This volume is a collection of essays by various contributors in honor of the late Laurence Berns, Richard Hammond Elliot Tutor Emeritus at St. John's College, Annapolis. The essays address the literary, political, theological, and philosophical themes of his life's work as a scholar, teacher, and constant companion of the "great books.".

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

In lieu of an abstract, here is a brief excerpt of the content:Criminal Law, Pain Relief, and Physician Aid in DyingFaye Girsh, Ed.D., Executive DirectorMadam:The article by Cantor and Thomas on “Pain Relief, Acceleration of Death, and Criminal Law” (KIEJ, June 1996) was a tortured attempt to develop criteria for the humane and compassionate physician who tries to serve the needs of a patient in unremitting pain. There are three areas that merit comment.The authors dealt with pain medications that (...) might have the undesirable side effect of causing death. They did not mention tolerance, which usually occurs in a patient with chronic pain and which would require exceedingly high doses to be effective. They did not mention barbiturates, which are preferable to opiates in causing death and which are used increasingly by compassionate physicians to answer the pleas of their suffering patients, but for which there is no excuse about alleviating pain. Would the criminal law just call the prescription of a lethal dose of barbiturates a reckless act and require conviction? Fortunately, juries, as the authors point out, do not balance these formulas on the head of a pin. When they see the circumstances under which these doctors do help their patients, they choose to nullify the law and would ignore the suggestions made in this article.The authors refer to the “consent” of the patient. They do not mention that these doctors are operating with more than consent; they are administering these medications because suffering patients are begging them to put them out of their pain and the indignities of the dying process. The article discusses the process as though it were the same as consent to elective surgery. These patients—with AIDS, cancer, and hopeless neurological degeneration—are not simply agreeing to a procedure; they are desperately seeking a way out. If the criminal law were to develop criteria for physician aid in dying, certainly one of them should be the active, witnessed, repeated request of such a patient for a certain, quick, painless, gentle death.The third point is that physicians do not face prosecution when administering no treatment to a patient whose life could be saved with treatment but who chooses not to receive it. This has been upheld by the U.S. Supreme Court; no criteria for suffering, alternatives, terminality, justifiability, or unbearability are required. The doctor can stand by while the patient dies what may be a protracted, difficult death, but it is within the law. What matters in this situation is the decision of the patient to refuse treatment, and that is absolute. How ironic that the patient who cannot refuse treatment must, with his or her doctor, jump through all these hoops to achieve a peaceful death.One worries that the physician who must balance the risk of “risky pain relief” against the accusation of recklessness will err in the direction of allowing [End Page 103] the patient to bear the pain rather than help with a dose of analgesics that might be considered beyond the stipulated “risk.”Why don’t we just be honest about it? Allow dying or hopelessly ill patients to make the decision about what they want by passing a carefully defined law allowing euthanasia rather than balancing the risk and trying to determine the “purpose” of the physician’s ministrations. We protect some patients from themselves while allowing others almost complete autonomy to make the choice of ending their lives. The authors were right in saying that proponents of rational laws consider this sheer hypocrisy.Faye Girsh, Executive DirectorHemlock Society U.S.A. Denver, COFaye Girsh misconceives our article’s objective as an “attempt to develop criteria” for the humane physician. Our goal in writing the paper was to analyze the current criminal law framework in the hope of sharpening debate and engendering normative critiques. We readily acknowledge tensions in the current legal framework and would support changes in that framework.Copyright © 1997 The Johns Hopkins University Press... (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add (...) Zermelo's choice function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Few have entertained the idea that Georg Cantor, the creator of set theory, might have influenced Edmund Husserl, the founder of the phenomenological movement. Yet an exchange of ideas took place between them when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively. Here their writings are examined to show how Husserl's and Cantor's ideas overlapped and (...) crisscrossed in the areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. (shrink)

This paper is concerned with the influence that the set theory of Georg Cantor bore upon the mathematical logic of Bertrand Russell. In some respects the influence is positive, and stems directly from Cantor's writings or through intermediary figures such as Peano; but in various ways negative influence is evident, for Russell adopted alternative views about the form and foundations of set theory. After an opening biographical section, six sections compare and contrast their views on matters of common (...) interest; irrational numbers, infinitesimals, cardinal and ordinal numbers, the axiom of infinity, the paradoxes, and the axioms of choice. Two further sections compare the two men over more general questions: the role of logic and the philosophy of mathematics. In a final section I draw some conclusions. (shrink)

The great influence of Georg Cantor's theory of sets and transfinite arithmetic has led to a considerable interest in his life. It is well known that he had a remarkable and unusual personality, and that he suffered from attacks of mental illness; but the ‘popular’ account of his life is richer in falsehood and distortion than in factual content. This paper attempts to correct these misrepresentations by drawing on a wide variety of manuscript sources concerning Cantor's life and (...) career, including the texts of some important documents. An appendix describes the most important collection of missing manuscripts, whose location would help further the preparation of a biographical study of Cantor. (shrink)

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and (...) articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets. (shrink)

Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

This chapter contains sections titled: Works cited.

Two thoughts about the concept of number are incompatible: that any zero or more things have a number, and that any zero or more things have a number only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any things have a number is Frege's; the thought that things have a number only (...) if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

RésuméCantor tenait pour impossible de fonder une science exacte « sans un brin de meta‐physique ». Il fonda sa théorie à l'aide de la doctrine platonicienne classique des Idées. Poussée par les antinomies, la mathématique moderne a formalisé la théorie cantorienne des ensembles et en a fait le fondement de l'interprétation structurelle de toute la mathématique. A la suite de certains physiciens, on propose aujourd'hui de revenir à un platonisme modifié . En allant dans le měme sens que Cantor, (...) on peut approfondir la compréhension moderne de la réalité mathématique en faisant appel à la doctrine classique des archétypes développée par C. G. Jung. (shrink)

Georg Cantor a cherché à assurer les fondements de sa théorie des ensembles. Cet article présente les differentiations cantoriennes concernant la notion d’infinité et une perspective historique de l’émergence de sa notion d’ensemble.

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct (...) the main points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry. (shrink)

Little is known of Edmund Husserl's direct encounter with Georg Cantor's ideas on Platonic idealism and the abstraction of number concepts during the late 19th century, when Husserl's philosophical orientation changed considerably and definitely. Closely analyzing and comparing the two men's writings during that important time in their intellectual careers, I describe the crucial shift in Husserl's views on psychologism and metaphysical idealism as it relates to Cantor's philosophy of arithmetic. I thus establish connections between their ideas which (...) have been until now been virtually unsuspected and contribute to a better understanding of the development of Husserl's thought and of the philosophical and metaphysical ideas within which Cantor chose to frame his theories. (shrink)

George Annas serves a critical function as an incisive commentator on the interactions between law and medicine and law and public health. Along with Alex Capron, Dena Davis, Rebecca Dresser, and Larry GostinProfessor Annas analyses legal aspects of a spectrum of medicolegal issues both in a forum and in a manner that makes them accessible and understandable to a broad community of healthcare providers. His latest book, SomeChoice, continues that valuable tradition. The bulk of the volume (17 out of (...) 22 chapters) is drawn from essays originally published as the feature of the NewEnglandJournalofMedicine. The topics include such staples of patients' rights debate as the boundaries of informed consent, authorization of medical experiments, and physician-assisted suicide. A treat is the inclusion of less frequently covered issues, such as access to health information concerning presidential candidates and constitutional bounds of limits on cigarette advertising. (shrink)

George Annas serves a critical function as an incisive commentator on the interactions between law and medicine and law and public health. Along with Alex Capron, Dena Davis, Rebecca Dresser, and Larry Gostin—to pinpoint a few—Professor Annas analyses legal aspects of a spectrum of medicolegal issues both in a forum and in a manner that makes them accessible and understandable to a broad community of healthcare providers. His latest book, Some Choice, continues that valuable tradition. The bulk of the (...) volume is drawn from essays originally published as the “Legal Issues in Medicine” feature of the New England Journal of Medicine. The topics include such staples of patients' rights debate as the boundaries of informed consent, authorization of medical experiments, and physician-assisted suicide. A treat is the inclusion of less frequently covered issues, such as access to health information concerning presidential candidates and constitutional bounds of limits on cigarette advertising. (shrink)

The importance of Georg Cantor’s religious convictions is often neglected in discussions of his mathematics and metaphysics. Herein I argue, pace Jan ́e (1995), that due to the importance of Christianity to Cantor, he would have never thought of absolutely infinite collections/inconsistent multiplicities,as being merely potential, or as being purely mathematical entities. I begin by considering and rejecting two arguments due to Ignacio Jan ́e based on letters to Hilbert and the generating principles for ordinals, respectively, showing that (...) my reading of Cantor is consistent with that evidence. I then argue that evidence from Cantor’s later writings shows that he was still very religious later in his career, and thus would not have given up on the reality of the absolute, as that would imply an imperfection on the part of God. (shrink)

In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik . In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said (...) to concern the defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege 'reckless' for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell-Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning. (shrink)

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and (...) its implications whatever. Against this account, I will argue here that in fact Cantor never saw any paradox at all, but that his conception of set at that time, and already as far back as 1883, was one in which the paradoxes cannot arise. (shrink)

The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in (...) class='Hi'>Cantor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of (...) space. Husserl’s Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

Georg Cantor, the founder of set theory, cared much about a philosophical foundation for his theory of infinite numbers. To that end, he studied intensively the works of Baruch de Spinoza. In the paper, we survey the influence of Spinozean thoughts onto Cantor’s; we discuss Spinoza’s philosophy of infinity, as it is contained in his Ethics; and we attempt to draw a parallel between Spinoza’s and Cantor’s ontologies. Our conclusion is that the study of Spinoza provides deepening (...) insights into Cantor’s philosophical theory, whilst Cantor can not be called a ‘Spinozist’ in any stricter sense of that word.Keywords: Georg Cantor; Baruch de Spinoza; Infinity in philosophy and mathematics; Transfinite numbers; Set theory; Infinity and pantheism. (shrink)

Abstract:In this article, Kundakunda's theory of omniscience is defended using formal principles derived from set theory. More precisely, analogous features in the work of Jain mystic Kundakunda and the German mathematician Georg Cantor are described, demonstrating that both thinkers demanded an independently existent, transcendental Absolute to render consistent their own systems of thought. Both of their projects entailed resolving the formal quandary of inaccessibility, or the inability for any sequential, determinate objectifications to ever mereologically sum up to a genuine (...) 'Absolute'—that is, a 'gestalt' that is 'more than the sum of its parts'. (shrink)

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)

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some (...) detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor's research. Throughout the paper, a special effort is made to place Cantor's scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the nineteenth century. (shrink)

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that (...) is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning. (shrink)