In the late 1810s and 1820s the Edinburgh phrenologists were largely concerned with trying to establish phrenology as the true science of mind. They challenged the accepted theories about the nature of mind and the brain; in turn, phrenology was attacked by the proponents of Scottish common-sense philosophy and by some medical men. The ensuing debate, which is discussed as an example of conflict between incommensurable world-views, involved a wide range of contentious theological, philosophical, scientific and methodological issues.

While many aspects of Shapin's historical thesis are accepted, this paper raises objections to specific parts of his historical account, and also to the historiographical assumptions underlying his sociological programme. In particular, Shapin's claim to have explained the Edinburgh phrenology debate in social terms is analysed and rejected.

Berkeley's "new theory of vision" and, In particular, His sensationalist solution to the problem of judging distance and magnitude were discussed by many eighteenth-Century authors who faced a variety of problem situations. More specifically, Berkeley's theory fed into the debate over whether the phenomena of vision were susceptible to mathematical analysis or were experientially determined. In this paper a variety of responses to berkeley are examined, Concluding with thomas reid's attempt to distinguish physical optics (which can be analyzed geometrically) from (...) the psychology of vision (in which experience plays a major role). (shrink)

This invaluable resource is the first one-volume, in-depth, comprehensive history of modern science ever published.

In this paper several problems are raised concerning Ian Barbour's four ways of interrelating science and religion—Conflict, Independence, Dialogue, and Integration—as put forward in such publications as his highly influential Religion in an Age of Science (1990) and widely adopted by other writers in this field. The authors argue that this taxonomy is not very useful or analytically helpful, especially to historians seeking to understand past engagements between science and religion.

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)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the (...) continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

Geoffrey Cantor and Chris Kenny have criticized attempts to classify various ways of relating science and religion. They hold that all typologies are too simple and too static to illuminate the complex and changing historical interactions of science and religion. I argue that typologies serve a useful pedagogical function even though every particular interaction must be seen in its historical context. I acknowledge the problems in making distinctions between categories of classification and examine some alternative typologies that have been (...) proposed. I leave as an open question whether my fourfold typology is applicable to differing religious traditions. Finally I consider some parallels between typologies for science‐religion interactions and typologies for relationships between religions. Can our discussions be both interdisciplinary and interreligious without the danger of imposing the conceptual framework of one discipline or religious tradition on another discipline or tradition? (shrink)

Geoffrey Cantor and Chris Kenny have criticized attempts to classify various ways of relating science and religion. They hold that all typologies are too simple and too static to illuminate the complex and changing historical interactions of science and religion. I argue that typologies serve a useful pedagogical function even though every particular interaction must be seen in its historical context. I acknowledge the problems in making distinctions between categories of classification and examine some alternative typologies that have been (...) proposed. I leave as an open question whether my fourfold typology is applicable to differing religious traditions. Finally I consider some parallels between typologies for science‐religion interactions and typologies for relationships between religions. Can our discussions be both interdisciplinary and interreligious without the danger of imposing the conceptual framework of one discipline or religious tradition on another discipline or tradition? (shrink)

This work is, in large part, a series of refutations; it is also the author's Ph.D. thesis. First to be refuted is Russell's vicious circle principle as a general remedy for the solution of the paradoxes. The author rejects the classification of paradoxes into syntactic and semantic, since in his view there are no purely syntactic paradoxes. The distinction in logic between the uninterpreted syntactical aspect of a system and the system when given a determinate interpretation is held to be (...) untenable. Tarski's distinction between object-language and meta-language and his concept of semantically closed language are considered irrelevant for the solution of the Liar paradox. The author claims that the usual versions of the Liar paradox have the same structure as the Barber paradox, viz., [S ↔ ~S]. The author solves the Liar paradox by pointing out that it does not have a proper reference. Cantor's diagonal argument for the indenumerability [[sic]] of the real numbers is labeled as unsatisfactory. Since the diagonal number is dependent upon the real numbers in the constructed list, the author claims that this makes the diagonal number to be of a different nature and status than the real numbers in the list; thus we have what the author calls the dependence fallacy. The author also refuses to accept Cantor's nested interval proof of the indenumerability [[sic]] of the real numbers. Within the proof, two infinite sequences are constructed, each of which converges to a limit. Because the proof does not give a "definite rule of convergence," the author is not satisfied that the infinite sequences converge. Also rejected is Cantor's theorem. Other paradoxes analyzed are the Berry, Richard, heterological, "Richardian", Russell, Cantor, and Burali-Forti paradoxes.--T. G. N. (shrink)

We show that there is a computable Boolean algebra $\mathscr{B}$ and a computably enumerable ideal I of $\mathscr{B}$ such that the quotient algebra $\mathscr{B}/I$ is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.

This volume is third in a series, Monuments of Western Thought, which Cantor and Klein are editing at Colgate. The bulk of this book consists of excerpts from the work of Dante and Machiavelli. Of the Dante material, seventy-five pages is from the Divine Comedy, the rest from De Monarchia. Of the Machiavelli material, thirty pages are from The Prince, the rest excerpted from various works and arranged under such heads as "Warfare" and "Fortune." The text is introduced by (...) a sketch of the intellectual, social, and political context in which Dante and Machiavelli lived, and by brief intellectual biographies. The book closes with short critical selections, assessing and interpreting the source material. On Dante: E. Gilson deals with the ideological frame of the Divine Comedy, social order under divine Authority: J. A. Mazzeo with the hierarchy of light in the Paradiso; G. Santayana with Beatrice as woman and symbol. On Machiavelli: J. W. Allen argues that Machiavelli's originality lies in looking at men and affairs without concerning himself with what they ought to be and do; H. Butterfield that Machiavelli admired ancient Rome and advocated close imitation of her; F. Chabod argues that the theme of The Prince is that society is amorphous, waiting the impress of a dominant individual of supreme virtù; F. Meinecke that virtù, fortuna, and necessità are the basic themes of Machiavelli's thought. Each section is followed by a list of questions designed to challenge the student's understanding of what he has read and to send him back to the text.--L. G. (shrink)

The central issues of regnum versus sacerdotium have been obscured by a concentration on personalities and a murder in a cathedral. Cantor is also concerned with personalities, but in this thorough study of church-state relations in Anglo-Norman England, he goes behind the legend and ably demonstrates that the controversies which were dramatized in blood in 1170 had already been settled by politico-ecclesiastical negotiations more than half-a-century earlier. The main interest of the study is in Cantor's discussion of St. (...) Anselm as an aging ecclesiastical statesman trying to avoid the extremes of that "fanatical high Gregorian" Paschal II, while insisting on the ending of lay investiture despite Henry I's opposition.--H. G. K. (shrink)

The ancient dualism of a sensible and an intelligible world important in Neoplatonic and medieval philosophy, down to Descartes and Kant, would seem to be supplanted today by a scientific view of mind-in-nature. Here, we revive the old dualism in a modified form, and describe mind as a symbolic language, founded in linguistic recursive computation according to the Church-Turing thesis, constituting a world L that serves the human organism as a map of the Universe U. This methodological distinction of L (...) vs. U helps to understand how and why structures of phenomena come to be opposed to their nature in human thought, a central topic in Heideggerian philosophy. U is uncountable according to Georg Cantor’s set theory but Language L, based on the recursive function system, is countable, and anchored in a Gray Area within U of observable phenomena, typically symbols, prelinguistic structures, genetic-historical records of their origins. Symbols, the phenomena most familiar to mathematicians, are capable of being addressed in L-processing. The Gray Area is the human Environment E, where we can live comfortably, that we manipulate to create our niche within hostile U, with L offering overall competence of the species to survive. The human being is seen in the light of his or her linguistic recursively computational mind. Nature U, by contrast, is the unfathomable abyss of being, infinite labyrinth of darkness, impenetrable and hostile to man. The U-man, biological organism, is a stranger in L-man, the mind-controlled rational person, as expounded by Saint Paul. Noumena can now be seen to reside in L, and are not fully supported by phenomena. Kant’s noumenal cause is the mental L-image of only partly phenomenal causation. Mathematics occurs naturally in pre-linguistic phenomena, including natural laws, which give rise to pure mathematical structures in the world of L. Mathematical foundation within philosophy is reversed to where natural mathematics in the Gray Area of pre-linguistic phenomena can be seen to be a prerequisite for intellectual discourse. Lesser, nonverbal versions of L based on images are shared with animals. (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a (...) sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an n (...) such that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)

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)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the (...) class='Hi'>Cantor space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0, the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X ⊆ BLR )} where BLR is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete. (shrink)