Mathematics is the study of nothing. We make something out of nothing, acting the creator in a world of nothing. Here’s the story: In the beginning is nothing. Not totally nothing because we’re there. But a blank page, a clear slate, a tabula rasa. We draw a distinction, a part of nothing. The indistinct blankness […]
The remarkable book Laws of Form by George Spencer-Brown was published in 1969 and is almost forgotten today. The best expositors have been William Bricken with his boundary mathematics, Louis Kauffman with his knot theory, and Francisco Varela with his work on self-reference. Otherwise it has become something of an underground classic but otherwise forgotten.
Before the 19th century it was commonly understood that only God (or perhaps the “gods”) were actually infinite. If one spoke about the actual infinite, one was doing theology. In mathematics infinity was considered a manner of speaking, which was clarified in the early 19th century with the careful definition of limits. In the late 19th century
Means and extremes in classical mathematics have to do with proportions. If A is to B as C is to D, we write A : B :: C : D. This is ordered so that A is greater than or equal to B and C is greater than or equal to D. A and D are
The dialectic of extremes and means is a method of reasoning whereby one begins with extremes and reasons to means or vice versa. If one begins with means, these are considered as unanalyzed entities, attributes, propositions, etc. The goal is to work out the implications of them as principles or to analyze them into their
A model is a realization of a mathematical formalism. So ordinary arithmetic is a model of ordinary algebra. That is, the algebra of the integers, the rational numbers, and the real numbers is realized by the arithmetic of the integers, the rational numbers, and the real numbers, respectively. Are there other models of ordinary algebra?
