Of the Geometrical Spirit


Book Description

Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("Of the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Petites-Ecoles de Port-Royal" ("Little Schools of Port-Royal"). The work was unpublished until over a century after his death. Here, Pascal looked into the issue of discovering truths, arguing that the ideal of such a method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up-first principles, therefore, cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true.Pascal also used De l'Esprit géométrique to develop a theory of definition. He distinguished between definitions which are conventional labels defined by the writer and definitions which are within the language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism. Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes.







Problem-Solving and Selected Topics in Euclidean Geometry


Book Description

"Problem-Solving and Selected Topics in Euclidean Geometry: in the Spirit of the Mathematical Olympiads" contains theorems which are of particular value for the solution of geometrical problems. Emphasis is given in the discussion of a variety of methods, which play a significant role for the solution of problems in Euclidean Geometry. Before the complete solution of every problem, a key idea is presented so that the reader will be able to provide the solution. Applications of the basic geometrical methods which include analysis, synthesis, construction and proof are given. Selected problems which have been given in mathematical olympiads or proposed in short lists in IMO's are discussed. In addition, a number of problems proposed by leading mathematicians in the subject are included here. The book also contains new problems with their solutions. The scope of the publication of the present book is to teach mathematical thinking through Geometry and to provide inspiration for both students and teachers to formulate "positive" conjectures and provide solutions.










The Natural philosophy of Emanuel Swedenborg


Book Description

Although Emanuel Swedenborg (1688–1772) is commonly known for his spiritual philosophy, his early career was focused unnatural science. During this period, Swedenborg thought of the world was like a gigantic machine, following the laws of mechanics and geometry. This volume analyzes this mechanistic worldview from the cognitive perspective, by means of a study of the metaphors in Swedenborg’s texts. The author argues that these conceptual metaphors are vital skills of the creative mind and scientific thinking, used to create visual analogies and abstract ideas. This means that Swedenborg’s mechanistic and geometrical worldview, allowed him to perceive the world as mechanical and geometrical. Swedenborg thought ”with” books and pens. The reading gave him associations and clues, forced him to interpret, and gave him material for his intellectual development.







An Essay on Man


Book Description

One of the twentieth century’s greatest philosophers presents the results of his lifetime study of man’s cultural achievements An Essay on Man is an original synthesis of contemporary knowledge, a unique interpretation of the intellectual crisis of our time, and a brilliant vindication of man’s ability to resolve human problems by the courageous use of his mind. In a new introduction Peter E. Gordon situates the book among Cassirer’s greater body of work, and looks at why his “hymn to humanity in an inhuman age” still resonates with readers today. “The best-balanced and most mature expression of [Cassirer’s] thought.”—Journal of Philosophy “No reader of this book can fail to be struck by the grandeur of its program or by the sensitive humanism of the author.”—Ernest Nagel, The Humanist “A rare work of philosophy and a rare work of art.”—Tomorrow




An Examination of Sir William Hamilton's Philosophy


Book Description

An Examination of Sir William Hamilton's Philosophy, first published in 1865, with a second edition in the same year, and third and fourth editions in 1867 and 1872, has long been out of print. The Examination was, for his contemporaries, a most significant and popular work, presenting an extended treatment of some matters central to empiricism that found little space in Mill's Logic, the best known being his treatment of matter and mind from a psychological viewpoint. Appearing just before his successful parliamentary candidature, the Examination, with its deliberate and explicit onslaught on the intuitionists who were, in Mill's view, allied with anti-progressive political and religious forces, brought his beliefs into the public arena in a new way. Some of those who supported him politically found themselves viciously attacked because they had associated themselves with one who assailed settled religious beliefs. Other religionists who rejected many of Mill's attitudes strong expressed their admiration of the Examination because of its exposure to what they, with him, saw as dangerous theological and moral positions. Alan Ryan's analytical and historial introduction dwells on the most significant philosophical elements in the work, placing them in perspective and showing their relations to other aspects of Mill's thought. The textual introduction, by John M. Robson, examines the treatise in context of Mill's life in the 1860s, outlines its composition, and discusses, among other matters, the importance of the extensive revisions Mill made, mostly in response to critics. These revisions appear in full in the textual apparatus. Also provided are a bibliographical index, which gives a guide to the literature on the subject, and a collation of Mill's quotations, an analytical index, and appendices giving the reading of manuscript fragments and listing textual emendations.




Seventeenth-Century Indivisibles Revisited


Book Description

The tremendous success of indivisibles methods in geometry in the seventeenth century, responds to a vast project: installation of infinity in mathematics. The pathways by the authors are very diverse, as are the characterizations of indivisibles, but there are significant factors of unity between the various doctrines of indivisible; the permanence of the language used by all authors is the strongest sign. These efforts do not lead to the stabilization of a mathematical theory (with principles or axioms, theorems respecting these first statements, followed by applications to a set of geometric situations), one must nevertheless admire the magnitude of the results obtained by these methods and highlights the rich relationships between them and integral calculus. The present book aims to be exhaustive since it analyzes the works of all major inventors of methods of indivisibles during the seventeenth century, from Kepler to Leibniz. It takes into account the rich existing literature usually devoted to a single author. This book results from the joint work of a team of specialists able to browse through this entire important episode in the history of mathematics and to comment it. The list of authors involved in indivisibles ́ field is probably sufficient to realize the richness of this attempt; one meets Kepler, Cavalieri, Galileo, Torricelli, Gregoire de Saint Vincent, Descartes, Roberval, Pascal, Tacquet, Lalouvère, Guldin, Barrow, Mengoli, Wallis, Leibniz, Newton.