In this new book, Michael Friedman argues that Kant's continuing efforts to find a metaphysics that could provide a foundation for the sciences is of the utmost ...

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he (...) maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

By a thorough study of the Posterior Analytics and related Aristotelian texts, Richard McKirahan reconstructs Aristotle's theory of episteme--science. The Posterior Analytics contains the first extensive treatment of the nature and structure of science in the history of philosophy, and McKirahan's aim is to interpret it sympathetically, following the lead of the text, rather than imposing contemporary frameworks on it. In addition to treating the theory as a whole, the author uses textual and philological as well as philosophical material to (...) interpret many important but difficult individual passages. A number of issues left obscure by the Aristotelian material are settled by reference to Euclid's geometrical practice in the Elements. To justify this use of Euclid, McKirahan makes a comparative analysis of fundamental features of Euclidian geometry with the corresponding elements of Aristotle's theory. Emerging from that discussion is a more precise and more complex picture of the relation between Aristotle's theory and Greek mathematics--a picture of mutual, rather than one-way, dependence. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

In Proclus' penetrating exposition of Euclid's method's and principles, the only one of its kind extant, we are afforded a unique vantage point for understanding the structure and strenght of the Euclidean system. A primary source for the history and philosophy of mathematics, Proclus' treatise contains much priceless information about the mathematics and mathematicians of the previous seven or eight centuries that has not been preserved elsewhere.

Paper mosaics, silk screen prints, fold-outs, silhouettes, and other types of cards to make yourself.

In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The author aims at developing a notion (...) of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Polish philosophy of science has been the beneficiary of three powerful creative streams of scientific and philosophical thought. First and fore­ most was the Lwow-Warsaw school of Polish analytical philosophy founded by Twardowski and continued in their several ways by Les­ niewski, Lukasiewicz, and Tarski, the great mathematical and logical philosophers, by Kotarbinski, probably the most distinguished teacher, public figure, and culturally influential philosopher of the inter-war and post-war period, and by Ajdukiewicz, the linguistic philosopher who was intellectually sympathetic with (...) the anti-irrationalist (as he would say), logistic and meta-theoretical inquiries of the Vienna Circle. Second was independent and lively Polish Marxism, with its fine development of social research under Krzywicki, a social anthropologist and younger contemporary of Engels, and then after the war the economist Lange, the philosophers Schaff, Kolakowski, Baczko, and many others. Finally there has been a wide range of philosophical, scientific and humanistic scholar­ ship which lends its various qualities to the understanding of both the logic of science and the historical situation of the sciences: we mention only that great and humane physicist Infeld, the phenomenologist with deep epistemological interest Ingarden, the historian of scientific ideas Zawirski, the historian of philosophy and aesthetics Tatarkiewicz, and the mathematical logicians such as Mostowski and Szaniawski. (shrink)

The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will (...) try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. (shrink)

The dynamics of the historical growth of mathematics as well as natural science is reflected by the interaction of the continuity principle and the principle of the identity of indiscernibles. Aristotle, for example on the one side is responsible for introducing the continuity principle into natural history. On the other hand he is regarded as the great representative of a logic which rests upon the assumption of the possibility of clear divisions and rigorous classification. In modern times the Aristotelian antithesis (...) presents itself in the problem of the concept of mechanical motion, and its mathematical formulation by means of the function concept. Leibniz' philosophy provides a differentiated and broad picture of the problem in which both principles play a major role. To talk of these means, according to Gueroult, "to take a central perspective from which one can perceive the unity of this colossal world of ideas as well as the sometimes contradictory components inherent in it". The contradictory elements alluded to here are I believe strongly related to the difference between philosophical and positive thought. Within positive scientific thought the continuity principle ranks very prominently, whereas the real importance of the principium indiscernibilium gets into view only from a philosophical perspective. (shrink)