(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that each have duals. Let us begin with the standard approaches to these three topics, and then define duals to each of them. To some extent, the original and the dual may be used together. (1) Set …
This revised post follows up on harmonic (or reciprocal or parallel) addition mentioned in a previous post here. Harmonic arithmetic is based on an automorphism that interchanges the zero with the infinite and the greater-than-one with the less-than-one: 0 ↔ ∞ and x ↔ 1/x. So zero becomes the new inaccessible number and infinity becomes the …
Given two equations with the same variable, how can they be combined? If the equations are consistent, they may be solved as simultaneous equations. But what if the equations are inconsistent? There are two ways to combine them in that case, one is OR, the other is AND. Consider the equations x = a and …
Begin with elements. Elements are a very general concept: they may be either members of sets or distinctions of classes. As a set is defined by its members, so a class is defined by its distinctions. So, the elements of sets are members and the elements of classes are distinctions. Sets may be divided into …
The power means are defined for a set of real numbers, a1, a2, …, an: The best-known of these are the arithmetic, geometric, and harmonic means, with p = 1, p = –1, and p → 0:
The following is a continuation and revision of the previous post, here. Based on the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A pdf version in parallel is here. Let a three-dimensional curve be expressed in parametric form as …
The following is slightly modified from the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A plane curve can be expressed in parametric form as x = x(t); y = y(t); where the coordinates of the point (x, y) of …
There’s a common understanding that most writings need to be interpreted — especially those of a religious or philosophical nature. But mathematical and scientific writings are similar and need to be interpreted, too. Consider that mathematicians and scientists write as if they were creating a world. Mathematicians say things like, “Let there be a line …
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. …
