The Politics of Logic

Badiou, Wittgenstein, and the Consequences of Formalism

Paul Livingston

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781136656736
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

In this book, Livingston develops the political implications of formal results obtained over the course of the twentieth century in set theory, metalogic, and computational theory. He argues that the results achieved by thinkers such as Cantor, Russell, Godel, Turing, and Cohen, even when they suggest inherent paradoxes and limitations to the structuring capacities of language or symbolic thought, have far-reaching implications for understanding the nature of political communities and their development and transformation. Alain Badiou's analysis of logical-mathematical structures forms the backbone of his comprehensive and provocative theory of ontology, politics, and the possibilities of radical change. Through interpretive readings of Badiou's work as well as the texts of Giorgio Agamben, Jacques Lacan, Jacques Derrida, Gilles Deleuze, and Ludwig Wittgenstein, Livingston develops a formally based taxonomy of critical positions on the nature and structure of political communities. These readings, along with readings of Parmenides and Plato, show how the formal results can transfigure two interrelated and ancient problems of the One and the Many: the problem of the relationship of a Form or Idea to the many of its participants, and the problem of the relationship of a social whole to its many constituents.

Part I

Introductory

1 Introduction

An Inquiry into Forms of Life

I begin with a formulation of Wittgenstein’s, as enigmatic today as it was when written more than fifty years ago, which nevertheless captures the central problem on which a post-“analytic” philosophical reflection on language and a critical theory of politics in the “continental” mode are today converging:

What has to be accepted, the given, is—so one could say—forms of life.¹

In the Philosophical Investigations, Wittgenstein uses the term Lebensform (or Lebensformen) only a handful of times. But in these few cases, Wittgenstein employs the term in a positively assertoric voice that is rare within a text dedicated almost wholly to criticizing what we may otherwise take to be the “givens” of our language and everyday world. Contemporary interpretations oscillate between two ways of understanding Wittgenstein’s elliptical invocation of forms of life as “the given.” One line of interpretation understands it as indicating a conventionalist anthropologism of practices or cultures. This is a doctrine of the communal determination of meaning by means of shared practical conventions and norms “implicit in practice.” Another takes it to suggest a biologism of adaptive forms, a “naturalist” reduction of language and meaning to broadly natural-scientific facts.² I shall argue, however, that Wittgenstein here gestures toward a different problematic altogether, one that actually tends to undermine the very terms in which this decision between culturalist and naturalist readings is normally couched.

Indeed, as I shall argue, the problem posed by Wittgenstein’s invocation of forms of life is not located simply either in the question of the nature of lives or of their forms, but rather in what lies between these two terms: in what it is for a form to be a form of life, what it means that something like form or forms shape a (human) life at all. This relationship crosscuts the distinction between “culturalist” and “naturalist” conceptions of forms, or of the lives they shape. This problematic is thus not one of criteriology or genealogy; it cannot be resolved by a taxonomy of empirically determined ways of life, whether sociologically or biologically defined. Nor is it, however, the old Platonic problem of the transcendent being of forms, which is sometimes answered today by a naturalism of forms that is simply the inverted image of the dominant conventionalism of lives.³ Rather, it is the problem of understanding the meaning of the connective “of” in forms of life: in what way does form inform a life, and what is it for a life to be lived in some determinative relation, obscure or clear, vague or explicit, to forms or to a form?⁴ What is implicated in this question is the obscure link between form and matter, the ideal and the real, the universal and the particular, or the transcendent and the immanent. This is the place of what Plato called “participation.” Contemporary thought has traced it as the place of the diagonal, the paradoxical, and the chora, but also the (history-making or supra-historical) “event,” and the fragile possibility of a radically clarified life to come.

The aim of this work is thus to consider the relationship between forms or form and collective life, under the condition of an age determined by the technicization of information made possible by the logicosyntactical formalization of language. As I shall argue, this requires an investigation into the consequences of formalism in two senses. First, it requires a consideration of the ways that collective life can be theoretically reflected in formal-symbolic theoretical structures and the extent to which such structures can illuminate the lived forms of community and social/political association. Second, however, it is necessary to consider the effects of the material and technological realization of some of these very same formal structures on the actual organization of contemporary politics. This includes, for instance, the actual communicational and computational technologies that today increasingly determine social, political, and economic institutions and modes of action around the globe.

There are, of course, many available definitions of “forms” and “formalism.” Here, though, I follow the suggestion of Georg Cantor in the inaugural definition of mathematical set theory, one of the most important developments of formalism in our time. As early as 1883, Cantor defined the notion of a set [Menge] in terms that already demonstrate his grasp of its profound philosophical significance:

By a ‘manifold’ or ‘set’ I understand in general any many [Viele] which can be thought of as one [Eines], that is, every totality of definite elements which can be united to a whole through a law. By this I believe I have defined something related to the Platonic eidos or idea.⁵

Cantor here recognizes the relevance of his new concept of the set to the ancient problem that most vexed Plato in accounting for the Idea as the One that unites the Many of its participants. This is the problem of the One and the Many itself, which divided Parmenides from Heraclitus and was already avidly pursued by the Pythagoreans.⁶

If the Platonic eidos thus captures the original philosophical thought of the unity of the Many as One, the comprehensive set theory that Cantor founded, as I shall argue, transfigures this ancient problem in two further and interrelated ways. First, as Cantor and Frege already grasped, we may consider the relationship of a set to its elements as capturing the relationship between a universal and the individuals that fall under it. Thus, the definition of a set can be understood as formally identical to the definition of a concept, or a general term. According to this attitude, we behold, in the structure of sets and of the otherwise undefined relation ∈, the actual underlying formal structure of the relationship that holds between an object and a predicate or property that is asserted to hold of it.⁷ This is nothing other than the relationship of the universal to the particular, the same relationship that Plato obscurely designated as “participation.” By studying what is embodied in the fundamental consequences of set theory’s axiomatic capture of the relationship between One and Many, we come to understand the possibility, range, limits, and structure of this relationship in a formally clarified light.

Second, Cantor’s discovery provides resources for addressing one of the most ancient problems of political philosophy: namely, the question of the relationship of the One of the state, social whole, or community to the Many of its members and constituents. Since we may consider the relationship of a set to its elements to formalize this relationship, we may also thus take the generality of the set to manifest the “common” structure of any community, however this structure may be further defined or articulated. Accordingly, the formal structures of sets will also illuminate the basis of claims to unity, wholeness, or sovereign power that bind political communities together as wholes, as well as the disruptive aspects of non-inclusion that can lead to their sundering or transformation.⁸ Over the course of the investigations of this book, I will therefore be interpreting formal results of set theory, and closely related results of computation theory, to discern their implications for the nature and structure of political life. Some of these connections will be direct, founded in what are arguably homologies of structure that result from the very inherent logic of the One and the Many as it finds application in the consideration of social unity and community. Others are less direct and more analogical in character. Even in these more analogical cases, however, I shall attempt to demonstrate, each time, the significant and illuminative bearing of the mathematical theory of multiplicity and unity, in connection with the metalogical theory of proof and truth, on the most important and central questions of the organization of collective life. The most important point of this bearing, I shall argue, is the application of these theories to the very idea of the (unified) meaning of a sign across the infinity of its possible iterations and the varied and multiple circumstances of its legitimate use.⁹ Since this idea of the unity of meaning underlies the coherence of every intelligible form of collective praxis, the formal theories here find, through their relevance to language and meaning, a direct bearing on the question of the underlying structure of political life.

As I shall argue, set theory and the other developments of contemporary formalism uniformly arise from a transformative experience of what Greek thought already grasped as logos. Today, we might translate logos as “language.” This translation misses the plurivocity of the original Greek term, which is ambiguous, for instance, between “language,” “word,” “meaning,” and “account.” And as is well known, the Greek experience of logos is the historical basis for all that develops afterwards as “reason” and “rationality,” as well as for what we today understand, using the techniques of formal logic, as norms for the meaningfulness of language or the integrity of thought. Here, at any rate, I shall attempt to preserve the profound continuity that runs between Socrates’ search for the logoi of the various phenomena with which he was concerned and today’s critical inquiry, developed in part by means of formal, symbolic logic, into the constitutive structure of language as such. In the following inquiry, the “linguistic turn” taken in the twentieth century by both the “analytic” and the “continental” traditions (though in different ways) thus has a certain methodological priority. The aim is not, however, to theorize the structure and possibilities of an everyday human life by means of an external description of the empirical phenomena of language or its use, but rather to discern the basis of these phenomena themselves in the broader and more enigmatic phenomenon of the logos.

According to a long-standing philosophical tradition originating with Aristotle, the possession of logos defines the distinctive form of a human life. Although Wittgenstein does not simply concur with this tradition, he understands “language” (Sprache) as the name for an unparalleled site of problems, whose place is also that of the everyday, or an ordinary life. His analysis of these problems considers the pictures we are prone to offer ourselves of an individual or collective life, critically reflecting on the temptations that lead us to these pictures, their (limited) satisfactions, and their (manifold) frustrations. This consideration is “political” in the broad sense that it investigates the ways that we live our lives in relation to our own (individual or collective) self-conceptions of them. And although the direct political implications of Wittgenstein’s arguments in the Investigations and other “central” texts are not always immediately obvious, it is nonetheless evident that he intended such analyses as the “private language argument” and the “rule-following considerations” to support a far-ranging critical reflection on the ordinary ways of life of the culture in which he found himself. This includes his critical engagement with the leading organizational structures and self-rationalizations of a twentieth-century industrial culture dedicated to (false or misleading, as Wittgenstein would suggest) guiding ideals of novelty and “progress” achieved through technical and organizational means.¹⁰

In the twentieth century, the material and historical “rationalization” of social life (for instance, in the widespread development and standardization of technologies and practices of communication, information exchange, and commodification) is in fact closely linked with developments arising from critical reflection on language and its formal structure or structures. Accordingly, both the concrete historical and the abstract critical consequences of formalism must be treated together if we wish to produce an analysis adequate to the most important social and political phenomena of our time.¹¹ Many existing analyses take into account the effects on social life of technology, progressive rationalization, and “instrumental reasoning.” But it is a substantial failing of many of these existing analyses that they do not consider, in any detail, the internal implications of the specific abstract and formal-logical structures that, on their own accounts, increasingly dominate social and political life. It is typical for these analyses, continuing in various styles the classical legacy of Kantian critique, to complain of the growing dominance of technical, instrumental, or calculative reason, while maintaining the possibility of a distinct, non-formal or “lived” modality of reason still accessible to critical thought and practice.¹² But if, as seems likely, the twentieth-century development of formal reflection on language and logic problematizes the very terms in which theorists have attempted to describe such an alternative modality of “lived” reason and reasoning, it may be that critical thought about technology and society must now continue explicitly in a formal mode, if it is to continue at all.

This immediately produces a new question, which nevertheless may be seen, retrospectively, as having been one of the most significant questions of a wide variety of philosophical projects (“analytic” as well as “continental”) in the twentieth century. This is the question of the formalization of formalism itself, of the reflection of formal-symbolic structures within themselves, and thus of the possibility of these structures coming to comprehend and articulate their own internal constitution and limits. Within the analytic tradition, this question is posed and pursued within the ill-defined field sometimes called “metalogic.” Its results are recognized as profound, but their larger significance has, so far at least, been difficult to place. In particular, despite the largely negative significance usually ascribed to them, the transformative results of Russell, Gödel, and Tarski have fundamentally articulated what we can expect from a critical reflection on the nature of language and our human access to it. On the “continental” side, as well, such transformative critical meta-reflection has resulted from the massive mid-century project of structuralism, especially as “post-structuralist” philosophers have subjected it to internal critique on its own terms.

One of my chief goals in the present work is thus to argue that these two strands of reflection on language—metalogical analysis on the “analytic” side, and post-structuralism in a deconstructive mode on the “continental”—can be allied, and thus can both be useful sources of critical reflection on the political implications of formalism as such. Their combination can yield, in particular, a formally clarified understanding of the constitution and structure of political communities, as well as of their possibilities of alteration and internal dynamics of change.

At Politics 1253a7–18, just after defining the human being in terms of its possession of the power of speech, or logos, Aristotle suggests an essential link between this definition of the human and the very possibility of politics:

Now, that man is more of a political animal than bees or any other gregarious animals is evident. Nature, as we often say, makes nothing in vain, and man is the only animal who has the...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Methodological Preface
• Acknowledgments
• Part I: Introductory
• Part II: Paradoxico-Criticism
• Part III: Badiou and the Stakes of Formalism
• Notes
• Bibliography
• Index