# mathematics

## From ratios to quotients

Ratios and proportions are symmetric. A:B ≡ B:A and A:B :: C:D iff C:D :: A:B. But when ratios are converted to quotients or fractions, they are no longer symmetric. There must be a convention as to which is the denominator and which is the numerator. In an ordinary fraction or quotient or rate the …

## Reciprocal derivative

The reciprocal difference quotient is or The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as x1 and x2 approach x: or as h approaches zero: The reciprocal derivative of a linear function, f(x) = ax + b, is The reciprocal derivative of a power function, …

## Harmonic vector realm

This post expands on Harmonic Algebra posted here. A vector space, or better a vector realm, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication. The operation + (vector addition) must satisfy the following conditions: Closure: If u …

## Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Reciprocal arithmetic. The vector inverse x−1 is defined as with positive norm. For a non-zero scalar k, The reciprocal (or harmonic or parallel) sum is symbolized in various ways, but I prefer a “boxplus” to maintain its relation with addition. The …

## Galileo’s method

Extracts about Galileo from Scientific Method: An historical and philosophical introduction by Barry Gower (Routledge, 1997): Galileo took great pains to ensure that his readers would be persuaded that his conclusions were correct. p. 23 The science of motion was then understood to be a study of the causes of motion, and to be, like …

## Mathematics and beauty

Extracts from Scientific Method in Ptolemy’s Harmonics by Andrew Barker (Cambridge University Press 2004): Mathematics is not the study of all quantities and all quantitative relations indiscriminately. It is the science of beauty. Its task, at the theoretical level, is to interpret, in terms of ‘rationally’ or mathematically intelligible form, the features, movements or states …

## Vectors and functions for space and time

A pdf version of this post is here. The time velocity of an n-dimensional vector variable or vector-valued function Δx per unit of an independent scalar variable t equals Similarly, the space lenticity with Δt and Δx, respectively: The rate of a scalar variable Δx per unit of an independent n-dimensional vector variable or vector-valued …

## 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. (1) Set …