Infinity in Mathematics and Theology

Abstract

“Infinity” and its derivatives are frequently used in mathematics and theology. Do these expressions denote the same thing in those distinct areas of scholarship? In this article the uses of “infinity” in mathematics and its uses in theology are examined and compared. One conclusion is that quite different concepts go under the heading of “infinity.” Although they must not be confused, there are some relations between mathematical and theological senses of infinity.     

Theology and Mathematics—Key Themes and Central Historical Figures

This article provides a survey of the history of the dialogue between the disciplines of mathematics and theology. It identifies three principal historical periods—the ancient, the scientific revolution, and modern. It identifies five key themes—the ontological status of mathematics; mathematics as a bridge to a noumenal world; God's relationship to the physical universe and how mathematics is part of this; truth—its nature, how we know it, certitude; and the role of mathematics in culture. The author develops these themes by exploring the thinking of several major historical figures.     

Philosophy as Anti-Religion in the Work of Alain Badiou

The Heideggerian rupture in the history of philosophy in the name of a phenomenological and poetic ontology has provided an opening which many of the key figures in twentieth century continental thought have exploited. However, this opening was marked by Heidegger himself as an ambiguous one, insofar as metaphysics was perhaps integrally ‘onto-theology,’ that is, ultimately continuous with the world-historical capture of the thought of being. This piece argues that the philosophy of Alain Badiou, which departs from the recognition that Heidegger is the ‘last universally recognised philosopher’, provides the means for a radical reconsideration of the philosophy-theology relationship in its specifically Heideggerian form, involving as it does further questions of science and technology, the status of the poem, and the nature of ontological thought as such. We argue that, through the deployment of mathematics as ontology, the Gordian knot of onto-theology and its legion of consequences can be cut, and a new assemblage of many of the key Heideggerian motifs can be put into play: the poem, history, and philosophy itself.

The errant name: Badiou and Deleuze on individuation,causality and infinite modes in Spinoza

Although Alain Badiou dedicates a number of texts to the philosophy of Benedict de Spinoza throughout his work—after all, the author of a systematic philosophy of being more geometrico must be a point of reference for the philosopher who claims that “mathematics = ontology”—the reading offered in Meditation Ten of his key work Being and Event presents the most significant moment of this engagement. Here, Badiou proposes a reading of Spinoza’s ontology that foregrounds a concept that is as central to, and celebrated in, his philosophy as it is strictly excluded by Spinoza: the void. In nuce, Badiou contends that for all of Spinoza’s efforts to offer an ontology of total plenitude, the void returns in his philosophy under the (at first sight) unlikely name of infinite mode. The presence of this errant name in Spinoza’s philosophy bears witness to the failure of his most profound intellectual endeavour. However striking Badiou’s reading of Spinoza, this paper argues that it fails to adequately grasp Spinoza’s metaphysics, particularly with respect to the central concept of modal essence, a concept which does not appear at all in the Badiouian text. By introducing a consideration of this concept, it becomes able to resolve the status of infinite modes, and to account for the move across the notorious finite–infinite divide. Thus the argument turns to the reading of Spinoza offered by Gilles Deleuze for a more thorough-going and nuanced approach, much superior to Badiou’s procrustean critique.

Badiou,Priest, and the Hegelian Infinite

Abstract

Hegel’s distinction between the bad and true infinites has provoked contrasting reactions in the works of Alain Badiou and Graham Priest. Badiou claims that Hegel illegitimately attempts to impose a distinction that is only relevant to the qualitative realm onto the quantitative realm. He suggests that Cantor’s mathematical account of infinite multiplicities that are determinate and actual remains an endlessly proliferating bad infinite when placed within Hegel’s faulty schema. In contrast, Priest affirms the Hegelian true infinite, claiming that Cantor’s formal mechanisms of boundary transcendence, such as ‘diagonalization’, are implicit in Hegel’s dialectic. While arguing that a clear dividing line can be drawn here between these two interpretations of the relationship between Hegel and Cantor, this paper also mounts a defence of the Hegelian true infinite by developing Priest’s suggestion that Cantorian diagonalizing functions are prefigured by Hegel’s dialectical overcoming of limits.     

The Problem of Patriotism: A Psychoanalytic and Theological Analysis

In this article, I examine patriotism from psychoanalytic and theological perspectives, arguing that it is a deeply problematic form of love and devotion. After providing a brief overview of the discourse vis-à-vis patriotism, I depict the characteristics of a dominant form of patriotism (self-state) in the U.S. Given this, I argue that, while patriotism has a variety of forms, the most prevalent form tends toward tragic consequences and it is this tragic tendency that I depict from psychoanalytic and theological perspectives. From a psychoanalytic perspective, the extant form of patriotism in the U.S. represents a self-state that signifies an idealized, omnipotent, and imaginary identification, which is accompanied by and contingent upon a devaluation of the Other. Moreover, the dominant form of patriotic self-states is secured by weak dissociation, omnipotent thinking and, in most cases, a collapse of the symbolic equation. From a Christian theological perspective, patriotism signifies absolutizing the relative and contingent. This form of idolatry manifests a distortion of reason and will, which contributes to corrupted and corrupting forms of love and devotion.     

Mathematical determinacy and the transferability of aboutness

Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “aboutness” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed.     

An Intriguing Third Way: Mapping Contextual Education for Curricular Integration

This article emerges from the experience of incorporating doctoral students into our Contextual Education (CXE) Program at Emmanuel College (Toronto). This change, I argue, helped us to distinguish more clearly among and thus distinctly orient the different kinds of relationships and theological practices that make up our program towards the often‐elusive goal of curricular integration. After outlining a definition of integration, I contextualize that definition in our particular practices at Emmanuel College using Kathryn Tanner's (1997) understanding of theology as a cultural practice as my guide. I then offer a brief overview of our CXE Programs to demonstrate how nurturing strategic partnerships within them has made certain forms of integration possible for our students. I close with some activities for practical application in other CXE contexts.     

On three arguments against categorical structuralism

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely autonomous from set theory.     

Phantom of consistency: Alain Badiou and Kantian transcendental idealism

Immanuel Kant is one of Alain Badiou’s principle philosophical enemies. Kant’s critical philosophy is anathema to Badiou not only because of the latter’s openly aired hatred of the motif of finitude so omnipresent in post-Kantian European intellectual traditions—Badiou blames Kant for inventing this motif—but also because of its idealism. For Badiou-the-materialist, as for any serious philosophical materialist writing in Kant’s wake, transcendental idealism must be dismantled and overcome. In his most recent works (especially 2006’s Logiques des mondes), Badiou attempts to invent a non-Kantian notion of the transcendental, a notion compatible with the basic tenets of materialism. However, from 1988’s Being and Event up through the present, Badiou’s oeuvre contains indications that he hasn’t managed fully to purge the traces of Kantian transcendental idealism that arguably continue to haunt his system—with these traces clustering around a concept Badiou christens “counting-for-one” (compter-pour-un). The result is that, in the end, Kant’s shadow still falls over Badiouian philosophy—this is despite Badiou’s admirable, sophisticated, and instructive attempts to step out from under it—thus calling into question this philosophy’s self-proclaimed status as materialist through and through.

The Argument From Scale Revisited

ABSTRACT

Several philosophers and scientists have claimed that modern cosmology forces us to think that humans are insignificant from the cosmic point of view. This argument for atheism derived from the scale of the universe became popular in the twentieth century, but its roots are long. Interestingly, it has not typically been considered as a very strong anti-theistic argument, and the size of the universe has also been used as proof for theism. This article offers a brief overview of how the argument from scale has been used in the past, and then discusses modern versions of it.     

Discerning the spirit(s) in the natural world: Toward a typology of “spirit” in the religion and science conversation

Recent developments in Christian theological thinking on the doctrine of the Holy Spirit (pneumatology) have raised questions about the relationship between “spirit” and science. Cursory review of the religion and science literature, however, yields a wide array of uses and understandings of the concept of spirit. Current attempts to assess the relationship of pneumatology to science require an inventory or basic classification scheme of the various notions of spirit that have been in circulation over the past generation. This essay presents a preliminary typology of such uses in order to enable scholars and researchers to chart courses for future research in this area.     

Wanting Nothing: imitation and production inthe economy of desire

Desire is examined here with a view to informing a theological response to capitalism. The notion of mimetic desire offered by René Girard is used as a starting point, Girard's ideas being brought into critical engagement with, first, other intersubjective accounts, namely those offered by Alexandre Kojève and by psychoanalysis in its Freudian and Lacanian variants; and then the account of desire as productive becoming offered by Gilles Deleuze and Félix Guattari, in which the link to capitalism is explicit. It is suggested that both mimetic desire and its productive counterpart can be understood as asserting the priority of desire itself over both subject and object. The possibilities for framing a coherent theological account of desire which is both mimetic and productive are then assessed with reference to Bernard of Clairvaux, Aquinas and Augustine, noting that accepting a metaphysics of participation makes plausible a productive desire which is nonetheless coded as a mimesis of the divine precedent. Such a conclusion brings into relief the inevitable difficulty of reconciling Christian and capitalist economies of desire.