The Kalām Cosmological Argument by William Lane Craig

The paper presents a new version of the "Cosmological Argument" – considered to be an ontological argument, since it exclusively uses ontological concepts and principles. It employs famous results of modern physics, and distinguishes between event-causation and agent-causation. Due to these features, the argument manages to avoid the objection of infinite regress. It remains true, however, that the conclusion of the argument (just like the conclusion of Thomas Aquinas’s causal argument) is too unspecific to be unambiguously considered an argument for the existence of God.

The Thomist 64 (2000): 101-25 CANTOR'S TRANSFINITE NUMBERS AND TRADITIONAL OBJECTIONS TO ACTUAL INFINITY JEANW. RIOUX Benedictine College Atchison, Kansas Georg Cantor was one of the most prominent mathematicians of the late nineteenth and early twentieth centuries. His development of a theory of transfinite numbers resurrected philosophical questions about infinity and led to a division of mathematics into schools of thought such as formalism and intuitionism. Cantor's published attempts to justify his mathematical theories were directed not only toward the mathematicians of his day but also toward philosophers, both ancient and contemporary. His efforts on the latter front were rooted in his desire to deal with objections to the very idea of the actual infinite in quantity, and he attached great importance to those objections that came from traditional philosophy.1 I intend to review the basic philosophical issues: Cantor's claims to a workable mathematics of real and actually infinite quantities, his response to Aristotelian objections to those claims, and my reflections on whether Cantor finally settled the matter, as he had hoped. 1 I shall focus upon Cantor's philosophical defense of transfinite numbers against this fundamental Aristotelian position as it is found in part 5 of his "Ueber unendliche, lineare Punktmannigfaltigkeiten," Mathematische Annalen 21 :545-86 (published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ei11 mathematisch-philosophischer Versuch in der Lehre des Unend.lichen [Leipzig: B. G. Teubner, 1883], hereafter Grumilage11). Cantor saw Aristotle as the source of the Scholastic position on infinity, and in the Gnmdlagen he addressed the basic error involved in all 'finitist' reasoning, as he saw it. A few years after writing the Grundlagen, Cantor published papers dealing with specific Scholastic arguments against the actual infinite in detail. For example, see Robin Small, "Cantor and the Scholastics," American Catholic Philosophical Quarterly 66 (1992): 407-28, where the question concerns the eternity of the world. 101 102 JEANW. RIOUX Cantor was well aware of the prevailing mathematical and philosophical climates of his day.2 As he drew nearer to a completed theory of transfinite numbers, he became increasingly interested in justifying it, not only as a consistent and practical3 exercise of mathematical thought, but also as one having a basis in the real world,4 as somehow providing as real a view of the natural world as did the relatively unproblematic theory of finite integers. While other mathematicians seemed unconcerned with such metaphysical questions, Cantor devoted much of his time and effort to addressing them, especially later in his life.5 Ironically, Cantor'sfascination with these metaphysical aspects of the theory, so foreign to his contemporaries in mathematics, turned out to be somewhat prophetic, since the dubious ontological character of transfinite numbers was to figure prominently in later developments in mathematics.6 I will begin by giving an overview of Cantor's transfinite number theory, focusing in particular upon his claim that transfinite numbers possess an objective, or real, infinity which is actual, not merely potential.7 I. SUMMARY OF CANTOR'S TRANSFINITE NUMBER THEORY Cantor discovered thatmerely 'potential' infinity, the sortwith which mathematicians were comfortable to that day, was not the 2 See Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, N.J.: Princeton University Press, 1979), 120-22. 3 As he notes in Grundlagen, § 8, the success in application of a mathematical concept is a major factor in whether it is accepted as legitimate or abandoned. 4 Cantor meant atleastthree thingsby 'real number': 'real' as distinguished from complex, rational, or irrational numbers; 'real' as existing in the understanding, as consistent and definite ideas in the mind; and 'real' as existing in the extramental world. Cantor held that all numbers that are real in the second sense are images of those that are real in the third sense. For more on the distinction between the last two types of reality, see Grundlagen, § 8. See also Dauben, Georg Cantor, 125-26. 5 Though many of the mathematicians of his day were able to look past Cantor's concern with the philosophical and theological implications of his transfinite theory (and in this way many came to defend the theory), to Cantor they were an inseparable part of his...

<i>The Parmenidean Ascent</i>

The axiom of choice asserts that if 41 is a set of non-null, disjoint sets mi, then there exists a set S, called a set of choice, which contains one and only one element from each of the mni. The existence of such a collection as S seems assured on the basis of intuitive reasoning, However, it must be recalled that the word set is undefined, and only attains meaning through its occurrences in the axioms of set theory. An equivalent assertion is that of the well-ordering theorem which states that every class can be so ordered that each of its non-empty subclasses has a first element. This article gives a. brief description of the significance of the axiom of choice and the well-orderingtheorem in the development of abstract set theory. The birth of Cantor's theory of sets. The use of the procedure of cuts by Dedekind [11i to define irrational numbers made it clear that the real numbers could only be attained (conceptually) by conceiving of an actual infinity of numbers previously introduced (rationals). Thus, it became necessary to admit reasoning which dared to use an actual infinity of propositions as premises, each corresponding in a one-to-one fashion to the rationals 1221. Manyof thepredecessors of Cantor had tried to reason on the infinite, but so many contradictions were met that the actual infinite was relegated to a place outside of mathematics. Cantor taught mathematicians to consider certain distinctions such as that between cardinal and ordinal, which only the grammarians had previously observed. Cantor also pointed out that, although there is no real distinction in the case of finite collections, it becomes necessary to distinguish between these conceptions in the case of infinite aggregates [41]. It can be shown that it is by means of the wellordering theorem (or, equivalently, the axiom of choice) that the concepts of cardinal and ordinal are united for the infinite case too. Many difficulties arose when analysis was made of objects whose

The object of this paper is to put Hume's criticism of the �fashionable theory� of the original contract within its proper contexts. This will enable us to show the inner coherence of Hume's arguments, but also their pronounced difference from those characteristically modern versions of contract theory which, like Rawls's, appeal to the notion of a hypothetical contract. We begin, however, with a few observations on the history of contract theories.

One of the principal lines of argument deployed by the friends of the kalām cosmological argument against the possibility of a beginningless series of events is a quite general argument against the possibility of an actual infinite. The principal thesis of the present paper is that if this argument worked as advertised, parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series must have a final member. Anyone who thinks that an endless series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite.

This chapter examines recent criticism of two arguments for the beginning of the universe similar to the kalām cosmological argument. These arguments are about the impossibility of the existence of an actual infinite and the impossibility of the successive formation of an actual infinite. This chapter argues in defense in these arguments and concludes that both arguments have yet to be found unsound. It explains that both arguments do seem to furnish philosophical grounds for holding to the finitude of the past and the beginning of the universe.

Large amounts of user behavior data provide opportunities for user behavior modeling and have great potential in many downstream applications such as advertising and anomaly detection. Compared with traditional methods, embedding-based methods are used more often recently because of their efficiency and scalability. These methods build a "behavior-entity" bipartite graph and learn static embeddings for nodes in the graph. However, behavior patterns in the real world could not be static because entity properties such as user interests usually evolve along with time. In this paper, we formulate user behaviors as a temporal event sequence and propose a stream network embedding approach to capture the evolving nature of user behaviors. Representation of each event is built and used to update the embeddings of nodes. Two contextual behavior modeling tasks are studied for dynamic user behaviors, and experimental results with real-world data demonstrate the effectiveness of our proposed approach over several competitive baselines.

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

Almost everyone who has reflected on theism is aware of the following argument: “The universe must have had a beginning, since an infinite series, or a set of infinite series, of causes and effects stretching back into the past ad infinitum is impossible”. I shall discuss that argument briefly at the end of this chapter. But before doing so, I shall consider a somewhat similar argument which originated with the 18th century English philosopher, Samuel Clarke. The latter argument does not deny the possibility of an infinite regress into the past; and, for that reason, a number of theists have found it more compelling than the anti–infinite regress argument. Now I think that in fact the anti–infinite regress argument has more merit than does Clarke’s argument. But, in order to facilitate an accurate assessment of the latter, I shall, for the time being, proceed on the assumption that an infinite causal regress into the past cannot be known to be impossible.

SCIENCE AND RELIGION: A New Introduction, 3rd edition by Alister E. McGrath. Hoboken, NJ: John Wiley and Sons, 2020. 272 pages. Paperback; $28.99. ISBN: 9781119599876. *Alister McGrath is a major international scholar who is prolific in his output. He has produced many popular books and academic tomes, and as a theological educator his output also includes many textbooks for students. Science and Religion: A New Introduction is now into its third edition and is an excellent introduction to the whole field of science and religion. The restructuring and inclusion of new material is designed to be helpful to the student, and reflects comments on the previous editions. The book introduces most of the areas of interaction between these bodies of thought, and I myself have used earlier editions in my own teaching, giving students a chapter of McGrath to start with for an essay, followed by more detailed material from elsewhere. *McGrath notes that science and religion are wide categories and serious study entails narrowing them down. He describes Ian Barbour's four models for interaction followed by what he calls four ways of imagining the relationship between them. The conflict model is rightly dismissed as a late nineteenth-century myth, and areas where conflict has been perceived, notably with Galileo and Darwin, are given the more nuanced treatment they deserve, thus dispelling the myths surrounding them. McGrath also gives a broader historical overview, refuting the further myth that the scientific revolution owed nothing to the medieval period. He describes the development of the Newtonian mechanistic model of the universe and brings us to the twentieth century with the development of the Big Bang theory. Regarding this last, it would have been good to note the pioneering work of Roman Catholic priest Georges Lemaître, often dubbed the "Father of the Big Bang," who, in contrast to Alexander Friedman, regarded solutions of Einstein's equations as physically realistic and not just mathematical curiosities. *McGrath moves on to a helpful chapter on religion and the philosophy of science. Some form of realism seems predominant and, indeed, the most rational position to take. It is interesting to note the adoption of "critical realism," including not only by science-religion scholars such as John Polkinghorne and others, but also such as the biblical scholar N. T. Wright and James Dunn. McGrath moves on to the role of explanation in science, noting how in science there are different methods for different sciences, and thus different levels of explanation across the different subdisciplines. Theology too has its own methods appropriate to its own object but there are differing views on the role of explanation. He discusses an important case study, that of "non-reductive physicalism" associated with Nancey Murphy and others. He also gives criteria for drawing an "inference to the best explanation." Various perspectives on the philosophy of science--logical positivism and the criteria of verification, falsificationism, and Kuhn's paradigm shifts--are discussed. Worthy of mention here would have been Imre Lakatos whose "methodology of scientific research programmes" has been applied to theology by Philip Hefner and Nancey Murphy. *Complementing the above there follows a useful chapter on science and the philosophy of religion. McGrath describes arguments for the existence of God, beginning with Aquinas's five ways. A section on the Kalām cosmological argument notes how this has been given a new lease on life by the Big Bang theory's postulation of a temporal origin to the universe, although it would have been good to note that the existence of the universe would demand an explanation even if it were to lack a temporal origin. He gives a careful analysis of Paley's natural theology, noting neglected aspects of Paley's work such as his responses to arguments of David Hume. He examines ways in which God may act in the world given the laws of nature uncovered by science, including through miracles, where he notes Hume's critique. However, as McGrath rightly says, Hume's critique needs to be qualified, since, on the one hand, he defines miracles as violations of laws of nature and yet, on the other, has a problem with inductive generalizations from past experience--which is just what laws of nature are. McGrath rightly sees evolutionary arguments debunking religion as committing the genetic fallacy and self-defeating if human rationality is flawed, since that could equally well affect judgments in areas other than religion, notably science. There is a good section on natural theology and the role of explanation. *In the next chapter, McGrath turns to models and analogies: first, as found within the natural sciences and then, within religion. After considering what the terms mean more generally, he gives specific examples for the sciences, including the kinetic theory of gases, wave-particle duality, Galileo's analogical reasoning which led him to postulate mountains on the moon, and Darwin's metaphor of "natural selection." In the theological sphere, he considers Aquinas's notion of analogia entis whereby the creation bears a likeness to its creator, and Ian Ramsey's model of the "divine economy" utilizing the Greek concept of oikonomia. He looks at Arthur Peacocke's theological application of models as linked to "critical realism," and Sally McFague's metaphors in theology--though he could perhaps have allowed more than one sentence on Janet Soskice. He then examines specific theological examples: creation and theories of the atonement. He has a helpful section on the notion of "mystery" in science and religion before returning to Ian Barbour on models. *McGrath's final chapter considers a number of contemporary debates. Noting Hume's distinction between "ought" and "is" he critiques the idea that science, say, evolutionary biology or neuroscience, can determine ethics and moral values. That leads to a more general critique of the imperialist stance that science can answer all interesting questions or that the only reality is that disclosed by science. An interesting example is mathematics, which discovers truths that do not belong to the natural sciences. It is also utterly astonishing that mathematics is effective in describing nature and very hard to explain on an atheistic view. *An important area considered is theodicy, which is arguably made more difficult by the long process of evolution, preceding the existence of humans by hundreds of millions of years. McGrath provides an overview of the helpful contributions of Christopher Southgate and his former student Bethany Sollereder. For these scholars, there is "no other way" for God to create such a rich diversity of creatures, with whom God suffers, and for whom God will bring eschatological fulfilment. On transhumanism, McGrath describes the approaches of Philip Hefner and Ted Peters who, while recognizing the creativity of technological enhancement, are also aware that, given fallen human nature, this can also be abused. *McGrath returns to the anthropic principle and fine-tuning. He says that fine-tuning is strongly consistent with a theistic perspective, but the debate about a multiverse as a possible explanation continues. He also considers the legitimacy of teleological language and directionality in biology. Simon Conway Morris's notion of convergent evolution may be the "best explanation" of what is observed and is resonant with a religious perspective but, like cosmological fine-tuning, does not prove that God exists. *McGrath concludes with two sections on the psychology of religion, considering whether this field can "explain away" religion. Religion may be "natural," but it is debatable as to whether that has any implication at all about the existence of God. Moreover, it is a long way from primitive apprehension of some vague supernatural agent to the systematic theology of, say, Thomas Aquinas or Karl Barth. To my mind, this is not unlike the difference--to give a scientific analogy--between the discovery of fire by early humans and the modern scientific understanding of combustion. *This is an excellent introduction to the field and very well suited to its pedagogic purpose. There are a few typographical errors (e.g., "magisterial" for "magisteria"). I also noticed that British cosmologist Paul Davies is mistakenly described as American. But these and my earlier minor points should not detract from a volume that provides a vital resource to educators and their students. *Reviewed by Rodney Holder, Emeritus Course Director, The Faraday Institute for Science and Religion, Cambridge, UK CB3 0UB.

Cybernetics, system(s) theory, information theory and Fuzzy Sets and Systems in the 1950s and 1960s

The cardinality of fuzzy sets and the continuum hypothesis

CHAPTER VII - AXIOMATIC SET THEORY

In this paper, we examine many real world examples of information granules, and construct a granular deductive reasoning system for these domains. Objects are ordered pairs; the first element is an assertion (to call logical formula) and the second is a semantic set corresponding to the assertion. So the granular language and its model involve both logic and set theory. So-called logic means that the reasoning obeys the syntax in logical language; So called set theory means that the operations corresponding to semantic sets of logical formulas obey the methods in set theory. The evaluation of truth values and the computation rule of granular formulas are established.