By Godehard Link
The essays accrued during this quantity concentrate on the position of formalist points in mathematical theorizing and perform, interpreting matters corresponding to infinity, finiteness, and facts systems, in addition to imperative historic figures within the box, together with Frege, Russell, Hilbert and Wittgenstein. utilizing smooth logico-philosophical instruments and systematic conceptual and logical analyses, the amount offers a radical, up to date account of the subject.
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Extra info for Formalism and Beyond: On the Nature of Mathematical Discourse
Example text
English translation in [30], 334-345. Page references are to this translation. [6] H. C. Brown. Review of [24]. Journal of Philosophy, Psychology and Scientific Method, 3:530–531, 1906. [7] H. C. Brown. Infinity and the Generalization of the Concept of Number. Journal of Philosophy, Psychology and Scientific Methods, 5:628–634, 1908. [8] J. Coolidge. The Elements of Non-Euclidean Geometry. Clarendon Press, Oxford, 1909. [9] H. S. M. Coxeter. Introduction to Geometry. Wiley, New York, 2nd edition, 1969.
This is well illustrated by the concept of “points at infinity” which is fundamental in any treatment of projective geometry. Let us now consider the . . two propositions: 1. Any two distinct points of a plane are on one and only one line. 1′. Any two distinct lines of a plane are on one and only one point. Either of these propositions is obtained from the other by simply interchanging the words point and line.... In view of the symmetry of these two propositions it would clearly add much to the symmetry and generality of all propositions derivable from these two, if we could regard them both as true without exception.
H. Smart. A First Course in Projective Geometry. , London, 1913. [29] E. Specker. Dualität. Dialectica, 12:451–465, 1958. [30] J. van Heijenoort. From Frege to Gödel: A source book in mathematical logic 1879–1931. Harvard University Press, Cambridge, 1967. [31] O. Veblen and J. W. Young. A set of Assumptions for Projective Geometry. Americal Journal of Mathematics, 30:347–380, 1908. [32] O. Veblen and J. W. Young. Projective Geometry, volume I. , Boston, 1910. [33] P. Weiss. The Nature of Systems.