The Game of Life

The late John Horton Conway had a gift for making complicated mathematics appear simple. One piece of his work helped to solve the major problem of group theory and led mathematicians to realize that if something holds for all simple groups, it has to be true for all groups. He also contributed the concept of surreal numbers and invented popular puzzles such as the Game of Life.

Contemporary Infinitesimalist Theories of Continua and Their Late Nineteenth- and Early Twentieth-Century Forerunners

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.

Generalized Elko theory

By using a totally antisymmetric spinor field, we generalize Elko theory. We compare our proposed theory with traditional totally antisymmetric spinor field theory based on the Dirac equation. As an application of our formalism, we comment about the possibility to link our generalized Elko theory with matroids, qubits and surreal numbers.

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II

In [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.