Category Archives: Philosophy

‘Patatime

Plakhova15 - ComplexitySimplified

[Art by Tatiana Plakhova. LaTeX version here.]

‘Pataphysics is the science of the trans-ontological: it finds imaginary solutions to bridge incompatible worlds (ontologies). As such, ‘pataphysics can only be a science of the particular; otherwise, these worlds would be subsumed under a more general ontology—thus intra-ontological, and not truly trans-ontological.

Its most radical concept is pataphor: a ‘metaphor-squared’ that leaps between ontologies, without any reduction or hierarchy.[1] So: if continental theory is an exercise in overcoming binary oppositions—conceptual distinctions that split our thinking into two separate ontologies—then it is deeply pataphorical in nature.

Pataphor and its many variations can be expressed by the following formula:

Here, A is one world (or ontology) and C is an incompatible world: A \neq C. Next, f is a metaphorical (or any meta-x) relation, and g is a non-figurative (or just x) relation. Last, B is an object that is metaphorically compared to something in world A, but which exists in world C. The concept is surprisingly general, occurring anywhere from economics and finance to various forms of art.

Not infrequently, some concept I have long found interesting turns out to have a pataphorical structure. Pataphor thus has a dual role, of both explaining why certain concepts are profound, and helping us know where to look—either to view old ideas in a new light, or even to synthesize bizarre new concepts.

This paper defines ‘patatime, by framing time travel as a ‘metatime’ relation. The first section shows how ‘patatime arises from the interaction of time and metatime in two well-known time travel paradoxes. The second section interprets Nick Land’s concept of templexity through ‘patatime. The last section identifies a pataphorical structure underlying many classic paradoxes and quasi-paradoxes.

1. Time Paradoxes & ‘Pataphysics

“It is fine to live two different moments of time as one: that
alone allows one to live authentically a single moment of
eternity, indeed all eternity since it has no moments.”
~Alfred Jarry – Days & Nights

There are two paradoxes associated with time travel. The grandfather paradox makes a change that creates a new timeline and annuls the original. The bootstrap paradox makes a change that is in fact continuous with the original timeline. Clearly, one and the same change cannot both annul and be consistent with the original timeline. Since a change must do one or the other, we get two paradoxes.

1.1 The Grandfather Paradox

In the grandfather paradox, someone goes back in time and kills their grandfather before he ever had children; yet, by so doing, the traveller could never have existed, and so could not have killed their grandfather. Killing one’s grandfather alters the course of history—the new future is incompatible with the original.

The grandfather paradox has the following ‘patatemporal structure:

Here, A is the present point in time where the time traveller begins; B is the point in the past where they kill their grandfather, annulling the original timeline to A; from the new timeline, C is the alternate present in which the time traveller was never born, and so could never kill their grandfather. Clearly, A belongs to a different ontology than C (i.e., A \neq C), since they belong to separate timelines.

This gives a relation A—ᶠ→B—ᵍ→C, where f is a ‘metatime’ relation (travelling to the past), while g is ‘time’. Thus, the grandfather paradox is a pataphor. The paradox holds for any change that rules out the future timeline that led to it.

It’s paradoxical because the metatime relation must still have a real effect even after the timeline it’s based on is annulled. So the grandfather paradox turns on the question of whether metatime can exist without an underlying time.

comic strip, by panistheman

1.2 The Bootstrap Paradox

‘Bootstrap paradox’ is from the phrase, ‘to pull oneself up by one’s bootstraps’. Someone is inspired by some object or information from the past—say, a poem. They travel back in time to see who created it, and it turns out that they write it themselves, from memory. Thus, this object or information has no origin: it is a causal loop. Time is changed, but in a way presupposed by the original timeline.

As self-reifying, the bootstrap paradox resembles hyperstition: fiction that makes itself real. Hyperstition is in fact a ‘literal’ pataphor. In a fictional ontology A, we speak ‘figuratively’ (f) of an object B, which we speak of non-figuratively (g) within a real ontology C. Written as a pataphor, hyperstition’s autopoiesis is ‘exogenous’, while in the bootstrap paradox it is precisely what’s at issue.

The bootstrap paradox is likewise a form of ‘patatime:

Here, A is the thing that inspires the traveller to go back in time—a metatime relation f—journeying to the point in time B where the thing was supposedly created. Last, the traveller ends up (re-)creating the thing C that later inspires them to travel back in time in the first place. (This occurs as a time relation g.) Here A \neq C, since A comes from an external source, but the traveller creates C.

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Lacan on the Number 13 and the Logic of Suspicion

[A pdf version is available here.]

Lacan once remarked on the Cartesian cogito that etymologically, “the French verb penser (to think)…means nothing other than peser (to weigh)” (1961-2: 14). Lacan’s mix of bad puns, abuse of notation, cryptic aphorisms & immense erudition has created a new style of writing, and even of thinking. A new language, in short, lacking any method for non-experts to weigh its words.

Of all things, Lacan’s earliest papers take the form of math puzzles—a lucid (albeit horrendously verbose) derivation, then reframing of the problem as metaphor. One such paper—“The Number Thirteen and the Logical Form of Suspicion”—has largely been forgotten. This post aims to recast the puzzle using discrete mathematics, and show how it bears upon Lacan’s later ideas.

What I like about this mathematical allegory is that even if one believes Lacan is a charlatan, here there’s no need to immerse oneself in psychoanalytic concepts, but only to think.

I hope that Lacanians will find working through my derivation a challenging exercise, and that mathematicians will be piqued at the idea of treating a math problem as a philosophical ‘text’. I for one would be glad to see more such texts.

1. Lacan’s Algorithm

We are given 12 identical pieces, and told one of them is ‘bad’—either lighter or heavier than the rest, we’re not sure which. Having only a scale with two plates, and no way to gauge numerical weight, we must find the bad piece in 3 weighings.

If we knew whether the bad piece was lighter or heavier, the problem would be easy: just split the pieces into two groups of 6, then split the ‘bad’ half into two groups of 3, then simply weigh two of the bad three. But we don’t.

Here, we’ll overview Lacan’s account for 12 pieces, and then in the next section we’ll consider n pieces, and try to explain why Lacan’s algorithm works.

Lacan begins by placing on the scales two groups of 4. Suppose they balance. Then the bad piece is in the remaining 4, so we can just weigh any 2 of the 4. If those balance, the 2 left-out pieces are bad; if they don’t, the 2 pieces on the scale are bad. So weigh one of the bad 2 against a good piece: if they balance, the other piece is bad; if they don’t, then the piece on the scale is bad. Simple.

Note how this was equivalent to the sub-problem of finding a bad piece out of 4 pieces, in 2 weighings. The sub-problem is embedded in the larger problem.

If the two groups of 4 don’t balance, we use the method of tripartite rotation.

Tripartite rotation

The scales don’t balance, so one is heavier (H), one lighter (L). So, we select 3 pieces from H, L, and the remainder (R), and rotate them: H → L → R → H.[1]

Case 1: Scales balance — the bad piece is in the 3 moved to R, and too light.
Case 2: Balance shifts — the bad piece is in the 3 moved to L, and too heavy.
Case 3: Unbalance doesn’t change — the bad piece is in the 2 unmoved pieces.

In cases 1 and 2, just weigh 2 of the bad pieces: if they’re equal, the remainder is bad; if not, we know the bad piece is the lighter (case 1) or heavier piece (case 2). For case 3, just pick one and weigh it against a good piece. And we’re done.

Lacan then considers the case of 13 pieces: 4 on each scale, 5 remainders. It’s clear that if the scales don’t balance, the problem is the same as with 12 pieces when the scale didn’t balance—the remainders are all good, whether 5 or 4.

Here, when the scales balance, we have a new problem. Recall how we could treat 4 pieces as a separate problem. So let’s examine the 5-piece sub-problem.

Start with 2 pieces on each scale and 1 remainder. If we’re lucky, the scales balance and the remainder is bad. If not, we have 4 pieces, but we know the 4-piece case takes two weighings, so the 5-piece case must take three weighings.

It’s the same even for 1 piece on each scale and 3 remainders. If we’re unlucky, the scales balance, giving a new sub-problem with 3 pieces—the smallest solvable version of Lacan’s problem. Weigh any 2. If they balance, the remainder is bad. If not, weigh a piece on the scale against the good piece. Total: three weighings.

So both the 3-piece and 4-piece cases take two weighings, 5 pieces takes three weighings, so it would seem that 13 pieces must take four weighings. Nope.

Actually, for 13 pieces, the 5 remainders aren’t truly a separate sub-problem. There’s a difference: we have 8 good pieces. For 3 or 4 pieces, this doesn’t matter, but for 5 pieces, Lacan can introduce a new trick: the ‘by-three-and-one’ position.

The ‘by-three-and-one’ position

Here, we have 2 pieces in each plate, with one of the 4 a good piece, and 2 remainders. If the scales balance, just weigh one remainder against a good piece and we’re done. If they don’t balance, here’s the trick: we can do the smallest possible tripartite rotation, H → L → R → H, where R is a good piece.

Case 1: Scales balance — the bad piece is in R.
Case 2: Balance shifts — the bad piece is in L.
Case 3: No change — the unmoved piece is bad.

Thus, the 5 remainders take two weighings, and the 13-piece case takes three.

In this case, treating the 5 remainders as a sub-problem was the wrong way to go, making it seem impossible to solve in 3 weighings. More pieces means more ways to divide between scales and remainder, increasing the risk of such pitfalls.

Thus, Lacan’s task is to find a general algorithm for any number of pieces, including a uniform way to divide them. The algorithm must minimize the maximum amount of weighings—i.e. find the minimum, assuming we don’t get lucky.

The problem also raises some new questions. The main one is: for a given number of pieces, how many weighings are needed? As in the solutions outlined above, Lacan answers this question, but fails to explain why his solution works.

Hence, the next section will diverge from Lacan’s exposition, using discrete mathematics to give an algorithm for n pieces. This will help us see how Lacan’s problem relates to the logic of suspicion, which we will outline in the final section.

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Pataphor in Mechanism Design

[All art from Michael Nesbit’s Phlataphor series; view his portfolio here. LaTeX version here.]

Game theory analyzes extant strategic situations to identify their equilibrium properties. Conversely, mechanism design (‘reverse game theory’) creates auctions, markets, or games whose incentive structures bring about pre-specified equilibrium properties.[1] Mechanism design is typically billed as the ‘engineering’ branch of economics (Roth, 2002). In a very real sense, then, it’s a science of imaginary solutions.

The latter phrase is one of numerous definitions for ‘pataphysics, a rigorously nonsensical philosophy claiming to be as far from metaphysics as metaphysics is from physics. If science proceeds toward ever greater levels of generality, ‘pataphysics views each phenomenon as a singularity — and hence, “examine[s] the laws governing exceptions” (Jarry, 1911: 21). As for the prefix, one could do worse than to think of it as a mix of meta (beyond) and para (beside).

Here, I’ll focus on pataphor, a figure of speech that purports to be as far from metaphor as metaphor is from non-figurative language. Pataphor is a fairly recent idea in ‘pataphysics, often misunderstood as merely a hyperbolic metaphor or a derogatory term. By clarifying its structure, I hope to make pataphor more accessible both as a writing exercise & as a concept.

Part 1 schematically defines pataphor based on several examples, using a notation adopted from category theory. Part 2 frames mechanism design as ‘economic pataphorology’, showing how pataphor (as well as meta-metaphor) can be applied in settings beyond literature. Part 3 outlines how the notation for pataphor allows an analogous definition of patonymy (cf. metonymy). Part 4 considers chains of pataphors. Part 5 raises questions for future research.

Pataphor

Pataphor was invented by the American writer Paul Avion, under the pseudonym Pablo A. Lopez. Below, we’ll use his own examples as illustrations. His definition runs as follows:

Pataphor – 1. An extended metaphor that creates its own context; 2. That which occurs when a lizard’s tail grows so long it breaks off and grows a new lizard.

Pataphor is typically viewed as a fun, if contrived, writing exercise — it’s not at all clear how one might apply it outside of literature. Further, the concept is often muddled, due mainly to the phrase ‘extended metaphor’. What’s needed is a schematic definition, both to clarify the concept and show where it’s applicable. Using a kind of math fittingly nicknamed ‘abstract nonsense’, we can define pataphor by the following formula:

pataformula

where A is the state of affairs in a first ontology (world #1), C is a second ontology (world #2), B (the ‘hinge’) is an object that A and C have in common, f is a metaphorical statement where something in A is compared to B, and g is a non-figurative statement in which the object B is implicated in the state of affairs C.[2]

A pataphor combines a metaphor and non-figurative statement; it is NOT a kind of metaphor.

We can see this in the following ‘canonical’ example of pataphor:

Non-figurative: Tom and Alice stood side by side in the lunch line.

Metaphor: Tom and Alice stood side by side in the lunch line, two pieces positioned on a chessboard.

Pataphor: Tom took a step closer to Alice and made a date for Friday night, checkmating. Rudy was furious at losing to Margaret so easily and dumped the board on the rose-colored quilt, stomping downstairs.

Here, A is Tom & Alice’s world and C is Rudy & Margaret’s world. B is the chessboard — metaphorical (f) in A, non-figurative (g) in C. We say that in metaphor, Tom won the game; in pataphor, Rudy lost.

Next we’ll look at a flawed pataphor (or quasi-pataphor):

Non-figurative: The moon rose over the sea.

Metaphor: The yellow eye rose over the sea.

Flawed Pataphor: The yellow eye rose over the sea: in time, a tear fell, beading along a whisker to fall into the blue porcelain dish.

Here, A is the world containing the moon at nightfall, B is the yellow eye, and C is the world containing the cat. The reason this pataphor is flawed is because we can interpret all of it as occurring within the cat’s world (C) by reading ‘sea’ as a metaphor for the milk.

Last, we’ll look at a more prolix pataphor:

Jenny is eleven years old. She lives on a farm in Luxembourg, West Virginia. Today Jenny is collecting eggs from the henhouse. It is 10 a.m. She walks slowly down the rows of cages, feeling around carefully for eggs tucked beneath clucking hens. She finds the first egg in number 6. When she holds it to the light she sees it is the deep tan of boot leather, an old oil-rubbed cowboy boot, creased with microscopic branching lines, catching the light at the swelling above the scarred dusty heel, curled at the cuff, bending and creaking as the foot of the cowboy squirms to rediscover its fit, a leathery thumb and index prying at the scruff, the heel stomping the floor. Victor the hotel manager swings open the door and gives Cowboy a faint smile.

Here, A is the world inhabited by Jenny, notably the egg she finds. The egg, due to its brown color, is metaphorically compared to a cowboy boot. Here, the boot acts as the ‘hinge’ B, opening onto the world C inhabited by the cowboy and Victor. (Note that without the final sentence of the paragraph, this would just be an extended metaphor.)

Exercise: Why is the following not a pataphor? “The sweaters are hanging in the closet, their profiles the silhouettes of elephants at the Municipal Zoo before Mr. Bigby’s five o’ clock show.”

Answer: It’s only an extended metaphor (A—ᶠ→C), since it lacks a non-figurative statement (g) made within the second ontology.

It should be clear now that pataphor is truly a novel and rich idea, and that to view it only as an extended metaphor destroys its most interesting quality, namely: being trans-ontological (spanning multiple ‘worlds’).

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Heideggerian Economics

the-fabric-of-time-by-nataliekelsey

[A pdf version is available here]

Lately I’ve had the poor judgment to start reading Heidegger’s Being and Time. I’ve been putting it off for years now, largely because it has no connection with the kind of philosophy I’m interested in. Yet, among my philosophical acquaintances there is a clear line between those who have read Heidegger and those who haven’t—working through this book really does seem to let people reach a whole new level of abstraction.

To my great surprise, in Being and Time (1927: 413), Heidegger remarks:

[E]ven that which is ready-to-hand can be made a theme for scientific investigation and determination… The context of equipment that is ready-to-hand in an everyday manner, its historical emergence and utilization, and its factical role in Dasein — all these are objects for the science of economics. The ready-to-hand can become the ‘Object’ of science without having to lose its character as equipment. A modification of our understanding of Being does not seem to be necessarily constitutive for the genesis of the theoretical attitude ‘towards Things’.

Curiously, no other sources I’ve found mention this excerpt. More well-known is a passage from “What are Poets for?” in which Heidegger denounces marketization (1946: 114-5):

In place of all the world-content of things that was formerly perceived and used to grant freely of itself, the object-character of technological dominion spreads itself over the earth ever more quickly, ruthlessly, and completely. Not only does it establish all things as producible in the process of production; it also delivers the products of production by means of the market. In self-assertive production, the humanness of man and the thingness of things dissolve into the calculated market value of a market which not only spans the whole earth as a world market, but also, as the will to will, trades in the nature of Being and thus subjects all beings to the trade of a calculation that dominates most tenaciously in those areas where there is no need of numbers.

Thus it’s very easy to appeal to Heidegger’s authority to support various Leftist clichés about capitalism. It’s far harder to bring Heidegger’s thought to bear on actual economic modelling—its ‘worldly philosophy’. In this post I’ll survey several of the less hand-wavey attempts in this direction. My main question is whether a Heideggerian economics is possible at all, and if so, whether there is a specific subfield of economics to which Heideggerian philosophy especially lends itself. My overview of each specific thinker sticks closely to the source material, as I’m hardly fluent enough in Heideggerese to give a synoptic overview or clever reinterpretation. I don’t expect to ever develop a systematic interpretation of my own, but I hope this post might prove inspiring to some economist with philosophical tastes far different from my own.

1. Schalow on ‘The Question of Economics’

Schalow’s approach is quite refreshing because he is both an orthodox Heideggerian and takes the viewpoint of mainstream economics, as opposed to Heideggerian Marxism such as Marcuse’s One-Dimensional Man. Schalow’s question is at once simpler and deeper: whether Heidegger’s thought leaves any room for economics. Here, ‘economics’ is minimally defined as theorizing the production and distribution of goods to meet human needs. (So in theory, then, this applies to any sort of economics, classical or modern.) The most obvious answer would seem to be ‘No’ — he notes: “It is clear that Heidegger refrains from ‘theorizing’ of any kind, which for him constitutes a form of metaphysical rationality” (p. 249).

Thus, Schalow takes a more abstract route, viewing economics simply as “an inescapable concern of human being (Dasein) who is temporally and spatially situated within the world” (p. 250). Schalow advocates a form of ‘chrono-economics’, where ‘scarcity’ is framed through time as numeraire. In a sense, this operates between ‘economic theory’ as a mathematical science vs. as a “humanistic recipe for achieving social justice” (p. 251); instead, “economic concerns are an extension of human finitude” (p. 250). Schalow makes various pedantic points about etymology which I’ll spare the reader, except for this one: “the term ‘logos’ derives its meaning from the horticultural activity of ‘collecting’ and ‘dispersing’ seeds” (p. 252).

It’s natural to interpret Being & Time as “lay[ing] out the pre-theoretical understanding of the everyday work-world in which the self produces goods and satisfies its instrumental needs” (p. 253). Similarly, “work is the self’s way of ‘skillful coping’ in its everyday dealings with the world” (p. 254). Hence Heidegger emphasizes production — which he will later associate with technē — over exchange, which he associates with the ‘they-self’ (p. 254). Yet, Schalow points out, both production and exchange can be construed as a form of ‘care’. Care, in turn, is configured by temporality, which forces us to prioritize some things over others (p. 256).

“The paradox of time…is the fact that it is its transitoriness which imparts the pregnancy of meaning on what we do” (p. 257). Therefore, “time constitutes the ‘economy of all economies’,” in that “temporality supplies the limit of all limits in which any provision or strategy of allocation can occur” (ibid.). We can go on to say that “time economizes all the economies, in defining the horizon of finitude as the key for any plan of allocation” (p. 258).

In his later thought, Heidegger took on a more historical view, arguing that the structure of Being was experienced differently in different epochs. In our own time, the strongest influence on our notion of Being is technology. Schalow gives an interesting summary (p. 261):

The advance of technology…occurs only through a proportional ‘decline’ in which the manifestness of being becomes secondary to the beings that ‘presence’ in terms of their instrumental uses.

In an age where the economy is so large as to be inconceivable except through mathematical models, one can say that “the modern age of technology dawns with the reduction of philosophical questions to economic ones” (p. 260). Thus, Heidegger is more inclined to view economics as instrumental (technē) rather than as “the self-originative form of disclosure found in art (poiēsis).” Yet, rather than merely a quantitative “artifice of instrumentality,” it is also possible to interpret economics in terms of poiēsis, as “a vehicle by which human beings disclose their immersion in the material contingencies of existence” (p. 262). Economics thus becomes “a dynamic event by which human culture adjusts to ‘manage’ its natural limitations” (ibid.). Framing economics in terms of temporality (as ‘chrono-economics’) allows it to remain open to Being, and thereby “to connect philosophy with economics without effacing the boundary between them” (p. 263).

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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 = 4 + 2 + 1 = 1×(2²) + 1×(2¹) + 1×(2⁰), and since each of the coefficients is 1, therefore the binary representation of 7 is (111). Conversely, 5 = 4 + 1 = 1×(2²) + 0×(2¹) + 1×(2⁰), 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: ☱ = (110) = 1×(2²) + 1×(2¹) + 0×(2⁰) = 4 + 2 = 6, ☵ = (010) = 0×(2²) + 1×(2¹) + 0×(2⁰) = 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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