# mathematics

## Home is the horizon

As there is a parallel algebra, so there is a parallel geometry. Home, the origin, is the horizon, the ends of the earth, and beyond that, the celestial equator, the heavens. We may attempt to journey to the centre of the earth with Jules Verne, but we’ll never make it because it is infinitely far […]

## Number and algebra and their dual

For the first post in this series, see here. (1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that have dual approaches. Let us begin with the standard approaches to these three topics, and then define duals to each of them. In some ways, the original

## Set theory and logic and their dual

(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. Set theory

## Harmonic operations

This revised post follows up on harmonic (or reciprocal or parallel) addition mentioned in a previous post here. See Grossman & Katz, Non-Newtonian Calculus (Lee Press, Pigeon Cove, MA: 1972) p.59f; and Kent E. Erickson, “A New Operation for Analyzing Series-Paralled Networks,” IRE Trans. on Circuit Theory, March 1959, pp.124-126. See also here and here.

## Combining equations

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

## Elemental inverse

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

## Means and operations

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:

## Curves for space and time, continued

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

## Curves for space and time

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

## Interpretation of math and science

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