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THE GAME OF PROBABILITY
Literature and Calculation from Pascal to Kleist
By Rüdiger Campe
STANFORD UNIVERSITY PRESS
Chapter One Theology and the Law
DICE IN THE AIR
A Prehistory of Probability: How Probability and Verisimilitude Came to Be Linked with Games of Chance
Jacob Bernoulli's Ars conjectandi (The Art of Conjecturing), which founded modern probability theory with its 1713 publication, includes a first and brief history of mathematical probability. According to the Basel mathematician, those who invented games of chance had, unbeknownst to themselves, already invented ways to measure probability. Bernoulli's point is that the idea of measuring probability seems implied in the invention of games of chance. If we look more closely, however, we see two different phases in his brief account. In a first step, Bernoulli's inventors prepare contrivances to engineer equally likely chance events (e.g., the hardware of regular-sided cubes—dice). In a second step, they come up with symbols for determining winners and losers (e.g., the software of the numbers carved on the cubes' sides). The two phases thus encompass the construction of strict contingency: first, the chance event; second, the additional strategic context in which chances appear as probabilities. How are the two phases connected to each other? In his redefinition of probability theory in the first decades of the twentieth century, Richard von Mises (1883–1953) developed the notion of "dice in the air" to illustrate how, in order to be subject to measurement, probability has to be radically removed from any everyday intuition of probability. The binary logic of events that are equally likely to happen or not to happen exists only in the "collectivity" of repeatable throws, and hence as if "in the air." For Bernoulli, however, the invention of dice is an ingenious act that detects and implements strict binary contingency and its measure within our everyday world. The second step, the strategic weighing of probability, seems to follow immediately from the first step of creating dice. The fact that the mathematical formula for probability had been discovered by inventing dice to produce chance events remained unnoticed, or at least unmentioned. Beyond the triumph of cracking pure chance—of being able to calibrate and measure probabilities—Bernoulli and his contemporaries do not seem even to consider how the everyday, the literary, or the logical meaning of the probable might weigh in on the use of dice as a measuring device.
Today's histories of probability have not solved the problem. In what sense can something that has no relation to anything beyond itself enter into a historical account—how can "dice in the air" emerge in the course of a story to be told? This book suggests that this very problem gives the literary history of probability—the history of probability and verisimilitude in rhetoric and poetics—a chance to intervene in the history of epistemological probability. Or to put it even more boldly: because of this problem, a literary history of probability has to supplement the history of probability in science. How did probability come to be associated with mathematical gaming theory? This inquiry, which is fundamental for understanding the emergence of probability, will lead back to probability in logic, rhetoric, and literary works. But only when an answer to this question can be provided does it make sense to reverse it: how did the emergence of mathematical probability influence the verisimilitude of the poets? This question then leads directly to the center of the traditional concept of poetry and its transformation into an understanding of literature as concerned with reality, an understanding that came about with the modern novel. In order to look into the effects of science in literary works, we must be able to account for the use of the poetological and logical category of probability within the history of mathematics. The two investigations work to complement each other, and differ only in respect to where we choose to begin the inquiry. They are both part of a history of writing that encompasses numbers and letters, the space of alphanumeric notation.
A cursory view of more recent histories of probability reveals the urgency of an investigation that traces the relationship between gambling calculations and their interpretation as a theory of probability and verisimilitude. In his reconstruction of the history of mathematics, Ivo Schneider clearly distinguishes the theory of chance in games of chance from the theory of probability. But this differentiation is merely a historical remark; the meaning of probability does not play a major role in Schneider's account. Ian Hacking and Lorraine Daston, however, in making the interpretation of calculation as probability the foundational moment of their comprehensive histories of probability, remain much closer to the eighteenth century and its way of arguing. But precisely because they are concerned with the mathematical and the semantic history of probability at the same time, they tend to overlook the historical difference between Pascal's wager and the interpretation of probability by Christiaan Huygens (1629–1695) and Jacob Bernoulli. Though the semantics of probability is finally receiving scholars' attention, the event of its inscription in the text of mathematical probability has nevertheless still been neglected. The persistence of this disregard is evidenced even by Hacking's and Daston's accounts, which are historical narratives in terms of discourses or ideas, based on a specific and unvoiced predicament for any history of science. The dilemma arises from the silent assumption that each and every semantic understanding of the mathematical term probability is a misunderstanding that, as such, affects and hampers its efficacy. If, as von Mises and others contended in the early twentieth century, a mathematical term such as probability is defined by its purely operational function within the calculation of probability—a development known as the axiomatic definition of mathematical concepts—then every paraphrase of such a term in ordinary language, every attempt to express its meaning in anything other than the strict formalism of an algorithm, is in conflict with its mathematical status. The axiomatic sense of a term is fundamentally linked to a lack of meaning according to a hermeneutical understanding. Even with
Hacking and Daston, the axiomatic definition of probability remains the hidden telos of the history of mathematical probability. Hacking, for his part, ties the origin of probability theory to the difficulties inherent in the very concept. Probability discussions in the early twentieth century had led to a controversy between the epistemic view (probability as a predicate of the observer's judgment) and the frequency view (probability as a predicate of the occurrence of certain phenomena). For Hacking, this opposition of interpretations forms the very criterion for the historical emergence of probability. In his view, probability emerges wherever the two interpretations and their mutual conflict can be shown to be in effect simultaneously. Daston, however, does not follow this ingenious formula. Instead, she frames the object of her studies historically as "classical probability." For her, probability as defined by Bernoulli represents the characteristic judgment of the Enlightenment, which incorporates reason in an ideal manner. Classical probability for Daston was doomed eventually to be surpassed by history simply because it is a semantic concept, a concept involving meaning, to begin with. For Daston, no less than for other historians of science, the ultimate understanding of probability in mathematics is the one developed in the twentieth century, which, however, is an axiomatic definition that strictly excludes any semantic understanding of the term.
Probability and verisimilitude thus do not simply constitute a case of what the philosopher and historian of scientific metaphors and concepts Hans Blumenberg (1920–1996) has called terminologization or mathematization. Terminologization for Blumenberg refers to a word's fixation as a perfectly defined term within the grammar of scientific expressions, particularly in mathematics. To become a term in this sense means to leave the realm of metaphors, interpretation, and ordinary language behind. Probability and verisimilitude instead pose the far more fundamental question of the relationship between scientific and nonscientific, precise and provisional, explicit and implicit knowledge: between true knowledge and its mere semblance. We must formulate our specific questions with careful emphasis: How did games of hazard come to function as the paradigm for the theory of probability and the calculation of chance in the first place? What allowed and what motivated this claim, and what did it entail? The important issue is no longer whether philosophers and mathematicians in the late seventeenth and early eighteenth centuries were on the right track, or were dealt a lucky hand, as they began to speak of probability within the theory of games of chance. Instead, presuming games of chance to be the unique example for conceptualizing probability in mathematical terms turns out to represent a simple gambit in the history of probability. At this time, the notion of probability began to be the object of a computational method—without, however, actually meaning anything other than what probability had always meant in logic, rhetoric, and literature. This moment—the ingenious discovery or bold declaration—was nevertheless a decisive stroke in a world of thinking and writing not yet characterized by the distinctions we know today, the distinctions between hard and soft science, between explaining and understanding.
An investigation into this foundational assumption necessarily means observing the involvement of topical probability and rhetorical-literary verisimilitude. These are categories that the question of the origin of probability theory zeroes in on, not ones it should take as given starting points. The task then is no longer to build a bridge from poetological terms and literary works to the idea or application of mathematical probability. Instead, the texture of probability theory and its semantic interpretation lead us back to literary works and their characteristic figuration of verisimilitude. The tension between term and metaphor already inheres in the moment when probability theory is defined and founded as the doctrine of chances and gaming. We can only gain access to this fundamental dichotomy between the calculation of chance and its interpretation as probability when we recognize the interpretation of probability as the attempt to provide mathematical probability theory with a semantic foundation. This foundation, then, is a foundation after the fact of the genuinely mathematical method.
In light of the belatedness of its foundation, a zone of conflict and paradox emerges within probability theory. On the one hand, we find Hacking's systematic paradox: for Hacking, the two opposing interpretations, the subjective-epistemic theory and the objective-frequency theory, can only establish probability when they coexist. On the other hand, we are again confronted with a paradox in Daston's historiographical argument. Probability theory in this case is only set in motion by the interpretation of probability as an act of reasonable human judgment, an interpretation that is, however, inadequate from the beginning, because all interpretation will prove to be incommensurate with the purely mathematical term. Finally, along with Blumenberg, we may characterize verisimilitude as essentially metaphorical and the calculation of chance in probability theory as scientific terminologization. In this case, modern probability becomes a terminally inconclusive back-and-forth between terminologization and re-metaphorization. Rhetorical and mathematical devices line up alongside each other and provide heterogeneous models for probability within probability theory.
This book contends that the invention of the modern novel, the novel of the eighteenth and nineteenth centuries, has to be seen through the lens of probability theory and, in particular, of statistics. The phantasmagorias of social normalcy and social laws based on probabilistic thinking indeed play major roles in the fabric of the modern novel. But before any attempt can be made to elucidate this relationship between the novel and probability theory, we must return to the interpretation of games of chance as a model of probability. We have to look even further back to the meaning of games before its interpretation within probability theory. In the pre-probabilistic understanding of gaming, the mutual relation between literary meaning and mathematical theory is negotiated in its most basic form. Even the relation between a semantic concept of the law and the law of large numbers finds its fundamental determination at this early stage. Today, we tend to see the nineteenth century's statistical norms and legalities as an overcoming of the semantic norm or an avoidance of the juridical concept of law. The semantics of games, however, which provided the origins of mathematical modeling, were themselves grounded in norms and laws of even older ancestry. The mathematical foundation of the theory already presupposed the fair game, lusus justus, before any norms could be deduced from its statistical application to morality and society. The very interpretation of gamblers' reckoning as the calculation of probability, which has been the subject of so much scrutiny, removed this older law of games, the law within probability theory, from our view.
A Brief Survey of the Semantics of Gaming: Theological and Juridical Politics
The semantic history of "game" provides clues for its use as a model example in probability theory, and this history is deeply connected to the politics of gaming in the early modern era. The church's fiats, the laws of the political sovereign, and police interventions against games of chance all worked together slowly and circuitously to carve games out as a sphere of calculable chance. An extensive literature of commentaries and tractates that separated the immanent world of chance from God's intervention in this world was inspired by the condemnation of games of chance by theologians and Christian jurists. A phenomenology of human behavior vis-à-vis chance events emerged as distinct from the Christian history of redemption and the divine sphere of miracles and providence. Jurists, especially Calvinist ones, picked up the debate in early modern times. They increasingly turned their focus away from the individual chance occurrence and concentrated on the contractual frame in which an event can first arise, or not arise, as the chance that decides the outcome of a game as a win or a loss. Concentrating on the rules of the game helped to neutralize the theological implications of the singular chance event.
In the sixteenth century, theological and moral themes found expression together in a fascinating allegory that relied on games of chance, not only for its colorful imagery, but also for its nature as allegory itself. The moral-theological elements can probably be traced back to the third-century Liber de aleatoribus, a pastoral pontifical decree that has come down to us misattributed to the Carthaginian Bishop Cyprian. The devil, as the oft-quoted text claims, is always present at the dice table. The game's diabolical nature reveals itself through the vices and crimes that follow more or less closely on its heels: the players' extreme passions, the squandering of fortunes, murder, and whoring. The fifth book of Saint Augustine's City of God is often cited for the dogmatic elements of the trope. In this book, the bishop of Carthage devotes himself to the struggle against the allegory of Fortuna and, by extension, the entire pagan pantheon. In condemning the divinatory arts, especially astrology, for their diabolical character, the very core of Christian dogma comes into play. The equilibrium between free will and divine power to decide the salvation of the individual and the whole world is eventually at stake in the game and the predictability of chance. In bulls and papal edicts of the sixteenth and early seventeenth centuries, therefore, indictments of divinatory writings and prohibitions of secular lot drawings always appear in near proximity to demands for executive measures against games of chance.
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