Duality, Matroids, Qubits, Twistors, and Surreal Numbers

The purpose of this chapter is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones. We will find that the contemporary theories, while offering novel and possible alternative visions of continua, need not be (and in many cases are not) regarded as replacements for the Cantor-Dedekind theory and its corresponding theories of analysis and differential geometry.

Download Full-text

Introduction to surreal numbers

Physica D Nonlinear Phenomena ◽

10.1016/0167-2789(87)90165-5 ◽

1987 ◽

Vol 28 (1-2) ◽

pp. 233

Author(s):

M.D. Kruskal

Keyword(s):

Surreal Numbers

Download Full-text

SURREAL TIME AND ULTRATASKS

The Review of Symbolic Logic ◽

10.1017/s1755020316000289 ◽

2016 ◽

Vol 9 (4) ◽

pp. 836-847 ◽

Cited By ~ 1

Author(s):

HAIDAR AL-DHALIMY ◽

CHARLES J. GEYER

Keyword(s):

Finite Length ◽

Ordinal Number ◽

Mathematical Structure ◽

Proper Class ◽

Real Numbers ◽

The Real ◽

Surreal Numbers

AbstractThis paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible.

Download Full-text