Introduction: Science and Common Sense Long before the beginnings of modern civilization, men ac- quired vast funds of information about their environment. ...

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy (...) the chance to satisfy their intellectual curiosity. (shrink)

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form . We generalize to some combinatorial principles that were shown by Jensen to hold in L. We show that satisfies the statement: “□κ holds whenever κ the least measurable cardinal λ of order λ++”. We introduce a hierarchy of combinatorial principles □κ, λ for 1 λ κ such that □κ□κ, 1 □κ, λ □κ, (...) κ□κ*. We prove that if holds in V. As an application, we show that ZFC + PFA Con. We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ+ for a singular, countably closed limit cardinal κ such that # exists; likewise for the failure of □κ* at such a κ. (shrink)

We present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

Provability, Computability and Reflection.

In this new edition are a detailed bibliography, index, and textual supplements, making it the perfect text for scholars and advanced students of Hume, ...

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. A gripping combination of science and accessibility, _Godel’s Proof_ by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and philosophy the chance to satisfy their intellectual curiosity. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for +, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös (...) cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property ↠ is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j is the least ω1-Erdös cardinal. (shrink)

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students (...) learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future. (shrink)

First published in 1956 The Language of Modern Physics gives a complete account of the concepts both of classical and quantum physics. It deals with themes like logic and semantics; basic ideas of physics and the methods scientists use for confirming their hypotheses.

HUME’S PHILOSOPHICAL SUBVERSION OF RELIGION, NATURAL AND REVEALED, WAS LIFELONG: THE "RELIGIOUS HYPOTHESIS" IS EMPTY. SO I HAVE ARGUED IN A NEW READING OF THE "DIALOGUES". THE ONLY HOPE FOR HUMANITY LIES IN MAN HIMSELF. HUME DISTINGUISHES BETWEEN THE "VULGAR" AND THE "ENLIGHTENED." AT THE APEX OF THE "ENLIGHTENED" STAND THE "HEROES IN PHILOSOPHY," OF WHOM ONLY GALILEO AND NEWTON ARE SPECIFIED. THE "ENLIGHTENED" PROVIDE LEADERSHIP AND KNOWLEDGE, A DUTY WE MAY VIEW AS THE "RELIGION OF MAN." QUITE POSSIBLY HUME (...) CONSIDERED HIMSELF THE THIRD "HERO IN PHILOSOPHY." EVIDENCE IS PROVIDED BY HIS TRIUMPHAL RECEPTION IN PARIS, 1763-1765: "THEY CONSIDER ME AS ONE OF THE GREATEST GENIUSES IN THE WORLD." THUS THE "RELIGION OF MAN" IRONICALLY CONSTITUTES HUME’S PERSONAL RELIGION. (shrink)

No account of the development of contemporary empiricism is adequate which neglects the writings and the influence of Charles Peirce. Although he is not easily pigeon-holed and can not be claimed as the exclusive property of any school or movement, it is appropriate that the hundredth anniversary of his birth should be commemorated at this Congress. For the movement of which it is a manifestation is engaged in a coöperative, intensive cultivation of the methods of the sciences with the help (...) of the most advanced tools of modern logic; and Peirce's intellectual career was also a single-minded devotion to that task. It is fitting, also, that his birth-year be celebrated at Harvard by an international congress. For although he was denied the privilege of teaching at this university, much of his influence was propagated by William James and Josiah Royce, two of its great teachers; and however much he may have suffered from neglect during his life-time, his work surely merits recognition from a movement not confined by national bounds. It is characteristic of the best established sciences that though individuals may pursue researches in them independently of one another, the conclusions reached tend to support each other and to converge toward a common stream of sound beliefs; and such convergence is indeed the sole identifiable warrant for the confidence that some measure of the truth has been attained. It is therefore a happy sign that so many of the central ideas of the present movement have been independently developed on both sides of the Atlantic. One is not minimizing the contributions of the Vienna Circle in pointing out that many of its recent views have been taken for granted for some time by American colleagues, largely because the latter have come to intellectual maturity under the influence of Peirce. (shrink)

William of Auvergne and His Treatise De Anima I. Introduction William of Auvergne, Bishop of Paris from until his death in, is of interest to us chiefly ...

Satire on politics, literature and art. James Joyce, Lenin, and Dadaist, Tristan Tzara come together in the memories of an obscure English diplomat (Henry Wilfred Carr) in Zurich.

We prove that if image is a Jensen extender model, then image satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model image satisfies the Gap-2 Cardinal Transfer Property → for all infinite cardinals κ and λ.

Professor Reichenbach's writings have repeatedly called attention to the important rôle which probability statements play in all inquiry, and he has made amply clear that no philosophy of science can be regarded as adequate which does not square its accounts with the problems of probable inference. Recently he has brought together in convenient form many reflections on the methodology of science familiar to readers of his earlier works, and at the same time he has set himself the task of solving (...) many well-known problems of epistemology in terms of his theory of probability. His latest book is therefore of great interest, both because of the light it throws on Professor Reichenbach's own views and because it reveals the power and limitations of one approach to the problems of science. In particular, while it does not add to the details of his theory of probability worked out elsewhere, the applications Professor Reichenbach now makes of it supply fresh clues for judging its import and adequacy. The object of the present essay, therefore, is to expound a number of his views on probability and epistemology, with a view to examining his conclusions and their relevance to the problems he aims to resolve. The discussion will try to determine whether several features of his present views do not follow from assumptions which he has not sufficiently considered; whether his logical constructions do not create new puzzles; and whether a different starting-point should not be taken if the clarification of scientific concepts and procedures, to which Professor Reichenbach's devotion is as unexcelled as it is well known, is to be successfully conducted. (shrink)

Hume's Dialogues Concerning Natural Religion are still much with us. What appears to be the definitive edition was published by Professor Norman Kemp Smith in 1935 with a learned introduction which, among other things, assembled a mass of evidence pointing to the conclusion that Philo is to be identified with Hume himself, and that Hume in the Dialogues is deliberately trying to undermine the religious hypothesis. Though these conclusions have been widely accepted, Dr. B. M. Laing, in the April issue (...) of Philosophy , strenuously attacked them in an ingenious argument based in part on his already published thesis that Hume's fundamental philosophy is not really the scepticism so long assigned him. Dr. Laing takes the opportunity in passing to indicate his sympathy with the recent position of a distinguished British poet that Voltaire, also, has been traditionally abused in the assumption that on the question of religion he wrote with his tongue in his cheek. The present examination is not designed to inquire into this new mode of making the wicked pious; nor to push the question back from the Dialogues to Hume's basic philosophy; nor to press the writer's conviction [“The Enigma of Hume,” Mind , XLV pp. 334–349] that not only is Philo to be identified as Hume, but in addition, Cleanthes as Joseph Butler, and Demea as Samuel Clarke. The present purpose is solely to indicate certain unhistorical bases of Dr. Laing's attack on the more traditional view so ably championed by Professor Norman Kemp Smith. (shrink)