Since their appearance in the late 19th century, the Cantor—Dedekind theory of real numbers and philosophy of the continuum have emerged as pillars of standard mathematical philosophy. On the other hand, this period also witnessed the emergence of a variety of alternative theories of real numbers and corresponding theories of continua, as well as non-Archimedean geometry, non-standard analysis, and a number of important generalizations of the system of real numbers, some of which have been described as arithmetic continua of one type or another.
With the exception of E.W. Hobson's essay, which is concerned with the ideas of Cantor and Dedekind and their reception at the turn of the century, the papers in the present collection are either concerned with or are contributions to, the latter groups of studies. All the contributors are outstanding authorities in their respective fields, and the essays, which are directed to historians and philosophers of mathematics as well as to mathematicians who are concerned with the foundations of their subject, are preceded by a lengthy historical introduction.
Part 0: General Introduction; P. Ehrlich. Part I: The Cantor—Dedekind Philosophy and its Early Reception. On the Infinite and Infinitesimal in Mathematical Analysis, Presidential Address to the London Mathematical Society, November 13, 1902, E.W. Hobson. Part II: Alternative Theories of Real Numbers. A Constructive Look at the Real Number Line; D.S. Bridges. The Surreals and Reals; J.H. Conway. Part III: Extensions and Generalizations of the Ordered Field of Reals: the Late 19th-Century Geometrical Motivation. Veronese's Non-Archimedean Linear Continuum; G. Fisher. Review of Hilbert's Foundations of Geometry; Henri Poincaré (1902); Translated for the American Mathematical Society by E.V. Huntington (1903). On Non-Archimedean Geometry, Invited Address to the 4th International Congress of Mathematicians, Rome, April 1908, Giuseppe Veronese; Translated by Mathieu Marion (with editorial notes by Philip Ehrlich). Part IV: Extensions and Generalizations of the Reals: Some 20th-Century Developments. Calculation, Order, and Continuity; H. Sinaceur. The Hyperreal Line; H.J. Keisler. All Numbers Great and Small; P. Ehrlich. Rational and Real Ordinal Numbers; D. Klaua.