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Metaphysics of natural science

This is the latest post in a series on science and metaphysics; the previous post is here. The one and only metaphysical postulate of natural science is this: Everything has a fixed nature. This postulate allows the study of classes or kinds or types of things with a common fixed nature. For example, it allows […]

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Harmonic sum of vectors

This post is a slight modification of section 2.0 “The Parallel Sum of Vectors” from W. N. Anderson & G. E. Trapp (1987) “The harmonic and geometric mean of vectors”, Linear and Multilinear Algebra, 22:2, 199-210. We will consider vectors in a real N dimensional inner product space, although some of the results given herein

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Rates of change

The difference quotient is the average rate of change of a function between two points: The instantaneous rate of change is the limit of the difference quotient as t1 and t0 approach each other, which is the derivative of f(t) at that point, denoted by f′(t). Derivatives are added by arithmetic addition, i.e., if f(t)

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Arithmetic of rates

Euclid wrote that “A ratio is a relation in respect of size between two magnitudes of the same kind.” Magnitudes of different kinds were not considered until Galileo. A ratio between magnitudes a and b is expressed as either a : b or b : a. A quotient is the result of dividing a dividend

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Two kinds of vector rates

This post builds on the previous one here. Vector rates rates of change are of two kinds. An ordinary rate for the vector change of f relative to a unit of x is defined as: The reciprocal vector rate is the vector reciprocal of an ordinary rate with a vector change of g relative to

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Two kinds of rates

Rates of change are of two kinds. An ordinary rate for the change of f relative to a unit of x is defined as: The reciprocal rate is the reciprocal of an ordinary rate with a change of g relative to a unit of x is defined as: An ordinary rate has its independent variable

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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

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Speed, pace and inverses

Legend: x-axis coordinate, x; length interval, Δx; independent length interval, x; t-axis coordinate, t; time interval, Δt; independent time interval, t. For scalars: Rates Speed/Pace Symbols Space Time Ordinary (Time) Speed Δx(t)/t measure given Inverse converse Inverse pace (space speed) x/Δt(x) given measure Converse (Space) Pace Δt(x)/x given measure Inverse Inverse speed (time pace) t/Δx(t)

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Duality of subject and object

This post reflects a previous one here. Color (or colour) is both subjective and objective. Objectively, the rays of color light from a glass prism are different wavelengths (or frequencies) of light. The colors we see are those that reflect from objects; the others are absorbed. Colors are additive. Primary colors are red, green, blue;

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Instantaneous speed and inverse

This post relates to the previous one here. Speed is the time rate of distance traversed. Pace is the space rate of elapsed time. The (time) speed of a body is the distance traversed per unit of independent time without regard to direction, Δx/t. The instantaneous speed is the speed at a point in space

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