This book uncovers the lost history of Christianity's encounters with Pythagorean ideas before the Renaissance. David Albertson skillfully examines ancient and medieval theologians, particularly Thierry of Chartres and Nicholas of Cusa, who successfully reconceived the Trinity and the Incarnation within the framework of Greek number theory. David Albertson challenges modern assumptions about the complex relationship between religion and science.

The aim of this paper is to show that John Wyclif’s theory of space is at once an interpretation of the Platonic theory of place and a Neopythagorean conception of magnitudes and numbers. The result is an original form of mathematical atomism in which atoms are point-like entities with a particular situation in space. If the core of this view comes from Boethius’ De arithmetica, John Wyclif is also influenced by Robert Grosseteste’s metaphysics, which includes the Boethian number theory (...) within the Christian tale of the creation of the world ex nihilo. John Wyclif, however, adds some novelty to this theory concerning the epistemological status of this hypothetical description of the creation of the world out of atoms. First, according to Wyclif, whereas geometry is concerned with sensible and imaginable beings, arithmetic, which is purely intellectual, has access to the deep mathematical structure of the universe. He then suggests a subordination of geometry under arithmetic, which he considers the most solid basis for his metaphysics. As a result, with the attribution of numbers and units to every level of reality, it becomes possible to reform our natural imagination, so that it can imagine the atomic structure of matter and space. (shrink)

Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for (...) example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications (For instance, Mycielski’s approach). Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration. (shrink)

Nicholas of Cusa was first of all a theologian but he was interested also in mathematic and natural sciences. In fact philosophico-theological and mathematical ideas were intertwined by him, theological and philosophical ideas influenced his mathematical considerations, in particular when he considered philosophical problems connected with mathematics and vice versa, mathematical ideas and examples were used by him to explain some ideas from theology. In this paper we attempt to indicate this mutual influence. We shall concentrate (...) on the following problems: the role and place of mathematics and mathematical knowledge in knowledge in general and in particular in theological knowledge, ontology of mathematical objects and their origin, in particular their relations to God and their meaning for the description of the world and physical reality, infinity in mathematics versus infinity in theology and their mutual relations and connections. It will be shown that—according to Nicholas—mathematics and mathematical thinking are tools of rationalization of theology and liberating it in a certain sense from the trap of apophatic theology. (shrink)

persuasive argument for the claim that we ought to evaluate mathematics from a mathematical point of view and reject extra-mathematical standards. Maddy considers the objection that her arguments leave it open for an ‘astrological naturalist’ to make an analogous claim: that we ought to reject extra-astrological standards in the evaluation of astrology. In this paper, I attempt to show that Maddy's response to this objection is insufficient, for it ultimately either (1) undermines mathematical naturalism itself, leaving us (...) with only scientific naturalism, or (2) leaves open the possibility of other unpalatable naturalisms. (shrink)

The paper is devoted to the philosophical and theological as well as mathematical ideas of Nicholas of Cusa. He was a mathematician, but first of all a theologian. Connections between theology and philosophy on the one side and mathematics on the other were, for him, bilateral. In this paper we shall concentrate only on one side and try to show how some theological ideas were used by him to answer fundamental questions in the philosophy of mathematics.

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer (...) to the recent debate on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

This chapter discusses the significance of mathematics in natural theology. It suggests that the existence of an independent noetic realm of mathematics should encourage an openness to the possibility of further metaphysical riches to be explored. Engagement with mathematics is only a part of our mental experience. In itself it can give just a hint of what might be meant by the spiritual. The realm of the divine is yet more distant still, but just as arithmetic may have led (...) our ancestors to begin an exploration that would eventually lead them into the riches of higher mathematics, so taking mathematical reality seriously might be the start of a journey which will lead to greater discoveries about the scope and nature of reality. In this way, mathematics has a modest preparatory part to play in the approach to natural theology, and may also contribute to natural theology through its role as a component in a more complex form of encounter with reality. (shrink)

In our contribution to this special issue on thought experiments and mathematics, we aim to insert theology into the conversation. There is a very long tradition of substantial inquiries into the relationship between theology and mathematics. Platonism has been provoking a consolidation of that tradition to some extent in recent decades. Accordingly, in this paper we look at James R. Brown’s Platonic account of thought experiments. Ultimately, we offer an analysis of some of the merits and perils inherent (...) in framing the use of thought experiments in mathematics and theology in terms of Platonism. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical OpticsAntoni MaletIntroductionIsaac Newton’s Mathematical Principles of Natural Philosophy embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of the mathematical (...) sciences.Barrow’s attitude towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. 1 In his last years he became a prolific [End Page 265] author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. 2 It is on certain of his sermons, as well as on the philosophical discussions contained in the Mathematical Lectures that our account of Barrow’s philosophy of the mathematical sciences will rest. 3 Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s Principia Mathematica. In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian [End Page 266] mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s Mathematical Lectures (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.Barrow on Matter and GodRecent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His... (shrink)

First published in Latin in 1699, John Craige’s _Theology _represents a rare early attempt to introduce mathematical reasoning into moral and theological dispute. Craige’s effort to determine the earliest possible date of the Apocalypse earned him ridicule as an eccentric and a crank. Yet, Richard Nash argues, the intensity of the response to Craige’s work testifies to how widely felt the conflict was between the old and newly emergent notions of probability.

Mathematics and the Divine seem to correspond to diametrically opposed tendencies of the human mind. Does the mathematician not seek what is precisely defined, and do the objects intended by the mystic and the theologian not lie beyond definition? Is mathematics not Man's search for a measure, and isn’t the Divine that which is immeasurable? The present book shows that the domains of mathematics and the Divine, which may seem so radically separated, have throughout history and across cultures, proved to (...) be intimately related. Religious activities such as the building of temples, the telling of ritual stories or the drawing of enigmatic figures all display distinct mathematical features. Major philosophical systems dealing with the Absolute and theological speculations focussing on our knowledge of the Ultimate have been based on or inspired by mathematics. A series of chapters by an international team of experts highlighting key figures, schools and trains of thought is presented here. Chinese number mysticism, the views of Pythagoras and Plato and their followers, Nicholas of Cusa's theological geometry, Spinozism and intuitionism as a philosophy of mathematics are treated side by side among many other themes in an attempt at creating a global view on the relation of mathematics and Man’s quest for the Absolute in the course of history. (Elsevier). (shrink)

In the present paper the roots of the Art of the Catalan philosopher Ramon Llull are examined. Moreover the impact of the Art on seventeenth mathematics is briefly discussed.

Theology and the Cartesian Doctrine of Freedom, now for the first time available in English,was Étienne Gilson's doctoral thesis and part of a larger project to show the medieval roots of Descartes at a time when the very existence of medieval philosophy was often ignored. Young Descartes was sent to La Flèche, one of the Jesuits schools that offered a complete philosophical program, and Descartes would have had the same philosophical training as a Jesuit. There is some controversy about (...) the exact dates of Descartes's stay at La Flèche and consequently about his philosophy instructor. By Gilson's calculations François Véron taught Descartes for three years. Véron eventually left the Jesuits to be free to engage in extraordinarily aggressive anti-Calvinist polemics. If anything, Véron's overbearing manner may have contributed to Descartes antipathy toward Scholastic philosophy. (Whatever Descartes's objections to its philosophy curriculum, later in life he recommended la Flèche as the best school in France.) Descartes,s great intellectual mission in life was not his mathematics but his physics, which was understood as a part of philosophy. We see him navigate the shoals of heated theological and religious strife in his attempt to articulate the metaphysical foundations(and in particular a philosophical vision of God) for his physics or theory of nature. As a layman, he always pleaded ignorance in technically theological matters. He presented himself as a loyal Catholic, quite sincerely in the portrait Gilson paints. (shrink)

The main idea consists in researching the existence of certain characteristics of nature similar to human reasonability and purposeful actions, originating and rigorously inferable from the postulates of quantum mechanics as well as from those of special and general relativity. The pathway of the “free-will theorems” proved by Conway and Kochen in 2006 and 2009 is followed and pioneered further. Those natural reasonability and teleology are identified as a special subject called “God” and studyable by “quantum theology”, a scientific (...) counterpart of classical theology as far as the latter endeavors to justify rationally God (without quotation marks) postulated by religion, but meaning in the present study, first of all, Christianity. The suggested “quantum theology” is situated in the historical opposition and conflict of science and religion in Modernity and its reflection in Heidegger’s conception of “ontotheology”. The conception of “ontomathematics” introduced in previous papers to unite and unify physics, mathematics, and philosophy is reinterpreted in the present context to be conservatively generalized in a way to include “quantum theology”. Two conceptions, “locality” and “nonlocality”, originating initially from physics and especially, from theory of entanglement and quantum information, are also generalized ontomathematically, and thus in relation to the newly introduced “quantum theology”. The “quantum theology” subject of “God” is juxtaposed with God of religion and billions of believers all over the world. Husserl’s conception of “philosophy as a rigorous science” is considered to be generalized now to theology as quantum theology, particularly by the phenomenon (in the sense of his “phenomenology”) of “God”, or by an “epoché” to whether God exists or not. The research of the phenomenon of “God” is extended to the conception of the “totality” being inherent for philosophy, especially for classical German philosophy preceding Husserl’s phenomenology. Those sequences for philosophy implied by the introduction of quantum theology are investigated further, on the pathway of ontomathematics, also in relation to what “God” should mean for today’s physics and mathematics. The option and research field of “experimental quantum theology” is discussed as a direct corollary from quantum theology though being sacrilegious as for classical theology. The concluding section reflects in a more loose way on the relation of “God” for quantum theology to God for religion and billions of believers all over the world. (shrink)

Mathematics and the Mind of God is the synopsis of a leture held at a symposium under this title at the Free University of Amsterdam in 1995. It takes a critical position with respect to the suggestion that there is a shortcut from the exact sciences to theology. It is true that mathematics is the pure form in which the exactness of these sciences can be expressed. The fundamental principle of it, however, the structurability of our world of experience, (...) is not expressible in mathematical form. It belongs to the subject matter of metaphysics, in the sense of rational relation of principles. Mystical experience and theological reflection are even beyond the realm of metaphysics, because they claim to transcend natural human reason. Jumping directly from science to theology therefore gives rise to many misunderstandings which are detrimental to both. (shrink)

Originally published in 1949. This meticulously researched book presents a comprehensive outline and discussion of Aristotle’s mathematics with the author's translations of the greek. To Aristotle, mathematics was one of the three theoretical sciences, the others being theology and the philosophy of nature. Arranged thematically, this book considers his thinking in relation to the other sciences and looks into such specifics as squaring of the circle, syllogism, parallels, incommensurability of the diagonal, angles, universal proof, gnomons, infinity, agelessness of the (...) universe, surface of water, meteorology, metaphysics and mechanics such as levers, rudders, wedges, wheels and inertia. The last few short chapters address ‘problems’ that Aristotle posed but couldn’t answer, related ethics issues and a summary of some short treatises that only briefly touch on mathematics. (shrink)

The problem of God's subordination to laws of logic and mathematics in jewish philosophy and theology.

Suppose we are asked to draw up a list of things we take to exist. Certain items seem unproblematic choices, while others (such as God) are likely to spark controversy. The book sets the grand theological theme aside and asks a less dramatic question: should mathematical objects (numbers, sets, functions, etc.) be on this list? In philosophical jargon this is the ‘ontological’ question for mathematics; it asks whether we ought to include mathematicalia in our ontology. The goal of this (...) work is to answer this question in the affirmative, by drawing on considerations on the the applicability of mathematics to natural science. (shrink)

Mathematics and the Divine seem to correspond to diametrically opposed tendencies of the human mind. Does the mathematician not seek what is precisely defined, and do the objects intended by the mystic and the theologian not lie beyond definition? Is mathematics not Man's search for a measure, and isn’t the Divine that which is immeasurable? The present book shows that the domains of mathematics and the Divine, which may seem so radically separated, have throughout history and across cultures, proved to (...) be intimately related. Religious activities such as the building of temples, the telling of ritual stories or the drawing of enigmatic figures all display distinct mathematical features. Major philosophical systems dealing with the Absolute and theological speculations focussing on our knowledge of the Ultimate have been based on or inspired by mathematics. A series of chapters by an international team of experts highlighting key figures, schools and trains of thought is presented here. Chinese number mysticism, the views of Pythagoras and Plato and their followers, Nicholas of Cusa's theological geometry, Spinozism and intuitionism as a philosophy of mathematics are treated side by side among many other themes in an attempt at creating a global view on the relation of mathematics and Man’s quest for the Absolute in the course of history. (Elsevier). (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That (...) series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

The publication of Copernicus’ On the Revolutions of the Heavenly Spheres marked the beginning of the Scientific Revolution. Christian doctrine played a key role for the emergence of the scientific turning point, that brought about the transition from a qualitative to a quantitative approach to natural phenomena. Although the Polish scientist was not a philosopher in the ordinary sense of the term, he shared with many other protagonists of modern science the idea of the universe as mathematical harmony created (...) by God. In this sense, modern scientific thought completed the development that took place since the Scholastic Age. In the historical period between the XII and the XVII century, indeed, Christianity proved a fundamental factor for a considerable growth in natural knowledge. (shrink)

A tension runs through the lucidly militant work of Alain Badiou. It takes various shapes, such as the tension between the rigorous ontology of mathematics and the structures of narrative, or between fiction and argument, image and formula, poem and matheme, or Anglo-American analytic rationalism and continental lyricism. However, the shape of that tension that interests me most is between the triumphant banishing of theology via mathematics and its perpetual recurrence in his thought. For all Badiou’s efforts to dismiss (...) theology as the philosophy of the ‘One’, for all his efforts to read Pascal, Kierkegaard or Paul as exemplars of the ‘event’ without buying into the belief system they purvey, for all his dismissals of the pious myth or fabulous core of Christianity, it seems as though he cannot avoid theology. The question then is whether this philosopher who is ‘rarely suspected of harbouring Christian zeal’ may actually provide an insight or two into theology. The texts on which I base my reading are Being an Event, the short book on Paul, Logiques des Mondes, especially the section on Kierkegaard, and parts of the disparate collection, Theoretical Writings. A rather formidable collection, to say the least. In what follows I begin by considering the absolute blockage of theology in Badiou’s philosophy, specifically through his banishment of the One. And yet, despite his best efforts to seal his system against theology, it has an uncanny knack of returning. I am particularly interested in the way theology has a ghostly presence in what appears for all the world like a fifth ‘procedure of truth’ (alongside the four pillars of art, science, politics and love), in his enthusiastic affirmations of Pascal and Kierkegaard, and the play between fable and truth in his engagement with the apostle Paul. (shrink)

Evangelista Torricelli is perhaps best known for being the most gifted of Galileo’s pupils, and for his works based on indivisibles, especially his stunning cubature of an infinite hyperboloid. Scattered among Torricelli’s writings, we find numerous traces of the philosophy of mathematics underlying his mathematical practice. Though virtually neglected by historians and philosophers alike, these traces reveal that Torricelli’s mathematical practice was informed by an original philosophy of mathematics. The latter was dashed with strains of Thomistic metaphysics and (...) theology. Torricelli’s philosophy of mathematics emphasized mathematical constructs as human-made beings of reason, yet mathematical truths as divine decrees, which upon being discovered by the mathematician ‘appropriate eternity’. In this paper, I reconstruct Torricelli’s philosophy of mathematics—which I label radical mathematical Thomism—placing it in the context of Thomistic patterns of thought.Keywords: Evangelista Torricelli; Thomism; Eternity; Indivisibles; Proportions. (shrink)

My discussion will concern itself with mathematics, medicine and the possible relations between the two. It will be an exercise in logical analysis, a review of some sad, sad facts, and in some sense a promise of glad tidings. In short, it will be an effort to bring the immortal inhabitants of the mathematical heaven into harmonious relations with the mortal ills of man's vale of tears.As to the curious role of mathematics with respect to the natural sciences, several (...) preliminary observations insist on prompt consideration. It is a matter of common knowledge, for example, that mathematics is being constantly applied to physics. With great success, Galileo wedded the two in his study of motion down an inclined plane. The mathematics he used to describe the physical phenomenon involved the equation of a parabola. His daring genius thus got far beyond the conventional circles that were permitted to motion by theology. His courageous use of the then “higher mathematics” was followed by centuries of application of still higher and higher mathematics, till now a small army of mathematicians is busy almost exclusively with the problem of inventing or discovering new varieties of mathematics which could serve physics better and better. The mathematical physicist of to-day is a specialist, nine-tenths mathematician, one-tenth physicist, with the nine-tenths serving the one-tenth. (shrink)