Implications of Knowing the Infinite for Knowing God: An Interdisciplinary Study Between Mathematics and Theology

WHEATON COLLEGE GRADUATE SCHOOL Implications of Knowing the Infinite for Knowing God: An Interdisciplinary Study Between Mathematics and Theology A ...
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WHEATON COLLEGE GRADUATE SCHOOL

Implications of Knowing the Infinite for Knowing God: An Interdisciplinary Study Between Mathematics and Theology

A Thesis Submitted To The Faculty Of The Graduate School In Partial Fulfillment Of The Requirements For The Degree Of Master Of Arts

Department of Biblical and Theological Studies

by Cara A. Tacoma Wheaton, Illinois May 2012

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Implications of Knowing the Infinite for Knowing God

by Cara A. Tacoma

Approved: ------------------------------------------------------Dr. Jeffrey Barbeau, First Reader

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------------------------------------------------------Dr. Kevin Vanhoozer, Second Reader

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------------------------------------------------------Dr. Robert Brabenec, Third Reader

-------------------------Date

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Disclaimer The views expressed in this thesis are those of the student and do not necessarily express the views of the Wheaton College Graduate School.

WHEATON COLLEGE

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To infinity and beyond . . . And to my professors, family, and friends who have supported me, taught me, and showed me the love of Christ.

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Abstract:

This study is a historical recovery of the theological contributions of mathematician Georg Cantor (1845-1918). I argue that Cantor’s mathematical work with the actually infinite led him to epistemological conclusions about God that are analogous to the conclusions drawn by Catholic theologian Erich Przywara (1889-1972) with his analogia entis, or analogy of being. Based on the connection I draw between their works, I further argue that Cantor’s work with the mathematically infinite is useful for informing a debate over the validity of the analogia entis begun in the twentieth century between Erich Przywara and Reformed theologian Karl Barth (1886-1968) and that continues into modern theology.

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Wheaton, Illinois Date______________20___

Implications of Knowing the Infinite for Knowing God

Wheaton College Department of Biblical and Theological Studies Master of Arts Degree

Permission is herewith granted to Wheaton College to make copies of the above title, at its discretion, upon the request of individuals or institutions and at their expense.

__________________________________________ Signature of Author

Extensive quotation of further reproduction of this material by persons or agencies other than Wheaton College may not be made without the expressed permission of the writer.

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CONTENTS INTRODUCTION

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LITERATURE REVIEW

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CHAPTER 1. ON NATURAL VS. REVEALED THEOLOGY Part 1. Setting the Stage: Historical Background of the Analogia Entis

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Part 2. Natural Theology: Erich Przywara and his Development of the Analogia Entis 15 Part 3. Revealed Theology: Karl Barth’s Response to the Analogia Entis

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CHAPTER 2. GEORG CANTOR AND SET THEORY Part 1. Introduction

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Part 2. Set Theory and the Mathematically Infinite

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Part 3. Cardinality of Infinite sets and the Continuum Hypothesis

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Part 4. Cantor and Theology

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CHAPTER 3. THE CONVERGENCE OF MATHEMATICS AND THEOLOGY Part 1. The Implications of Cantor’s Thoughts for his View of Human Knowing

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Part 2. Connecting Cantor’s Actually Infinite and Przywara’s Analogia Entis

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Part 3. The Importance of Cantor’s Voice for the Debate between Barth and Przywara 63 CONCLUSION

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BIBLIOGRAPHY

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Introduction “For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. For what can be known about God is plain to them, because God has shown it to them. For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse” (Romans 1:18-20, ESV). Can humans know God? If so, to what extent can we know him? And how can humans know God? Can we know Him through creation, or only by revelation in Christ and scripture? These have been perennial questions for millennia. And answering these questions is divisive; responses are varied not only between Orthodox, Catholic, and Protestant theologians but also within each of these church traditions. This thesis is concerned not with the question of our ability to know about God,1 but rather with the extent and means by which we can come to know of him. In this thesis, I argue that the work of German mathematician Georg Cantor contributes profoundly to a discussion that is presently taking place among theologians: the debate over the analogia entis, or analogy of being. In the early twentieth century, German Jesuit theologian Erich Przywara (18891972) published his work on the theological doctrine of the analogia entis. Przywara’s analogy of being allows that humans can know something about God on the basis of our own existence (i.e. our being). Przywara claimed that we humans are similar to God because we exist, as God exists. However, we are dissimilar to God because we are 1. I accept the presupposition that humans can, in fact, know something of God.

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created and he is not. Przywara viewed human existence as in tension with our essence; the fact that we exist does not define our essential humanness. God’s being, on the other hand, is in unity with his essence. Because God is uncreated, his being is his essence and his essence is his being. That is to say, God’s essence is “I am.” Because we are both similar and dissimilar to God in these ways, we are analogous to him by our being. Because of this analogy, we can know something of God. Przywara’s analogy of being thus provides a claim that God is knowable by creation (human beings). Przywara asserted an epistemological claim about the knowability of God based on human ontology. Decades before Przywara had published his work on the analogy of being,2 German mathematician Georg Cantor (1845-1918) proposed the existence and definition of the actually infinite, an idea that had been denied by mathematicians, philosophers, and theologians alike since the time of Aristotle. Cantor came to his theory of actual infinity through set theory, the branch of mathematics that he fathered. He argued that his theory allowed humans to know something about God on the basis of the existence of the actually infinite. Cantor claimed that the actually infinite is similar to God in that the infinite set of actually infinite sets is an absolutely infinite set, which can only be comprehended by God and is itself associated with God.3 The actually infinite is, at the same time, dissimilar to God because God has allowed humanity to know and comprehend the actually infinite. The actually infinite is only partial knowledge, whereas the absolutely infinite is complete (the set of all actually infinite sets) and beyond human 2. I have found no evidence that Cantor himself referenced the Catholic concept of analogy with regards to his work. Cantor was certainly not aware of Przywara’s work on the analogy of being because Cantor had died more than a decade before Przywara published his work on the analogia entis. 3. Cantor is vague on what precisely it means for the absolutely infinite to be associated with God; I am not sure whether Cantor would claim that the absolutely infinite is God or is merely known by God alone.

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comprehension. Absolute infinity, Cantor claimed, exists only in the mind of God. However, Cantor described actually infinite sets as a “staircase” leading to the absolutely infinite, i.e. God.4 Therefore, the actually infinite allows humanity to know something about God. By asserting that the actually infinite is a “staircase” to God, Cantor has provided an epistemological claim about the knowability of God based on mathematical ontology. In this thesis, I draw an analogy of my own between what Przywara has claimed with his analogy of being and what Cantor has claimed with his theory of the actually infinite. Their claims are similar in that they both enable humans to say something about God on the basis of creation. Their claims are dissimilar in that Przywara uses human existence as the basis of our ability to know God, whereas Cantor uses the existence of the actually infinite as the basis of our ability to know God. Thus, although their separate enterprises have very different starting points (Przywara’s is theology and Cantor’s is mathematics), they have essentially the same end: the claim that God is not only knowable but knowable on the basis of the ontological nature of creation (for Przywara, the existence of humans, for Cantor, the existence of actual infinity). Therefore, this work is a historical recovery of the theological contributions of Cantor. Not only did Georg Cantor revolutionize mathematics with his theory of the actually infinite, but also he made profound theological claims about the knowability of God and about how humans might come to know God through mathematics. In order to demonstrate the significance of Cantor’s theological claims, I place him in a conversation that began between Erich Przywara and Reformed theologian Karl Barth (1886-1968) in 4. Rüdiger Thiele, Mathematics and the Divine: A Historical Study, ed. T. Koetsier and L. Bergmans (Amsterdam: Elsevier, 2005), 532.

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the twentieth century and that continues into present-day theology. This debate centers on the validity of the analogy of being. Barth rejected Przywara’s analogy of being because he believed it inevitably led to idolatry, the idolization of human beings. The debate over the analogy of being also touches on issues of human nature because the ability to accept this analogy is based upon fundamental beliefs regarding the nature of humanity (its capacity to know and do good) after the Fall. This study addresses these issues of human nature but remains focused on the epistemological consequences of the analogy of being and of Cantor’s work. I argue that Cantor’s innovative mathematical claims about the infinite have something to contribute to this modern debate surrounding the analogy of being. I bring Cantor into the debate on the basis of the analogy I have already drawn between his work and Przywara’s. Because Cantor’s work is like Przywara’s in claiming the knowability of God through creation, Cantor becomes a meaningful voice for the validity of the analogia entis. If we accept Cantor’s claims about the actually infinite in relation to God to be true, what must we accept about the knowability of God? If we accept that we can reflect on the actually infinite, which is suggested by creation, and know something about God, then must we also accept that Erich Przywara was correct to claim that we can say something about God on the basis of our own created being? Or may we still side with Barth and say that attempting to know God by philosophical reflection upon creation inevitably leads to idolatry? What do Cantor’s claims about the infinite mean for the knowability of God? Although I do not comment upon where I stand in the debate between Erich Przywara’s analogy of being and Karl Barth’s rejection of it, nor on the validity of

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Cantor’s theological claims about the infinite, I offer the work of Georg Cantor for the consideration of those interested in the analogy of being debate. What does Cantor’s work with the infinite imply for the knowability of God? In order to explore Cantor’s contribution to this debate, I must provide rather detailed historical background of Przywara and his analogy of being, Karl Barth and his rejection of it, the current discussion about the analogy of being, Cantor’s work with infinite sets, and Cantor’s theological claims about the infinite. Thus, the study will open with a review of relevant literature in the disciplines of both theology and mathematics. The literature review will set the stage of the existing conversation in each of these fields in order to elucidate the relevance of this study. The first chapter begins with a description of the historical context of Erich Przywara’s theological doctrine of the analogia entis. It then explains the doctrine of the analogia entis itself and how it is foundational for a Roman Catholic natural theology. The first chapter concludes with a detailed explication of Karl Barth’s revealed theology and his rejection of the analogy of being and natural theology. I explore how Barth’s theology differed not only from Przywara’s natural theology but also from a fellow Protestant’s view of revealed theology as articulated by Swiss Reformed theologian Emil Brunner (1889-1966), which accepts general and special revelation. The second chapter expounds Georg Cantor’s work with the mathematically infinite by first explicating Cantor’s set theory, which is the branch of mathematics fathered by Cantor and the foundation of his work with the infinite. I then provide detail on the development of Cantor’s transfinite ordinal numbers, transfinite cardinal numbers, and the Continuum Hypothesis, all of which is background to his epochal discovery

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regarding the actually infinite. Cantor’s work with the infinite then led him to theological conclusions regarding mathematical ontology and human knowing. The third and final chapter discusses the implications of Cantor’s mathematical and theological claims concerning the knowability of God. It then relates the work of Georg Cantor and the theology he proposed as a result of his work to the claims of Erich Przywara’s analogia entis. Finally, it explains how Cantor is a helpful, though unexpected, voice in this epistemological dialogue because of the relation between his work and Przywara’s. By engaging Cantor, this study offers a different perspective on discussion provoked by the work of Erich Przywara and the response of Karl Barth. It adds a voice to the validity of the analogia entis from an “outsider’s” perspective; a mathematician demonstrated that issues of theology are not esoteric but are, in fact, relevant to a discipline often considered disparate from theology.

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Literature Review Relevant theological literature This study centers on a discussion begun by Erich Przywara and Karl Barth in the early twentieth century that is lively in current literature. This debate is focused on Przywara’s analogia entis (Religionsphilosophie katholicher Theologie, 1927) and is concerned with the validity of the analogia entis as a way of allowing us to know something about God on the basis of creation and human existence. Perhaps the best overview of this current discussion of the analogia entis is The Analogy of Being: Invention of the Antichrist or Wisdom of God? (2011), a compilation of essays by Catholic, Protestant, and Orthodox theologians who provide perspectives on the legitimacy of the analogy of being as a theological doctrine. The authors of the essays, including David B. Hart, John R. Betz, Thomas Joseph White, and Bruce L. McCormack, discuss whether the source of revelation is Christ alone or if people can know of God philosophically through human reason. This work draws out the distinct sides of this debate, demonstrating how most Catholic theologians will affirm the analogy of being whereas most Protestants will not. A number of articles have also been written about this debate between Przywara and Barth. Princeton Theological Review devoted an entire issue to this conversation, with articles ranging from defense of Barth, to claims that Barth and Przywara have similar ontological premises, to Przywara’s motivation for proposing the analogia entis (Princeton Theological Review, Vol. XV, No. 1, Spring 2009).

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A few theologians stand out in this modern conversation over the analogia entis. Keith Johnson has written several pieces on perspectives on the analogy of being, including his recent monograph, Karl Barth and the Analogia Entis (2010). His other publications on this topic include “Reconsidering Barth’s Rejection of Przywara’s Analogia Entis” (Modern Theology, 26:4, October 2010) and “Erich Przywara’s Early Version of the Analogia Entis” (Princeton Theological Review, XV:1, Spring 2009). John Betz stands out as another particularly important voice in this debate, with a chapter in the aforementioned book, The Analogy of Being: Invention of the Antichrist or Wisdom of God? as well as publications “Beyond the Sublime: The Aesthetics of the Analogy of Being” (Parts One and Two, Modern Theology, 21:3, 2005 and 22:1, 2006, respectively), and his current projects of translating Przywara’s Analogy of Being with David B. Hart and completing his own monograph on Erich Przywara. Yet another author involved in this discussion is Neils Neilson, who contributed to understanding Przywara’s rendering of the analogia entis in his article, “Przywara’s Philosophy of the Analogia Entis” (The Review of Metaphysics, V:4, 1952) and to the discussion between Przywara and Barth with his article, “The Debate Between Karl Barth and Erich Przywara: A New Evaluation of Protestant and Roman Catholic differences.” Relevant mathematical literature Crucial to this thesis is an understanding of set theory, the branch of mathematics fathered by nineteenth-century mathematician, Georg Cantor (1845-1918). The seminal biography on Georg Cantor (Georg Cantor: His Mathematics and Philosophy of the Infinite, 1979) was written by Joseph Dauben, the premier Cantor scholar. This biography

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not only describes Cantor’s life but also details his work with mathematics as well as the philosophical and theological implications Cantor himself saw in his work with the infinite. Dauben also wrote an article on Georg Cantor from a talk he presented in 1993 entitled “Georg Cantor and the Battle for the Transfinite Sets” (2004), which is a shorter version of some of the basic information provided in his monograph on Cantor. Before Cantor, infinity was an ill-defined concept. Cantor was the first to distinguish between different infinities and define them precisely. The history of his development of different infinities will be described later in greater detail. Ever since Cantor defined different infinities, many authors have been fascinated with concepts of infinity and its consequences for philosophy, as demonstrated in Infinity and the Mind (1982) and Roads to Infinity (2010). Other authors have drawn connections between the mathematically infinite and its historical relevance for religion (Naming Infinity, 2009) or the term “infinity” as it is used in the disciplines of mathematics and theology (“Infinity in Mathematics and Theology,” Theology and Science, 9:1, 2011). Two authors who draw clear connections between Cantor and the theological implications of his work are Rüdiger Thiele, in his chapter on Cantor in Mathematics and the Divine: A Historical Study (2005), and Bruce Hedman, in his paper “Cantor’s Concept of Infinity: Implications of Infinity for Contingence” (Perspectives on Science and Christian Faith, 46, 1993). Hedman, a minister and mathematician, examines the implications of Cantor’s theory of the transfinite for a modern worldview of contingence. Four other authors contribute to relating the infinite to philosophy and theology in the recent compilation of essays (Infinity: New Research Frontiers, 2011). Graham Oppy, David Bentley Hart, Robert John Russell, and Denys A. Turner provide perspectives on the infinite and areas

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for future research in their essays relating God and infinity. Russell’s contribution, in particular, is especially relevant to this study because his chapter provides theological insights on Cantor’s work specifically and how it relates to Pannenberg’s doctrine of God. Still other works have related mathematics to Christian faith generally, offering a Christian perspective on the discipline of mathematics. One such example of this type of literature includes Mathematics in a Postmodern Age: A Christian Perspective (2001), a compilation of essays that connects the nature of mathematics to God, discusses the influences of mathematics on Christian outlook, and speaks to the teaching of mathematics from a Christian perspective. Another example of a book relating mathematics to the Christian faith includes James Bradley and Russell Howell’s recent book Mathematics Through the Eyes of Faith (2011), which relates mathematics to the Christian faith through topics such as ontology, epistemology, and infinity. Within mathematics, this research fills a void in the literature on the infinite. While previous literature connects the infinite to philosophy, to religion (broadly), or to mathematics from a Christian perspective, this study asks what Cantor’s definition of the actually infinite means for the knowability of God. Within theology, it provokes interdisciplinary discussion by providing an important perspective for the debate between Barth and Przywara from someone outside the academic theological community. The questions of the knowability of God are truly at the core of the Christian faith; thus, this study is not only uniquely and decidedly Christian, it is specifically related to fundamental Christian issues, rather than relating mathematics to the fringes of theology.

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Chapter 1 – On Natural vs. Revealed Theology Setting the Stage: The History of the Analogia Entis In order to understand the disagreement concerning natural and revealed theology, the issue around which this study centers, one must recognize the background of the analogia entis and its importance for modern theology. The following section provides some historical background of the roots and origin of the analogia entis. The doctrine of the analogia entis was not original to Erich Przywara, although he was certainly one of its greatest proponents. The term itself originated sometime in the seventeenth century among Jesuits such as Mendoz and Arriaga, as well as in the work of Cajetan and Suárez.5 Although the term did not come into use until the seventeenth century, the concept of an analogy of being dates back to the beginnings of Greek philosophy. According to Eberhard Jüngel, analogy is already found in the work of Heraclitus and Parmedides.6 John Betz argues that although the concept of analogy may be found in the work of Heraclitus and Parmedides, “the explicit philosophical use of analogy clearly arises out of an attempt to solve the metaphysical conundrum inherited from them. And it arises, specifically, in the worlds of Plato and Aristotle.”7 Both Plato and Aristotle claimed that human experience is not an illusion, but it is also not ‘real’ in an ultimate sense; this ‘middle ground’ regarding human being can be described by analogy. Aristotle writes that “some things are one in number, some in species, some in genus, and some analogically or proportionally. Those things are one in number which

5. John Betz, The Analogy of Being: Invention of the Antichrist or the Wisdom of God? ed. Thomas Joseph White (Grand Rapids, MI: Eerdmans, 2011), 49. 6. Betz, Analogy of Being, 44. 7. Ibid., 45.

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have one matter; in species, which have one intelligible structure; in genus, which have the same figure of predication; and proportionally, which are related to each other as some third thing is to a fourth.”8 Thomas Aquinas provides a commentary on this portion of Aristotle’s Metaphysics that references analogy. Thomas explains that Aristotle means that analogy can be taken in two different ways: two things may be related to a third thing, or two things might be related to two other things by the same proportion. Thomas gives examples for each of these. For the first type of analogy, Thomas writes, “For example, the term healthy is predicated of urine because it signifies the relationship of a sign of health [to health itself]; and of medicine because it signifies the relationship of a cause to the same health.” In this example, health is related to urine and medicine by analogy; two things (urine, medicine) are related to the third (health). For the second type of analogy (that of proportionality), Thomas gives the example, “Tranquillity [sic] to the sea and serenity to the air; for tranquillity is a state of rest in the sea, and serenity is a state of rest in the air.”9 Here Thomas has related two things (tranquility and serenity) to two other things (sea and air) through the analogy of rest. These two types of analogies are referred to as analogy of attribution (analogia attributionis) and analogy of proportionality (analogia proportionalitatis), respectively. Both of these analogies demonstrate something like Aristotle’s doctrine of the mean—a middle ground between clarity and ambiguity.10 The analogy allows for two things to be at the same time both similar and dissimilar.

8. Aristotle, Metaphysics, 1016b3-1017a6. 9. Thomas Aquinas, Commentary on Aristotle’s Metaphysics, trans. John P. Rowan (Notre Dame, IN: Dumb Ox Books, 1961), 317. 10. Betz, Analogy of Being, 48.

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These philosophical roots of the analogia entis are crucial to the modern theological debate surrounding the doctrine. These questions of analogy in philosophy are invested in the same concern as that of the analogy of being: how do we determine the relationship between two things? In his examples, Thomas relates things of the same ontological status: created. But how can one relate two things of a different ontological status, such as creation and Creator? This was the question of concern for theologians and was the question that the analogia entis answers. In the case of philosophy, the concern was the relationship between being and nonbeing, whereas theology concerned itself particularly with the relationship between God and his creatures, between his being and ours. Thus, Przywara’s basis for this analogy is the metaphysical work already accomplished by St. Augustine in his Confessions and by Thomas Aquinas.11 As Betz rightly claims, “The all-important question, then, that was passed down from Greek philosophy to Christian theology, answered in some way by every great theologian of the church from Augustine to Aquinas, and revisited in the modern debate concerning the analogia entis, was what to make of the relationship between philosophy and theology— indeed, what also to make of the relationship between reason and faith, nature and grace.”12 The further details of the development of the analogia entis before Przywara’s rendering of it are not of particular relevance to this study; it is important to note, however, that Przywara took an existing concept, reworked it, and brought it to its current status in theology through his masterful presentation. As a result, Przywara is a crucial partner to engage in discussion of the analogy of being and is the most important Catholic 11. White, Analogy of Being, 5-6. 12. Betz, Analogy of Being, 49.

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voice in this study. As Keith Johnson wrote, “While the analogia entis has a long history in Christian theology, the person responsible for bringing it to prominence in modern thought is Erich Przywara. He is the pivotal figure in the contemporary debate about the analogia entis.”13 Understanding God’s relationship to the world was crucial to Przywara, not only because he was working out of the context of the Great War that had caused the disillusionment of much of Europe, but also because the First Vatican Council had, in the recent past, asserted the ability of humans to know God by human reason. This declaration, officially accepted in 1870, was called the Dei Filius, which declared anyone anathema who denied that God could be known with certainty from nature through human reason.14 The Dei Filius states, “The same Holy mother Church holds and teaches that God, the source and end of all things, can be known with certainty from the consideration of created things, by the natural power of human reason: ever since the creation of the world, his invisible nature has been clearly perceived in the things that have been made.”15 The Dei Filius thus claims that truths about God can actually be known by consideration of creation through human rational. The disillusionment after the Great War, coupled with the certainty provided by the declaration of the Dei Filius, contributed to an intellectual environment in which Przywara’s analogia entis could receive especially welcome reception in Catholic quarters. Przywara went beyond the Dei Filius’ claim and asserted that not only could the supernatural be known by consideration

13. Keith Johnson, “Erich Przywara’s Early Version of the Analogia Entis,” The Princeton Theological Review XV. no 1 (Spring 2009), 7. 14. White, Analogy of Being, 20-21. 15. Dei Filius, Chapter 2, On Revelation, #1. Accessed online on April 13, 2012. http://www.ewtn.com/library/councils/v1.htm#4

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of the natural, but also Przywara provided a reason why we could know God by reflection on creation: the intrinsic relationship between our being and God’s. Such certainty and means of knowledge about God were surely welcome in a time when human ability to truly know anything was being questioned. The analogy of being addressed a distinctively modern, epistemological concern: can humans know God, and if so, how? White writes, “What it seemed possible to know in the Enlightenment concerning God, morality, or human nature became increasingly uncertain or dubious in modernity.”16 In the wake of World War II, humanity’s disillusionment with its own abilities, and Schleiermacher’s argument for experience as the basis of the Christian faith, a theological doctrine that answered the question of human ability to know God and how humanity relates to him found fertile ground. As the modern world struggled to find certainty, the analogia entis offered a way for humanity to know about God and find its being grounded in him. So despite the somewhat opaque nature of the discussion, it is unsurprising that the debate became as fervent as it did. Having set the historical stage of the development of the analogia entis over time and the ethos in which Przywara developed his articulation of the analogy, this chapter will now trace Przywara’s initial motivation to his mature rendering of the analogia entis, followed by Karl Barth’s response and his motivation for rejecting the doctrine. Natural Theology: Erich Przywara and His Development of the Analogia Entis In the early twentieth century, German Jesuit Erich Przywara (1889-1972) put forward the clearest and most explicated articulation of the theological concept of an analogia entis, his explanation of a relation of being between God and humanity. Born 16. White, Analogy of Being, 2.

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October 12, 1889, in Kattowitz, Germany, Przywara was a brilliant theologian as well as mentor to now-renowned theologians Karl Rahner and Hans Urs von Balthasar. He was respected by many revered theologians, including one of the greatest theologians of the twentieth century—Karl Barth, who opposed him. Przywara wrote mostly in the genre of essay, with his topics ranging from sacramentality, to Catholic life, to the interaction between God and creation.17 O’Meara wrote of Przywara: “Some have compared him to Origen or Thomas Aquinas, calling him a seer and a herald; he was not a researcher or professor, and certainly not a pompous academic or ambitious ecclesiastic. Choosing not to condemn the cultural world in which he worked as a theologian, he was a missionary, one seeking God’s deep, primal revelation not among African tribes but in horizons of European culture.”18 So why do so few people know Przywara’s name if he was so widely respected? Przywara’s lack of acknowledgement is largely due to linguistic and stylistic difficulties; Przywara’s writing was dense and difficult to follow.19 But Przywara’s theology is worth recovering and understanding, and his work on the analogia entis is critical to modern discussions regarding natural theology. To better understand Przywara’s analogy of being, one should first consider Przywara’s motivation behind the doctrine and then move to the doctrine itself. Przywara developed the doctrine with ecclesiological interests in a post-Great War world; Przywara wanted the church to better relate to the world through his

17. Thomas O’Meara, Erich Przywara, S.J: His Theology and His World (Notre Dame, IN: University of Notre Dame Press, 2002), 15-16. 18. O’Meara, Erich Przywara, 1-2. 19. Ibid., 2.

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description of how God relates to the world.20 Przywara’s motivation for his work was to develop a Catholicism that engages with the world rather than retreats from it or becomes enmeshed with it. Keith Johnson, in his article “Erich Przywara’s Early Version of the Analogia Entis,” argues that Przywara can only be rightly understood in the context of his ecclesiological concerns. Johnson writes, “Przywara developed the analogia entis with the hope that it would prompt the Roman Catholic Church to engage the secular world with the truth of the Gospel in a new way, and that this engagement would, in turn, transform and renew the Catholic Church itself.”21 So Przywara began his work in order to provide the Catholic Church a way to engage the created world and thereby start a rejuvenation of the Church itself. In Przywara’s time, the analogia entis was hotly debated throughout Germany, because the doctrine related to a fundamental Christian question: what is the relationship between God and his creation? Thomas Joseph White, O.P. considers the debate over the analogia entis to “have given rise to the single most important ecumenical controversy of the twentieth century.”22 Przywara himself deemed the doctrine fundamental because the analogia entis was, as John Betz writes, “a succinct way of stating the Catholic understanding of creation over against what he perceived to be the dialectical extremes of ‘pantheism’ (ancient and modern) and Lutheran-Reformed ‘theopanism’ (as he saw exemplified in the theology of early Barth).”23 Przywara believed the analogy of being offered a way of talking about the relationship between God and his creation without entangling and confusing them nor 20. For further development of this argument, see Keith Johnson’s “Erich Przywara’s Early Version of the Analogia Entis” in the Spring 2009 edition of The Princeton Theological Review, 7-19. 21. Johnson, “Erich Przywara’s Early Version of the Analogia Entis,” 7. 22. White, Analogy of Being, 1. 23. John Betz, The Analogy of Being, 36.

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wholly separating them. As Przywara writes, “Catholicism finds no contradiction between the supernatural and the Incarnation, on the one hand, and the natural forms of Divine revelation and impartation on the other, and yet again finds that in God, as ‘all in all’, there is no duality of grades, but a unity.”24 What precisely was the analogy of being that Przywara had presented that caused such a fervent debate in Germany, particularly between Przywara and Karl Barth? As Przywara himself explains it, the analogy of being is that the human is like [God], through the possession of a unity of essence and existence, but even in this similarity it is unlike Him, because in Deity the unity of essence and existence is that of identity, whereas in that of creation the unity is one of tension. Since now the relationship of essence and existence is of the esse of ‘being’, Deity and the creation are in ‘being’ like and unlike, i.e. analogous to one another: analogia entis = analogy of being.25 All human beings are similar to God in that they receive their being from God and are analogous to God in that being, yet are always transcended by God because of human createdness. Thus, humanity’s being is one of tension with its essence whereas God’s being is one of unity with his essence: God is being, and we are not. Przywara calls God’s unity of essence and existence the mystery of the pure “Is.”26 We are because God is. Thus, humanity is analogous to God in our being, but he is always above us as the source of our being. In using the analogia entis, Przywara related creation to God while neither falling into pantheism nor deism. Because God is the source of humanity’s being, our being is both from him and in him.27 Yet at the same time, God, as the source of our being, is over

24. Erich Przywara, Polarity (London: Oxford University Press, 1935), 33. 25. Przywara, Polarity, 32. 26. Ibid., 32. 27. It is not clear to me how precisely Przywara considers humanity “in” God. What is clear is that this unity with God is a result of God’s condescension, not human action. Przywara writes, “Analogia entis

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us and we are utterly dependent on him. For Przywara, this implies that the relationship between God and humanity is one of both immanence and transcendence: humans are only understood in relationship to God but he always transcends them: “God in us and over us . . . the synonym of analogia entis.”28 Hans Urs von Balthasar, in his book on Karl Barth’s theology, notes the importance of remembering that Przywara was writing on the analogy of being during Barth’s early theological development and thus was addressing Barth’s dialectical theology, which was, as von Balthasar put it, “quite abstract and schematic, as therefore was Przywara’s analogy.”29 So although Przywara’s analogy was abstract, and provided a way to connect creation to Creator, he was very careful to not confuse the two, in order to address Barth’s dialectical “yes and no.” This was, in fact, his entire purpose—to relate nature and grace, creation and God, without confusing them. As O’Meara writes, There was no confusion of philosophy and theology in Przywara’s mind or in his writings because he found around him no confusion of God’s creation and grace. A Jesuit education as well as the Roman Catholic mentality of distinguishing before uniting nature and grace kept those two varied horizons distinct, while the theme of participation in divine life, important to Augustine and Aquinas, drew them together. Philosophy as a reflection on being and society could be the language of the expression of faith but not its total source or judge.30 O’Meara is correct in noting that Przywara’s theology here is consistent with his Jesuit education, which would have taught him that nature and grace must first be distinguished and then brought into union through participation in Christ, rather than starting united and

bespeaks just this, that in the self-same act in which man, in the likeness of a creature, becomes ‘within’ God, he also becomes ‘within’ Him who, in likeness, is above all comparison. Thus the ultimate unity of the creaturely is not within itself, but with That Which is above itself.” Przywara further explains that this unity is achieved “from above downwards” and not “from below upwards.” However, the nature of the unity itself remains unclear (Przywara, Polarity, 34). 28. Przywara, Polarity, 33. 29. Hans Urs von Balthasar, The Theology of Karl Barth, 1st ed. (New York: Holt, Rinehart and Winston, 1971), 39. 30. O’Meara, Erich Przywara, 66.

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attempting to be differentiated. Moreover, Przywara claimed that philosophical reflection on our being provides one mode of speaking of God but not the ultimate or sole mode. So not only was Przywara careful not to enmesh creation and Creator (so as to safeguard against pantheism), but also he made note of the limits of natural theology and human knowledge. As Keith Johnson writes, “Przywara argues that while humans can obtain some knowledge of this relationship through philosophical reflection, they cannot recognize the full extent of it on their own. He illustrates this fact by examining the failure of the philosophical tradition to determine the nature of the relationship between the human consciousness and God without collapsing God into creation or completely divorcing God from creation.”31 Therefore, the relationship between the analogy of being and epistemology, or more specifically, the human ability to know God, is clear. The analogia entis is the basis of human knowledge of God; it is the reason we can know of him. Because we are analogous to God in our very being, we have a means by which we may come to know about God. According to Przywara, humans can have some knowledge of God and their relationship to him internally; that is, derived philosophically from their own mind. However, this knowledge is always limited. Although human knowledge can give testimony to God because it was created by God, it is always transcended by God, who is above it. This is the analogy of being. As Przywara writes, “The analogia entis is a relation between God and creation given hitherward from God, and not constructed thitherward from man, since all experience of God is an experience that we are in Him.”32 Humanity’s relationship to God is totally dependent on God, so human knowledge of 31. Keith L. Johnson, “Reconsidering Barth's Rejection of Przywara's Analogia Entis.” Modern Theology 26, no. 4 (2010): 632-650. 32. Przywara, Polarity, 95.

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God will always be limited. Humans can know something of God since our being is analogous to his because he created us, but our knowledge will always be transcended by God because of his utter otherness as Creator. Przywara describes this idea as the human “open upwards”33 to God: the human is separate from God, but is open up to him in testimony to him because of our very existence. Johnson explains the tension of Przywara’s analogy as follows: “On the one hand, the very fact of the creature’s existence testifies to God as its source, and, on the other hand, God is utterly distinct from the creature because the creature remains dependent upon God for its existence at every moment.”34 Thus, Przywara proposes a natural theology—a theology that allows a means by which humans can come to know truths about God from creation (nature, human, universe). Sin has affected human nature, but for Przywara, it did not destroy the imago Dei entirely; thus, we are still constitutionally open to receiving grace from God (hence Przywara’s adherence to obediential potency, explained later) and can still truly know something of God. For Przywara, the analogia entis was the grounding of natural theology: “Thus all movements towards God, all illumination by God of the human experience which seeks to enlighten itself, presuppose a tranquil condition of ‘God in me and I in God’, because precisely by reason of the nature of the analogia entis, the relationship between God and man is not a function of man’s activity but by God’s condescension” [his emphasis].35 So the analogy of being is the foundation for a natural theology; all movement to God in knowledge through human reason and from experience in nature is based on this relationship with God (the analogy of being) and God’s condescension to humanity. 33. Przywara, Polarity, 32. 34. Johnson, “Reconsidering Barth's Rejection of Przywara's Analogia Entis,” 632-650. 35. Przywara, Polarity, 39-40.

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How does Catholic doctrine of natural theology differ from a Protestant view of revealed theology? Although the difference between natural theology and Barth’s particular rendering of revealed theology will be addressed later, the distinctions between the natural theology just described and another Protestant view of revealed theology are worth noting.36 Natural theology has officially been a part of Catholic doctrine since 1870 at the First Vatican Council, but within the Roman Catholic Church, the concept can be traced back to Thomas Aquinas.37 To claim that Catholic belief in natural theology negates the need for scripture or special revelation would be inaccurate; faith, grace, and revelation are still crucial to the life of a believer, but natural theology provides a common ground between all people by providing all people a means by which to know God: nature. Natural theology allows the human to know some truths about God from creation and through reason and rationality. These truths are not apart from God’s revelation, but rather they are truths in line with scripture and accessible by human reason. Natural theology may at first sound like a Protestant view of general revelation, which claims that creation can be a source of knowledge about God from which humans may know nonsalvific truths about God, such as his existence or his goodness. However, general revelation differs from natural theology in that general revelation is a source of knowledge about God, whereas natural theology is a method for knowing of God. General revelation, as a source of knowledge, has two basic parts. The first aspect is 36. The views of revealed theology within Protestant circles vary greatly and provide different views of general revelation; some in liberal Protestantism even hold that general revelation and reflection on God is sufficient for salvation. This section is meant to provide a brief overview of general revelation as held by a majority of Protestants, Emil Brunner being an outstanding example of this view. The theology of Emil Brunner will be explained in greater detail in later in this chapter. 37. J. Van Engen, Evangelical Dictionary of Theology, 2nd ed. (Grand Rapids, MI: Baker Academic, 2001), s.v. “Natural Theology,” 815.

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internal—within each human there is a “seed of religion,” or something within the person that calls out to know of a higher, divine being. This is what John Calvin called the sensus divinitatis, or sense of the divine, which prevents all of humanity from, on the final day, claiming unfamiliarity with God. Calvin writes, “There is within the human mind, and indeed by natural instinct, an awareness of divinity. . . . To prevent anyone from taking refuge in the pretense of ignorance, God Himself has implanted in all men a certain understanding of his divine majesty.”38 The second facet of general revelation is external—the evidence of nature and providential history for understanding truths about God such as his existence, qualities of his character, and his moral law for humanity.39 However, for many Protestants, general revelation is incomplete without special revelation and is also prone to misunderstanding because of human falleness. Special revelation provides order and opportunity for salvation that the general revelation in creation lacks. Special revelation is salvific revelation in the person of Christ and in the scriptures; it allows humans to know saving truths about God, such as the incarnation and atonement. A good articulation of general and special revelation is found in the Belgic Confession, a seminal confessional document in many mainline Protestant churches. The Belgic Confession states: “We know [God] by two means: First, by the creation, preservation, and government of the universe, since that universe is before our eyes like a beautiful book. . . . Second, he makes himself known to us more openly by his holy and

38. John Calvin, Institutes of the Christian Religion, ed. John McNeil (Philadelphia: The Westminster Press, 1960), 43 (I/III). 39. Bruce Demarest, Evangelical Dictionary of Theology, 2nd ed. (Grand Rapids, MI: Baker Academic, 2001), s.v. “General Revelation,” 1019-1020.

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divine Word, as much as we need in this life, for his glory and for the salvation of his own.”40 The distinctions between the Catholic natural theology and Protestant general revelation are important. Again, natural theology is a method by which humans may know of God, whereas general revelation is one source of truths about God. Not only are natural theology and general revelation different in that the former is a method of knowledge and the latter is a source, the viewpoints are also founded in essentially different beliefs about human nature and the imago Dei in humanity after the Fall. Roman Catholics hold that the imago Dei still functions even after the Fall; that is, humans, postFall, still have the capacity to do good. Many Catholics, including Przywara, hold to obediential potency—that humanity still has the capacity to be obedient to God through his divine intervention, like a tool in God’s hands41—a concept that denies total depravity. Although this Catholic articulation of natural theology, which is supported by the analogia entis, confesses that knowledge from creation is not salvific, it does allow that that knowledge is true and sufficient in and of itself, and the human still maintains a capacity to do good, despite the damage of the imago Dei. As a result, what atheistic philosophers, such as Aristotle, come to know through their God-given, rational mind is not invalidated because it is unaccompanied by belief. Although general revelation (as previously articulated and held by many mainline Protestant denominations) admits nature as a source of knowledge of God’s existence and character, it claims that this knowledge is inadequate without God’s special revelation in Christ and scripture. This view holds that the imago Dei is corrupted, and that total depravity is in effect. Although 40. The Belgic Confession, article 2. 41. Przywara, Polarity, 33.

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Przywara does not, when talking about the analogy of being, directly reference the imago Dei, he does refer to obediential potency, which connects the analogia entis to the issue of human nature. It is worth noting that Przywara can hold the analogy of being because of his view of a human nature that still has a positive ordering toward God; however, human nature is not the main focus of this study. Unlike Przywara’s classically Roman Catholic view of obediential potency, this Protestant view of general and special revelation argues for the capacity of the human to still know something of God, despite the human’s inability to do good. However, some Protestant traditions (such as the Reformed tradition) hold that even knowledge and ability to know God is fundamentally corrupted by sin; this is why general revelation has the capacity to be misconstrued because of fallen minds, and must be accompanied by special revelation (this is referred to as the noetic effects of the fall). Whereas many Protestants will draw a distinction between general revelation and special revelation, many Catholics, such as Przywara, would see a continuance or continuity between natural theology and revealed theology, the natural and the supernatural, the finite and the infinite. As Van Engen writes, “Catholics are in fact inclined to see a continuum between natural theology, that which is known of God by the light of natural reason, and revealed theology, that which is known by the light of faith.”42 In line with his Catholic tradition, Przywara sought to create a doctrine of analogy which related humanity to God so that we can know of God through our very being, while never confusing nature and grace, humanity and God. Human ability to know God rests upon the analogy of being; the analogia entis allows for humans to know God through reason, while maintaining the limits of human knowledge. 42. Van Engen, Evangelical Dictionary, 816.

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While Catholics hold to a continuum of natural and revealed theology, for Reformed theologian Karl Barth the difference between natural and revealed theology was not at all small and the distinctions he drew between the two were sharp. Barth, who held a dramatically different view from those mentioned, denied general revelation altogether and affirmed the complete destruction of the imago Dei in humanity, and soundly rejected natural theology and Przywara’s analogia entis. Barth believed the principles of the Reformation themselves (such as sola gratia, sola fide) were at stake in the discussion of the analogy of being, principles which he was determined to uphold. Although Przywara sought to maintain a complicated and specific balance of the immanence and transcendence of God in relation to his creatures so as not to confuse God with his creation, Karl Barth believed this analogy was unfounded, and was, in fact, the very invention of the anti-Christ. Revealed Theology: Karl Barth’s Response to Przywara’s Analogia Entis Reformed theologian Karl Barth (1886-1968) fervently disagreed with Przywara on the analogia entis. Barth, who was concerned with the relation between God and humanity, vehemently opposed the analogia entis, famously calling it the “invention of the anti-Christ” in his preface to the Church Dogmatics, and added that the doctrine was a fundamental reason for not becoming Catholic.43 Barth was not being melodramatic in this statement, nor was he saying it simply polemically. Barth literally believed the analogy of being was the invention of the “not Christ” because, for Barth, the doctrine represented abstract thinking of reflecting on one’s self to God, which eliminated the

43. Karl Barth, Church Dogmatics, I/Preface.

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need for the concrete revelation in Christ. Thus, the doctrine is without Christ, or the antiChrist. In his seminal Church Dogmatics, Karl Barth expounds on his doctrine of the knowability of God. He begins by pointing out that to ask if God can be known is irrelevant; God has already been revealed and known in the person of Christ. Therefore, to ask about our ability to know him would be to ask about other means of knowing him apart from Christ (read, natural theology), which is fundamentally incorrect. Barth writes, “The type of thinking which wants to begin with the question of the knowability of God and then to pass on from that point to the question of the fulfillment of the knowledge of God is not grateful but grasping, not obedient but self-autonomous. It is not theological thinking.”44 Barth believes starting with the question of whether or not God is knowable is to begin with an abstract principle rather than concrete reality. Hans Urs von Balthasar described Barth’s concern with working from an abstract concept like the analogy of being: “In the hands of a sinner, the concept of being becomes the most dangerous instrument for disobedience, because with it the sinner purports to produce something from within himself that can only come as a gift from God. In this respect, it is the most direct expression of disobedience.”45 Because the question is already answered in the concrete person of Christ, we can only begin theology by asking about the mode and extent to which God can be known. Barth believes we must start our thought gratefully and obediently, such that we are seeking to follow God rather than be autonomous and seek other ways to know him than his self-revelation in Christ.

44. Karl Barth, Church Dogmatics, CD II/1. 45. Von Balthasar, Theology of Karl Barth, 162.

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Thus, Barth felt the analogia entis exemplified the core problems with a natural theology (and thus, the Catholic Church)—that it eliminated the need for Christ altogether and allowed humans to depend on their own reason, rather than on God’s sovereignty. Such thinking would undoubtedly lead to idolatry (by way of Ludwig Feuerbach’s theory of anthropological projection), or to a belief that humanity can come not only to know God, but through that knowledge come to possess God. Barth viewed such thinking as abstract and dangerous, and claimed that only Jesus Christ offers concrete revelation, and revealed theology maintains God’s utter sovereignty. Benjamin Heidgerken summarized well Barth’s concerns about natural theology: Karl Barth argues that any continuity between the natural and the supernatural is unthinkable for the Christian theologian. Because of the infinite qualitative difference between what is time-bound and finite on the one hand, and what is eternal and infinite on the other, there can be no proven path from the creation to the Creator. God is God; we are not. Barth strongly opposes every human attempt to grasp or control the Divine, for in doing so one automatically slips into idolatry, worshipping what one believes God to be rather than the absolutely free and gracious Creator of the world. For Barth, when the created is considered analogous to the divine, when philosophy is thought to be continuous with theology, when Athens has anything to do with Jerusalem, “God” can be nothing more than a Feuerbachian projection of idealized humanity. Even in God’s definitive self-revelation in Christ, God cannot be revealed to us in such a way that we are able to possess God. If this were so, God would cease to be the Deus abscondus, the One whose thoughts are not like our thoughts, whose ways are not like our ways.46 Heidgerken provides an outstanding summary of Barth’s reasons for rejecting natural theology. For Barth, it was of utmost importance that the utter sovereignty and freedom of God be protected, and that we humans not try to possess God. Barth held that God was “wholly other”47 and that the Catholics’ claim of some sort of continuum between humanity and God was certainly idolatry. Thus, the analogia entis must be rejected 46. Benjamin Heidgerken, “Prolegomena.” The Princeton Theological Review XV, no. 1 (2009): 5. 47. Barth’s early work held that God was “wholly other.” Later in his life, Barth spoke of the “humanity of God,” but only on the foundation of the person of Jesus Christ.

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because it serves as a base for allowing such a continuum. To relate the created to Creator, the finite to the infinite was to undermine God’s otherness and to imply that we still have a capacity for good, which Barth rejected in his claim that the imago Dei was completely destroyed at the Fall. Barth was committed to upholding the Reformation principles of sola gratia, sola fide, solo Christus. Barth wrote that in order to protect the principles that the Reformers stood for, we must say what they said, but even more strongly because the questions posed to modernity were not even thought of at the time of the Reformers. He wrote: If we really wish to maintain the Reformers’ position over against that of Roman Catholicism and Neo-Protestantism, we are not in a position to-day to repeat the statements of Luther and Calvin without at the same time making them more pointed than they themselves did. This applies to the doctrine of grace, e.g. in the definition of the relation between justification and sanctification. But the practical non-existence of St. Thomas in the sixteenth century has had even graver consequences, in that the Reformers could not clearly perceive the range of the decisive connection which exists in the Roman Catholic system between the problem of justification and the problem of the knowledge of God, between reconciliation and revelation.48 Therefore, Barth cannot accept the analogia entis because he believes it goes against the Reformation principle of sola gratia—justification by grace alone. Barth was concerned that Thomistic thought had seeped into Protestant theology in a way it had not at the time of the Reformers. As a result, these Catholic influences needed to be addressed and the Reformers’ claims needed to be said even more strongly in order to combat the problem. So Barth sought to protect the reformation principles against the threat of Thomistic strains of thought. If humans, by their createdness, are intrinsically connected to God (as the Catholics claimed) and can come to a true knowledge of God by reflecting upon their

48. Emil Brunner and Karl Barth. Natural Theology: Comprising "Nature and Grace" by Professor Dr. Emil Brunner and the Reply "No!" by Dr. Karl Barth (Eugene, OR: Wipf & Stock Publishers, 2002), 101.

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own being, then justification by grace alone is undermined. This is what Barth believed to be the Catholic mistake and, likewise, Przywara’s mistake. Barth’s position regarding natural theology was radically different than not only the Catholics’ view, but also most Protestants’. Dr. Emil Brunner (1889-1966), a professor of theology at Zurich, wrote a letter to Karl Barth entitled Nature and Grace: A Contribution to the Discussion with Karl Barth. In this letter, Brunner, a Reformed theologian like Barth, attempted to show that Barth and he were really intending to convey the same theological ideas. Brunner believed that although his desires were correct, Barth drew the wrong conclusions. Moreover, Brunner stated that Barth should not implicate as Catholics all Protestants who do not join Barth in these conclusions.49 Brunner then exposits his views of two forms of revelation: that in creation and that in Jesus Christ.50 He goes on to propose a view that is essentially what was explained as the Protestant view of general and special revelation previously. Brunner first explains that the imago Dei has not been destroyed in humanity, as Barth claims, but rather it has been dulled or corrupted. In saying this, Brunner believed he was in line with Calvin, who wrote, “Man’s nature [is] deformed; yet his soul bears, though almost obliterated, the image of God.”51 Thus, Brunner writes, because of this deformity, natural revelation is no longer sufficient for humans to know God (as mentioned before, this was the noetic effect of the Fall). Instead of desiring God, humans long for other gods, and thus require special revelation through Christ to come to true knowledge of God. Brunner writes, “Only the Christian, i.e. the man who stands within the revelation in Christ, has the true natural

49. Brunner and Barth. Natural Theology, 16. 50. Ibid., 26. 51. Calvin, Institutes of the Christian Religion, 183.

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knowledge of God.”52 So Brunner supported a view that he believed allowed for some natural knowledge of God, but a knowledge that was fundamentally incomplete without special revelation in Christ. For Brunner, this general revelation or natural knowledge provided a “point of contact” between God and creation, while avoiding natural theology, which Brunner (along with most Protestants) feared would undermine the unique revelation of Jesus Christ. Barth found this view repulsive, as a lukewarm stance which refused to side with the Catholics’ natural theology on one end, nor Barth’s selfperceived true Protestant revealed theology on the other.53 Clearly displeased by Brunner’s attempt to relate their two theologies, Barth responded to Brunner’s letter with a resounding “Nein!” and therein discredited both Brunner’s claims and those of the Catholics, such as Przywara. Since Brunner had conceded a form of partial natural theology, he had, in Barth’s mind, conceded the entire point. By speaking of grace and revelation in such a way, Barth believes Brunner has joined the Catholics in their abstractions and has abandoned Reformation principles of sola gratia, sola fide, solo Christus.54 Thus, Barth soundly rejects Brunner’s argument as essentially Catholic and utterly incorrect, and with him, rejects natural theology in its entirety.55 As Barth demonstrates throughout his theology, the discussion of human nature in relation to grace, and thus the discussion of God’s knowability, is fundamentally a discussion of all theology. Where one stands on these issues, and on the analogia entis, directs the rest of one’s theological thinking. Revelation and knowledge of God are critical to the base of

52. Brunner and Barth. Natural Theology, 27. 53. Ibid., 15. 54. Ibid., 85. 55. Later in his life, Barth writes about a natural theology of sorts in which nature is the grand theater in which God enacts his drama of redemption; as such, the theater points to the glory of God, but only with first the revelation of God in Jesus Christ.

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one’s theology. As Calvin demonstrated by beginning his Institutes of the Christian Religion with the section on knowledge of God and knowledge of self, these issues must be resolved first when forming a coherent theology. Brunner was not the only theologian to try to make ties with Barth and find some theological common ground. Some theologians argue that Barth fundamentally misunderstood Przywara. Hans Urs von Balthasar even argued the two theologians were making the same point from different perspectives in his book The Theology of Karl Barth. 56 In his book, von Balthasar expounds on Barth’s theology and Catholic theology to demonstrate their similarities. Von Balthasar has an ecumenical concern; he hoped to begin a dialogue between Barth and Catholic theology that would draw the two together.57 Although he respected von Balthasar’s work, Barth never conceded his point. This thesis also does not concede von Balthasar’s point and assumes that Barth correctly understood Przywara and should have maintained his rejection of the analogy of being because of his own theological premises.58 Thus, Barth’s disagreement with Brunner and Przywara set the stage for a debate between natural and revealed theology that has continued into the present-day. Przywara proposed the analogy of being as a way to understand the relationship between God and his creation, a relationship which allows us to know something of God and maintains a positive view of human nature’s capacity post-Fall. Barth, affirming the priority of revealed theology, asserted that any true knowledge of God must start with the person of

56. Von Balthasar, The Theology of Karl Barth. 57. Ibid., 3-9. 58. For a basis for this assumption, please see Keith L. Johnson’s article, “Reconsidering Barth’s Rejection of Przywara's Analogia Entis.” Modern Theology 26, no. 4 (2010).

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Christ, not from creation or the human. He thereby rejected the analogia entis and natural theology and affirmed the destruction of the imago Dei in human nature after the Fall.

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Chapter 2: Georg Cantor and Set Theory Introduction At this point, this study will take what at first may seem like a grand departure from the topic developed thus far. Until now, it has explored the historical debate between Erich Przywara and Karl Barth over the analogy of being. It has provided context for the debate, elaborated on the theology of Przywara with respect to the analogia entis and how that analogy provides the foundation of the Roman Catholic’s natural theology and how that contrasts to Protestant theology of general revelation and special revelation, and finally, expounded on the revealed theology specific to Karl Barth. And now the study turns to a mathematician’s revolutionary work that has theological implications for the modern debate over the analogy of being. In the next chapter, I explore the value of the work of mathematician and lay theologian Georg Cantor (18451918). Cantor fathered the branch of mathematics known as set theory and pioneered work with the mathematically infinite. The first part of chapter two will explore Cantor’s set theory and his theory of the actually infinite, including the mathematics that led him to his conclusions. Next, having provided the background of Cantor’s mathematical thought, especially his work with what he called transfinite numbers, I will elucidate Cantor’s theological conclusions. The third and final chapter will provide the synthesis of the work of Erich Pryzwara and Georg Cantor and show how Cantor’s work can inform the debate over the analogy of being, bridging the gap between mathematics and theology and demonstrating the importance of this interdisciplinary work.

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Set Theory and the Mathematically Infinite Set theory is a specialized area of mathematics fathered by nineteenth century mathematician Georg Cantor. Set theory, put quite simply, is the branch of mathematics that deals with mathematical sets. In set theory, the primary concern is to represent a collection of objects as a single object, that is, a set.59 A mathematical set is a collection of mathematical objects, such as numbers.60 For example, consider set A defined as A = {0, 1, 2, 3}. It can be said that the number 1 is an element of A, mathematically written 1 A. Say set B is defined as follows: B = {0, 1, 2}. Thus, B is a subset of A, written B A. Said another way, the elements of B are contained within the set A. Further, A has cardinality of four; that is, there are four “objects,” or numbers, in A, whereas B has a cardinality of three, written |A| = 4 and |B| = 3. The issue of cardinality will be discussed in greater depth later. That brief introduction to set theory is sufficient to set up the basis of the mathematical discussion of Cantor’s transfinite numbers.61 To understand Cantor’s theory of transfinite numbers, first consider a set of limitless size. Some examples of sets of limitless or infinite size would include the natural numbers, N = {1, 2, 3, 4, 5…} or the integers, Z = {…-3, -2, -1, 0, 1, 2, 3…}. Cantor went beyond this simple discussion of infinity by defining transfinite ordinal numbers.62 The definition of transfinite ordinal numbers will be fleshed out and then transfinite cardinal numbers, which are crucial to this study, will be addressed. Before describing the transfinite ordinals and cardinals at 59. Keith J. Devlin, Fundamentals of Contemporary Set Theory (New York: Springer, 1979), 2. 60. Martin M. Zuckerman, Sets and Transfinite Numbers (New York: Macmillan, 1974), 1. 61. For Cantor’s original research on transfinite ordinal numbers, please see his work Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (Leipzig: B. G. Teubner, 1883). 62. For succinct, mathematically rigorous definitions, please see Borowski and Borwein’s Collins Dictionary of Mathematics, 2nd ed. (London: Collins, 1989).

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length, however, it is worth clearly addressing the history of the development of Cantor’s definitions of different types of infinity: the potentially infinite, the actually infinite, the transfinite, and the absolutely infinite. Aristotle was one of the first predominant thinkers to talk about infinity. He thought of the infinite as limitlessness. Aristotle wrote, “[T]he infinite has this mode of existence: one thing is always being after another, and each thing that is taken is always finite, but always different.”63 However, Aristotle never spoke of infinity as something which truly existed, but only ever as a concept. He spoke only of the potentially infinite, but not of a completed infinite, or actually infinite. For Aristotle, “the only infinity considered possible [was] potential infinity: something capable of being endlessly divided or added to, but never fully actualized as infinite.”64 For centuries, this was the primary thought about the infinite, not only in mathematics, but also in philosophy and theology. In theology, infinity was considered to be associated with God in some sense, but “infinity” itself was ill-defined because it was thought to be only a concept, not something which actually existed. Theologians such as Augustine, Gregory of Nyssa, and Thomas Aquinas all spoke of God as infinite and as creation as finite.65 Through Cantor’s work with infinite sets, he postulated that the infinite did actually exist and that he could speak of it as a completed set. Cantor then clearly defined the infinite (by a process of one-to-one correspondence, explained later). Cantor called this the actually infinite, and thereby claimed that the infinite could be known. Because Aristotle’s explanation of the potentially infinite had been the prevailing theory for so long, Cantor faced a great deal of

63. Aristotle, Physics, III.6.206a26-30. 64. Robert John Russell, Infinity: New Research Frontiers, ed. Michael Heller and W. Hugh Woodin (New York: Cambridge University Press, 2011), 278. 65. Ibid., 278.

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opposition from mathematicians, theologians, and philosophers alike (opposition that will be explained in greater detail later in this chapter). Cantor himself, when he found his astonishing results regarding the actually infinite, could not believe what he had discovered. In 1877, he wrote in a letter to his colleague Richard Dedekind, “I see it, but I do not believe it.”66 Because of the controversy surrounding his claims, Cantor defined his new classes of numbers (ordinal numbers and cardinal numbers, described later) as transfinite—that is, a set which is actually infinite, but not absolutely infinite. Cantor coined his numbers with the term “transfinite” to avoid implications associated with “infinity” and in order to distinguish from the absolutely infinite. Cantor further defined the absolutely infinite to be the infinite set of all actually infinite sets, and reserved the term “absolutely infinite” for God alone. Therefore, with his discovery and definition of actual infinity, Cantor transformed the study of the infinite. Today, Cantor’s work is accepted and is taught in mathematics classes without controversy. However, the modern idea of considering all of the points on a line to be a complete, infinite set would have been unthinkable to Aristotle or other philosophers or mathematicians before Cantor.67 So Cantor’s work was truly revolutionary; it changed mathematics forever. With clear definitions of different “types” of infinity (potential, actual, transfinite, and absolute) established, I now turn to a description of Cantor’s transfinite ordinal numbers.68 Ordinal numbers are a measure of a set that considers both size (cardinality)

66. Robert Bartle and Donald Sherbert, Introduction to Real Analysis. 3rd ed. (New York: Wiley, 1999), 21. 67. Russell Howell and James Bradley, Mathematics Through the Eyes of Faith (New York: Harper One, 2011), 19. 68. The ordinals were much less clearly defined by Cantor than his cardinals (described later) were. I leave it to the interested reader to explore ordinal numbers at greater depth. I describe the relevant work with ordinal numbers and then address the cardinal numbers, which are much more integral to this study.

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and order. Specifically, ordinal numbers are well-ordered; that is, they are a set of numbers that has a ‘beginning,’ or a smallest element. Moreover, every subset of a wellordered set has a smallest element. For example, the natural numbers N are well-ordered because the set of natural numbers, as well as every subset of the natural number system, has a ‘beginning,’ or smallest number. Consider, for example, the even numbers within the natural number set: {2, 4, 6, . . .}. Although the set has no limit, and is therefore infinite, it is still well-ordered because it has a smallest element: {2}. Every member of the set has a successor, but not every member has a predecessor; that is, two has no predecessor. As a counterexample, consider the integers Z. Although the integers too are an infinite set, they have no smallest term. The integers have no beginning and no end. For example, consider the subset of even integers: {. . . -4, -2, 0, 2, 4 . . .}. There will be no smallest term in the set. Every element in the set has both a predecessor and a successor. Thus, the set is not well-ordered.69 Joseph Dauben, the foremost Cantor scholar, notes that the issue of well-orderedness is crucial to Cantor’s transfinite numbers and their arithmetic,70 which follows. Cantor defined finite ordinal numbers, finite cardinal numbers, transfinite ordinal numbers, and transfinite cardinal numbers. First consider Cantor’s transfinite ordinal numbers, which are not only transfinite (neither finite, nor absolutely infinite), but also ordinal, and are thus are not only concerned with size but also order, and are also wellordered. Cantor defined this special class of numbers (the transfinite ordinal numbers) by extending the set of natural numbers. The natural numbers themselves are unbounded, but 70. Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, Mass.: Harvard University Press, 1979), 101.

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Cantor considered a number which would follow the natural number sequence and this he defined as the first transfinite ordinal number, which he denoted using the Greek letter, ω. (Finite ordinal numbers, mentioned before, are the natural numbers that precede ω). From ω, Cantor moved further through the set by adding one unit to each successive term. Following that sequence of numbers exists yet another sequence which begins with ω + ω, or 2 ω, and continues on by the addition of one unit to each successive term, and so on. Thus, the ordinal numbers are defined as follows (with the naturals comprising the finite ordinal numbers and ω and beyond comprising the transfinite ordinals): 1, 2, 3, 4, …, ω, ω + 1, ω + 2, ω + 3, … , ω + ω, ω + ω + 1, ω + ω + 2, …, ω + ω + ω, ω + ω + ω + 1, …71 The transfinite ordinal numbers are a mathematically interesting number system, beyond the natural numbers and distinct from other number systems such as the integers or reals. Cantor’s ordinal numbers are additively associative, but interestingly, are not additively commutative. For finite ordinal numbers, addition is related to counting. For example, consider finite ordinal numbers two and three; the addition of two and three means that one counts to two and then count three more, ending at five. Similarly, if one counts to three and then counts two more, the addition ends at five. So 2 + 3 = 3 + 2; thus, the numbers are commutative. However, the commutative property of addition does not hold true for the transfinite ordinal numbers. 72 Although Cantor defined addition of transfinite ordinals as with finite ordinals (that is, to count to the first number and then count more units beyond that), the transfinite ordinals will not hold to commutativity. That is, 4 + ω ≠ ω + 4, because to count four units in the naturals and then to count ω 71. Thiele, Mathematics and the Divine, 531. 72. The commutative property of addition states that a + b = b + a for all real numbers a and b.

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units more would be equivalent to ω itself.73 4 + ω = {1, 2, 3, 4, 1, 2, 3, 4, 5 . . .} whereas ω + 4 = {ω, ω + 1, ω + 2, ω + 3, ω + 4}. Thus, Cantor’s transfinite ordinal numbers are associative under addition, but not commutative. Cantor’s most important discovery regarding the ordinals for this study is that the ordinal numbers do not hold to the commutative property of addition. This discovery was key to Cantor’s work with the infinite because it discredited Aristotle’s earlier argument that if a completed infinity did exist, it would obliterate, or “annihilate,” any finite number.74 According to Aristotle, any finite number added to infinity was so insubstantial compared to the infinite that it was “annihilated.” That is, for any finite number a, a + ∞ = ∞. However, Cantor showed that his transfinite ordinal numbers did not “annihilate” finites in this way. When added on the right, a finite number actually did change the transfinite number ω. (As previously shown, ω + 4 = {ω, ω + 1, ω + 2, ω + 3, ω + 4}). Thus, Cantor had shown Aristotle’s work with the infinite to be incorrect. As Dauben writes, “Cantor believed that Aristotle was quite mistaken in his analysis of the infinite, and that his authority was exceedingly detrimental.”75 This was one step in Cantor’s progress in revolutionizing the infinite. Although this discovery is important for Cantor’s work, it was Cantor’s work with the cardinality of infinite sets that is of crucial to this study. Specifically, Cantor’s proof that different cardinalities of actually infinite sets exist is of particular interest here. Therefore, the following section provides further detail on cardinal numbers and Cantor’s famous Continuum Hypothesis.

73. Thiele, Mathematics and the Divine, 531. 74. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 122-3. 75. Ibid., 123.

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Cardinality of Infinite Sets and the Continuum Hypothesis The concept of cardinality became an integral part of Cantor’s work with infinite sets. Cardinality, unlike ordinality, is concerned with only the size of the set rather than the size and order of the elements of a set. That is, a cardinal number is defined as “a measure of the size of a set that does not take into account the order of its members.”76 Recall sets A and B as defined previously as A = {0, 1, 2, 3} and B = {0, 1, 2}. Sets A and B do not have the same cardinality because for each element of B, there did not correspond one and only one element in A; that is, there is not a one-to-one correspondence between the elements of A and B, because they are not the same “size.” An example of two sets that do have a one-to-one correspondence follows (Figure A):

1 2 3 4 5

3 6 9 12 15

Set C

Set D Figure A

For every element of C there is an element of D such that each element of C is mapped to an element of D by the rule n  3n. Therefore, because there exists a one-toone correspondence between set C and set D, they have the same cardinality, or size. Using this concept of mapping and one-to-one correspondence, Cantor proved that the set N of natural numbers had the same cardinality as, say, the set of odd natural numbers, 76. E. J. Borowski and Jonathan M. Borwein. Collins Dictionary of Mathematics. 2nd ed. (London: Collins, 1989), 65.

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although both sets are infinite (mapped by the rule 2n – 1). Later, Cantor proved that other sets, such as the integers Z and the rationals Q (that is, numbers which can be written as a fraction), all have the same infinite cardinality—the countably infinite, or denumerable.77 Cantor proved the denumerability of the rationals Q in a now-famous diagonal proof which is helpful in understanding the nature of a denumerably infinite set. An informal example of his diagonal proof follows.78 First, consider the set N × N, which consists of all the ordered pairs (m, n), where m, n

N. We can write the ordered pairs as follows: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1),

(1,4), (2,3), (3,2), (4,1), . . . , as an increasing sum m + n, and increasing m. This proof can be demonstrated visually as shown in Figure B. • (1, 4)

• (2, 4)

• (3, 4)

• (4, 4)

• (1, 3)

• (2, 3)

• (3, 3)

• (4, 3)

• (1, 2)

• (2, 2)

• (3, 2)

• (4, 2)

• (1, 1)

• (2, 1)

• (3, 1)

• (4, 1)

Figure B Extending on this proof, consider the rationals to be similar to ordered pairs because each ordered pair could be written as a fraction to represent a rational number. For example, ordered pair (1, 2) represents the rational number ½ and ordered pair (3, 4)

77. Thiele, Mathematics and the Divine, 531. 78. Bartle and Sherbert, Real Analysis, 18-19.

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represents the rational number ¾. Therefore, a similar diagonal mapping can be created to prove the denumerability of the rationals. This proof demonstrates the denumerability of the positive rational number system Q by showing how rationals can be systematically counted and shows how, in this sense, the naturals N, the integers Z, and the rationals Q, are all alike in that they are countably infinite. While each of these discoveries is impressive in its own right, likely Cantor’s most provocative discovery is his proof that real numbers, R, are non-denumerable. That is to say, the real number system (which includes the rationals and the irrationals) is not the same cardinality as the aforementioned infinite sets. Thus, Cantor proved there are different “sizes” or “levels” of infinity—those which are countably infinite and those which are not. The significance of this discovery was profound. Before Cantor, no one had even precisely defined the infinite. Because of Cantor’s work, not only was the actually infinite specifically defined, but now various sizes of the actually infinite were defined, as well as the absolutely infinite (as the infinite set of all actually infinite sets). Cantor had revolutionized mathematics with his theories of the infinite; mathematics after Cantor would never be the same. As Hedman wrote, “Cantor’s epochal discovery was that the integers and the real numbers were of different cardinality.”79 A non-rigorous explanation of this proof follows.80 Cantor proved by contradiction that the reals R were not countable (or nondenumerable) using a subset of the reals (the set of real numbers between zero and one). Begin with the hypothesis that every real number x in the interval between zero and one, 79. Bruce Hedman, Cantor’s Concept of Infinity (Wheaton, IL: ACMS talks, 1991), 64. 80. The following proof is a paraphrase of the rigorous proof found in Bartle and Sherbert’s Introduction to Real Analysis, 50.

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inclusive (that is, 0 ≤ x ≤ 1) has a decimal representation x = 0.b1b2b3…, where bi = 0, 1, . . . , 9. Cantor then claimed every real number could be enumerated as in his diagonal argument for the rational numbers Q. However, Cantor showed that he could find yet another real number y = 0.y1y2y3 . . . yn . . . by setting y1 = 2 if the first decimal place of b1were set to be greater than or equal to five (that is, b11 ≥ 5) and y2 = 7 if the first decimal place of b1 were less than or equal to four (that is, b11 ≤ 4). Thus, y could be said to be in the interval 0 ≤ x ≤ 1. (All that to say that even if all of the real numbers had seemed to be enumerated, there could always be found another real number within the interval. There is an infinite number of real numbers within any interval of the reals). So the original hypothesis that all real numbers x in the interval [0, 1] had been enumerated is false. Therefore, the real numbers are said to be uncountable, or non-denumerable. Having proved that the real number system could not achieve one-to-one correspondence with a denumerable set and that it actually formed a different “size” of infinity, Cantor termed these various “levels” of infinity different transfinite cardinal numbers. Recall that cardinality refers to the sizes of a set. So set E = {5, 6, 7} has finite cardinal number three (3) because there are three elements in the set. However, when referring to the infinite sets, Cantor had now proved that there were different sizes of infinite sets. The first “level” of infinity (e.g. the naturals) he denoted Aleph-Null (‫א‬0) and the second “level” (e.g. real numbers) he denoted ‫א‬1. Thus, ‫א‬0 and ‫א‬1 denote the first two infinite cardinal numbers. Much like the ordinal numbers, Cantor defined a number system of the cardinal numbers with its own arithmetic (such as addition and multiplication).81 This number system consisted of the infinite cardinal numbers as 81. See Theodore G. Faticoni, The Mathematics of Infinity (Hoboken, N.J.: Wiley-Interscience, 2006), 145162, on the arithmetic of cardinals.

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shown: { ‫א‬0, ‫א‬1, ‫א‬2, ‫א‬3, …}, with each cardinal number representing a larger and larger size of actually infinite set. The cardinal numbers, unlike the ordinal numbers, had a hierarchical nature. There was a progression to the cardinals; with each increasing cardinal number, there was an increase in the size of the infinite set. Cantor defined exponentiations as 2‫ א‬, which defined as the cardinality of the set of all subsets (which is 0

called the power set)82 of the naturals N.83 Cantor proved that a set like the naturals N always has a smaller cardinal number (‫א‬0) than its power set, 2‫ א‬.84 For example, the 0

power set of the natural numbers is an uncountable set, whereas the naturals themselves form a countable set. So although N has cardinal number ‫א‬0, its power set (or its set of all sets), P(N), has cardinal number ‫א‬1. Recall that Cantor denoted the cardinal number of the set of all real numbers as ‫א‬1. Cantor proposed a hypothesis, called the Continuum Hypothesis, that there exists no cardinal number between the cardinal number for the set of the naturals, ‫א‬0, and the cardinal number for the set of all real numbers, ‫א‬1. That is, there is no cardinal number c that satisfies ‫א‬0 < c < 2‫ א‬. Put another way, the Continuum Hypothesis states that 2‫א = א‬1. 0

0

As Zuckerman expressed it in his Theorem 6.12.1: “(1) |2|‫א = א‬1 implies the Continuum Hypothesis. 0

(2) The Continuum Hypothesis implies either |2|‫א = א‬1 or |2|‫ א‬and ‫א‬1 are 0

0

incomparable.

82. In 1901, Bertrand Russell developed what is now known as “Russell’s Paradox,” which shows the internal inconsistency of set theory if one considers all sets that do not include themselves as members. 83. Thiele, Mathematics and the Divine, 531. 84. Joseph Dauben, "Georg Cantor and the Battle for Transfinite Set Theory." Journal of the Association of Christians in the Mathematical Sciences (2004). Available online at http://www.acmsonline.org/journal/2004/dauben93.html, 9.

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(3) The Continuum Hypothesis together with the Axiom of Choice85 imply |2|‫= א‬ 0

‫א‬1.”86 Years after Cantor’s life, the Continuum Hypothesis was proved to be neither provable nor disprovable; that is to say, the Continuum Hypothesis is, in fact, axiomatic to set theory, although it is separate from the ZFC Axioms of set theory. The ZFC Axioms, or Zermelo-Franklin-Choice Axioms, were established in the 1920s as statements which are accepted truths, or axioms, in set theory and are the foundation for most set mathematics today.87 (The ZFC Axioms for set theory are akin to Euclid’s axioms for geometry). Paul J. Cohen proved that the Continuum Hypothesis was axiomatic to set theory but was separate from the ZFC Axioms. He was awarded the Field’s Medal in mathematics for solving the long-standing mathematical hypothesis. Specifically, Cohen proved that “‫א‬1 = card(R) is independent of the ZFC Axioms of Set Theory.”88 That is, we can neither prove nor disprove that the cardinality of the reals R is the second infinite cardinal number (‫א‬1) after the cardinality of the naturals N (‫א‬0), and that the statement is independent, or separate from, the traditional axioms of set theory. Thus, if the Continuum Hypothesis is held as axiomatic, it can be used to build a whole new system of mathematics. Because the Continuum Hypothesis was proved to be neither provable nor disprovable until approximately eighty years after Cantor’s lifetime, such a new system of mathematics is not relevant for this discussion of Cantor. Cantor himself was never able to prove or disprove the Continuum Hypothesis. In fact, he was somewhat tortured by his inability to prove the Continuum Hypothesis, 85. The Axiom of Choice of set theory states that for any group of non-empty sets, at least one item can be chosen from each set. 86. Zuckerman, Sets, 349. 87. Faticoni, The Mathematics of Infinity, 187. 88. Ibid., 186.

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partially because of the theological implications he saw for it. As R. Thiele wrote, “The question was whether the order of infinity of the real numbers was the next after that of the natural numbers. In Cantor’s picture of set theory as a staircase of alephs leading to God’s throne (the absolute and complete infinity) the first step of the stairs is already missed.”89 If Cantor could show the hierarchy of the cardinal numbers by resolving the Continuum Hypothesis, he could inductively prove his “staircase of alephs leading to God’s throne.” Mathematical induction is a process of proof that considers that if a statement is true for some k and that whenever it is assumed true for n, it can be proven true for n + 1, then the statement is true for all successive numbers greater than k in that well-ordered set. The issue of well-orderedness is crucial to mathematical induction. Without knowing whether or not ‫א‬1 was certainly the second cardinal number after ‫א‬0, Cantor could not prove his way to absolute infinity, associated with God. Cantor saw his work as an endeavor not only related to mathematics, but also to philosophy and theology as well. The following section expounds upon how Cantor’s work with the infinite and the different cardinalities of infinite sets (‫ )א‬led him to profound, though controversial, theological claims. Cantor and Theology Cantor himself recognized the theological implications of his work with the infinite. Epistemological questions arose from Cantor’s revolutionary mathematics, namely, can one know something of the infinite? And if so, how? And how does the actually infinite relate to the absolutely infinite (i.e. God)? The following section includes

89. Thiele, Mathematics and the Divine, 532.

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a brief biography of Cantor to provide context for his mathematical and theological thought90 and then describes the theology he developed in answer to these questions. Georg Cantor was born in St. Petersburg on March 3, 1845 to a Roman Catholic mother, Marie Böhme (1819-1896), and a Protestant father, Woldemar Georg Cantor (1813?-1863). Georg himself was baptized a Lutheran and remained a devout Protestant throughout his life,91 although most of his interlocutors regarding the infinite were Roman Catholic.92 In his youth, Cantor’s family moved from Russia to Germany where he later began to study mathematics, receiving his doctorate from the University of Berlin in 1868 at the young age of twenty-two. In 1870, he accepted a position as an instructor of mathematics at the University of Halle. By twenty-seven, Cantor had already published a paper on trigonometric series, a paper that also contained his theory of real numbers, which was the beginning of his later work with transfinite numbers.93 In 1874, Cantor published a short article entitled “On a Property of the Collection of All Real Algebraic Numbers” in Crelle’s Journal that introduced his revolutionary discovery of the non-denumerability of real numbers.94 Cantor was a brilliant but controversial mathematician. The mathematical community of Cantor’s day largely rejected his theory of transfinite numbers (ordinals and cardinals). Many of the well-known mathematicians at the time believed Cantor’s approach to concepts of infinity to be unorthodox and dangerous. Leopold Kronecker (1823-1891) called Cantor “a corrupter of youth”95 while Henri Poincaré (1854-1912) 90. For a complete treatment of the life and work of Georg Cantor, please see the works of the foremost Cantor scholar, Joseph Dauben. 91. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 1. 92. Ibid., 144. 93. Dauben, “Georg Cantor and the Battle for Transfinite Set Theory,” 3. 94. Ibid., 3. 95. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 1.

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believed Cantor had polluted mathematics with his set theory.96 Both Kronecker and Poincaré publicly decried Cantor’s work with the infinite. Kronecker fought to keep Cantor’s work from being published because he, like many mathematicians and philosophers of the nineteenth century, still held to the Aristotelian claim that we could speak of the potentially infinite but not an actually infinite. 97 Recall that Aristotle rejected the idea of the infinite as a complete set, as Cantor claimed integers, the naturals, and his transfinites to be. At first, even Cantor was resistant to his mathematical research findings because he ascribed to the long held Aristotelian view that the actually infinite could not exist.98 Despite the negative feedback from much of the mathematical community, there were a few mathematicians, such J.W. Richard Dedekind (1831-1916)99 and David Hilbert (1862-1943), who believed set theory to be an insightful addition to mathematics. Hilbert believed Georg Cantor to be “a scholar who is unrivalled by all mathematicians from Euler to Einstein.”100 Bertrand Russell (1872-1970) called Cantor one of the greatest minds of the nineteenth century.101 The questions that Cantor’s work provoked about the nature of mathematics and the infinite was divisive for the mathematical community of his time. While mathematicians were divided in their views of Cantor’s set theory, the theological community’s reception of Cantor’s work was similarly mixed. Cantor did his set theoretic work around the time of the papal encyclical Aeterni Patris issued by Pope

96. Thiele, Mathematics and the Divine, 525. 97. Howell and Bradley, Mathematics Through the Eyes of Faith, 55. 98. Dauben, “Georg Cantor and the Battle for Transfinite Set Theory,” 2. 99. Russell, Infinity: New Research Frontiers, 279. 100. Thiele, Mathematics and the Divine, 525. 101. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 1.

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Leo XIII. Leo XIII believed Thomas Aquinas to be the authoritative theologian, and therefore, “The goal of the encyclical was a revival and modernization of Christian philosophy along Thomistic lines.”102 Thus, the encyclical resulted in a growth of concern for science and its connection to Christian philosophy, especially among Roman Catholic intellectuals.103 This growing interest in the sciences in the Catholic community put Cantor into conversation with neo-Thomist theologians such as Tillman Pesch, Thomas Esser, Joseph Hontheim, Ignatius Jeiler, Cardinal Johannes Baptist Franzelin, and Constantin Gutberlet.104 Further detail on the conversations between Cantor and a few of these theologians will be provided later. As a Lutheran, Cantor did not separate the knowledge of the infinite from what he believed to be the source of all knowledge: God. As he considered his different “sizes” of infinite sets in light of his belief in the creator-God, Cantor began wrestling with epistemological questions: can people know the infinite, or is the infinite something only existent in the mind of God? And if humans can know something of the infinite, how do they come to know it? Cantor was deeply interested in the theological implications his mathematics of infinity had. Cantor wrote in a letter to Father Thomas Esser that he believed there was “an inseverable bond which links metaphysics with theology” and that “the foundation of the principles of mathematics and natural sciences devolves upon metaphysics.”105 When Cantor referred to metaphysics, “he meant the philosophical study of the relations between the constructs of mind and the objects of the external world.”106 In mathematics, Cantor was relating the patterns he observed in the world to the construct 102. Thiele, Mathematics and the Divine, 534. 103. Ibid., 534. 104. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 142-144. 105. Letter to Father Esser February 1, 1896. 106. Dauben, Cantor: His Mathematics and Philosophy of the Infinite, 125.

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of the infinite in his mind. Philosophers and theologians too were trying to make sense of what they observed in the world to their ability to know truths. With this definition of metaphysics, it is clear why Cantor saw the intricate and deep connections between mathematics, philosophy, and theology. As Thiele writes: “For Cantor . . . scientific knowledge and religious belief were untearably connected, both were rolled into one, united in the end. Understanding the (earthly) theory of infinite sets was for Cantor a necessary condition to begin understanding the infinity of God.”107 Cantor saw his study of the mathematically infinite as utterly inseparable from a discussion of God, who he considered to be associated with the absolutely infinite. However, recall that since antiquity, philosophers and theologians alike had been rejecting the actually infinite as knowable, and theologians had long been associating potential infinity with God.108 As a result, theologians of Cantor’s time felt that Cantor’s claim of human ability to know the actually infinite, or speak of the infinite as a complete set, put the uniqueness of God as the infinite being at risk. Cantor’s study of the infinite also raised questions regarding the ontology of mathematics and, with it, called into question the knowability of the actually infinite and of God. What is mathematics? Cantor held that mathematics was an observable reality that God had inscribed into creation. However, the ontology of mathematics is certainly a matter of debate. One mathematician, Keith Devlin, has defined mathematics to be the study of patterns.109 As such, mathematics is abstract by its very nature. Mathematics itself cannot be observed in the same way biological truths can be; rather, mathematics

107. Thiele, Mathematics and the Divine, 526. 108. Ibid., 528. 109. Keith Devlin. The Language of Mathematics: Making the Invisible Visible (New York: W. H. Freeman, 1998).

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exists in the world but as a reality of patterns instead of a touchable, tangible biological truth. As a study of patterns, mathematics is real and discoverable110 but exists in rational thought as observation of physical existence. As one author observed, “Mathematics is about human beings recognizing and abstracting numerical and spatial patterns they find in the world around them, and systematically putting their discoveries into logical and symbolic formulations. This view emphasizes that mathematics is more than pure deduction; the concepts, axioms, and definitions used in mathematics derive ultimately from the human experience of creation.”111 Cantor recognized this connection between mathematics of rational thought to its suggestion by creation, and he defined three different “levels” of existence of knowledge: that which exists in the mind of God, which he named Intellectus Divinum; that which exists in the human mind (in abstracto); and finally, that which exists in the physical universe (in concreto).112 Cantor always believed that absolute infinity (as distinguished from transfinite earlier in this paper) existed only in the mind of God, Intellectus Divinum, and considered the set of all of his cardinal numbers (denoted ‫ )ת‬to be absolutely infinite and beyond human comprehension. While Cantor recognized the finitude of the human mind and its inability to know the absolutely infinite, he still gave credence to the God-given capacity of the mind to comprehend the transfinite. (To say the transfinites can be “comprehended” means “that their properties can be formally described and these descriptions can be used to distinguish one transfinite from another.”)113 As Hedman notes, “Cantor argued that God

110. The debate between mathematicians about whether mathematics is discovered or invented is another issue in and of itself. For the sake of clarity, this paper assumes that mathematics is discovered as a reality created by God rather than invented by humans. 111. Bradley and Howell, Mathematics Through Eyes of Faith, 230. 112. Hedman, Concept of Infinity, 65. 113. Russell, Infinity: New Research Frontiers, 284.

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instilled the concept of number, both finite and transfinite, into the mind of man. Cantor frequently appealed to their existence as eternal ideas in the mind of God as the basis for the existence of the transfinites in the mind of man.”114 This is not to say that Cantor believed his theory of transfinite numbers to be merely a construction of his own mind. Rather, Cantor believed that mathematics and the infinite are evidenced in the physical world created by God and discovered by the human mind.115 Hedman wrote: “Kant argued that the mind was not informed by the physical world, but only imposed its own patterns upon its perceptions. Cantor was explicitly opposed to any Kantian interpretation of science, and maintained passionately that the transfinites were not his mind’s own invention, but were suggested to him through physical considerations.”116 Cantor held that while the absolutely infinite could only exist in Intellectus Divinum, the transfinite could exist not only in abstracto but also in concreto (for example, Cantor believed that the atoms of the world consisted of an actually infinite set in concreto).117 It was precisely this claim that the transfinite could exist in abstracto and in concreto that put Cantor at odds with the Catholic interlocutors mentioned earlier in this paper. While several of them were supportive of Cantor’s work with the infinite (notably Cardinal Johannes Baptist Franzelin and Constantin Gutberlet), they were all

114. Hedman, Concept of Infinity, 65. 115. Robert John Russell, in a footnote, helpfully addresses the metaphysical implications of Cantor’s claim that the transfinite is actually existent in creation. Although the claim of the infinite (associated with God) in creation (traditionally considered finite), may seem to fall into either pantheism, God in the world, or atheism, no need for God, the concern is not valid. Russell responds to the charge, “Is Cantor correct in believe that the world is actually infinite?” by noting that “the God-world distinction is founded on an ontological distinction that does not require the traditional finite/infinite distinction as its basis. Even a world filled with actual transfinites could still be considered a created world and not divine” (Infinity: New Research Frontiers, 282-3). 116. Hedman, Concept of Infinity, 67. 117. Thiele, Mathematics and the Divine, 535.

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wary of Cantor’s assertions regarding the existence of the transfinite in abstracto and in concreto. Cardinal Franzelin and his student Gutberlet were particularly concerned about the possibility of Cantor’s view leading to pantheism. Because God had been, in philosophy and theology, associated with the concept of infinity, to speak of the actually infinite existing both in the mind of the human (in abstracto) and in nature itself (in concreto) sounded to the Catholics to be claiming that God existed in the human mind and in nature. However, for Cantor, his concept of the transfinite was not at all enmeshed with his concept of God; rather, he believed that the transfinite was suggested to him through creation and through Cantor’s rational thought by the absolutely infinite God. This accusation, in part, explains Cantor’s concern to distinguish the absolutely infinite (again, associated with God) from his transfinite number discovery. Eventually, the Cardinal conceded the possibility of Cantor’s statements regarding the existence of the infinite in abstracto and in concreto but would not accept the necessity of them. Thiele writes: “The Cardinal agreed that Cantor deduced the possibility (his emphasis) of a Transfinitum from the concept of the all-mighty of God, however, he disagreed when Cantor deduced the necessity of the Creation of a Transfinitum from the infinite Goodness and Glory of God.”118 With the interest of protecting the freedom of God and safeguarding against pantheism, Cantor’s Roman Catholic conversation partners could not agree with Cantor’s insistence that humans necessarily know the actual, concrete transfinite. To gain agreement with his claims, Cantor proposed again a distinction between an “‘Infinitum aeternum increatum sive Absolutum,’ an infinity reserved for God and his attributes, and an ‘Infinitum creatu sive Transfinitum,’ infinity evidenced throughout created nature and 118. Thiele, Mathematics and the Divine, 536.

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exemplified in the actually infinite number of objects in the universe. Cantor added that the important difference between absolute infinitum and actual transfinitum should not be forgotten.”119 That is, Cantor wanted to make it abundantly clear that he recognized the distinction between his transfinite numbers and absolute infinite, i.e. God. Given this distinction, Cardinal Franzelin endorsed Cantor’s view, a fact about which Cantor was very proud and would remind his friends of in letters.120 Thus, Cantor drew a connection between mathematical ontology and human epistemology through his claim that the transfinite is knowable by suggestion through creation and the physical universe. This study suggests that Cantor’s theory of the knowability of the actually infinite and its relation to the knowability of God is actually in agreement with the work of the Catholic theologian discussed in chapter one who lived and worked toward the end of Georg Cantor’s life: Erich Przywara (1889-1972). As the reader may have noticed in the discussion of Cantor’s theological conclusions from his work with the mathematically infinite, much of Cantor’s language regarding the knowability of the infinite resonates with the language Przywara used regarding the knowability of God. The third and final chapter provides a synthesis of the work of these two extraordinary individuals who separately came to similar conclusions regarding the knowability of God.

119. Dauben, Cantor: His Mathematics and Philosophy, 145. 120. Ibid., 145.

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Chapter 3: The Convergence of Mathematics and Theology The Implications of Cantor’s Thought for his View of Human Knowing As mentioned in the previous section on Cantor and his theology, Cantor was very aware of the theological implications of his work. Drawing theological conclusions from Cantor’s theory of the actually infinite is therefore a fair treatment of his work. As Cantor himself wrote, “Every extension of our insight into the origin of the creatively-possible therefore must lead to an extension of our knowledge of God.”121 Cantor had a strong belief in the connectedness of mathematics, metaphysics, and God. After a hospital stay, Cantor wrote in a letter that he had received his “inspiration from above” to develop his mathematics. He claimed that inspiration had guided him to the Bible and to his mathematical work. Cantor even held that he was himself a medium of revelation, a vessel God had used to allow the world to know of transfinite numbers.122 He viewed himself as like a prophet, sent to bring knowledge of the infinite from God to humanity. The persecution Cantor received from both the mathematical and theological communities of his time only served to confirm his own view of himself as a prophet of the infinite. Cantor had such a strong view of the importance of his message, that he held that to deny the existence of actual infinity was to limit God’s power.123 Cantor’s claims about the actually infinite shed light on his view of human knowing. Cantor obviously had a very strong view of human ability to know: know mathematics, know the actually infinite, and, to know something of God’s nature. He put

121. Quotation found in Mathematics and the Divine, 535. Originally quoted from the Archive for History of Exact Sciences 2 (1965), 511. 122. Dauben, “Georg Cantor and the Battle for Transfinite Set Theory,” 13. 123. Bradley and Howell, Mathematics through Eyes of Faith, 55.

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a great deal of emphasis on the power of human reason, although he certainly credited God as the source of human reason and therefore as the ultimate source of our ability to know. At every turn, Cantor gave credit to God, as the source of the transfinite and all actually infinite sets, as the ultimate source of knowledge. But Cantor also believed that this knowledge of the transfinite and of God was attainable, that the two were connected by way of his “staircase” of alephs: the actually infinite sets that lead to the absolutely infinite set, associated with God. So Cantor believed he could come to know of God because of the actually infinite sets God had inscribed in creation. Although Cantor was raised a Lutheran, his theology of human nature seems to be much closer to a Catholic theology. Recall the Roman Catholic doctrine of obediential potency—the capacity of human nature, post-Fall, to be used for good by God (that is, human nature was not completely corrupted at the Fall). Cantor believed that not only could he know something of the transfinite and God, but also that he was himself the vessel of this knowledge to humanity. This demonstrates his fundamental belief that human nature maintained its positive ordering to God so that God could act through him. This belief indicates a sense of structural integrity to the nature of the human being, even after the Fall. His belief implies that there remains a capacity for humans to know and do good, and that this knowledge is not utterly corrupted because of human fallenness. Przywara shared this positive view of human capacity to know and do good after the Fall. The result of Przywara’s and Cantor’s respective works evidence their belief in the obediential potency of human nature: that God can still supernaturally work through human nature. Because of this, Przywara and Cantor come to similar claims about the knowability of God. Cantor’s assertions about the knowability of the infinite and of God

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are analogous, although he was presumably unaware of it, to the epistemological claims Przywara made in his analogy of being between humanity and God. Both hold that God is in a relationship of immanence and transcendence with his creation—“God in us and over us.”124 This connection is explored further in the following section. Connecting Cantor’s Actually Infinite and Przywara’s Analogia Entis What justifies bringing Georg Cantor, a professional mathematician, into a theological debate over the analogia entis? In this section, I explain the parallels between Cantor’s and Przywara’s claims about the knowability of God that justify Cantor as a viable voice in the modern discussion about Przywara’s analogy of being. I argue that Cantor’s claim that the actually infinite is knowable and provides an avenue by which humans can come to know about God is analogous to Przywara’s claim that humans can reflect upon their own being as a way of knowing of God. These claims are analogous because they are both similar and dissimilar: they are similar in that they both make an epistemological claim based on ontology, and they are dissimilar in that they begin from different ontological bases. I expand on these similarities and dissimilarities below. First, Przywara was concerned to avoid pantheism with his analogy of being. As drawn out in the first chapter of this study, Przywara’s analogy was meant to provide humanity a means of speaking of God without collapsing God into his creation. Przywara avoided conflating God and creation by emphasizing God’s utter transcendence of creation. While God is like humanity by way of our being, God is always completely distinct from his creation. 124. Przywara, Polarity, 33.

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Similarly, Cantor was concerned with avoiding confusing the absolutely infinite and the transfinite. As discussed in “Cantor and Theology,” Catholic theologians were concerned that Cantor’s work implied a type of pantheism. By speaking of the transfinite (actual infinity) as being suggested to him by creation and even existent in creation (in concreto), Cantor seemed to suggest that God also existed in creation. Catholic theologians worried that Cantor’s ideas supported the view of God in creation, or the absolutely infinite in creation, which is the heresy of pantheism. Cantor was adamant that his view on the knowability of the transfinite did not collapse into pantheism. In his work, Cantor drew a connection between mathematical ontology and human epistemology through his claim that his transfinite numbers (an actually infinite set) is knowable by suggestion through creation and the physical universe. He had a sense of the immanence and transcendence of the infinite—the actually infinite exists in creation and is knowable by human reason, while the absolutely infinite is above us in the mind of God.125 Thus, the existence of actually infinite sets in creation did not collapse into pantheism because God was associated with the absolutely infinite set, which is not existent in creation but transcends it. Przywara’s and Cantor’s theories are also alike (and most importantly alike) in their resulting claims regarding the knowability of God. Przywara claimed that one mode of knowing of God is philosophical reflection upon our own being based on an analogy between our existence and God’s—the analogy of being. Przywara’s analogy of being provided a way for humans to say something about God based on creation. In allowing this avenue by which humans may know God, Przywara made a claim about the 125. As mentioned in a footnote at the beginning of this study, it is unclear whether or not Cantor would also affirm that the absolutely infinite is associated with the very nature of God.

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knowability of God: God is not only knowable, but one mode of his knowability is human ontology. Similarly, Cantor claimed that God is not only knowable, but one way in which he is knowable is mathematical ontology: the existence of the actually infinite. Cantor further explained that the actually infinite provided a “staircase” to the throne of God. Cantor associated the absolutely infinite with God and claimed it was beyond human comprehension. While it is on one hand true that the absolutely infinite is incomprehensible because it cannot be uniquely characterized by a single property, it is at the same time comprehensible because it is defined by the set of actually infinite sets. Thus, there is some ontological connection between the actually infinite and the absolutely infinite (much like there is for Przywara’s ontological connection between God’s being and humanity’s). Although the absolutely infinite is on a different ontological basis from the actually infinite (as God is from humanity), there is still a connection between the existence of each. Moreover, this ontological connection between the absolutely infinite and actually infinite provides one avenue by which the absolutely infinite is knowable—on the basis of the actually infinite. As Russell writes, “We do, in fact, know something about the Absolutely Infinite: all of the properties it possesses must be shared with and disclosed to us through the properties of the transfinites. The Absolute Infinite is in this sense knowable, comprehensible; each of its properties must be found in at least one transfinite number.”126 Therefore, the absolutely infinite is partially knowable by means of the actually infinite. In the same way that Przywara’s analogy of being claimed that God had provided something inherent in creation (our being) by which we can come to know him, so too did Cantor’s theory of the actually infinite claim that God, 126. Russell, Infinity: New Research Frontiers, 284.

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as the only knower of the absolutely infinite, had provided something given in creation (the actually infinite) by which humanity could come to know him. God himself is not existent in creation, but the actually infinite sets that God has inscribed in creation allow humans a means of knowing him. In Cantor’s theory, the proof of the Continuum Hypothesis would allow him to inductively conclude that the infinite set of actually infinite sets (Cantor’s ‘alephs’) was an absolutely infinite set, the infinite associated with God, therefore allowing the actually infinite to provide him a way of knowing about God. But in the same way that Przywara claimed that knowledge based on philosophical reflection by the analogy of being is always limited, so too Cantor claimed that knowledge of the infinite was limited—God is the only knower of absolute infinite. There is also dissimilarity between the analogia entis and Cantor’s theory of the actually infinite. While Przywara argues for the knowability of God on the basis of human existence, Cantor argues on the basis of the existence of actual infinity. Although the two have different starting points, their enterprise is essentially the same: both are concerned with a mode by which humans can come to know of God, and both argue that God is knowable based on something inherent in creation, rather than by divine revelation alone. Thus, Cantor’s theological claims regarding the knowability of the infinite, indeed, do not collapse into pantheism but instead are in line with Catholic theologian Erich Przywara’s view of human knowing as presented in his analogia entis. Moreover, the most provocative similarity between Cantor’s and Przywara’s work is the way in which they each speak of the knowability of God based on something inscribed in creation by God as a means of revelation. In the similarities and dissimilarities I have

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drawn out, Przywara’s analogy of being and Cantor’s theory of actual infinity are themselves analogous to one another. Cantor’s claim that the actually infinite is created by God and suggested through nature puts actual infinity in a relation of similarity and dissimilarity to God, something created that testifies to God and is always transcended by him. The actually infinite is from God, in creation, and knowable by humanity, yet always transcended by God, the only knower of the absolutely infinite (the set of all actually infinite sets). This is like Przywara’s view that creation and human beings are in a relation of similarity and dissimilarity to God as they are created by God; but God is always above them: the relationship of both immanence and transcendence. Therefore, both Przywara’s and Cantor’s claims hold that creation is “entirely from God, yet utterly distinct from God, as like God and yet radically dissimilar to God, as given to be itself and yet as wholly ordered toward God.”127 I close this section with a quotation from Robert John Russell, from his very recent chapter relating God and the infinite. In his chapter, Russell touches briefly on the important analogy between the knowability of the infinite and the knowability of God that I have argued for throughout this study. Russell writes, “Just as the incomprehensibility of Absolute Infinity in mathematics is safeguarded by its reflection in the transfinites, so too the mystery of God is vouchsafed by the revelation of God in our lives and in the universe that God creates and science explores. Mathematically, Absolute Infinity is known through the transfinites, and yet being so, it remains unknown in itself. Theologically, the God who is known is the God who is unknowable.”128

127. White, Analogy of Being, 5-6. 128. Russell, Infinity: New Research Frontiers, 285.

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The Importance of Cantor’s Voice for the Debate Between Barth and Przywara Georg Cantor can enter into the modern debate over the analogia entis as an “outside” voice from the discipline of mathematics and help inform the debate with his mathematics, theology, and philosophy of the infinite. Although Cantor himself did not relate his work to Przywara’s (primarily because he was deceased before Przywara ever published his work on the analogy of being), I argue that Cantor still has a meaningful part to play in this debate because of the analogy I have drawn in the previous section between his work and Przywara’s. As already mentioned, although Cantor and Przywara have different starting points, they have similar conclusions regarding the knowability of God. Both of these thinkers are interested in a natural mode by which humans can come to say something about God. Both Cantor and Przywara claim that something can be known of the Creator based on his creation. But what use are Cantor’s claims in this regard for the debate over the analogy of being? Cantor’s theory of the actually infinite does not relate humanity to God by way of human existence in the same way the analogy of being does, so what use is his work to a discussion of essence and existence of humanity? Cantor has something to add to this debate because although his work with the infinite is dealing with a soulless creation (i.e. mathematics), he still relates the existence of the actually infinite, a created entity, to God by way of epistemology. He uses creation as one mode of knowing God. Therefore, introducing Cantor’s work with the infinite into the debate begun by Przywara and Barth provokes questions for those interested in the relationship of created to Creator. In what ways can Cantor’s work with actually infinite sets inform our theology? What does Cantor’s work imply for the knowability of God? If one accepts

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Cantor’s mathematical work on the infinite to be true, must one also accept his theological claims regarding the knowability of God in order to remain consistent? Ultimately, does Cantor’s work with the infinite imply that Przywara was correct, and that humans can know of God by something inherent in creation? Or are the theological premises of both Przywara and Cantor essentially faulty, and Barth was correct to say that any continuity between created and Creator, finite and infinite, constitutes idolatry? Conclusion: I do not attempt to answer these questions in this thesis, as the discussion of all of these points would be too unwieldy for a single study. However, I have argued that there is an analogy between Przywara’s claims with the analogy of being and Cantor’s work with the actually infinite. I draw this analogy based on the similarity and dissimilarity of their claims. They are similar in their epistemological assertion: each of these thinkers has claimed that one mode of knowing God is by something given in, or inherent in, creation. They are dissimilar in that Przywara has claimed that humans can say something about God based on the relationship of immanence and transcendence between his being and ours, whereas Cantor has claimed humans can say something about God based on the relationship of immanence and transcendence between the absolutely infinite and the actually infinite. Neither Przywara nor Cantor claim that these means of speaking of God are the exclusive way in which humans can know of God; rather, each presents their work as one mode of coming to know something about God. I further argue that this analogy between Przywara’s and Cantor’s epistemological claims justifies Cantor as a worthwhile

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voice to engage in the modern debate over the analogy of being that was begun by Przywara and Barth. In order to make this argument, I have provided the historical background surrounding Przywara’s development of the analogia entis and have elucidated the doctrine of the analogy of being itself. I next placed the analogy of being in the context of the debate that surrounds it—a debate begun by Przywara and Barth that continues into present-day theology. In order to do that, I described Barth’s revealed theology, how it differs from natural theology, and how Barth’s disagreement with the doctrine is founded in his view of the imago Dei being destroyed at the Fall (rather than maintaining that humanity retains a capacity for obedience as in the Catholic view of obediential potency). I then expounded Cantor’s innovative set theory and his work with the mathematically infinite. I distinguished between his definitions of the potentially, actually, and absolutely infinite (as well as his transfinite) and went on to explain the theological significance Cantor saw for his work and the controversy that surrounded his claims. Finally, I drew together Przywara’s analogy of being and Cantor’s theory of the actually infinite by an analogy between their claims about the knowability of God by something inherent in creation. Based on this analogy, I argue that Cantor is a profound voice in the debate over the relation between Creator and creation as presented in the analogia entis, and should be engaged in this modern debate. This study is the beginning of exploring the connection between the infinite and its implications for theology. At the end of his chapter on the connections between Cantor’s work and its contribution to Pannenberg’s theology, Robert John Russell writes that “the distinction between divine essence and existence” would be a promising area of

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study regarding the infinite and theology.129 My study provides the connection point between Cantor’s work and a leading discussion of God’s essence and existence in relation to humanity’s—the debate over the analogy of being. I now invoke others to join in this discussion by attempting to answer the questions this work raises for that debate. I add that this connection between Cantor’s work and Przywara’s is of interest because Cantor came to these conclusions not strictly through theology, but rather through his work with the mathematically infinite. Moreover, not only did Cantor approach the topic from a different discipline but also came to these conclusions several decades before Przywara first put forth the analogia entis in writing. Thus, presumably (and to the best knowledge of this author), the two came to their conclusions regarding the knowability of God independently of one another. That two disciplines apparently disparate come to such similar conclusions independently of one other regarding perennial questions of the Christian faith affirms the importance of interdisciplinary study. This research demonstrates that connections between disciplines such as mathematics and theology are not tangential, but rather are connected at some of the deepest questions of each discipline. Ultimately, both mathematics and theology are trying to answer the question of why certain aspects of the world are the way they are. In attempting to answer the questions within their own discipline, theology and mathematics intersect here, at the question of how humans may come to know about God, which demonstrates that the disciplines are connected in meaningful ways. It also affirms the orderliness of God’s creation and implicitly defends the importance of interdisciplinary study.

129. Russell, Infinity: New Research Frontiers, 289.

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Whether one affirms natural theology or revealed theology, whether one sides with Przywara or with Barth on the topic of the relation between humanity and God, whether one accepts or rejects Cantor’s conclusions about the infinite, this study forces the reader to reckon with the questions not only from within theology but also from within mathematics. Thus, this study validates, as Barth claimed, that the discussion of the analogia entis is really a debate about everything.130

130. White, Analogy of Being, 49.

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