Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

. . . Unlike Dewey, he has provided detailed incisive argumentation, and has shown just where the dogmas and dualisms break down." -- Richard Rorty, The Yale Review.

Kurt Godel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computation theory, as well as for the strong individuality of his writings on the philosophy of mathematics. Less well-known is his discovery of unusual cosmological models for Einstein's (...) equations, permitting "time-travel" into the past. This second volume of a comprehensive edition of Godel's works collects together all his publications from 1938 to 1974. Together with Volume I (Publications 1929-1936), it makes available for the first time in a single source all of his previously published work. Continuing the format established in the earlier volume, the present text includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, a facing English translation of the one German original, and a complete bibliography. Succeeding volumes are to contain unpublished manuscripts, lectures, correspondence, and extracts from the notebooks. Collected Works is designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science. These volumes will also interest scientists and all others who wish to be acquainted with one of the great minds of the twentieth century. (shrink)

"Sir Hermann Bondi, NatureThe late S. Chandrasekhar was best known for his discovery of the upper limit to the mass of a white dwarf star, for which he received ...

We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...) algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem. Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity. (shrink)

I argue (contra Guyer et al.) that in the Critique of Judgment Kant espouses a formal, intentional theory of pleasure, and reconstruct Kant's arguments that this view can both identify what all pleasures have in common, and differentiate among kinds of pleasure. Through his investigation of aesthetic experience in the Critique of Judgment, I argue, Kant radically departs from his views about pleasure as mere sensation in the Groundwork and the Critique of Practical Reason, and provides a view of pleasure (...) whereby we can understand pleasure itself to be ruled by an a priori principle. (shrink)

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or (...) as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics. This I will explain below. The second statement expresses what looks like a rather wild thesis. (shrink)

Against the assumption that aesthetic form relates to a harmonious arrangement of parts into a beautiful whole, this book argues that reason is the real theme of the Critique of Judgment as of the two earlier Critiques. Since aesthetic judgment of the beautiful becomes possible only when the mind is confronted with things of nature, for which no determined concepts of understanding are available, aesthetic judgment is involved in an epistemological or, rather, para-epistemological task. The predicate “beautiful” indicates that something (...) has minimal form and is cognizable. This book explores this concept of form, in particular the role of presentation (Darstellung) in what Kant refers to as “mere form,” which involves not only the understanding, but also reason as the faculty of ideas. Such a notion of form reveals why the beautiful can be related to the morally good. On the basis of this reinterpreted concept of form, most major concepts and themes of the Critique of Judgment—such as disinterestedness, free play, the sublime, genius, and beautiful arts—are examined by the author and shown in a new light. (shrink)

Berlin's intense consciousness of the plurality of values, the nature of historical understanding, and the fragility of human freedom premeates essays ranging from his early debates on logical positivism to his later work.