On the occasion of the 150th birthday of Georg Cantor (1845â1918), the founder of the theory of sets, the development of the logical foundations of this theory is described as a sequence of catastrophes and of trials to save it. Presently, most mathematicians agree that the set theory exactly defines the subject of mathematics, i.e., any subject is a mathematical one if it may be defined in the language (i.e., in the notions) of set theory. Hence the nature of formal (...) definitions plays an important role within the logical foundations of mathematics. Its study is also helpful to answer the question of how it is possible that the set theory as a universal new ontology for the subject of mathematics (as people hoped around 1900) totally failed but nevertheless the language of set theory is successful in all the mathematical practice. (shrink)
The uniqueness of the parallel lines is independent from the analogous statement on parallel planes and the usual further axioms of three-dimensional affine geometry. MSC: 51A15, 03F65.
Starting from the thesis that the history of mathematics, for saving its duration and support as a scientific discipline, has to look for the needs and problems of contemporary mathematics, six main problems of contemporary mathematics are listed (concerning partly its social state) and subsequently, 16 questions about the history of mathematics. Moreover, some analogues between the work-sharing of the historians of mathematics and that of the scientists are stated.
_Jakob the Liar_ (1999) Director: Peter Kassovitz. Screenplay: Jurek Becker (adapted from his novel), Peter Kassovitz and Didier Decoin. Producers: Nick Gillott, Steven Haft, Robin Williams, Marsha Garces Williams. Cast: Alan Arkin, Bob Balaban, Zoltan Barabas, Michael Jeter, Liev Schreiber, Robin Williams.
Presenting the first book-length study in English of Aristotle's Sophistical Refutations, this work takes a fresh look at this seminal text on false reasoning.
Olaf Müller presents a supposedly empirically equivalent theory to Newtonian optics, which in his view is therefore threatened by underdetermination. This threat could even be expanded to modern physics, since this branch of physics is partly based on Newton’s theory. In this paper, I will show that Müller’s alternative theory contains an ill-defined concept, viz. the definition of whiteness as the absence of optical causal factors. This results from a fundamental property of whiteness: for every source of white light there (...) exist metameric sources. I further argue that this cannot be reconciled by borrowing other concepts from modern physics, as is, I will show, tacitly presupposed in Müller’s argument. As a consequence, his alternative theory is not empirically equivalent to Newtonian optics and his argument in favour of underdetermination fails. (shrink)
Interpreting the world through a social lens is a central characteristic of human cognition. Humans ascribe intentions to the behaviors of other individuals and groups. Humans also make inferences about others’ emotional and mental states. This capacity for social attribution underlies many of the concepts at the core of legal and political systems. The developing scientific understanding of the neural mechanisms used in social attribution may alter many earlier suppositions. However, just as often, these new methods will lead back to (...) old conundrums. Cognitive neuroscience will not dispel the hard problems of social judgment. (shrink)
In an oft-quoted passage from The Principles of Morals and Legislation, Jeremy Bentham addresses the issue of our treatment of animals with the following words: ‘the question is not, Can they reason? nor, can they talk? but, Can they suffer?’ The point is well taken, for surely if animals suffer, they are legitimate objects of our moral concern. It is curious therefore, given the current interest in the moral status of animals, that Bentham's question has been assumed to be merely (...) rhetorical. No-one has seriously examined the claim, central to arguments for animal liberation and animal rights, that animals actually feel pain. Peter Singer's Animal Liberation is perhaps typical in this regard. His treatment of the issue covers a scant seven pages, after which he summarily announces that ‘there are no good reasons, scientific or philosophical, for denying that animals feel pain’. In this paper I shall suggest that the issue of animal pain is not so easily dispensed with, and that the evidence brought forward to demonstrate that animals feel pain is far from conclusive. (shrink)
Peter Singer is probably the best-known and most controversial ethicist in the world today. He rigorously applies utilitarian moral theory to issues such as world poverty, the environment, abortion, euthanasia and, most famously, animal welfare. He has also written a book about his grandfather, David Oppenheim, who died in Theresienstadt concentration camp. He is Ira W. DeCamp Professor of Bioethics at Princeton University.
Shortly before G. E. Moore wrote down the formative for the early analytic philosophy lectures on Some Main Problems of Philosophy (1910–1911), he had become acquainted with two books which influenced his thought: (1) a book by Husserl's pupil August Messer and (2) a book by the Greifswald objectivist Dimitri Michaltschew. Central to Michaltschew's book was the concept of the given. In Part I, I argue that Moore elaborated his concept of sense-data in the wake of the Greifswald (...) concept. Carnap did the same when he wrote his Aufbau, the only difference being that he spoke not of sense-data but of Erlebnisse. This means, I argue, that both Moore's sense-data and Carnap'sErlebnisse have little to do with either British empiricists or the neo-Kantians. In Part II, I try to ascertain what made early analytic philosophy different from all those philosophical groups and movements that either exercised influence on it, or were closely related to it: phenomenologists, Greifswald objectivists, Brentanists. For this purpose, I identify the sine qua non practices of the early analytic philosophers: exactness; acceptance of the propositional turn; descriptivism; objectivism. If one of these practices was not explored by a given philosophical school or group, in all probability, it was not truly analytic. (shrink)
Histories of philosophy frequently depict the later eleventh century as the scene of a series of bouts between dialecticians and anti-dialecticians — Berengar vs. Lanfranc, Roscelin vs. Anselm — preliminaries to the twelfth century welterweight contest between Abelard and St. Bernard and — dare one say? — the thirteenth century heavy-weight championship between St. Thomas and St. Bonaventure.The bouts took place — no question about that — but whether the contestants can properly be characterized as dialecticians and anti-dialecticians is less (...) certain. Dialectics is logic, the third part of the trivium, and increasingly cultivated in the eleventh century; men like Berengar and Roscelin were plainly eager to apply the logical tools with which they had been equipped to the solution of intellectual problems. In particular they undertook the solution of certain central problems of theology — Berengar that of the Eucharist and Roscelin that of the Trinity — and it was this, we are told, that aroused the ire of the anti-dialecticians: if the aim of the dialecticians was to lay bare the mysteries of faith to the light of reason that of the anti-dialecticians was to protect those same mysteries from profanation. (shrink)
In this paper, we explore Peirce's work for insights into a theory of learning and cognition for education. Our focus for this exploration is Peirce's paper The Fixation of Belief (FOB), originally published in 1877 in Popular Science Monthly. We begin by examining Peirce's assertion that the study of logic is essential for understanding thought and reasoning. We explicate Peirce's view of the nature of reasoning itself—the characteristic guiding principles or ‘habits of mind’ that underlie acts of inference, the dimensions (...) of and interaction between doubt and belief, and his four methods of resolving or ‘fixing’ belief (i.e., tenacity, authority, a priori, and experimentation). The four methods are then juxtaposed against current models of teaching and learning such as constructivism, schema theory, situated cognition, and inquiry learning. Finally, we discuss Peirce's modes of inference as they relate educationally to the resolution of doubt and beliefs and offer an example of belief resolution from an experienced teacher in a professional development environment. (shrink)
