Cambridge: Cambridge University Press. Constituting Objectivity. Dordrecht: Springer. Brouwer, L. Carson, Emily, New York: Springer. Ewald, William, ed. Oxford: Clarendon Press. Folina, Janet, Frege, Gottlob, The Foundations of Arithmetic, translated by J. Friedman, Michael, The Dynamics of Reason.

Friedman, Michael and Alfred Nordman, eds. Guyer, Paul, ed. New York: Cambridge University Press. Hamilton, William Rowan, During a long, productive, and often turbulent life, he published more than 70 books and about 2, articles, married four times, became involved in innumerable public controversies, and was honoured and reviled in almost equal measure throughout the world.

## Truth and lies essay topics

His mother and sister died when he was two years old, and his father died some 18 months later. During his childhood Bertrand Russell was educated at home. That year he briefly attended lectures in economics at the University of Berlin. As a founding figure of the analytic movement in philosophy, Bertrand Russell helped to transform the substance, character, and style of philosophy in the English-speaking world. He was also one of the greatest logicians of the 20th century.

Russell was born in Ravenscroft, the country home of his parents, Lord and Lady Amberley. In , after a long and distinguished political career in which he served twice as prime minister , Lord Russell was ennobled by Queen Victoria, becoming the 1st Earl Russell. Bertrand Russell became the 3rd Earl Russell in , after his elder brother, Frank, died childless.

By the time he was age six, his sister, Rachel, his parents, and his grandfather had all died, and he and Frank were left in the care of their grandmother, Countess Russell. Though Frank was sent to Winchester School, Bertrand was educated privately at home, and his childhood, to his later great regret, was spent largely in isolation from other children. This led him to imagine that all knowledge might be provided with such secure foundations, a hope that lay at the very heart of his motivations as a philosopher.

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His earliest philosophical work was written during his adolescence and records the skeptical doubts that led him to abandon the Christian faith in which he had been brought up by his grandmother. There he made lifelong friends through his membership in the famously secretive student society the Apostles , whose members included some of the most influential philosophers of the day.

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Inspired by his discussions with this group, Russell abandoned mathematics for philosophy and won a fellowship at Trinity on the strength of a thesis entitled An Essay on the Foundations of Geometry, a revised version of which was published as his first philosophical book in In Russell published his first political work, German Social Democracy.

Though sympathetic to the reformist aims of the German socialist movement, it included some trenchant and farsighted criticisms of Marxist dogmas. The book was written partly as the outcome of a visit to Berlin in with his first wife, Alys Pearsall Smith, whom he had married the previous year.

In Berlin, Russell formulated an ambitious scheme of writing two series of books, one on the philosophy of the sciences, the other on social and political questions. Shortly after finishing his book on geometry, he abandoned the metaphysical idealism that was to have provided the framework for this grand synthesis. A much greater influence on his thought at this time, however, was a group of German mathematicians that included Karl Weierstrass , Georg Cantor , and Richard Dedekind , whose work was aimed at providing mathematics with a set of logically rigorous foundations.

In arguing for this view with passion and acuity , Russell exerted a profound influence on the entire tradition of English-speaking analytic philosophy , bequeathing to it its characteristic style, method, and tone. Inspired by the work of the mathematicians whom he so greatly admired, Russell conceived the idea of demonstrating that mathematics not only had logically rigorous foundations but also that it was in its entirety nothing but logic. The philosophical case for this point of view—subsequently known as logicism —was stated at length in The Principles of Mathematics There Russell argued that the whole of mathematics could be derived from a few simple axioms that made no use of specifically mathematical notions, such as number and square root , but were rather confined to purely logical notions, such as proposition and class.

The contradiction arises from the following considerations: Some classes are members of themselves e. If it is, then it is not, and if it is not, then it is.

## René Descartes

At first this paradox seemed trivial, but the more Russell reflected upon it, the deeper the problem seemed, and eventually he was persuaded that there was something fundamentally wrong with the notion of class as he had understood it in The Principles of Mathematics. Frege saw the depth of the problem immediately. Whereas Frege sank into a deep depression, Russell set about repairing the damage by attempting to construct a theory of logic immune to the paradox. Like a malignant cancerous growth, however, the contradiction reappeared in different guises whenever Russell thought that he had eliminated it.

In particular, Russell came to the conclusion that there were no such things as classes and propositions and that therefore, whatever logic was, it was not the study of them. By the time he and his collaborator, Alfred North Whitehead , had finished the three volumes of Principia Mathematica —13 , the theory of types and other innovations to the basic logical system had made it unmanageably complicated. Very few people, whether philosophers or mathematicians, have made the gargantuan effort required to master the details of this monumental work. Figure 2 Suppose that the angle of parallellism for P Q is one half a right angle.

Projective geometry Today projective geometry does not play a big role in mathematics, but in the late nineteenth century it came to be synonymous with modern geometry. Figure 3 To obviate such irksome exceptions, projective geometry added to each straight line in space an ideal point, shared by every line parallel to it. Figure 4 In the new setting, the projective properties of figures can be defined unexceptionably.

Klein's Erlangen program In a booklet issued when he joined the faculty at Erlangen , Felix Klein b. From this standpoint, the task of a branch of geometry can be stated thus: Given a manifold and a group of transformations of the manifold, to study the manifold configurations with respect to those features which are not altered by the transformations of the group.

Klein , p. Each particular value system x 1 , … , x n is called an element of the manifold. Axiomatics perfected According to Aristotle, scientific knowledge episteme must be expressed in statements that follow deductively from a finite list of self-evident statements axioms and only employ terms defined from a finite list of self-understood terms primitives. All that need be considered are the relations between the geometric concepts, recorded in the statements and definitions. In the course of deduction it is both permitted and useful to bear in mind the meaning of the geometric concepts that occur in it, but it is not at all necessary.

Indeed, when it actually beomes necessary, this shows that there is a gap in the proof, and—if the gap cannot be eliminated by modifying the argument—that the premises are too weak to support it. Hilbert replied on 29 December Every theory is only a scaffolding or schema of concepts together with their necessary mutual relations, and the basic elements can be conceived in any way you wish.

## An essay on the foundations of geometry

If I take for my points any system of things, for example, the system love, law, chimney-sweep, … and I just assume all my axioms as relations between these things, my theorems—for example, the theorem of Pythagoras—also hold of these things. Lie Groups For a philosopher, the most satisfying feature of the tremendous complication attained by 19th-century mathematics was perhaps the promptness with which the newly created or discovered?

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Bibliography Primary Sources Bolyai, J. Scientia absoluta spatii. Appendix to Bolyai, F. Maros Vasarhely: J. English translation by G. Halsted printed as a supplement to Bonola Cayley, Arthur, Ehresmann, Ch. Einstein, A. Euclides, Elementa , I. Heiberg ed. Teubner, 5 volumes. For English translation, see below under Heath. Gauss, C. English translation by A. Hiltebietel and J. Hilbert, D. Teubner, pp. Grundlagen der Geometrie , mit Supplementen von P.

Zehnte Auflage.

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Stuttgart: Teubner. Tenth, revised edition of Hilbert Klein, F. Revised version of Klein Lie, S. Theorie der Transformationsgruppen 3 volumes , Unter Mitwirkung von F. Engel, Leipzig: Teubner. Lobachevsky, N. Locke, J. Published anonymously; the author's name was added in the second edition.