Category Archives: Philosophy

Chinese Logic: An Introduction

zao-wou-ki-1

[LaTeX version here; Chinese version here]

Introduction

As late as 1898, logic was seen by the Chinese as “an entirely alien area of intellectual inquiry”: the sole Chinese-language textbook on logic was labeled by Liang Qichao (梁启超)—at that time a foremost authority on Western knowledge—as “impossible to classify” (无可归类), alongside museum guides and cookbooks (Kurtz, 2011: 4-5). This same textbook had previously been categorized by Huang Qingcheng (黄庆澄) as a book on ‘dialects’ (方言). The Chinese word for logic (luójí 逻辑) itself is, according to the Cihai (《辞海》/ Sea of Words) dictionary, merely a transliteration from the English—the entire Chinese lexicon had no word resembling it (Lu, 2009: 98). Hence it never occurred even to specialists that this esoteric discipline might have close affinities with the roots of Chinese philosophy, from the I Ching (易经) to the ancient Chinese dialecticians (辩者), as well as the famous paradoxes of Buddhism.

With the advent of computers, “there is now more research effort in logics for computer science than there ever was in traditional logics” (Marek & Nerode, 1994: 281). This has led to a proliferation of logical methods, including modal logic, temporal logic, epistemic logic, and fuzzy logic. Further, such new logical systems permit multiple truth values, semantic patterns based on games, and even logical contradictions. In light of these possibilities, research in ‘Chinese logic’ aims to reinterpret the history of Chinese thought by means of such tools.

This essay consists of three parts: the mathematics of the I Ching, the debates within the School of Names, and the paradoxes of Buddhism. The first section will, through examining the binary arithmetic of the I Ching, provide an introduction to basic logical notation. The second section will explore Gongsun Long’s famous bái mǎ fēi mǎ (白马非马) paradox, as well as the logical system of the Mohist school. The third section will explain the seven-valued logic of the Buddhist monk Nāgārjuna by way of paraconsistent logic.

yijing-lattice

1. The I Ching (易经) & Binary Arithmetic

The I Ching is one of the oldest books in history. Throughout the world, there is no other text quite like it. Its original function was for divination, giving advice for future actions; yet, after centuries of commentary, it has taken on a fundamental role in Chinese culture. In part, this is because its commentaries became (apocryphally) associated with Confucius, thereby establishing it as a classic.

Its survival of the ‘burning of books and burying of scholars’ (焚书坑儒) in 213-210BC has magnified the I Ching’s importance. Historically, the Zhou dynasty was marked by hundreds of years of war and dissension. Finally, Qin Shi Huang united the nation in 221BC, to become China’s first emperor. According to the standard account, in order to unify thought and political opinion, Emperor Qin Shi Huang ordered that all books not about medicine, farming, or divination be burned. And so, the vast majority of ancient Chinese knowledge has been lost to history. Yet, since the I Ching was about divination, it avoided sharing the same fate. In a sense, then, the I Ching has come to represent the collective wisdom of ancient China—it embodies their entire philosophical cosmology.

Confucius’s interest in the I Ching is well known. In verse 7.16 of the Analects, he says: “If some years were added to my life, I would give fifty to the study of the Yi [I Ching], and then I might come to be without great faults.” Curiously, this appears at odds with the rest of his philosophy. After all, the Analects elsewhere says: “The subjects on which the Master did not talk, were—extraordinary things, feats of strength, disorder, and spiritual beings.” (7.20). That is, Confucius had no interest in oracles. Hence we can conclude that for Confucius, the main content of the I Ching was not divination, but philosophy.

The core tenet of the I Ching is deeply metaphysical, namely: the complementarity of Yin (阴) and Yang (阳). Yin represents negativity, femininity, winter, coldness and wetness. Yang represents positivity, masculinity, dryness, and warmth. Accordingly, the gua (卦) or fundamental components of the I Ching’s hexagrams, are two lines: ‘⚋’ for Yin, ‘⚊’ for Yang.

The trigrams, made up of three lines, have 8 combinations (2³ = 8), and so are called the bagua (八卦), where (八) means 8. The bagua and its associated meanings are: ☰ (乾/天: the Creative/Sky), ☱ (兑/泽: the Joyous/Marsh), ☲ (离/火: the Clinging/Fire), ☳ (震/雷: the Arousing/Thunder), ☴ (巽/风: the Gentle/Wind), ☵ (坎/水: the Abysmal/Water), ☶ (艮/山: Keeping Still/Mountain), ☷ (坤/地: the Receptive/Earth). The I Ching’s commentaries revolve around 64 hexagrams of six lines (2⁶ = 64 combinations). There are multiple ways of ordering the hexagrams: the most well-known is the King Wen (文王) sequence, but the most important for our purposes is the Fu Xi (伏羲) sequence.

diagram-of-i-ching-hexagrams-owned-by-german-mathematician-and-philosopher-gottfried-wilhelm-leibniz

diagram of the I Ching’s hexagrams owned by Leibniz

In the 17th century, the mathematician Gottfried Wilhelm Leibniz attempted to develop a system of arithmetic using only the numbers 0 and 1, called binary arithmetic. Binary arithmetic is in base 2: its key point is that any integer can be uniquely represented as a sum of powers of two. For example, 7 = 1 + 2 + 4 = 1×(2⁰) + 1×(2¹) + 1×(2²), and since each of the coefficients is 1, therefore the binary representation of 7 is (111). Conversely, 5 = 1 + 4 = 1×(2⁰) + 0×(2¹) + 1×(2²), where the middle coefficient is 0, so that 5 in binary is (101). For larger numbers, we simply include larger powers of two: 2³ = 8, 2⁴ = 16, etc.

Leibniz corresponded with various Christian missionaries in China, and had received a poster containing the Fu Xi sequence. To his astonishment, by letting ⚋ = 0 and ⚊ = 1, the Fu Xi sequence of 64 hexagrams exactly corresponds with the binary numbers from 0 to 63! Using the trigrams as a simplified example, from top to bottom we read: ☱ = (011) = 0×(2⁰) + 1×(2¹) + 1×(2²) = 2 + 4 = 6, ☵ = (010) = 0×(2⁰) + 1×(2¹) + 0×(2²) = 2, and so on. Thus, according to the Fu Xi and binary sequence, the bagua are ordered as: ☷, ☶, ☵, ☴, ☳, ☲, ☱, ☰.

Further, since we can treat the trigrams as numbers, we can also perform on them arithmetic operations such as addition and multiplication. To do this involves modular arithmetic, which for pedagogical purposes is occasionally called ‘clock arithmetic’. Its main feature is that it is cyclical: after arriving at the base number (‘mod n’, in our case: mod 2), we start up once again at zero. So in mod 2 arithmetic, 1 + 1 = 0: we only use the numbers 0 and 1. In the same way, a 12-hour clock only involves the numbers 1 to 12, and so is ‘mod 12’; hence, 15:00 is the same as 3:00, and so on. Therefore, the mod 2 addition of the I Ching’s trigrams can be represented by the following table:

logic-table1

Note that this is equivalent to the ‘⊻’ (exclusive or) operation in Boolean logic. (Boolean logic simply uses 0 for ‘false’ and 1 for ‘true’.) This logical point of view comes most in handy for defining multiplication, since binary multiplication is equivalent to the logical ‘∧’ (and) operation (Schöter, 1998: 6):

logic-table2

The advantage of logic over modular arithmetic is that we can define complements (¬). For example, Fire (☲/101) and Water (☵/010) are complementary, and so are Sky (☰/111) and Earth (☷/000). The use of logic is actually quite helpful in analyzing the trigrams’ associated meanings. Using the slightly different terminology of lattice theory (Schöter, 1998: 9):

  1. The Creative [乾/☰] is the union (⊻) of complements.
  2. The Joyous [兑/☱] is the union (⊻) of the Arousing [震/☳] and Abyss [坎/☵].
  3. Fire [火/☲] is the union (⊻) of the Arousing [震/☳] and Stillness [艮/☶].
  4. The Gentle [巽/☴] is the union (⊻) of the Abyss [坎/☵] and Stillness [艮/☶].
  5. Arousing [震/☳] is the intersection (∧) of the Joyous [兑/☱] and Fire [火/☲].
  6. Abyss [坎/☵] is the intersection (∧) of the Joyous [兑/☱] and Gentle [巽/☴].
  7. Stillness [艮/☶] is the intersection (∧) of Fire [火/☲] and the Gentle [巽/☴].
  8. The Receptive [坤/☷] is the intersection (∧) of complements.

In a beautiful essay, Goldenberg (1975) uses a branch of mathematics called group theory to unify the above points. A group is an algebraic structure with two operations (e.g. addition and multiplication). It turns out that the I Ching’s hexagrams satisfy many of the conditions for a group, which are as follows. 1) Closure: any operation between two hexagrams produces a new hexagram. 2) Associativity: in arithmetic operations, the order of the hexagrams does not matter, e.g. (☵ + ☴) + ☳ = ☵ + (☴ + ☳) = ☲. 3) Identity Element: there exists a hexagram (the identity element) such that an operation with it and any other hexagram produces that same hexagram, e.g. ☷ + ☱ = ☱, as well as ☰ × ☱ = ☱. 4) Inverse: for every hexagram, there exists another hexagram, such that an operation combining them produces the identity element; here, for the addition operation, every hexagram is its own inverse, e.g. ☶ + ☶ = ☷. Note, however, that there does not exist a multiplicative inverse. Further, addition and multiplication both satisfy the property that a ⋅ b = b ⋅ a, so that the hexagrams are commutative. So while the hexagrams’ lack of a multiplicative inverse precludes them from being a group, since they satisfy the remaining properties they are thus a ‘commutative ring’.

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Élie Ayache’s The Medium of Contingency – A Review

Plakhova5

[All art by Tatiana Plakhova. Review in pdf here]

Élie Ayache, The Medium of Contingency: An Inverse View of the Market,
Palgrave-Macmillan, 2015, 414pp., $50.00 (hbk), ISBN 9781137286543.

Ayache’s project is to outline the ontology of quantitative finance as a discipline. That is, he wants to find what distinguishes it as a genre, distinct from economics or even stocks and bonds—what most of us associate with ‘finance’. Quantitative finance, dealing with derivatives, is a whole new level of abstraction. So Ayache has to show that economic and social concerns are exogenous (external) to derivative prices: the underlying asset can simply be treated as a stochastic process. His issue with probability is that it is epistemological—a shorthand for when we don’t know the true mechanism. Taleb’s notion of black swans as radically unforeseeable (unknowable) events is simply an extension of this. Conversely, market-makers—those groups of people yelling at each other in old movies about Wall Street—don’t need probability to do their jobs. Ayache’s aim is thus to introduce into theory the practice of derivatives trading—from within, rather than outside, the market. And it’s reasonable to think that delineating the ontology of this immensely rich field will yield insights applicable elsewhere in philosophy.

This is not a didactic book. People coming from philosophy will not learn about finance, nor about how derivatives work. Ayache reinterprets these, assuming familiarity with the standard view. Even Pierre Menard—Ayache’s claim to fame—is only given a few perfunctory mentions here. People coming from finance will not learn anything about philosophy, since Ayache assumes a graduate-level knowledge of it. Further, Ayache’s comments on Taleb’s Antifragile are limited to one page. The only conceivable reason to even skim this book is that you’d like to see just how abstract the philosophy of finance can get.

I got interested in Ayache because I write philosophy of economics. I wanted to learn what quantitative finance is all about, so several years ago I read through all his articles in Wilmott Magazine, gradually learning how to make sense of sentences like “Only in a diffusion framework is the one-touch option…replicable by a continuum of vanilla butterflies” (Sept 2006: 19). I’ve made it through all of Ayache’s published essays. Now I’ve read this entire book, and I deserve a goddamn medal. I read it so that you don’t have to.

Much of Ayache’s reception so far has been quite silly. I recently came across an article (Ferraro, 2016) that cited Ayache’s concept of ‘contingency’ as an inspiration behind a game based on sumo wrestling. (You can’t make this stuff up.) Frank Ruda (2013), an otherwise respectable philosopher, wrote a nonsensical article comparing him to Stalin![1] Philosophy grad students occasionally mention his work to give their papers a more ‘empirical’ feel (which is comparable in silliness to the sumo wrestling), especially Ayache’s clever reading of Borges’ short story on Pierre Menard—from which these graduate students draw sweeping conclusions about capitalism and high-frequency trading.

Ayache expects the reader to have already read The Blank Swan, which itself is not understandable without reading Meillassoux’s After Finitude. Thus, for most readers, decreasing returns will have long set in. My goal here is to summarize the main arguments and/or good ideas of each chapter, divested of the pages and pages of empty verbosity accompanying them. I try to avoid technical jargon from finance and philosophy except as needed to explain the arguments, though I do provide requisite background knowledge that Ayache has omitted. So first, let’s cover the most important concepts that the reader may find unfamiliar.

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Combinatorial Game Theory: Surreal Numbers and the Void

Chess, by Andrew Phillips (small)

[A pdf version is available here; LaTeX here]

Any number can be written as a tuple of games played by the void with itself.

Denote the void by the empty set ∅. We write: {∅|∅} = 0, with | as partition. ‘Tuple’ signifies ordering matters, so that {0|∅} = 1 and {∅|0} = −1. Then recursively construct the integers: {n|∅} = n + 1. Plug {∅|∅} into {0|∅} to get {{∅|∅}|∅} = 1, then this into {1|∅} to get {{{|}|∅}|∅} = 2…

So if games exist, numbers exist. Or rather: if games exist, numbers don’t have to.

Mixed orderings generate fractions, e.g. {0|1} = {{∅|∅}|{{∅|∅}|∅}} = ½. Games with infinity (written ω) or infinitesimals (ε = 1/ω) permit irrationals, and thus all reals. Further, it is valid to define {ω|∅} = ω + 1, etc. Once arithmetic operations are defined, more complex games can define and use such quantities as ∛ω and ωω.

Therefore: by defining numbers as games, we can construct the surreal numbers.(1)

∗                                   ∗                                   ∗

As well as defining numbers as games, we can treat games like numbers.

{∅|∅} can be played as the zero game. Simply: player 1 cannot move, and loses. Any game where player 2 has a winning strategy is equivalent to the zero game. Take two games G and H, with G a 2nd player win. The player with a winning strategy in H can treat the games separately, only moving in G to respond to the opponent. Player 2 wins G, but does not affect H’s outcome. Conversely, given G′ and H′, with G′ a 1st player win, player 1 is last to move in G′, giving player 2 an extra move in H′, potentially altering its outcome. In terms of outcomes, we say G + H = 0 + H → G = 0.

For a game G, −G is G with the roles reversed, as by turning the board around in chess.

G = H if G + (−H) = 0, i.e. is a player 2 win, and so is equivalent to the zero game.

Two more properties are clear: G + (H + K) = (G + H) + K (associativity) and G + H = H + G (commutativity).

We can see that G + (−G) = 0 by a clever example called the Tweedledum & Tweedledee argument. In the game Blue-Red Hackenbush, players are given a drawing composed of separate edges. Each turn, player 1 removes a blue edge, plus any other edges no longer connected to the ground, and player 2 does likewise for red edges. Since in G + (−G) the number of pieces is the same, player 2 can just copy the moves of player 1 until all pieces are taken. Player 1 will not be able to move, and so will lose. Hence player 2 has a winning strategy.

tweedledum and tweedledee

Thus games are a proper mathematical object—namely, an Abelian group.(10)

∗                                   ∗                                   ∗

A new notation links all this to surreal numbers. For any set of games GL and GR, there exists a game G = {GL|GR}. Intuitively, the Left player moves in any game in GL, and likewise for Right in GR. The zero game {∅|∅} = 0 is valid, and we may construct the surreals as before. Now we can write the surreals more easily with sets: {1, 2, … , n|} = n + 1.

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Avant-Garde Philosophy of Economics

by Tatiana Plakhova (2011)

To most people, the title of this post is a triple oxymoron. Those left thoroughly traumatized by Econ 101 in college share their skepticism with those who have dipped their toe into hybrid fields like neuroeconomics and found them to be a synthesis of the dullest parts of both disciplines. For the vast, vast majority of cases, this sentiment is quite right: ‘philosophy of economics’ tends to be divided between heterodox schools of economics whose writings have entirely decoupled from economic formalism, and—on the other side of the spectrum—baroque econophysicists with lots to say about intriguing things like ‘quantum economics’ and negative probabilities via p-adic numbers, but typically within a dry positivist framework. As for the middle-ground material, a 20-page paper typically yields two or three salvageable sentences, if even that. Yet, as anyone who follows my Twitter knows, I look very hard for papers that aren’t terrible—and eventually I’ve found some.

Often the ‘giants’ of economic theory (e.g. Nobel laureates like Harsanyi or Lucas) have compelling things to say about methodology, but to include them on this list seems like cheating, so we’ll instead keep to scholars who most economists have never heard of. We also—naturally—want authors who write mainly in natural language, and whose work is therefore accessible to readers who are not specialists in economic theory. Lastly, let’s strike from the list those writers who do not engage directly with economic formalism itself, but only ‘the economy’. This last qualification is the most draconian of the lot, and manages to purge the philosophers of economics (e.g. Mäki, McCloskey) who tend to be the most well-known.

The remaining authors make up the vanguard of philosophy of economics—those who alchemically permute the elements of economic theory into transdisciplinary concoctions seemingly more at home in a story by Lovecraft or Borges than in academia, and who help us ascend to levels of abstraction we never could have imagined. Their descriptions are ordered for ease of exposition, building from and often contradicting one another. For those who would like to read more, some recommended readings are provided under each entry. I hope that readers will see that people have for a long time been thinking very hard about problems in economics, and that thinking abstractly does not mean avoiding practical issues.

Category Theory, by j5rson

M. Ali Khan

Khan is a fascinating character, and stands out even among the other members of this list: by training he is a mathematical economist, familiar with some of the highest levels of abstraction yet achieved in economic theory, but at the same time an avid fan of continental philosophy, liberally citing sources such as De Man (a very unique choice, even within the continental crowd!), Derrida, and similar figures on the more literary side of theory, such as Ricoeur and Jameson. It may be helpful to contrast Khan to Deirdre McCloskey, who has written a couple of books on writing in economics: McCloskey uses undergraduate-level literary theory to look at economics, which (let’s face it) is a fairly impoverished framework, forcing her to cut a lot of corners and sand away various rough edges that are very much worth exploring. An example is how she considers the Duhem-Quine thesis to be in her own camp, which she proudly labels ‘postmodern’—yet, just about any philosopher you talk to will consider this completely absurd: Quine was as modernist as they come. (Moreover, in the 30 years she had between the first and second editions, it appears she has never bothered to read the source texts.) Khan, by contrast, has thoroughly done his homework and then some.

Khan’s greatest paper is titled “The Irony in/of Economic Theory,” where he claims that this ‘irony’ operates as a (perhaps unavoidable) literary trope within economic theory as a genre of writing. Khan likewise draws from rhetorical figures such as synecdoche and allegory, and it will be helpful to start at a more basic level than he does and build up from there. The prevailing view of the intersection of mathematics and literary theory is that models are metaphors: this is due to two books by Max Black (1962) and Mary Hesse (1963) whose main thesis was exactly this point. While this is satisfying, and readily accepted by theorists such as McCloskey, Khan does not content himself with this statement, and we’ll shortly see why.

Consider: a metaphor compares one thing to another on the basis of some kind of structural similarity, and this is a very useful account of, say, models in physics, which use mathematical formulas to adequate certain patterns and laws of nature. However, in economics it often doesn’t matter nearly as much who the particular agents are that are depicted by the formulas: the Prisoner’s dilemma can model the behaviour of cancer cells just as well as it can model human relations. If we change the object of a metaphor (e.g. cancer cells → people), it becomes a different metaphor; what we need is a kind of rhetorical figure where it doesn’t matter if we replace one or more of the components, provided we retain the overall framework. This is precisely what allegory does: in one of Aesop’s fables, say “The Tortoise and the Hare,” we can replace the tortoise by a slug and the hare by a grasshopper, but nobody would consider this to be an entirely new allegory—all that matters here is that one character is slow and the other is fast. Moreover, we can treat this allegory itself as a metaphor, as when we compare an everyday situation to Aesop’s fable (which was exactly Aesop’s point), which is why it’s easy to treat economic models simply as metaphors, even though their fundamental structure is allegorical.

The reason this is important is because Khan takes this idea to a whole new level of abstraction: in effect, he wants to connect the allegorical structure of economic models to the allegorical nature of economic texts—in particular, Paul Samuelson’s Foundations of Economic Analysis, which begins with the enigmatic epigraph “Mathematics is a language.” For Khan: “the Foundations is an allegory of economic theory and…the epigraph is a prosopopeia for this allegory” (1993: 763). Since I had to look it up too, prosopopeia is a rhetorical device in which a speaker or writer communicates to the audience by speaking as another person or object. Khan is quite clear that he finds Samuelson’s epigraph quite puzzling, but instead of just saying “It’s wrong” (which would be tedious) he find a way to détourne it that is actually quite clever. He takes as a major theme throughout the paper the ways that the same economic subject-matter can be depicted in different ways by using different mathematical formalisms. Now, it’s fairly trivial that one can do this, but Khan focuses on how in many ways certain formalisms are observationally equivalent to each other. For instance, he gives the following chart (1993: 772):

correspondence between probability & measure theoretic terms (in Khan, 1993; 772)

Correspondence between probability & measure theory

That is to say, to present probabilistic ideas using the formalism of measure theory doesn’t at all affect the content of what’s being said: it’s essentially just using the full toolbox of real analysis instead of only set notation. What interests Khan here is how these new notations change the differential relations between ideas, creating brand new forms of Derridean différance in the realm of meaning—which, in turn, translate into new mathematical possibilities as our broadened horizons of meaning let us develop brand new interpretations of things we didn’t notice before. Khan’s favorite example here is nonstandard analysis, which he claims ought to make up a third column in the above chart, as probabilistic and measure theoretic concepts (and much else besides) can likewise be expressed in nonstandard terms. To briefly jot down what nonstandard analysis is: using mathematical logic, it is possible to rigorously define infinitesimals in a way that is actually usable, rather than simply gestured to by evoking marginal quantities. While theorems using such nonstandard tools often differ greatly from ‘standard’ theorems, it is provable that any nonstandard theorem can be proved standardly, and vice versa; yet, some theorems are far easier to prove nonstandardly, whence its appeal (Dauben, 1985). In economics, for example, an agent can be modelled as an infinitesimal quantity, which is handy for general equilibrium models where we care less about particulars than about aggregate properties, and part of Khan’s own mathematical work in general equilibrium theory does precisely this.

To underscore his overall point, Khan effectively puts Samuelson’s epigraph through a prism: “Differential Calculus is a Language”, “Convex Analysis is a Language”, “Nonsmooth Analysis is a Language”, and so on. Referring to Samuelson’s original epigraph, this lets Khan “interpret the word ‘language’ as a metonymy for the collectivity of languages” (1993: 768), which lets him translate it into: “Mathematics is a Tower of Babel.” Fittingly, in order to navigate this Tower of Babel, Khan (following Derrida) adopts a term originating from architecture: namely, the distinction between keystone and cornerstone. A keystone is a component of a structure that is meant to be the center of attention, and clinches its aesthetic ambiance; however, a keystone has no real architectural significance, but could be removed without affecting the rest of the structure. On the other hand, a cornerstone is an unassuming, unnoticed element that is actually crucial for the structural integrity of the whole; take it away and the rest goes crashing down. Read the rest of this entry

Proofs and Calibrations: An Interview with Élie Ayache

Swans, by Divine-Anarchy

Only Élie Ayache could take something as tedious as plugging variables into a formula and turn it into something charming. The costs of entry to his corpus are high—readers must be familiar with avant-garde Continental philosophy plus actively interested in the materiality of options markets. Nevertheless, Ayache earns a place alongside such thinkers as Bataille, Klossowski, Baudrillard, Deleuze/Guattari, and Lyotard, who smash the concepts of political economy into brick walls to see what remains intact—the concepts or the wall. And yet, The Blank Swan is so much more. The syntax of options (‘optionetics’, to pilfer a lovely phrase) lies entirely outside the purview of post-Marxist ‘critical’ theory that has grown crusty at best, procrustean at worst. “Cantor’s transfinite seems to be materially operative in our derivatives world,” notes Ayache (après Meillassoux), as derivatives create new intensive ‘surfaces’ on which yet more exotic derivatives can be written. The market is therefore untotalizable, im-probable (beyond the very category of probability); it is not meaningful to speak of ‘capitalism’ as such.

As with Nietzsche or Niels Bohr, to write ‘about’ Ayache places the preposition in conflict with itself. “The market proposes a way of thinking of the future that is no longer mediated by knowledge” (2006: 34). One tries to find a position from which to describe, or critique, but finds the ground pulled out from underneath: “The market never starts. You are immediately in the middle of it or you are nowhere.” (Local, 12:27–12:44). Philosophical ‘depth’ has no meaning for the surface of the market, in which “the infinite is often the best approximation of the finite” (2007: 262). Thus the following interview is not an introduction to Ayache’s work, but outlines some less obvious aspects that help to illuminate the whole.

In one of your essays you said that Meillassoux referred to your notion of the market as an ‘arché-market’, but it’s not clear to me how it’s analogous to the arché-fossil. Could you explain the link?

If I understand Meillassoux well, arché-fossil is what provides evidence to science that dates back to days when thought and as a matter of fact life didn’t exist and when givenness of being to thought didn’t exist; so arché-fossil is the light reaching from the outermost recesses of the universe or the fact that decaying isotopes can help science to establish contact with periods of time that predate thought, etc. Arché-market is something different. In conversation with Meillassoux, I once pointed to him that the ‘market’ wasn’t limited to my eyes to the financial market or even to the market in the sense of exchange of goods against prices. Rather, the market was a new logic or a new category of thought, a medium that conducts contingency ‘instantaneously’ without the apparatus of possibility and probability. Ideally, I wanted to convince him that my ‘market’ is the register where his whole factual (i.e. non metaphysical) speculation should be conducted. He then advised me to no longer call this category by the name of ‘market’ but, in order to avoid confusion, by the more venerable ‘arché-market’. For one thing, a contingent event can make the ‘market’ disappear; however, the arché-market as higher category and register cannot disappear as it is the very medium of contingency.

How does your philosophical position account for the fact that relativistic effects cause minute differences in the ‘same’ price in different regions of the world, noticeable only on the nanosecond scale in HFT? This strikes me as a crucial issue for your own theory, especially since you focus on the market as ‘surface’, whereas relativistic arbitrage would imply ‘ruptures’ in this surface.

HFT is not really my cup of tea. It is a necessary and unavoidable development of the technology and this is all that I have to say. Financial theory holds that prices should verify arbitrage instantaneously and I can only welcome a technology that now applies this ideal requirement of financial theory. Doubtless financial theory understands ‘instantaneous’ arbitrage in pre-relativistic terms and doubtless there must be interesting extensions of arbitrage to relativistic physics, and doubtless the HFT technology may be hitting on that limit. However, all this is of no interest to me; the market is not equal to HFT. Sadly, HFT is distracting the attention of thinkers and of philosophers away from the hard problem of the market, which is the real metaphysical and ontological problem that derivatives pose. Surely HFT is attracting money and investment from the banks and surely the sociology of finance should look into it. However those banks are (in my opinion) investing in HFT because they have abandoned the thought of derivatives. The hard problem of the market is the smile problem. To solve the smile problem you need something else than probability; you need a new metaphysics. This is what I am trying to develop both technologically in my company (ITO 33) and philosophically in my personal research and writing. The smile problem is simply that statistics and the corresponding paradigm have to be replaced by the prices of contingent claims. The smile problem is that we imply volatility from the option instant prices and not from the historical series of prices of the underlying. Why this is essential and not accidental, why this is a crucial problem and not just an ‘approximation’ or a temporary defect of the theory/technology, is a question that I am still amazed that neither the bankers nor the quants nor the philosophers of probability have started to tackle. And why are derivatives so important? Because the definition of the market to me is the place where underlying and derivative trade on the same level and floor. Why a surface? Because of this identity of levels and absence of depth or hierarchy between underlying and derivative. There are no possibilities and states of the world underlying the prices of the underlying and consequently evaluating the derivatives. All there is is the surface of prices of derivatives and derivatives on derivatives. While derivatives can certainly be traded by HFT as proxys of the underlying, the problem which they pose really, or the smile problem, is a very ‘slow’ problem in the sense that it requires calibration and recalibration to all the prices of all derivatives written on that underlying at once. To repeat, time and time series are not the proper dimension here. Place and writing is.

What is your opinion of Taleb’s latest book Antifragile?

I think Antifragile is a very clever concept. Taleb is trying to generalize convexity (of options) to life and beyond the strictly financial realm. But with this he is becoming less and less of a dynamic trader and more and more a fan of static hedging (take care of your losses and your profits will take care of themselves). By contrast, I advocate dynamic hedging and the dynamic trading of derivatives. There is a constant battle between convexity and time decay (the cost of convexity) which Taleb seems to (want to) ignore. This battle is what the dynamic market is all about. My work is to try to generalize the matter or the category of the market beyond the financial realm.

How did you discover philosophy? When did you become interested in writing?

At the age of six, on my way to school, I once wondered whether the pedestrian crossing the street in front of me would have accomplished the same act and crossed the street if, for some reason that day, say because of illness, I had not gone to school. Then I realized that I wouldn’t have been there in the first place to even notice the pedestrian and even conceive of his being.

When I found myself stuck in a military camp in Lebanon in 1982 with nothing really interesting to do and wasn’t allowed to travel to France to study. I then discovered how writing was there and had nothing to do with time.

Like many readers, I have a hard time getting my bearings in Part III of your book, despite your insistence that it’s the most important part. You claim that the virtual cannot be theorized, only narrated, which is understandable (and reminiscent of Lyotard’s Libidinal Economy), but your writing often reads more like a Hegelian bildungsroman than like Deleuze. Could you perhaps spell out what you’re trying to do in Part III? Why did you choose Barton Fink, of all films?

Barton Fink is the key to my philosophy. From possibility (Barton Fink in his room) to the total of possibilities (Karl Mundt) to the writing surface (the liberation of Barton Fink at the end). Also notice that he ends up writing the same play as in the beginning of the movie, in true Menard fashion.

Part III: The book is the arché-arché-market

Most of your essays over the past few years have been revisions to The Blank Swan. Have you thought of writing another book, perhaps a sequel of sorts? (Or does your book place under erasure any attempt at doing so?) If so, what sort of problems and material would you want it to address?

I am currently completing a book. More strictly critical of financial theory. More metaphysical. Better. Harder.

“There is no economic world.”

There is no economic world. There is only an abstract economic description. It is wrong to think that the task of economics is to find out how the economy is. Economics concerns what we can say about the economy…

This thesis (adapted from Niels Bohr, the father of quantum theory[1]) is, to anyone not thoroughly debauched by philosophy, clearly nonsensical—the sort of postmodern tripe that embodies everything wrong with ‘theory’. Yet, it is quite the opposite. François Laruelle argues that any notion of ‘world’—as a priori/mnemotechnic cognitive mapping—is a product of philosophical thinking; in fact, he often uses the words ‘philosophy’ and ‘world’ interchangeably. Therefore, if the corpus of economics has a ‘world’, this implies that any worthwhile statements it makes are translatable into philosophy, which thus becomes privileged as a meta-discourse in relation to the ‘regional knowledge’ of economics. Such a role has been traditionally claimed by Marxism, as well as obliquely by disciplines such as psychoanalysis, whose proponents believe that they can have knowledge of the economy by imposing their concepts a priori upon whatever data is at hand (regardless of whether said theorist knows minutiae such as the difference between stocks and bonds…). To subvert this hierarchy—to argue that economics is properly non-philosophical, thus eliminating all grounds for the use of postmodern tripe—the thesis that ‘there is no economic world’ becomes essential. This paper presents a unified theory of economics and philosophy, arguing that economics consists of nonknowledge rather than knowledge (episteme/technē), that economics operates through unwriting or deconceptualizing the material of the other social sciences, and that economic models should not be viewed as attempts to represent the world, but as a radically non-Bayesian method of framing events in their contingency.

§1. World versus ‘World’

There is a famous story involving the British analytic philosopher A.J. Ayer and the French continental philosophers Georges Bataille and Georges Ambrosino, in a midnight conversation in January 1951 (Bataille, 2001: 111-3). Ayer introduced the simple proposition that “the sun existed before man,” which as a scientific realist he saw no reason to doubt. Ambrosino, a physician steeped in French phenomenology, insisted that “certainly the sun had not existed before the world.” Bataille, on the other hand, was agnostic. As he wrote afterwards (111):

This is a proposition that indicates the perfect non-sense that a reasonable proposition can assume. A common meaning must have a meaning within all meaning when one asserts any proposition that in principle implies a subject & an object. In the proposition: there was the sun and there were no humans, there is a subject without an object.

The easy way out of this dilemma (or as Bataille put it, this “abyss between French philosophers and English philosophers”) is to say that while Ayer was talking about the sun (as a well-defined scientific object composed of various elements, etc.), Ambrosino and Bataille were talking about ‘the sun’ (as ideal representation of the Real).[2] While Ambrosino had taken a purely idealist position, Bataille’s stance is much more interesting: he had, in fact, hit upon a problem that would later become known as the ‘arché-fossil’. This idea would be central to Quentin Meillassoux’s attempt to philosophize in a way that avoids what he calls ‘correlationism’—that is, the idea that “we only ever have access to the correlation between thinking and being, and never to either term considered apart from the other” (2008: 5), with ‘thinking’ and ‘being’ meant in the sense of ‘models’ and ‘objects’. In more visual terms, Meillassoux is searching for a way of doing philosophy that doesn’t just involve the imposition of a ‘grid’ of concepts (or ‘syntax’) upon the mass of data comprising the world—as has been the norm in philosophy since Kant’s Critique of Pure Reason.[3] An arché-fossil is any sort of scientific object or datum describing the state of the universe prior to the existence of subjects (e.g. humans) that could experience it—or, recalling the above anecdote: the arché-fossil describes the state of the world prior to ‘the world’. After introducing this concept, Meillassoux goes on to outline the ‘mechanics’ of why this idea is so immediately absurd to philosophers in the phenomenological tradition. The existence of ‘ancestral’ data implies (15):

  • that being is not co-extensive with manifestation, since events have occurred in the past which were not manifest to anyone;
  • that what is preceded in time the manifestation of what is;
  • that manifestation itself emerged in time and space, and that consequently manifestation is not the givenness of a world, but rather an intra-worldly occurrence;
  • that this event can, moreover, be dated;
  • that thought is in a position to think manifestation’s emergence in being, as well as a being or a time anterior to manifestation;
  • that the fossil-matter is the givenness in the present of a being that is anterior to givenness; that is to say, that an arché-fossil manifests an entity’s anteriority vis-à-vis manifestation.

DCP_0086 by Phillip Stearns

The notion of the arché-fossil underscores the tension between the world and ‘the world’. From the perspective of ‘the world’ there is either ‘world’ or ‘non-world’, whose boundary is set by the existence of an experiencing subject. Yet, by carbon-dating a meteorite (for example), it is possible to state that the ‘non-world’ and the world existed simultaneously (or: co-extensively), and moreover, that the evidence for this is given to us within ‘the world’. Philosophically, this is clearly unacceptable. Yet, it sheds some light upon an old Daoist koan:

庄子:“如果把天下就藏在天下,就不会被丢失,这是一般事物的通理”

“Hide the world in the world and the world will never be lost—this is the eternal truth.” ~Zhuangzi[4]

Zhuangzi is the same person who, upon waking up from a dream that he was a butterfly, wondered if he was actually a butterfly dreaming that he was a man. The anecdote is no doubt as popular as it is because of its stark opposition of ‘world’ (dream) and world (reality). A dream, after all, proceeds according to an internal logic where any sort of (arché-)hints that it is a dream, e.g. words on a page changing the second time you look at them, somehow don’t count. The most absurd events may occur in the most bizarre of settings, but any sense of contingency (the idea that it could be otherwise) is lost. If we take the lack of contingency in dreams as a principle, however, the very fact that Zhuangzi can ask whether he’s a butterfly or a man proves he isn’t dreaming! Zhuangzi’s query creates a false partition—with ‘dream’ and ‘non-dream’ as the only members of the state space—and is thus self-defeating: nonknowledge is in fact the most useful kind of knowledge he can have. So in order to avoid a performative contradiction, Zhuangzi must accept that the principle can’t be psychologically necessary. This gives rise to a fundamental contingency, where in order to make a convincing case that he is a butterfly, Zhuangzi has to argue that the current rules of psychology (and perhaps even of nature) would have to be able to be other than they are—the same position as Meillassoux!

For Meillassoux, this division of world and ‘world’ is the problem, and ought to be gotten rid of; Zhuangzi’s stance is similar, though his method eliminates this opposition in an entirely different way—which is the same as that of economics. Anyone accustomed to think in philosophical terms may be inclined, on reading the following sections, to suppose that the argument rests on a tacit assumption of this dyad. If such a supposition is found helpful, there is no harm in the reader’s adopting it as a temporary working hypothesis. In fact, however, no such division is made.

§2. Econo-fiction

To verify the claim ‘oil prices are manipulated by the USA’, a researcher could (in theory) physically go to each stage of the oil production/distribution process, from oil wells to spot or futures markets, to various nodes along logistical networks, to gas stations, etc. In the above claim, ‘oil price’ is well-defined as a variable; moreover, its role as subject of the sentence makes the former claim ‘economic’ in its genre. (Cf. the political statement ‘the USA manipulates oil prices’, with its focus on agency.) ‘USA’ is of course vague, but suffices for the problem at hand. The verb ‘to manipulate’ reifies (in this context), but is in principle observable. Our researcher could measure the ‘value added’ in each stage as it is expressed in price, then perform an (unavoidably qualitative) analysis of how fluctuations in the magnitude of this value-added (with respect to production costs, etc.) can be causally traced to the USA. In this context, economic methods would not per se be needed, only mercantile arithmetic. Economics is often thought of as simply an armchair version of our poor researcher’s task (implying that an ideal model is one that is just as complex as the real world). Yet, in the above statement economics acknowledges not the subject, verb, or object, but the preposition ‘by’: in a sort of econo-fiction, it shows the numerical properties that make ‘manipulation’ meaningful.

Economics can be defined as the science of non-discursive social relations, with a broad definition of ‘discourse’ such that one could equally say ‘non-conceptual’.[5] In fact, economics takes place through a process of deconceptualizing the findings of business, finance, and politics. As soon as you think you can understand an economic notion (e.g. an algebraic relation) intuitively and talk about it lucidly, economists develop a way to formalize it (via econometrics and so on) so as to make it entirely untranslatable into normal language. John von Neumann once remarked: “in mathematics you don’t understand things. You just get used to them.” This is exactly what Bohr was saying! By continually deconceptualizing its former results economics systematically prevents itself from creating a ‘world’. As in Roland Barthes’ famous formulation, the task of economics is to inexpress the expressible.[6]

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Exogeneity – Economics and Nonknowledge

2009_07_07, by Tas Vicze

An economic fact is structured as follows: “consumers in the sample place a premium on liquidity β = 0.73.” This serves its task in economic models and allows economists to draw conclusions that are correct for all practical purposes. But in everyday life such a number is meaningless. The reason for this is that this number cuts across all discourse, all affect, and the social conditions that engender it, rendering these causalities exogenous to the non-conceptual statements of economics. As such, an economic fact’s epistemological scope is not sufficiently expressed by the all-too-philosophical categories of episteme (‘know-what’) and technē (‘know-how’)—though obviously these cannot help but play a significant part—but can be better characterized as a form of nonknowledge. This use of the term ‘nonknowledge’ is, however, not reducible to the Confucian dictum “To know what you know and to know what you do not know, this is knowledge”—which merely redoubles epistemology on itself in a transcendental begging of the question. To know what we don’t know would require that we know what we don’t know we don’t know, ad infinitum. The nonknowledge of economics is, as it were, the last instance of the Confucian limit statement. This nonknowledge takes place in a single number, which unifies (without being unitary) and commensurates (without commensurability) disparate orders of causality. The purpose of such a number is, borrowing a phrase from Roland Barthes (a far better political economist than Althusser ever was), to inexpress the expressible. Exogeneity is the reason why economics must necessarily be (in)expressed by numbers, not words—as well as why philosophy, mired in discourse, is unable to speak of economics.

In its disjunction from Knowledge proper, economics is non-paranoiac precisely to the extent that it is okay not to know. This has in the past led to accusations that economics is a form of religion, by analogy with the latter’s suppression of questions via dogmatism. This is not internal to economics, however, but rather takes place in its traversal (and subsequent travesty) in(to) discourse, where causality is truncated and contingency forgotten. Economics deals in facticity without fact (Heidegger), which as such remains open, ‘closable’ in the last instance only. The clause “for all practical purposes” helps to underscore its (non-)answerality, its indecisionality between practice and theory. Yet, economics tra(ns)verses into discourse precisely by supplying this ‘last instance’—by endogenizing it, as best illustrated by Milton Friedman and the money supply. Philosophy cannot handle exogeneity. Its own limit statement is that economics become a Theory of Everything.

2009_06_30 by Tas Vicze

A concept is a model. This implies that the only form of ‘concept’ in economics is an economic theory itself, in its entirety. This goes unnoticed because the first idea associated with economics in the public mind is “supply and demand”—this despite the fact that real economists never use these terms in practice. An introductory course in economics (many people’s only exposure to the discipline) teaches students to play around with such concepts as supply and demand, interest rates, and so on. Philosophy ‘of’ economics likewise proceeds by attaching to an economic ‘object’ such as ‘labour’, then trying to relate it to other concepts. But those few who take an intermediate economics course find that they are being taught the same information, but without concepts. ‘Objects’ are replaced with exogenous variables, or rates of change x/y (read: “derivative of x with respect to y”). An economic model is an elaborate tautology, in the extended Wittgensteinian sense of the term where pq is a tautology, since the concept of p is contained in the concept of q. Its conclusions arise from the syzygy (roughly, coalignment) of its variables along with its presuppositions (made for the sake of simplification). In an elaborate process of synergy, this syzygy creates a concept that may be imposed at will. Conclusions (prescriptive and descriptive) seem obvious to an economist but are not so to a layperson, and philosophy students pontificate about neoliberalism while econometrics students are incapable even of articulating what they don’t understand in class. Yet, it’s precisely this para-conceptual syzygy that constitutes all that is valuable in economics. What separates a good model from a bad one is that in the latter, a specific presupposition may be shown to do all the ‘work’ in providing the model’s conclusion.

Semantically, all of the interesting statements of economics take place within the preposition of a philosophical statement. As Laruelle writes, “The identity of the with (the One with the One, God with God), is the true ‘mystical’ content of philosophy, its ‘black box’.” The armchair philosopher is forced to engage in an amphibological attempt to render (pseudo-)exogeneity as endogenous, forced to autopositionally posit black boxes in the form of virtus dormitiva (Molière). Philosophy creates names for the Real, and by these purports to have explained it. Conversely, an economic variable is a name that does not name (Lao Tzu: 名可名,非常名), but non-conceptually gestures toward radical ( absolute) exogeneity.

**Thanks to Tas Vicze for the artwork; you can view the rest of his portfolio here.