigm shift; each generation’s new perspective is meant to be more encompassing, as if mathematicians were collectively climbing and simultaneously building a ladder that at each rung offers a broadening panorama and the growing conviction that the process will never end—“knowledge and indifference,” as Weil wrote, alluding to the Gita; or ataraxia, the absence of worry at which Pyrrhonian skepticism aimed in conceding the fruitlessness of the quest for ultimate truth.

EVERYTHING

Faust: Ich fühl’s, vergebens hab ich alle Schätze

Des Menschengeists auf mich herbeigerafft,
Und wenn ich mich am Ende niedersetze,

Quillt innerlich doch keine neue Kraft;

Ich bin nicht um ein Haar breit höher,

Bin dem Unendlichen nicht näher.*

Physicists like Steven Weinberg can “dream of a final theory,” but mathematicians can realistically dream only of an endlessly receding horizon. String theory, for example, the preferred approach at the IAS to unification of fundamental physics, is often described as a theory of everything. But what is “everything”? Those of us living in a different part of the universe or in an alternate, possibly virtual, reality have the leisure to reflect on this question, even though we may be ignorant of the physical laws governing your universe. Philosophy begins either by distinguishing things from one another or by affirming an underlying unity behind the diversity of appearance.²² Thus Aquinas devotes questions 30 and 31 of Summa Theologica, I, to the triune unity of three persons in the divine essence; Vedantic monists identify atman with brahman, the personal self with the universal principle; and Nāgārjuna, whose Mūlamadhyamika-kārikā (Fundamental Stanzas on the Middle Way, hence-forth abbreviated MMK) is a founding document of the Madhyamaka school of Buddhist philosophy, propounded the identity of samsāra, the cycle of deaths and rebirths in the world of suffering, with nirvana, the escape from that very cycle. Are there lots of different things or just One Big Thing (an elephant, for example, or the universe) viewed under different guises? Keep asking this question for a few thousand years and you are liable to invent set theory; and you might be tempted to call set theory the theory of theories of everything, but it’s really only a first baby step toward such a theory.

Figure 7.2. The complete graph on five vertices, from Wikimedia Commons (labels added): http://en.wikipedia.org/wiki/File:RamseyTheory_K5_no_mono_K3.svg.

The definition of a set as a collection of elements raises more ontological questions than it answers; it would be more forthright to define a set as a “collection” of “elements.” When the sets are finite, though, the scare quotes can be safely discarded. Over 400 of Erdős’s 1417 papers indexed by MathSciNet, the biggest single chunk of his papers, are in the field of combinatorics, the branch of mathematics largely concerned with counting elements of finite sets. Many of these papers study properties of graphs: a graph has a collection of points (vertices), the elements of the set, and lines (edges) connecting the points, like the one in figure 7.2. This graph has five vertices and 10 edges; it is a complete graph, which means that there is an edge connecting each pair of points, like a road running between any pair of cities on a map. Here is Ramsey’s theorem about complete graphs:

Theorem. For every positive integer k, there is a positive integer R(k) such that if the edges of the complete graph on R(k) vertices are all colored either black or gray, then there must be k vertices such that all edges joining them have the same color.

Tim Gowers discusses Ramsey’s theorem at length in his article The Two Cultures of Mathematics, on the differences between problem solvers and theory builders, which is convenient for our purposes because Erdős and Grothendieck exemplify the two cultures almost to the point of cari-cature. As it happens, “one of Erdős’s most famous results,” in Gowers’s estimation, is that the Ramsey number R(k) has to be at least 2^k/2. If you know R(k), you have solved the party problem: Erdős’ theorem implies that if you want to hold a dinner party where either (at least) k people already form a clique or (at least) k people don’t know each other yet—don’t ask why—then you need to invite at least 2^k/2 people. This is not very informative for small numbers: if k = 3 it says you need at least 2.828 guests, which you already knew; but for k = 20. Erdős tells you to invite at least 2¹⁰ = 1024 guests to your party.23

Most graphs are not complete. If we erase the gray edges in figure 7.2, we’re left with a graph in the form of a pentagon, where each partygoer knows his or her neighbors but not those sitting across the table. Given a big enough blackboard, we can draw a graph whose vertices are all the mathematicians who have ever lived; mathematicians A and B are connected by an edge if they have collaborated by publishing a paper together. Four hundred ninety-eight edges radiate from Erdős, making him the most connected mathematician in history. No edge connects Erdős to Grothendieck, who hardly collaborated at all; but you can get from Grothendieck to Erdős by a link of three edges, by way of Serre and Sarvadaman Chowla. This is the shortest such link; in other words, Grothendieck’s Erdős number is 3—he has 3 degrees of separation from Erdős. My colleagues mostly have Erdős numbers and know what they are: your Erdős number is 1 if you have collaborated with Erdős, 2 if you have collaborated with someone with Erdős number 1 but not with Erdős himself, and so on; only Erdős has Erdős number 0. My own Erdős number is 3, as illustrated by the graph in figure 7.3.

Figure 7.3 is a graphic representation of a communications network: mathematicians who have collaborated are considered to be in communication. Results as apparently pointless as Ramsey’s theorem have extensive applications in communications and computer science. Did the animators working on James Cameron’s Avatar consult graph theorists in designing the organic network of branches and vines with ports to plug in the pigtails built into every living being on the moon Pandora?

The message of Gowers’s lucid and thoughtful article is not merely that problem solvers, and Erdős in particular, are as deserving of respect as theory builders,²⁴ but also that problems whose statements are simple enough to be understood by readers with no specialized training in mathematics, even problems about finite sets, can nevertheless reveal a rich underlying structure (his word). Gowers makes this structure “obvious” in an eight-line proof of Erdős’ theorem that R(k) ≥ 2^k/2. It’s a proof many mathematicians would be tempted to call beautiful. But can such a proof be sublime in the sense of Kant and Edmund Burke? Can meditating about finite sets inspire awe and terror? Or are these sentiments reserved for the contemplation of infinity?

Figure 7.3. A small part of Erdős’s collaboration graph, including Grothendieck and the author of this book. Chowla and Murty are among the 498 mathematicians with Erdős number 1.

Cantor’s diagonalization trick, mentioned in the last dialogue, shows how every infinite set S engenders a set P(S) of an even bigger infinity … bigger in the precise Cantorian sense that their elements can’t be matched side by side like two decks of fifty-two cards. Think it over for a moment and you’ll realize that this fundamental property of Cantor’s transfinite arithmetic means no theory of everything is even remotely possible… because a gambler employing Cantor’s trick will see your everything and raise you with an incommensurably bigger everything. Like many of his distinguished predecessors, Cantor resolved this hopeless paradox by giving names to the One Big Everything that you meet when you climb this endless ladder of everythings to its unattainable summit. He called it “Absolute”—“the ‘Actus Purissimus’ which by many is called ‘God.’ ” Joseph Dauben’s study of Cantor, from which this quotation is taken, also records how the inventor of set theory maintained an extensive correspondence with Catholic theologians in a largely successful effort to convince them that, precisely because any human attempt to comprehend the full sequence of infinities collapses in paradox, his transfinite arithmetic does not detract from the sublimity of the Absolute and therefore poses “no danger to religious truths.”25

Voevodsky, worrying that the foundations of mathematics may be inconsistent, has occasionally advised mathematicians to welcome potentially contradictory foundations. Artificial intelligence finds this less problematic than one might think. Automated expert systems have been designed to take contradictory input—for example, conflicting diagnoses of a cancer patient—and to use them as guides to concrete action, rather than to submit to the paralysis of conventional logic, which de-duces from any contradiction that every statement is simultaneously true and false.26 Suspending attention to logical niceties when faced with a life or death decision needs no justification, but when is concrete action urgent in mathematics?

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

The question of identity would seem to have been settled long ago in mathematics by the adoption of the = sign as a standard item in the lexicon used to construct meaningful mathematical propositions. But there is a rich philosophical literature on this question. Leibniz’s principle of the identity of indiscernables states roughly that A = B if everything that is true of (or can be predicated of) A is true of B, and vice versa—if A and B have the same attributes in any sense that can be given to this term. Cantor’s Absolute, impervious to meaningful predication, resembles the God of the church fathers on the grounds of Leibniz’s principle:

According to the classical theism of Augustine, Anselm, Aquinas and their adherents, God is radically unlike creatures in that he is devoid of any complexity or composition, whether physical or metaphysical…. There is also no real distinction between God as subject of his attributes and his attributes. God is thus in a sense requiring clarification identical to each of his attributes, which implies that each attribute is identical to every other one.²⁷

By insisting that “The essence of entities is not present in the conditions, etc.” and that “nirvana is uncompounded. Both existents and nonexistents are compounded,” the MMK excluded nirvana from Leibniz’s world system. Nāgārjuna’s commentator Candrakīrti considered “nirvana … cognitive nonsense … a ‘scandal’ to logic. In order for nirvana to be cognitively affirmed or denied, it must be reduced to samsāra, to existence in causes and conditions.”²⁸ Nāgārjuna’s equation of samsāra and nirvana has been translated as the identity of “the Phenomenal” and the “Absolute.”²⁹

Identity in Leibniz’s sense is the property captured by the equal sign in set theory. But since A is called A and B is called B, this does not quite suffice to unravel A = B. In his 1892 article Über Sinn und Bedeutung, Frege introduced the distinction between sense and reference, giving the example of the morning star and the evening star, both of which turn out upon inspection to be the planet Venus: two senses for a single reference. Frege’s attempt to solve the problem of identity remains influential among philosophers but is not nearly subtle enough to account for the gradations of identity that inevitably arise in mathematical practice.

Heidegger, meanwhile, was able to spin the tautological equation A = A—which we were just claiming was unproblematic for set theory—into an essay on fundamental ontology, starting with the claim that the proper formulation of “A = A” is that “every A is itself the same with itself.”30 Heidegger’s doubts about the transparency of identity were anticipated (by nearly 2000 years) by Nāgārjuna, who wrote that “I do not think those who teach the identity or difference of self and things are wise in the meaning of the teaching.”³¹

“Sameness,” according to Heidegger, “implies the relation of ‘with,’ … the unification into a unity.” For Voevodsky (following the traditions of homotopy theory; see the following), the equation A = A is a space, the space of all ways A can be identical to itself. In this he is faithful to Grothendieck, the first mathematician to be guided by the principle that knowing a mathematical object is tantamount to knowing its relations to all other objects of the same kind, meaning objects that belong to a well-defined class of structures. The formal language for putting together objects of the same kind is that of categories: objects of a given kind—such as sets, or groups, or topological spaces—form a category, and the relations between them are called morphisms. The language of categories, with its deliberate echo of Aristotle and Kant, was invented by Saunders MacLane and second-generation Bourbakiste Samuel Eilenberg by abstracting common features of methods of reasoning that were simultaneously appearing all over mathematics, especially in algebra and topology. Chapter 4 of Bourbaki’s treatise on set theory, written by Bourbaki’s first generation, had attempted to make structures into a rigorous notion at the center of mathematics by proposing an axiomatic treatment of “structures” as such. But mathematicians of the postwar period paid no attention, preferring by far the categorical way of thinking, especially as expanded by Grothendieck, so that today when mathematicians refer to structures, they are usually thinking of objects in a specific category.³²

Like any mathematical formalism, that of categories is based on a list of axioms; for example, every object, among its many self-relations, enjoys the relation of identity with itself; if you have a relation (morphism) from A to B and a relation from B to C, then you can string them together to obtain a relation from A to C; and that’s practically all you need. So, according to the principle that Grothendieck used to great effect, “knowing” a set A means “knowing” where A fits in the category of sets, which amounts to “knowing” the structured web of its relations (the morphisms in this case are just set-theoretic functions) to all other sets, including itself.

This principle, valid in any category, is known as Yoneda’s lemma, and it is so formulated as to be obvious to prove, but the experienced will be aware that we are skating on the edge of paradox—of Russell’s paradox, more precisely—by talking about things like “all other sets.” Russell showed early on that careless talk about all sets—talking about the “set of all sets that don’t contain themselves,” for example—leads to instant fatal contradiction. That’s why one refers to the category of sets rather than the set of all sets; in the former, Russell’s noxious deductions are not permitted. But this comes at a cost: unless one makes additional restrictions, we have lost the principle of identity; it is not appropriate to say that two sets A and B are equal. “The essential lesson taught by the categorical viewpoint” according to Barry Mazur “is that it is usually either quixotic, or irrelevant”33 to insist on equality in a category. Yoneda’s lemma is in this sense quite different from Leibniz’ principle. If we think of the properties of relation with a given object in the category as a predicate, then to say that two objects are indiscernable with respect to Yoneda’s lemma is not to say that they are identical but that they are isomorphic—which means A and B are equal to all intents and purposes but not really equal.

For one thing, as their distinct names make clear, A and B are not in a tautological relation of identity, unlike A and itself. You can put A and B into a relation of identity, like two decks of cards, placing them side by side, one card at a time, in a long row of fifty-two cards. But you can pick up the second deck, shuffle it, and then lay the cards down alongside A again: the relation of identity has been shuffled, and the new one is just as good as the old one, at least if you ignore what’s on the faces of the cards. With this version of identity, a deck of cards is identical to any other fifty-two-element set (for example, the set of letters of the alphabet, upper-and lowercase). When identifying two decks of cards, you can also insist on paying attention to structure; you might insist that cards in deck A be matched with cards in deck B in the same suit or that number cards be matched with number cards and face cards with face cards. This is the idea behind the Galois group, as we explained in chapter β: there is no uniform way to label the roots of a general polynomial, but a labeling is allowed only if it accounts for some underlying structure.

Different notions of equality are reviewed on the nlab Web site, at http://ncatlab.org/nlab/show/equality, with a warning that “it hasn’t all been said here yet.” So far so good; but this is where things begin to get complicated—or interesting, depending on your viewpoint. From the equality page you can link to the identity type page, where you learn that in homotopy type theory, the “incarnation of equality” is “an identity type … ‘the type of proofs that x = y’ or ‘the type of reasons why x = y.’ ”³⁴

“All things have one nature,” wrote Nāgārjuna, “that is, no nature.” This is interpreted to mean that “everything is empty of essence and of independent identity,” that the nature of all things is to be “dependently co-arisen,” in other words, to be in relation to other things.³⁵ We can read this as a paradox “arising at the limits of thought” or as a precursor of Yoneda’s lemma. The MMK and its commentators seem not to have realized that they may have anticipated higher-category theory. If identity is subordinated in ordinary category theory to relatedness, higher-category theory takes relations between relations, and relations between relations between relations, and so on, to infinity if you choose to go there. Higher categories, such as n-categories, are central in one way or another to most of Grothendieck’s work after he left the IHES and are the framework of Voevodsky’s univalent foundations. The n-Category Café is a blog run jointly by mathematicians, physicists, and philosophers, all higher-category enthusiasts; and the nLab is a hyperlinked online “collaborative wiki” that records information relevant to the “n point of view” (nPOV), which is that “category theory and higher category theory provide … a valuable unifying point of view for the understanding of” concepts involved in mathematics, physics, and philosophy.³⁶ We will soon accompany them a short way up that ladder, but we’re not quite ready yet.

EXISTENCE

[I]t’s a race to see who can venture out furthest into the borderlands of the nonexistent (Pynchon, Against the Day, p. 535).

Facing an audience of philosophers, Langlands confided that

[i]t is … very difficult even to understand what, in its higher reaches, mathematics is, and even more difficult to communicate this understanding, in part because it often comes in the form of intimations, a word that suggests that mathematics, and not only its basic concepts, exists independently of us. This is a notion that is hard to credit, but hard for a professional mathematician to do without.37

Professional mathematicians can hardly avoid using the word exist in specific contexts, in a particular way, that does not readily lend itself to resolving differences between the standard options of platonism (or realism) and nominalism (or conventionalism). Philosopher Emily Grosholz reads Leibniz’s thesis of the existence of intelligibles to mean, in particular, that mathematical objects “must exist if other things are to be possible for thought.”³⁸ For the nominalist philosopher Jody Azzouni, the mere use of the word exist, in particular in mathematical practice, does not entail adherence to a specific philosophical position: “Ordinary speakers find the word ‘exist’ useful—even indispensable—in contexts where they either aren’t committed to the subject matter, or where ontic commitment isn’t at issue.”³⁹ Foundations do provide some protection against ontological existence proofs, along the lines of Descartes’ proof that God must exist because (1) Descartes has a (clear and distinct) idea of a supremely perfect being; (2) necessary existence is a perfection; and, therefore, (3) a supremely perfect being, that is, God, exists. Ontological proofs had already been proposed by St. Anselm in the eleventh century, and by the tenth century, Hindu logician Udayana, who reasoned that “ ‘God exists’ makes sense because there is no means to establish ‘God exists’ makes no sense.” Kant’s rejection of Descartes’ version, on the grounds that existence is not a predicate, is very similar to mathematicians’ refusal to admit notions like the set of all sets, or Cantor’s Absolute, in spite of any clear and distinct ideas one likes to think one has on the subject.⁴⁰

Virtual reality offers an appealing metaphorical alternative to the futile opposition between platonism and nominalism, but it will not soon make the word exist obsolete in mathematics. Paradigmatic of its acceptable and indispensable use are the so-called existence and uniqueness theorems for various kinds of differential equations (see chapter 4). Con-ceptually speaking, such a theorem typically asserts that a (system of) differential equations of a certain type has (or fails to have) exactly one solution when something like initial conditions are specified, and this solution includes information about behavior in the long run. Thus when you toss a ball in the air or launch a rocket, the initial conditions, which in this case amount to the projectile’s initial speed and angle, determine whether it will either fall back to earth at a predictable location (like the V-2 rockets in Gravity’s Rainbow), or orbit around the earth at a predictable height (as Venus orbits the sun in Mason & Dixon), or hurtle off endlessly into boundless space (like the Chums of Chance at the end of Against the Day). In practice one has to specify more conditions, such as weather variables and properties of the materials constituting the rocket, in order to obtain an accurate answer, but the rocket’s trajectory and the unique solution to the equations determining its motion are the same thing; so if you fail to bring the space shuttle home safely, it’s the fault of the equipment, not the equations.⁴¹

When mathematicians establish the existence of objects, we usually want to understand their uniqueness at the same time. Most of us would agree the rocket’s trajectory is unique, the path it really takes—that’s how we understand the reality of reality, and this understanding must account in no small measure for why mathematicians are attached, as Langlands says, to the sense of the independent existence of mathematics. But branches of mathematics that cannot rely on the physical world for guidance are subject to the problems of identity. When we solve a problem—in algebraic geometry, for example—we want to be able to point to the answer unambiguously. In the categorical framework, the best we can do is say that the solution is “unique up to unique isomorphism.” This means that when you and I set out to solve the same problem, we know we will be making choices at the outset, based on our individual perspectives; unique means there is a way to translate my solution into yours, and up to unique isomorphism means there is only one way to do it. But finding that translation means solving another problem!

Problems of translation pile up as one climbs the n-categorical ladder:

[O]ne seems caught at first sight in an infinite chain of ever ‘higher,’ and presumably, messier structures, where one is going to get hopelessly lost, unless one discovers some simple guiding principle.42

Although we are free not to climb it, this ladder forces itself upon our attention when we think carefully about questions of identity and relations in a fixed mathematical context. The avatar ladder, in contrast, emerges when we observe mathematicians looking not down for reassurance as to the solidity of their foundations, but rather into the future in the search for ultimate meaning, which may well entail the scrapping of the foundations that are conventionally assumed to have allowed us to reach this point of reflection. But the two ladders run epistemologically parallel in the practice of categorification, to which we turn at the end of the chapter.

Erdős is sometimes called a platonist because he frequently referred to a register of optimal proofs, called THE BOOK, which could be consulted only by God, in whom he did not believe and whom he called SF, “the supreme fascist.” But Erdős did not mean this dogmatically; “in a way,” he said, “mathematics is the only infinite human activity.”⁴³

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

But I’m now suddenly in a position to tell you what the Langlands program is about, and since this may not happen again, we had better take advantage of the opportunity. Very briefly, then: the Langlands program postulates a correspondence between Galois representations—structures derived from the symmetries of roots of polynomial equations, normally studied by number theorists (cf. chapter β)—and automorphic forms (or automorphic representations)—structures derived from solutions to differential and difference equations with an unusual degree of symmetry (cf. chapter 4), normally studied by geometers or mathematical physicists. Langlands conjectured (and in many cases proved) such tight relations between the two kinds of structures that one can only hope to be successful in either field today by learning to see each side of the Langlands correspondence as the avatar of the other—as exhibiting the other side’s otherwise inaccessible properties.⁴⁴

This is probably also my last chance to say something about the ideas Langlands calls “reckless,” and I really ought to, because Grothendieck has been getting far more good lines than Langlands. The ideas themselves are too technical to present here—they have been attracting a lot of attention because they are the only proposal with the potential, however remote, of resolving all of Langlands’s conjectures simultaneously—but I can say a word about just how reckless they are. How Langlands estimates their likelihood of success can be inferred from the anecdote that provided the epigraph for his first exposition of these ideas. Langlands has studied and written in many languages, among them Turkish, to which he has shown a special attachment. So it is not so surprising that Langlands would have chosen for his epigraph the expression Ya tutarsa, associated with one of the stories about the Turkish folk hero Nasreddin Hoca (pronounced “hodja”). Nasreddin is staring at a lake. When his neighbors ask him what he is doing, he explains that he has put some yogurt in the lake and that he is now waiting for it to ferment so that the whole lake will turn to yogurt. His neighbors laugh at him and tell him that the lake will never turn to yogurt. “Ya tutarsa,” Nasreddin replies—“What if it did?”

Even when one is not in a position to prove Langlands’s conjectures, a standard procedure is to reason on one side of the correspondence to deduce interesting and surprising consequences on the other side and then to prove these consequences directly. Specialists call this a “guide to intuition,” as if to confirm that a “non-logical cognitive phenomenon”45 is involved. The most spectacular example was undoubtedly the series of deductions that led to the discovery by Gerhard Frey and Ken Ribet, in the mid-1980s, that Fermat’s Last Theorem could be proved by establishing one part of the Langlands correspondence. Since the Langlands program had already shown such exceptional explanatory power that it had grown “too big to fail,” this dramatically intensified the belief among number theorists in the truth of Fermat’s Last Theorem and gave Andrew Wiles the confidence he needed to spend seven years in an ultimately successful effort to prove as much of the Langlands correspondence as he needed.

The Langlands program, meanwhile, flourishes as never before, in large part thanks to the new ideas Wiles brought to bear on a problem that had motivated number theorists for centuries but that is now well advanced along Weil’s cycle of knowledge and indifference. One of the most fruitful approaches to proving Langlands’s conjectures was developed by the Soviet-born mathematician Vladimir G. Drinfel’d, using the full range of techniques developed by … Grothendieck …, which is fitting, since the Galois representation side of the Langlands correspondence is naturally interpreted as an avatar of Grothendieck’s motives, and vice versa.46 Having absorbed earlier work of Goro Shimura, Deligne, and Langlands, Drinfel’d defined several new (Grothendieck-style) geometries to bridge the two apparently unrelated “structures” that matched Langlands’ predictions and, in so doing, launched the geometric Langlands program and incidentally began the process of aligning these structures into categories. Drinfel’d and Alexander Beilinson, and later Ed Frenkel and his collaborator Dennis Gaitsgory, were among the first to construct a Langlands correspondence as a relation between categories.

Drinfel’d’s geometries looked extremely strange when he defined them; did they preexist his definition? Are they avatars of a platonist Langlands correspondence or do they bring the Langlands correspondence into being? In designing his geometries, Drinfel’d was clearly guided by the hope of applying Weil’s topological insight; his definition’s merit was that it provided just the fixed points he needed. In 1990 I was sitting in Manin’s seminar in Moscow when Beilinson stood up and interrupted the speaker to explain that the objects studied in algebraic geometry were an illusion, maya, to hold fixed points together. Of course he said no such thing, but that’s what I heard, perhaps because I had been reading secondhand Indian metaphysics—and whatever he actually said, this was the perspective behind a dream I had two years later….

SECOND LIFE

Möge der Traum den wir das Leben nennen … * (Gauss, quoted by Mehrtens, p. 26)

Jaron Lanier, who invented the term virtual reality, described the “uncanny moment” when “your brain starts to believe in the virtual world instead of the physical one.” Jerry Garcia and Timothy Leary had compared early VR to a psychedelic experience. “The body and the rest of reality no longer have a prescribed boundary,” writes Lanier. “So what are you at this point?”47

Many readers will have first experienced a single-sense version of this transition when they put on 3D glasses to watch a film called Avatar. The VR of mathematics is no less dependent on the experience of bodies in space. Gilles Châtelet uses the spatial metaphor of gestures to account for what Grothendieck called “obvious”: “ ‘Obviousness’ for Grothendieck is not linked to the proximity of two terms in a deductive chain but rather to the effect of ‘naturalness’ linked to the abolition of the space between the symbol that captures and the gesture that is captured.”⁴⁸ One need add only that this “abolition” is a never-ending process. The socialization outlined in chapter 2 had less to do with assimilating norms, values, and respect for a transhistorical hierarchy than with learning to use the word exist appropriately and effortlessly, to display a pragmatic “belief in” the world of mathematical objects adequate for seamless interaction with the community. When you don the gloves and goggles of contemporary geometry, differential equations, number theory, or algebraic geometry, boundaries with the rest of reality no longer matter.

Maybe Châtelet’s geste is a literal description in VR, not a metaphor:

[T]he participant in Second Life is not inventing a fiction about his avatar; he is performing fictional actions through his avatar…. No wonder, then, that animating an avatar in Second Life would come so naturally to you…. You have been animating an avatar all along: … the avatar consisting in your body, and you have been doing it whenever you act.⁴⁹

Mehrtens thought the early-twentieth-century Foundations Crisis was not about foundations at all; rather, it marked “a shaking [Erschütterung] of the concepts of truth, meaning, object, and existence in mathematics. These concepts are not so much epistemological foundations of the science as the overall orientation markers by means of which the … identity of the science defines itself…. Mathematics is a language and the work thereon, and so its identity cannot be established [herstellen] on an ‘object’….”⁵⁰

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

Science today should be entrusted to men of great spirituality, to prophets and saints. But instead it’s been left to business talents, chess--players, athletes. They don’t know what they’re doing.

—V. Grossman, Life and Fate (modified from Robert Chandler’s translation, p. 693)

Grothendieck, like Serre, Tate, and Godement, was (briefly) a leading member of Bourbaki’s second generation and agreed with the claim in Bourbaki’s “programmatic manifesto” that (thanks largely to Bourbaki) mathematics “possesses the powerful tools furnished by the theory of the great types of structures; in a single view, it sweeps over immense domains, now unified by the axiomatic method, but which were formerly in a completely chaotic state.”51 In Récoltes et Sémailles, Grothendieck takes Bourbaki’s structuralism a step further, revealing himself as a kind of structural platonist—a structure is not something to be invented, “we can only bring it to light patiently, humbly make its acquaintance, ‘discover’ it…. [structures] didn’t wait for us to be, and to be exactly what they are!”⁵²

But ruthless elimination of superfluous assumptions in the quest for maximum generality is no less important than structures in the Bourbaki legacy. (Ontological?) minimalism is the hallmark of the Bourbaki style. Godement reports that, as a young man, “Grothendieck wore only a shirt, pants and sandals winter and summer, and these were very worn-out (as he was totally penniless), but always incredibly clean, as was his room.” All this struck Godement as “forming part of a search for total purity,”⁵³ the search that in mathematics led Grothendieck to “strip […] away just enough,” as Tate put it.

The hypothetical category of motives is supposed to be built out of pure motives. Grothendieck suggested a specific recipe to define pure motives in terms of algebraic geometry, and this is the recipe we have in mind when we think about motives, though after more than forty years, it has been shown to work as predicted only in the very simplest cases. Motives that are not pure are mixed (rather than impure—Deligne reports that Grothendieck chose this terminology). Current theories of mixed motives—Voevodsky’s is the most influential⁵⁴—are a stunning monument to their creators’ persistence and are rich enough to have solved outstanding problems, but they fall far short of Grothen dieck’s original expectations. They are still avatars of the desired theory: one has access to various derived categories but not to the ulterior category.

A human being using categories inevitably has to make choices that would be redundant from an Absolute standpoint. We can still prove that the properties of interest don’t depend on these choices: they are “unique up to unique isomorphism.” But then, we’ve seen, new problems arise: how the properties don’t depend on choices. As Plato reminds us in the Phaedo, “when the soul makes use of the body for any inquiry… then it is dragged by the body to things which never remain the same, and it wanders about and is confused and dizzy like a drunken man.”55

Grothendieck contrasted Serre’s elegance—on at least four occasions in Récoltes et Sémailles, Serre is depicted as the “incarnation of elegance”—in applying the “hammer-and-chisel method” to solve a problem, with his own approach: making the problem disappear by letting the sea rise to “submerge and dissolve” it. “Grothendieck creates truly massive books with numerous coauthors, offering set-theoretically vast yet conceptually simple mathematical systems adapted to express the heart of each matter and dissolve the problems.”⁵⁶ Grothendieck lets the sea rise by eliminating dependence on a choice of perspective. As in Einstein’s theory of special relativity, all perspectives are equally valid. Choices have been eliminated in the process, axiomatizing the possibility of choice as a new and higher level of mathematics up the category ladder. “When one follows Grothendieck’s work throughout its development, one has exactly [the] impression of rising step by step towards perfection. The face of Buddha is at the top, a human, not a symbolic face, a true portrait and not a traditional representation.”⁵⁷

In an all too familiar trade-off, the result of striving for ultimate simplicity is intolerable complexity; to eliminate too-long proofs we find ourselves “hopelessly lost” among the too-long definitions. The Madhyamaka school thematized this very trade-off and rejected infinite regress without resorting to an Aristotelian Prime Mover. Nor did they have patience with those who hoped to take refuge in comfortable illusions—for example, with the help of mind-altering drugs. The Prasannapada, Candrakīrti’s seventh-century commentary on MMK, “considers as sick that form of life which seeks expression through metaphysical thinking … the metaphysics of ‘is’ and ‘not-is’ [is] not a solution to the problem of suffering but a drug invented by those who love to ‘get high.’ ”58

Grothendieck lived in communes off and on during the 1970s, but he reportedly wasn’t interested in drugs either. In fact, I’ve found only a few documents suggesting that drugs can enhance mathematics, and the evidence is ambiguous. The Irish mathematician William Rowan Hamilton, the inventor of quaternions whose spirit haunts Pynchon’s Against the Day, was an alcoholic for the last third of his life. Contemporary biographers claimed that he drank for the sake of his work:

To continue to the end a task, in which good progress had been made, required, as he was convinced, support and stimulus for the brain, and this he administered to himself in the injurious form of porter taken in small sips as he felt fatigued. The need thus experienced, connected as it was, with his disinclination to be disturbed at his work by regular meals, was, according to his son’s testimony, the principal cause of his recourse to alcoholic stimulant, for which he admits that his father had besides a constitutional proclivity, as well as a disposition, arising from his genial nature, to conform to the prevailing custom of the time when he first entered into social life.⁵⁹

When his friend Ronald Graham bet Erdős $500 he couldn’t quit amphetamines for a month, Erdős won the bet but complained, “I didn’t get any work done … I’d have no ideas, like an ordinary person. You’ve set mathematics back a month.” Erdős went back to taking pills and writing papers.

Hamilton and Erdős indulged their drug habits, it’s said, to maintain their stamina. California chaos theorist Ralph Abraham is the only mathematician I know who has claimed drugs actually affected the contents of mathematical research, and for the better. In a 1991 interview with the style magazine GQ, Abraham claimed, “In the 1960s a lot of people on the frontiers of math experimented with psychedelic substances. There was a brief and extremely creative kiss between the community of hippies and top mathematicians. I know this because I was a purveyor of psychedelics to the mathematical community.” The interview is maddeningly short on specifics. Molecular biologist Kary Mullis “seriously doubt[s]” that he would have invented the PCR technique for which he won the Nobel Prize if he hadn’t taken LSD.⁶⁰ Timothy Leary wrote in 1977 that he expected “the new wave of turned on young mathematicians, physicists, and astronomers … to use their energized nervous systems … to provide new correlations between psychology and science.”61 Can any Fields Medals be traced to psychedelic inspiration? Will we ever know?

Pharmacological enhancement may be redundant if mathematics is itself the drug. Mathematicians stress the addiction even more than the buzz:

Marcus du Sautoy: “Doing mathematics is like taking a drug. Once you have experienced the buzz of cracking an unsolved problem or discovering a new mathematical concept, you spend your life trying to repeat that feeling.”

Marie-France Vignéras: “ … it happens suddenly: one direction becomes more dense, or more luminous. To experience this intense moment is the reason why I became a mathematician.”

Joan Birman: “[T]he moment I learned about an unanswered question that involved braids, I was hooked!”

Grothendieck: C’est en termes cohomologiques,… que Serre m’a expliqué les conjectures de Weil, vers les années 1955—et ce n’est qu’en ces termes qu’elles étaient susceptibles de m’ “accrocher” en effet. [It’s in cohomological terms … that Serre explained the Weil conjectures to me, around 1955—and it was indeed only in these terms that they could have “hooked” me.]

Misha Gromov: “You become a mathematician, a slave of this insatiable hunger of your brain, of everybody’s brain, for making structures of everything that goes into it.”⁶²

Grothendieck was briefly drawn to Buddhism, but his affinity was with Buddhist pacifism rather than its metaphysics. According to the mathematician Winfried Scharlau, who has spent years assembling documents and interviewing Grothendieck’s acquaintances in an effort to reconstruct his life, he “was never especially interested in [Buddhist] scriptures … and never studied them in depth.” He “understood meditation to be pretty much the exact opposite of what it means to Buddhism.” Far from seeking to empty his mind to realize Nāgārjuna’s “no nature,” or the ataraxia of the skeptics, or “the rich treasure of inner peace” that the seventeenth-century Spanish mystic Miguel de Molinos promised to those contemplating la nada, Grothendieck used meditation as a form of “intellectual work, penetration of one’s own ego, revelation, perception, knowledge, and understanding of psychological events”63—miring himself ever more inextricably in samsāra.

Erdős, meanwhile, comes across in Hoffman’s book as “a weirdo,” one of the most eccentric humans ever to walk the earth. “He lived out of a shabby suitcase and a drab orange plastic bag from … a large department store in Budapest…. He was twenty-one when he buttered his first piece of bread … ‘I took him to the Johnson Space Center to see rockets,’ one of his colleagues recalled, ‘but he didn’t even look up.’ ” But his theology is refreshingly down to earth. Had Jesus paid him a visit, the Jewish atheist Erdős would have asked him whether Cantor’s continuum hypothesis is true—whether Cantor’s procedure for creating an endless string of increasingly big infinities doesn’t leave any out. Kurt Gödel and Paul Cohen proved that the question as stated is meaningless: it can’t be decided within the standard axioms of set theory (ZFC: Zermelo-Fraenkel + axiom of choice). Yes and no are, therefore, both acceptable, but Erdős thought Jesus might have a nicer answer: “The Father, the Holy Ghost, and I have been thinking about that long before creation, but we haven’t yet come to a conclusion.”⁶⁴

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

FAUST: … Da muß sich manches Rätsel lösen.
MEPHISTOPHELES: Doch manches Rätsel knüpft sich auch.*

Not only is it OK to write A = A in a category, it’s even one of the rules that any A has a special identity relation to itself, usually just called “identity.” Writing A = B is another matter, unless B happens to be A. In a category there can be no end to the Bs that are indistinguishable from—isomorphic to—A but are not the same as A. This is the price you pay to be able to talk about the category of all sets: it contains not just one set of real numbers, for example, but (at least) one for each time the set of real numbers is invoked—two new ones just as a result of this sentence—all equivalent for all intents and purposes, all (uniquely) isomorphic, no two equal.

Figure 7.4. From Voevodsky’s lecture on September 25, 2010. Voevodsky is in the lower-right corner (courtesy of the Institute for Advanced Study).

The next step is to replace A = A by something more complicated, like a space. If you add the right axioms, the result is called an ∞-category.65 Actually there are many ways to define ∞-categories, each useful for different purposes, none of them prior to the others, so that they can all be seen as avatars—the word incarnation is actually used in the literature—but when one asks avatars of what? the only sensible answer is that they are avatars of the theory of ∞-categories mathematicians need.

∞-categories serve as a model for Voevodsky’s new foundations. Recall Manin’s observation that two kinds of foundations are at issue in mathematics: the foundations that represent the starting point for building what one wants to build and the (logical and philosophical) Foundations, without which the entire structure will supposedly crumble. Grothendieck had strong feelings about the former, alluding in a 1983 letter to Daniel Quillen to “people such as yourself … who … may not think themselves too good for indulging [sic] in occasional reflection on foundational matters and in the process help others….”66 But it’s only a matter of habit to call these foundations as if they were necessarily below our feet; since the sole stipulation is that we actually work at eye-level, there’s no reason not to situate foundations of this sort up above our heads.⁶⁷

What Grothendieck’s letter to Quillen calls “foundations” is really a common language sufficiently rich and precise to address the problems that may arise in the development of the theory that one was going to develop in any case. Weil (and Bourbaki more generally) used the word in the same way. Such a language has more in common with the satellites and transmitters that carry the signals permitting electronic communication than with the 120 meters of concrete attached to the base of the 452-meter-high Petronas Towers in Kuala Lumpur to protect them from destabilization by the forces of nature.⁶⁸ The pragmatist philosopher C.S. Pierce understood this well:

[T]he intellectual powers essential to the mathematician [are] “concentration, imagination, and generalization.” Then, after a dramatic pause, he cried, “Did I hear someone say demonstration? Why, my friends,” he continued, “demonstration is merely the pavement on which the chariot of the mathematician rolls.”⁶⁹

It can be argued whether, in the long war at the heart of the Indian epic Mahabharata, Arjuna the archer or his charioteer Krishna, avatar of the god Vishnu, is the true hero. The case could even be made⁷⁰ that the chariot and Arjuna’s bow, not to mention the horses, should be granted equal standing. But only a certain philosophical cast of mind would think of giving top billing, Foundational as it were, to the pavement….

The philological exegesis of foundation is not meant to reveal a buried secret history but rather to emphasize that foundation is a metaphor, one of several possible, even for the Euclidean axiomatic approach. Metaphors are useful as reminders of turning points in the development of a field, but most of the time mathematical practice speaks for itself. A starting point for the development of higher categories is the branch of topology called homotopy theory. Whereas two spaces are topologically equivalent (homeomorphic) if they can be transformed into one another by stretching or shrinking without tearing and without loss, they are homotopy equivalent if one can be transformed to the other, still without tearing, but certain kinds of loss are allowed. Thus in figure 7.5, the two spikes on the empty-set sign can be retracted (in the first step), and the vertical line can be pinched to a point (in the third step) without changing the homotopy type; the obwarzanek in figure 7.6, which is three dimensional (and edible), is homotopy equivalent to an imaginary one-dimensional infinity sign making a circuit around the two holes. Two homotopy equivalent spaces are not considered topologically the same, but they have the same cohomology—the two holes in the cases illustrated in figures 7.5 and 7.6—which is to say they have the same algebraic structure. This is one reason homotopy theory provided the source of Voevodsky’s intuition in constructing his category of mixed motives.

Figure 7.5. A homotopy from the empty set sign to the infinity sign.

But working rigorously with homotopies presents new problems. To make it a mathematical theory, we have to imagine each homotopy as a stretching/shrinking procedure that takes place in a fixed time interval, say one second. A problem is already apparent in figure 7.5: each of the three steps is supposed to last one second, but the whole procedure is also a homotopy and, therefore, wants to last one second. The traditional solution is to speed up the intermediate steps to make the total come out to one second. But there are (infinitely) many ways to do this, and the set of all such ways is itself a topological space: how do we choose the right one? The traditional answer is that there is no right answer: for the riddles topology was originally invented to solve, it doesn’t matter which way you choose.

Figure 7.6. An obwarzanek (Polish soft pretzel), emblem of the sixth European Congress of Mathematicians held in Krakow in 2012, homotopy equivalent to the infinity sign. (© Polish Mathematical Society)

As Mephistopheles reminds us, however, solving one set of riddles leaves us knotted in a whole new tangle of riddles. Starting in the 1960s, topologists concerned with keeping track of all the intermediate choices sought (small-f) foundations for higher category theory. “Ontic commitment” was not at issue; it rarely is for mathematicians. “Usually, number-theorists (like me) neither understand, nor care about such foundational matters, and questions about them are normally met with a shrug.” Thus spoke a prominent number theorist on MathOverflow, using the word foundations with an implicit capital F. Even as he took care to avoid triggering a Russell paradox in his massive opus on ∞-categories, Jacob Lurie referred to such questions as “a nuisance.” “[T]he open secret,” writes Barry Mazur, “is that, for the most part, mathematicians who are not focussed on the architecture of formal systems per se … somehow achieve a sense of utterly firm conviction in their mathematical doings, without actually going through the exercise of translating their particular argumentation into a brand-name formal system.”⁷¹

To steer clear of set-theoretic paradoxes, Lurie chose to work in the setting of a Grothendieck universe. This gadget, one of Grothendieck’s rare gifts to Foundations as such, is designed to outrun Cantor’s trick for making a bigger set out of any given set, and it guarantees that one never meets an inherently contradictory concept like the “set of all sets.” But it requires a special axiom of its own that (unfortunately?) cannot be proved consistent with the ZFC axioms: the existence of what set theorists call an inaccessible cardinal. Even if you don’t know ZFC, it should be obvious that you can’t use it to deduce the existence of an inaccessible cardinal; otherwise it would be accessible! Voevodsky’s Univalent Foundations require not just one inaccessible cardinal but an infinite string of cardinals, each inaccessible from its predecessor.72

In a move that is standard when Foundational matters briefly peep through the mathematical scenery, Lurie reassures his readers that “none of the results” of his book “will depend on this assumption in an essential way.” This already qualifies inaccessible cardinals as “disputed objects,” in the terminology of Jessica Carter. Before laying a foundation, wrote Kronecker, “a rational builder” will want to be carefully informed about the structure for which it is intended; he reminds us that “with the richer development of a science the need arises to alter its underlying concepts and principles” [die ihr zu Grunde liegenden Begriffe und Prinzipien], but there is no risk to rationality: “important results turn out to be completely independent” of “explanations of basic mathematical concepts.” Jody Azzouni agrees:

Once we’ve established the usefulness of a mathematical system, and ensured (to the best of our ability) its consistency both internally, and with respect to the mathematics and the empirical subject matter it’s to be, respectively, applied with and applied to, we’re done—epistemically speaking. There is nothing more to find out.⁷³

Voevodsky shares Kronecker’s and Azzouni’s flexible attitude to Foundations but not their epistemic confidence. Even if you do not worry, like Voevodsky, that mathematics may be inconsistent, you have to wonder: is the Univalent Foundations program motivated by attachment to the notion of the independent existence of mathematics, and our consequent obligation to render it faithfully, or by the sense that its existence is contingent on our providing adequate Foundations?

Bourbaki’s “manifesto” mentioned earlier, written by Jean Dieudonné (later Grothendieck’s scribe), compares mathematics to a “big city, whose outlying districts and suburbs encroach incessantly … on the surrounding country, while the center is rebuilt … each time in accordance with a more clearly conceived plan and a more majestic order, tearing down the … labyrinths of alleys and projecting toward the periphery new avenues, more direct, broader and more commodious.” The text’s title is The Architecture of Mathematics, but the word foundation is conspicuously absent; the image is modernist and echoes Wittgenstein’s comparison in the Philosophical Investigations of language to an “ancient city,” notably in its emphasis on horizontal expansion rather than vertical growth. A Bourbaki who grew up in today’s parallel networked world, at ease with the blue Na’vi inhabitants of the distant moon Pandora of Cameron’s film, where mountains float free of gravity, might trade in Dieudonné’s image of a “universe” centered around a “nucleus” of “mother-structures” (groups, ordered sets, topological structure) for a three-dimensional network of intercommunicating roots, branches, and ladders.⁷⁴

Figure 7.7. Historical Monument of the American Republic, by Erastus Salisbury Field. (Michele and Donald D’Amour Museum of Fine Arts, Springfield, Massachusetts. The Morgan Wesson Memorial Collection. Photography by David Stansbury)

My private images of the intertwined avatar and categorical ladders remind me insistently of the Pandoran rainforest. Like the imperialist adventurers from a ruined earth, philosophers of Mathematics seek their “unobtainium” in the form of reliable Foundations, seemingly unaware that, just as Pandora’s unobtainium neutralized gravity, upward movement along the ladder of avatars abolishes foundations—not as a common reference, which will indeed be retained as long as it facilitates communication—but as a restraint on our imagination, or as anything other than a “nuisance.”

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

The danger pinpointed by Ivor Grattan-Guinness as “royal road to me”–type history consists in projecting present notions abusively on historical texts, a danger to which he finds some mathematicians particularly insensitive.75 The “royal road to me” is abusive retrospection; from the historian’s perspective, on the other hand, the avatar is a (possibly abusive) projection of a speculative future mathematics on current practice. It would be odd to do history without any consideration for the past—but perhaps not for the mathematician, whose understanding is not directed to history but rather to other mathematicians.

Jamie Tappenden hints at the avatar perspective in his chapters for The Philosophy of Mathematical Practice—a book dedicated, as its title makes clear, to philosophy of small-m mathematics.⁷⁶ Recounting how a property of prime numbers migrated to algebra, he revives Aristotle’s distinction between nominal and real definitions, so that the avatar would correspond to versions of the latter. “The core motivation is that in mathematics (and elsewhere) finding the proper principles of classification can be an advance in knowledge.” “Putting together a useful, metaphysically uncontentious doctrine of real definition seems promising if we take the relevant improvement to partially involve finding ‘a key that will explain a mass of facts.’ ” (Note Tappenden’s type 4 quotation marks!)

Tappenden refers to the problem of “carving [reality?] at the joints.” In 1900 Hilbert wrote:

If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems (my emphasis; note the definite article).

But is this recognition or creation? Grothendieck’s revolutionary rethinking of what mathematicians mean by “space” is often compared to Bernhard Riemann’s nineteenth-century invention of a new framework for geometry, of which his non-Euclidean geometry is only the best-known illustration. Tappenden asserts that, in contrast to the competing Weierstrass school, for Riemann, “What is to count as fundamental in a given area of investigation has to be discovered.”⁷⁷

The avatar perspective blurs the border between ontology and epistemology; it would be of no interest were it not for the possibility of convincing the audience of its fruitfulness. Good examples are regularly provided by string theory, whose practitioners discover often astonishing properties of familiar mathematical objects on the basis of something called “physical intuition,” only later confirmed rigorously by mathematical proof. Here I suppose the inaccessible mathematical theory (yet to be formalized within algebraic geometry, for instance) can be intuited through its avatar in physics. For example, Maxim Kontsevich’s homological mirror symmetry conjecture postulates an equivalence between the category of B-branes on a complex manifold M and the category of A-branes on a symplectic manifold M′. The category of B-branes is a familiar object from Grothendieck’s geometry, whereas the category of A-branes is usually said to be related to the Fukaya category, for which there is, unfortunately, no rigorous definition in general. This may or may not trouble physicists, but it makes communication between physicists and mathematicians problematic.78

ŚŪNYATĀ

When I was a Princeton undergraduate, the vast woods behind the IAS were a favorite place to explore alternate realities. These days it’s done more officially in engineering labs on campuses all over the world, even on the IAS campus itself. All mathematicians spend our working lives exploring the virtual realities gradually furnished and decorated by generations of our predecessors. True creators like Grothendieck design not only our virtual surroundings, but also the high-level heuristics we use to perceive them; the rest of us play with their avatars.

Insofar as the A-branes of the previous section are well-defined objects of mathematics rather than of physical intuition, this is largely due to the work of Andreas Floer. Before his suicide at the age of 34, Floer had taken several steps up the ladder, inventing a version of homology in which holes like those in figures 7.5 and 7.6 are seen as avatars of infinite-dimensional holes in a family of infinite-dimensional spaces originally studied in the settings of the differential equations of mathematical physics.

When he was a professor at Berkeley, Floer reportedly lived at Barrington Hall, a student cooperative famous for its periodic “wine parties,” where the punch was laced with LSD.79 Most mathematical venues offer more conventional refreshments. By next morning’s breakfast, mysterious hands will have whisked away the dozens of beer bottles that still littered each table when the Oberwolfach conference center’s last guests retired to their rooms, between 1 and 2 a.m. The often-prodigious consumption of alcohol starts just after dinner and certainly contributes to social integration—and, thus, over the long term to fruitful scientific collaboration; but its direct effect on mathematical creativity is questionable. On my last visit I pegged 10 p.m. as the tipping point beyond which scientific coherence can no longer compete with the beer drinker’s propensity to entertain increasingly controversial and less clear and distinct topics. It’s much the same at the endless round of champagne receptions in France: mathematical notes are compared for the first glass or two, after which conversation reverts to university politics and gossip.

Nāgārjuna dissolves phenomenal reality with his doctrine of the emptiness of all phenomena, “including, most radically, emptiness itself.”⁸⁰ The MMK uses the Sanskrit word śūnyatā for emptiness, and since śūnya is the term for zero in early Indian mathematics, it has been speculated that the Indian invention of zero can be traced to classical Indian metaphysics of emptiness, specifically that of Nāgārjuna.⁸¹ From this standpoint, it is appropriate that (as recalled in chapter α) all of set-theoretic mathematics is founded on the act of counting to zero.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

One might attempt to dismiss the avatar perspective in mathematics as merely a fashionable name for the practice of assuming a familiar conjecture (including it as an axiom) and deriving consequences. For example, it is standard in analytic number theory to assume the Riemann hypothesis (see chapter α) and see where it leads. As with Grothendieck’s hypothetical motives, reasoning on the basis of an unproved hypothesis can be naturalized as a reflex brought to bear in specific circumstances, and the practice can be derided as philosophically irrelevant on the grounds that reasoning with the unproved hypothesis follows the same rules as reasoning without it. That only begs a more interesting question: what circumstances lend themselves to the recourse to this reflex? From the standpoint of solving the problem, this is irrelevant; but the admission of a hypothesis or the adoption of the avatar perspective is seen in a different light if we allow that the goal is to make the solution obvious.

The practice I am describing here, working back from the avatar to the “underlying principle” (quotation marks of type 4) is precisely opposite to the practice of deduction.82 In this it resembles, and overlaps, the increasingly familiar practice called categorification—rooted in Grothendieck’s work on the Weil conjectures—in which one seeks structures one or more steps up the categorical ladder whose reflections are the familiar structures one seeks to understand.⁸³ This is a creative act, not a deductive procedure, since the categorified object is more complex, or richer, or somehow more meaningful than the avatar it is meant to explain; and, of course, the act can, in principle, be repeated indefinitely. If you were to ask for a single characteristic of contemporary mathematics that cries out for philosophical analysis, I would advise you to practice climbing the categorical and avatar ladders in search of meaning, rather than searching for solid Foundations. The philosophy of Mathematics that dominates university departments dispenses with meaning and sees actually existing mathematicians as avatars of formalized theorem provers. Thus, the ulterior entity of which a given mathematician is an incarnation is a sort of divine Turing machine, and the Turing test for mathematicians is beside the point.

“Without the capacity for symbolic transcendence,” writes Robert Bellah, “for seeing the realm of daily life in terms of a realm beyond it … one would be trapped in a world of what has been called dreadful immanence.”⁸⁴ But remembering Erdős, I need to qualify what I just wrote about the search for meaning. Erdős sought symbolic transcendence not at the apex of an endless categorical ladder, but rather in the inexhaustible activity of being a mathematician: for Erdős “the aim of life … is to prove and conjecture.”

In other respects, Erdős had more than a few things in common with Grothendieck. Both men were extraordinarily devoted to their mothers. Both were Central European Jews displaced, irreversibly, by World War II: Erdős left Hungary and just kept traveling, while Grothendieck remained stateless for many years by choice.⁸⁵ While Grothendieck’s pre-monition of the avatar ladder reaches ceaselessly skyward, Erdős built a no less tangled horizontal network of collaborations.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

Voevodsky’s Univalent Foundations program was the focus of a highly successful yearlong program at Princeton’s IAS in 2012–2013. Some of the participants will be reconvening in Paris in the spring of 2014 for a three-month program on “certified mathematics”; Voevodsky is slated to make an appearance. It’s impossible to overestimate the consequences for philosophy, especially the philosophy of Mathematics, if Voevodsky’s proposed new Foundations were adopted. By replacing the principle of identity by a more flexible account modeled on space, the new approach poses a clear challenge, on which I cannot elaborate here, to the philosophy underlying “identity politics”; it also undermines the case for analytic philosophy to seek guidance in the metaphysics of set theory, as in W.V.O. Quine’s slogan “to be is to be the value of a variable.” By a ricochet effect, the new Foundations might also destabilize the currently fashionable computational theory of mind. When a cognitivist like Steven Pinker confidently asserts that “the brain processes information, and thinking is a kind of computation,”86 he is not claiming that the brain works like an electronic computer but rather drawing on an established repertoire in philosophy of mind, especially in English-speaking countries. Information, as cognitivists understand it, is native to set theory and propositional logic; type theory, on the other hand, like the logic underlying Voevodsky’s univalent foundations, opens the way to a potentially endless expansion of logic along the lines of geometric intuition and is often described as a plausible foundation for intutition-istic logic. To say, for example, that “the brain is a topos⁸⁷ and thinking is a way of creating new Types” may in the end have the same meaning as Pinker’s sentence quoted earlier, but it would signal a different intention.

Within mathematics itself, Voevodsky’s proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory—the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodsky’s motives—then we could easily see Grothendieck’s program absorbing the Langlands program within Voevodsky’s new paradigm. As a number theorist, I find such a scenario unlikely—in number theory, prime numbers are the main players, and they are so different from one another that their aggregate behavior is more statistical than categorical—but there’s no denying that curiosity about homotopy type theory is growing in other branches of mathematics. The topic has appeared in several guises on MathOverflow; proof theorists as well as algebraic topologists are paying close attention.88

Sets, categories, and higher categories are all types in Voevodsky’s system; but propositions, and even “true” and “false,” the bottom rungs of logic, are types as well, and one reason some find the program appealing is that it encompasses the language and the matter of mathematics—syntax and semantics—in a common framework. I’m intrigued by this, and even by the suggestion (see chapter 3) that it will facilitate “computerization of mathematics,” but I can’t say it appeals to me. I would like to say that the ideal-type of a mathematical proposition is not a string of symbols in a formal language but a declarative sentence, in natural language, in which every word means exactly what it says, so that quotation marks are disallowed; and this may even be a characterization of mathematical propositions. So in the statement of a theorem, you are not allowed to say If X is a “motive” but only If X is a motive, and this motive has to mean or to be able to mean the same thing to any potential reader. But I immediately have to qualify this attempt at a characterization, because I want the sentences about “motives” to be included in mathematics as well. Mathematics unrolls on two tiers at least, the one we know and the one where what we know is revealed as an avatar of what we expect to know—but beyond that there is always another tier.

We have learned to see the roots of a polynomial as an avatar of the Galois group, which is, in turn, an avatar of the (category of) Galois representations, which is an avatar of the category of motives, which is an avatar of the automorphic side of the Langlands correspondence. Some suggest that Langlands will find his correspondence is itself an avatar of a still more abstract correspondence with other avatars in algebraic geometry, in physics (where it overlaps with mirror symmetry, whose A-models and B-models can be treated as avatars of one another), and even in homotopy theory. My research on this chapter provided unexpected preparation for the latest developments in the geometric Langlands correspondence, which cannot be formulated without reference to ∞-categories. So it’s not absurd to speculate that something like Voevodsky’s Univalent Foundations might reveal all the programs now associated with Langlands’ name to be avatars of a common theory, itself soon to be caught up in the cycle of “knowledge and indifference.” Bourbaki’s insistence on the unity of mathematics, taken to its grimly logical conclusion, suggests that if the development is sufficiently tight, the whole infinite cascade of Langlands dualities, and every other mathematical relation yet to be conceived, could be derived step by step from One Big Theorem at the infinite level—something on the order of samsāra = nirvana. “Buddha is said to have remarked that śūnyatā is to be treated like a ladder for mounting up to the roof of prajña [wisdom, understanding]. Once the roof is reached, the ladder should be discarded.” Did Wittgenstein have Buddha in mind when he wrote in the Tractatus that “anyone who understands me eventually recognizes [my propositions] as nonsensical, when he has used them—as steps—to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it.)”?89 The metaphor of foundations does not suit mathematical concepts; they are rather suspended from more remote concepts, and when after a long and arduous climb “suddenly … one direction becomes … more luminous” and we finally find our way to the concepts we had been seeking, we see that they are in turn the avatars of concepts we glimpse only dimly, and so on without end.

---

* These different cohomological theories would be like so many different thematic developments, each in its own “tempo,” “key,” and (“major” or “minor”) “mode,” of the same “basic motif” (called a “motivic cohomology theory”), which would at the same time be the most fundamental, or the “finest,” of all these different thematic “incarnations” … (that is, of all these possible cohomological theories) (Grothendieck, ReS, p. 60).

* Faust: ‘Tis true, I feel! In vain have I amass’d

Within me all the treasures of man’s mind,

And when I pause, and sit me down at last,

No new power welling inwardly I find;

A hairbreadth is not added to my height,

I am no nearer to the Infinite (from the 1865 Theodore Martin translation).

* May the dream we call life …

* FAUST: surely there / Will many a riddle be untied.

MEPHISTOPHELES: And many a riddle be knotted too (from the 1865 Theodore Martin translation).