One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets.

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

This article discusses the history of the concepts of potential infinity and actual infinity in the context of Christian theology, mathematical thinking and metaphysical reasoning. It shows that the structure of Ancient Greek rationality could not go beyond the concept of potential infinity, which is highlighted in Aristotle's metaphysics. The limitations of the metaphysical mind of ancient Greece were overcome through Christian theology and its concept of the infinite God, as formulated in Gregory of Nyssa's theology. That is how the (...) concept of actual infinity emerged. However, Gregory of Nyssa's understanding of human rationality went still further. He said that access to the infinity of God was to be found only in an asymptotic ascension, as expressed in apophatic theology, and this view endangered the rationality of theology. Deeply influenced by the apophatic tradition, Nicholas of Cusa avoided this danger by showing that infinity could be accessed by symbolic representation and asymptotic mathematical reasoning. Thus, he made an immense contribution in making infinity rationally accessible. This endeavor was finally realized by Georg Cantor. This rational accessibility revealed discernible structures of infinity, such as the continuum hypothesis, and the cardinal numbers. However, the probable logical inconsistency of Georg Cantor's all-encompassing Absolute Infinity points to an intuitive understanding of infinity that goes beyond its rational structures – as in apophatic theology. (shrink)

1st ser. The four historical conceptions of being.--2d ser. Nature, man, and the moral order.

This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.

This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. "Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics." --The American Mathematical Monthly.

I ARGUE THAT THE NOTION OF THE BEST POSSIBLE WORLD IS MEANINGLESS AND THEREFORE A CHIMERA, BECAUSE FOR ANY WORLD WHICH MIGHT BE SO DESIGNATED, THERE COULD ALWAYS BE ANOTHER WHICH WAS BETTER, EITHER IN BEING POPULATED BY BEINGS WITH BETTER OR A GREATER QUANTITY OF GOOD CHARACTERISTICS, OR ELSE BY BEING MORE OPTIMIFIC.

This book, one of the first full-length studies of the modalities to emerge from the debate to which Saul Kripke, David Lewis, Ruth Marcus, and others are contributing, is an exploration and defense of the notion of modality de re, the idea that objects have both essential and accidental properties. Plantinga develops his argument by means of the notion of possible worlds and ranges over such key problems as the nature of essence, transworld identity, negative existential propositions, and the existence (...) of unactual objects in other possible worlds. He also applies his logical theories to the elucidation of two problems in the philosophy of religion: the problem of evil and the ontological argument. (shrink)

Sets of contingent objects, perhaps, are as contingent as their members; but properties, propositions, numbers and states of affairs, it seems, are objects whose non-existence is quite impossible. If so, however, how are they related to God? Suppose God has a nature: a property he has essentially that includes each property essential to him. Does God have a nature? And if he does, is there a conflict between God's sovereignty and his having a nature? How is God related to such (...) abstract objects as properties and propositions? These are the questions I want to explore. - Introduction. (shrink)

In a previous paper, Thomas V. Morris and I sketched a view on which abstract objects, in particular, properties, relations, and propositions , are created by God no less than contingent, concrete objects. In this paper r suggest a way of extending this account to cover mathematical objects as well. Drawing on some recent work in logic and metaphysics, I also develop a more detailed account of the structure of PRPs in answer to the paradoxes that arise on a naive (...) understanding of the structure ofthe abstract universe. (shrink)

God is traditionally taken to be a perfect being, and the creator and sustainer of all that is. So, if theism is true, what sort of world should we expect? To answer this question, we need an account of the array of possible worlds from which God is said to choose. It seems that either there is (a) exactly one best possible world; or (b) more than one unsurpassable world; or (c) an infinite hierarchy of increasingly better worlds. Influential arguments (...) for atheism have been advanced on each hierarchy, and these jointly comprise a daunting trilemma for theism. In this paper, I argue that if theism is true, we should expect the actual world to be a multiverse comprised of all and only those universes which are worthy of creation and sustenance. I further argue that this multiverse is the unique best of all possible worlds. Finally, I explain how his unconventional view bears on the trilemma for theism. (shrink)

Theism can be defended against the Philosophical Problem of Evil, provided one rejects the Principle of Perfectionism, without relying on the Greater Good Defence or, unless one is a libertarian, the Free-Will Defence.A corollary of the All Good Possible Worlds Defence and the No Best Possible World Defence, is that God’s goodness need not determine God’s choice to create. The reasons, if any, which God has are relevant to the Theological Problem of Evil but not to the Philosophical Problem of (...) Evil. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

Most process theologians have rejected the creation of the world out of nothing, holding that our universe was created out of some antecedent universe. This article shows how on process grounds, and with faithfulness to much of what Whitehead had to say, process theologians can and should affirm the creation of our universe out of nothing. Standard process objections to this are refuted.

These selections from Le système du monde, the classic ten-volume history of the physical sciences written by the great French physicist Pierre Duhem (1861-1916), focus on cosmology, Duhem's greatest interest. By reconsidering the work of such Arab and Christian scholars as Averroes, Avicenna, Gregory of Rimini, Albert of Saxony, Nicole Oresme, Duns Scotus, and William of Occam, Duhem demonstrated the sophistication of medieval science and cosmology.

Wherever we look, we notice complexity. Philosophically, the concept constitutes a tangled web of problems, in theory as well as daily life. Complexity and Analysis is a meticulous rendering of these problems, tackling the seldom considered nature of complexity that confronts ontological analysts and holists alike. Stewart Umphrey expertly describes the limits of analysis as they have come to light within mathematics, the natural sciences, and analytic philosophy, explaining how Aristotle came upon, and sought to move beyond, the limits of (...) ontological analysis. In trying to understand any complex entity, Umphrey argues, one succeeds in meeting the criterion of metaphysical adequacy only if one fails to meet the crietrion of epistemological adequacy. Ranging across an array of subjects including Kantian and Hegelian idealism, this book provides a superb account of how our own complexity presents not only theoretical problems, but ethical and political dilemmas of great practical significance. (shrink)

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.