Postcards from the edge, or Snapshots of the theory of generalised Moonshine

In this essay I will consider a sequence of questions. The first questions concern the biological function of intelligence in general, and cognitive prostheses of human intelligence in particular. These will lead into questions concerning human language, perhaps the most important cognitive prosthesis humanity has ever developed. While it is traditional to rhapsodize about the cognitive power encapsulated in human language, I will emphasize how horribly limited human language...

Actual infinity in its various forms is discussed, searched and not found.

The objects of the great Nonlinear Revolutions - Catastrophes and Chaos in the 1960s-70s (henceforth, CT); and, small-world and scale-free Network Theory (NT), emerging quite recently - will be spliced together by a New Kind of Number Theory, focused on digitizations (i.e., binary strings). NT nodes then become feature-rich representations (nodules in a "rhizosphere") of CT contents. The "Box-Kite" formalism of zero-divisors (ZD's) - first showing in the 16-D Sedenions, then ...

We answer a question of Zadrozny.

What happens when mathematics realizes infinity. When are mathematical definitions actually useful?

The concept of infinity took centuries to achieve recognized status in the field of mathematics, despite the fact that it was implicitly present in nearly all mathematical endeavors. Here I explore the idea that a similar development might be warranted in physics. Several threads will be speculatively examined, including some involving nonstandard analysis. While there are intriguing possibilities, there also are noteworthy difficulties.

The so-called problem of grue was introduced by Nelson Goodman in 1954 as a "riddle" about induction, a riddle which has been widely thought to cast doubt on the validity and rationality of induction. That unnecessary doubt in turn is partly responsible for the reluctance to adopt the view that probability is part of logic. Several authors have pointed out deficiencies in grue; nevertheless, the "problem" still excites. Here, adapted from Groarke, is presented the basis of gr...

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem.

For more than fifty years, taxonomists have proposed numerous alternative definitions of species while they searched for a unique, comprehensive, and persuasive definition. This monograph shows that these efforts have been unnecessary, and indeed have provably been a pursuit of a will o' the wisp because they have failed to recognize the theoretical impossibility of what they seek to accomplish. A clear and rigorous understanding of the logic underlying species definition lea...

This article, dedicated, with admiration to Reuben Hersh, for his forthcoming 90th birthday, argues that mathematics today is not yet a science, but that it is high time that it should become one.