This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.

It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that in particular (...) projective absoluteness is equiconsistent with the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem. (shrink)

Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.

One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.

We give an optimal lower bound in terms of large cardinal axioms for the logical strength of projective uniformization in conjuction with other regularity properties of projective sets of real numbers, namely Lebesgue measurability and its dual in the sense of category . Our proof uses a projective computation of the real numbers which code inital segments of a core model and answers a question in Hauser.

We show that form, n≧1 the existence of a∏ n m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a∏ n m indescribable cardinal κ together with the Generalized Continuum Hypothesis.

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

In explaining his concept of set Cantor intimates a connection with the metaphysical scheme put forward in Plato’s Philebus to determine the place of pleasure. We argue that these determinations capture key ideas of Cantorian set theory and, moreover, extend to intuitions which continue to play a central role in the modern mathematics of infinity.

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...) way of recognizing objects of any kind. (shrink)

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core (...) models. (shrink)

It is shown how certain generic extensions of a fine structural model in the sense of Mitchell and Steel [MiSt] can be reorganized as relativizations of the model to the generic object. This is then applied to the construction of Steel's core model for one Woodin cardinal [St] and its generalizations.

The paper exposes the philosophical and mathematical flaws in an attempt to settle the continuum problem by a new class of axioms based on probabilistic reasoning. I also examine the larger proposal behind this approach, namely the introduction of new primitive notions that would supersede the set theoretic foundation of mathematics.

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and (...) Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. (shrink)

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown how sufficiently iterable fine structure models recognize themselves as global core models.

Erratum to: Axiomathes DOI 10.1007/s10516-014-9234-yIn the original publication of the article, some of the references were published incorrectly. Please find below the corrected version of these references.

We extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain (...) all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques. (shrink)

We prove the following consistency results about indescribable cardinals which answer a question of A. Kanamori and M. Magidor .Theorem 1.1 . CON.Theorem 5.1 . Assuming the existence of σmn indescribable cardinals for all m < ω and n < ω and given a function : {: m 2, n } 1} → {0,1} there is a poset P L[] such that GCH holds in P and Theorem 1.1 extends the work begun in [2], and its proof uses an iterated (...) forcing construction together with master condition arguments. By combining these techniques with some observations about small forcing and indescribability, one obtains the Easton-style result 5.1. (shrink)

The paper exposes the philosophical and mathematical flaws in an attempt to settle the continuum problem by a new class of axioms based on probabilistic reasoning. I also examine the larger proposal behind this approach, namely the introduction of new primitive notions that would supersede the set theoretic foundation of mathematics.

EDITORIAL NOTE.—Arnold Hauser died in February 1978 shortly after returning to his native Hungary; he had lived nearly half of his 85 years in a kind of self-imposed exile. He is considered, by those who know his work, to be perhaps the greatest sociologist of art, though his last years were spent in comparative neglect and obscurity. We present here as a testament to the importance of both the critic and the discipline he helped shape a section from the (...) translation of his Sociology of Art . Hauser's work draws on the influences of his teachers, Simmel, Bergson, Lukács, Mannheim, Sombart, and Troeltsch. He developed in his immense Social History of Art the groundwork for a sociological analysis of art ranging from prehistoric cave painting to film. In his later works—The Philosophy of Art History , Mannerism: The Crisis of the Renaissance and the Origins of Modern Art , and The Sociology of Art—he continued to redefine his brilliant defense of art for society's sake. The theory of l'art pour l'art wants nothing to do with this contradiction and denies not only the moral and social usefulness of art but its every possible practical function as well. "Nobody would write poetry," says Eugenio Montale, "if the problem of literature consisted in making oneself understood." It has been doubted whether the capacity of making oneself understood, the unambiguous communication of feelings and experiences, even lies within the power of art. What Eduardo Hanslick asserts about music in Von musikalischen Schönen, namely, that its relation to everything which is nonmusical, everything which has emotional content, is vague and noncommittal, is to some extent true for all arts. Just as music expresses something which cannot be translated into any other form, so literature expresses something which is eminently literary, linguistic, something locked into words and syntactic structures. In the same way the untranslatable content of a painting, a pictorial idea, a vision, can only be seized and held onto in optical forms. The composer thinks in tones, the painter in lines and colors, the poet in words, tropes, and rhythms. Indubitable as this is, it is not the whole truth; side by side with the content which can only be expressed adequately in a particular form, there is an intrinsic value which can be translated into any form. Kenneth Northcott, translator of this essay, is professor of older German literature and of comparative literature at the University of Chicago. He is currently engaged in the translation of The Sociology of Art and in the study of late medieval satire in Germany and the works of Harold Pinter, in whose plays he also delights to act. (shrink)

Thomistic Hylomorphists claim that we human persons have rational or intellective souls which can continue to exist separately from our bodies after we die. Much of the recent scholarly discussion of Thomistic Hylomorphism has centered on this thesis and the question of whether human persons can survive death along with their souls or whether only their souls can survive in this separated, disembodied, post-mortem state. As a result, two rival versions of Thomistic Hyomorphism have been formulated: Survivalism and Corruptionism. This (...) chapter makes a new contribution to the debate between Survivalists and Corruptionists by identifying a heretofore undiscussed problem for Corruptionist Thomistic Hylomorphism. In particular, it is argued that if Corruptionists were right that human persons cannot survive their deaths along with their souls, this would undermine the grounds on which all Thomistic Hylomorphists, including Corruptionists, rely to justify their belief that our souls can continue to exist after our deaths. Given this, it is argued that Thomistic Hylomorphists have grounds for thinking that our souls can continue to exist after our deaths only if they allow, as Survivalists do and Corruptionists do not, that we human persons can continue to exist after our deaths, with souls but no bodies. (shrink)

It is not quite as easy to see that there is in fact no formula of this modal language having the same truth conditions (in terms of S5 Kripke semantics) as (1). This was rst conjectured by Allen Hazen2 and later proved by Harold Hodes3. We present a simple direct proof of this result and discuss some consequences for the logical analysis of ordinary modal discourse.

We see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed by some combination the axioms d, t, b, 4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and (...) contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure. (shrink)

This paper examines the impact of the recent financial crisis on the relation between a firm’s risk and social performance using a sample of non-financial U.S. firms covering the period 1991–2012. We find that the relation between SP and risk is significantly different in the crisis period compared to the pre-crisis period. SP reduces volatility during the financial crisis. The risk reduction potential of SP is mainly due to the strengths component of SP. Since the relation of risk is stronger (...) with SP strengths than SP concerns, this implies an asymmetric relation between these SP components and a firm’s risk. Specifically, strengths act as a risk reduction tool during an adverse economic environment. (shrink)

This chapter contains sections titled: The Experiential Roots of Religion The ARE in the Twentieth Century The Decline of Traditional Foundationalism and Stock Objections to RE The ARE via the Principle of Critical Trust (PCT) RE and TE Conceptual Coherence of TE Intracoherence of TE The Structure of the CTA The Impartiality Argument for the PCT Objections to the ARE The ARE in the Twenty‐First Century References.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 33.1 (2000) vii-ix [Access article in PDF] In Memory of Henry I first met Henry W. Johstone Jr. during the spring of 1968. I was a doctoral candidate at the University of Wisconsin and Henry was in Madison as part of a distinguished visitor series hosted by my mentor, Lloyd Bitzer. Lloyd had invited a group of graduate students to his home to meet the guest (...) of honor. I was particularly excited because Henry had accepted my paper on Aristotle's theory of exemplification to appear in the new international journal Philosophy and Rhetoric, which he edited. This was my first publication and I wanted to thank him, which seemed to this novice the appropriate thing to do. I remember that evening for Henry's seeming reserve, which I later learned was a deep shyness that masked his capacity for uncanny directness and a wicked sense of humor, and for his generosity in deflecting my naiveté about the basis for editorial judgments. I also remember it for his deft way of engaging a room of young rhetoricians with questions that conveyed his doubt about their subject (he later changed his mind) without clouding his sincere openness to what we had to say. The following fall, we met again during my interview for a position in the Speech Department at Penn State University. I was offered the position and became Henry's colleague as Book Review Editor of P&R. For the next thirty years, I was blessed to serve as Henry's partner-in-crime in the joys of life that may remain today only among the professoriate. We shared in the editorial work of the journal, in co-teaching our graduate seminar in Philosophy and Rhetoric, in our intellectual passion for the bonds and wars between philosophy and rhetoric, and in the pleasure of choosing the right bottle of wine. But more than that, we shared these common pursuits in contexts of luncheons, dinners, and social occasions that mixed intellectual work with the pleasure of treasured company.My son phoned today to tell me that Henry had died. His passing leaves a void in my life that goes beyond the loss of a friend and colleague. Henry was the person who, through patient selection of manuscripts for me to referee and guidance, taught me how to read journal submissions and offer critiques that might help the author to improve and the journal to prosper. Henry was the person who taught me the value of generosity with junior colleagues by his respectful and serious engagement of my own work. He had a marvelous capacity to entertain any idea seriously, to consider the possibility that even a far-fetched conjecture might have value rather than rejecting it out of hand. He lived the ideas he wrote about as a philosopher, arguing always ad hominem and con amore. Henry was the person who took me in tow as a young Turk and suffered my impatience with the conservative ways of our established institution [End Page vii] by taking me to lunch, engaging me in discussions of ideas, and sharing the delights of intellectual work, and in the process helping me retain my ideals and my focus.Lunch for Henry was more than a noontime meal. It was the way he sustained bonds with those he valued as friends and colleagues. He would call two weeks in advance to arrange a date at the Allen Room of the State College Hotel. He would arrive at my office ten minutes early and we would take a leisurely stroll along the Old Main Mall to the hotel. Our arrival was announced by the host with "Good afternoon Dr. Johnstone," that showed in its formality deep respect and in its tone equally deep affection. Henry always took a cocktail with lunch, inviting his companion to join him, and unless you were specific about the bill beforehand, Henry usually insisted on paying. Settling the bill was never an issue for Henry. He often included a graduate assistant from the journal or an assistant professor in our party and seemed to... (shrink)

I thought the paper by Kai-yee Wong and Chris Fraser was fascinating and insightful. Two things I especially appreciated are the clarity with which they summarize my views. I think they are quite fair and accurate. Second, I appreciate their suggestion that the way to deal with the practical problem of weakness of will has much to do with the role of the Background in shaping our actions. I think they are especially on the right track when they say that (...) the improvement of Background skills may actually narrow the range of real options for action, (p. 21) nonetheless, they do not decrease freedom. As they say, “It is a process of strengthening the self, and the agent is likely to experience the concomitant restriction of ‘live’ options not as a limitation but as strength of character.” (p. 21). That seems to me very much on the right track. What they are suggesting, and it is a powerful addition to my own writings, is that we should not just think of the Background as facilitating languages, games and social practices generally, but for morality as well (p. 23). (shrink)

This study examined lexical bundles used differently by Chinese and American PhD students of linguistics in their dissertation abstracts. Two corpora were built, with each having 700 dissertation abstracts produced by Chinese and American PhD students of linguistics, respectively. The study then used lexical analysis software to retrieve frequently used three-word LBs, from which LBs having different frequencies at a significant level across the two corpora were identified and termed as bundles used differently. BUDs were then categorized and analyzed manually (...) in terms of structure, function, and distribution in rhetorical moves. The major findings were: 57.14% of the frequently used LBs were BUDs, of which 90.67% had occurrences in both corpora. The BUDs distributed inequivalently across categories and moves, with the text-oriented category and the move of Result having the most BUDs; and BUDs exhibited two major patterns: the Chinese and American students filled different constituents into structurally and functionally similar constructions, and used LBs of dissimilar functions to fulfill the same communicative purposes. These findings indicate that variations in LB use have a high pedagogic value and confirm the need for using corpora to identify and teach core genre-specific vocabularies to second/foreign language learners. (shrink)

The demise of behaviorism has made ethologists more willing to ascribe mental states to animals. However, a methodology that can avoid the charge of excessive anthropomorphism is needed. We describe a series of experiments that could help determine whether the behavior of nonhuman animals towards dead conspecifics is concept mediated. These experiments form the basis of a general point. The behavior of some animals is clearly guided by complex mental processes. The techniques developed by comparative psychologists and behavioral ecologists are (...) able to provide us with the tools to critically evaluate hypotheses concerning the continuity between human minds and animal minds. (shrink)

We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cut-elimination procedure and also no known non-trivial bound on the proof-depth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside these trees. Thus we call this system “deep” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows one to give (...) a straightforward syntactic cut-elimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cut-elimination procedure for the shallow system. For both systems we thus obtain an upper bound of φ20 on the depth of proofs, where φ is the Veblen function. (shrink)

The sense of agency is a basic feature of our subjective experience. Experimental studies usually focus on either its attributional aspects or on its motoric aspects. Here, we combine both aspects and focus on the subjective experience of the time between action and effect. Previous studies [Haggard, P., Aschersleben, G., Gehrke, J., & Prinz, W.. Action, binding and awareness. In W. Prinz, & B. Hommel, Common mechanisms in perception and action: Attention and performance. Oxford: Oxford University Press] have shown a (...) temporal attraction in the perceived times of actions and effects, but did directly not study the relation between them. In three experiments, time estimates of an interval between an action and its subsequent sensory effect were obtained. The actions were either voluntary key press actions performed by the participant or kinematically identical movements applied passively to the finger. The effects were either auditory or visual events or a passive movement induced to another finger. The results first indicated a shortening of the interval between one’s own voluntary action and a subsequent effect, relative to passive movement conditions. Second, intervals initiated by observed movements, either of another person or of an inanimate object, were always perceived like those involving passive movements of one’s own body, and never like those involving active movements. Third, this binding effect was comparable for auditory, somatic and visual effects of action. Our results provide the first direct evidence that agency involves a generalisable relation between actions and their consequences, and is triggered by efferent motor commands. (shrink)

The purpose of this article is to examine empirically the impact of environmental certification on firm financial performance (FP). The main question is whether there is a "green premium" for certified firms, and, if so, for what kind of certification. We analyze the short-run and the long-run stock price performance using an event-study methodology on a sample of Canadian and U.S. firms. The results of short-run event abnormal returns indicate that forest certification does not have any significant impact on firm (...) FP regardless of the certification system carried out by firms. Unlike the short-run results, the long-run post-event abnormal returns suggest that forest certification has, on average, a negative impact on firm FP. However, the impact of forest certification on firm FP depends on who grants the certification, since only industry-led certification (Sustainable Forestry Initiative, Canadian Standards Association and ISO14001) are penalized by financial markets, whereas non-governmental organizations—led Forest Stewardship Council certification is not. (shrink)

Successful innovations are deemed to be necessary requisites for enterprise success. On the other hand, works council participation and employee participation are judged differently as either fostering employee and enterprise benefits or only the former or even none. Both forms of participation have found diverging theoretical and empirical argumentations regarding innovations. Here, we argue and show empirically that both forms of participation deliver positive contributions to innovation success, economically and employee-related, substantiated with qualitative reports from 36 process innovation cases and (...) quantitative data from 44 cases. Qualitative case analyses reveal different profiles of works council participation depending on the innovation type. Independent of the innovation types, more successful innovations are marked by more intensive participation. Quantitative examinations of a causal model with path analysis specify how this is achieved: works council and employee participation further the growth of appropriate knowledge and the former also raises the coordination capability; both are essential preconditions for innovation success. A direct impact of works councils on innovation success complements the indirect effects. The slightly modified path analysis explains 53% of the innovation success variance. (shrink)

The positivists defined analyticity as truth in virtue of meaning alone and advocated the view that the notion of analyticity so defined is co-extensive with both the notion of an a priori truth an...

This pathbreaking work argues that the major intellectual trend in China from the seventeenth through to the early nineteenth century was Confucian ritualism, as expressed in ethics and classical learning. Through the performance of rites, the early Qing scholars believed they could cultivate Confucian virtues and achieve social order. The author shows how Confucian ritualism, with its emphasis on lineage, became a broad movement of social reform that stressed conformity and clearly prescribed rules of behavior, expressed notably in the growing (...) cult of female chastity. (shrink)

This book, first published in 2002, is a systematic critical overview of German aesthetics from 1750 to the present. It begins with the work of Baumgarten and covers all the major writers on German aesthetics that follow, including Kant, Schiller, Schelling, Hegel, Nietzsche, Heidegger, Gadamer and Adorno. The book offers a clear and non-technical exposition of ideas, placing these in a wider philosophical context where necessary. Such is the importance of German aesthetics that the market for this book will extend (...) far beyond the domain of philosophy to such fields as literary studies, fine art and music. (shrink)