iSoul In the beginning is reality.

Category Archives: Science

Science particularly as related to creation and the creation-evolution controversy

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; secondary colors are cyan, magenta, yellow.

But wavelengths go beyond the visible spectrum. Color is a visual phenomenon so human perceptions are what makes the different colors.

People perceive different colors differently. This is part cultural, part individual. Color symbolism varies over the world. See here. Pigment colors are subtractive. See here. Primary colors are cyan, magenta, yellow; secondary colors are red, green, blue.

Color is both subjective and objective. It has a subjective side and an objective side. There is a color duality.

The arts and the sciences form a duality. The arts are (ultimately) subjective. The sciences are (ultimately) objective. Style and taste are important in the arts but not in the sciences. Standards and measurements are important in the sciences but not in the arts.

Communication is a duality of subjectivity and objectivity. A subject expresses themselves but only subjectively unless they attend to the meanings of words and sentences so that others can objectively understand.

The arts have forms and expectations to facilitate communication while emphasizing the subjective. The sciences allow flexibility for subjects to express the objective content of their work.

History is an art because it expresses a narrative that emphasizes some people and events over other people and events. Science avoids giving undo emphasis to anything other than the elements of a science. Science and history are a duality.

Dualities are not dualisms. A dualism divides reality into two separate classes. Dualities distinguish two but do not see them as conflicting. Dualities combine two into one.

Reciprocal derivative, part 1

The reciprocal difference quotient is

\frac{\Delta x}{\Delta f(x)}=\frac{h}{f(x+h)-f(x)}

The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as h approaches zero:

\grave{}f(x)=\lim_{h\rightarrow 0}\frac{\Delta x}{\Delta f(x)}= \lim_{h\rightarrow 0}\frac{h}{f(x+h)-f(x)}

An alternate limit form of the reciprocal derivative definition of f(a) is

\grave{}f(a)=\lim_{x\rightarrow a}= \frac{x-a}{f(x)-f(a)}

The reciprocal derivative of a linear function, f(x) = ax + b, is

\grave{}f(x)=\lim_{h\rightarrow 0}\frac{h}{f(x+h)-f(x)}= \lim_{h\rightarrow 0}\frac{h}{a(x+h)+b-ax-b}= \lim_{h\rightarrow 0}\frac{h}{ah}=\frac{1}{a}

The reciprocal derivative of a power function, f(x) = xn, is

\grave{}f(a)=\lim_{x\rightarrow a}\frac{x-a}{f(x)-f(a)}= \lim_{x\rightarrow a}\frac{x-a}{x^{n}-a^{n}}= \frac{1}{na^{n-1}}=\frac{1}{n}a^{1-n}

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Adding and averaging rates

A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged.

Addition

Rates with the same units in their denominator are added using ordinary addition, which will be called arithmetic addition since addition takes place in the numerator. For example, if x1 and x2 are lengths, and t is a given time interval, then the time speed rates v1 and v2 are added by arithmetic addition:

v_{1}+v_{2}=\frac{x_{1}}{t}+\frac{x_{2}}{t} =\frac{x_{1}+x_{2}}{t} = v_{3}

If x1 and x2 are displacements, and t is a given time interval, then the time velocity rates v1 and v2 are added by arithmetic addition:

\mathbf{v}_{1}+\mathbf{v}_{2} =\frac{\mathbf{x}_{1}}{t}+\frac{\mathbf{x}_{2}}{t} =\frac{\mathbf{x}_{1}+\mathbf{x}_{2}}{t} = \mathbf{v}_{3}

where vector addition means addition of dimensions, i.e., parallelogram addition.

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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 and v are any vectors in V, then the sum u + v belongs to V.
(1) Commutative law: For all vectors u and v in V, u + v = v + u
(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
(3) Additive identity: The set V contains an additive identity element, usually denoted by 0, such that for any vector v in V, 0 + vv and v + 0 = v.
(4) Additive inverses: For any vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by −v.

The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If v is any vector in V, and c is any real number, then the product c · v belongs to V.
(5) Distributive law: For all real numbers c and all vectors uv in V, c · (u + v) = c · u + c · v
(6) Distributive law: For all real numbers c, d and all vectors v in V, (c + d) · v = c · v + d · v
(7) Associative law: For all real numbers c, d and all vectors v in V, c · (d · v) = (cd) · v
(8) Multiplicative identity: The set V contains a multiplicative identity element, usually denoted by 1, such that for any vector v in V, 1 · v = v

Consider the non-zero real numbers together with the element ∞ as components of Euclidean vectors, with · as the usual dot product, and vector addition defined as harmonic addition:

\mathbf{u}\oplus \mathbf{v}=\left [ \mathbf{u}^{-1}+\mathbf{v}^{-1} \right ]^{-1}

which is undefined for zero vectors, but has the additive identity ∞ (infinity). It is isomorphic to the vector space (or realm) with 0 as the additive identity.

The independent variable is usually in the denominator but if the independent variable is in the numerator, then the denominator contains a dependent variable. See here for what this looks like. The addition of quotients with a dependent vector in the denominator follows the above.

Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Harmonic algebra.

The vector inverse x−1 is defined as

\mathbf{x}^{-1}=\frac{\mathbf{x}}{\|x\|^{2}}=\frac{\mathbf{x}}{\mathbf{x}\cdot \mathbf{x}}

with positive norm. For a non-zero scalar k,

(k\mathbf{x})^{-1}=\frac{k\mathbf{x}}{\|k\mathbf{x}\|^{2}}=\frac{\mathbf{\mathbf{x}}}{k\|\mathbf{x}\|^{2}}=\frac{1}{k}\mathbf{x}^{-1}

\|\mathbf{x}^{-1}\| = \|\mathbf{x}||^{-1}

The harmonic (or parallel) sum is usually symbolized by a colon (:), but I prefer a circle plus to maintain its relation with addition. The harmonic sum of vectors x and y is defined as

\mathbf{x}\oplus \mathbf{y}=\left [ \mathbf{x}^{-1}+\mathbf{y}^{-1} \right ]^{-1}

if x0, y0, and x + y0; otherwise the sum equals 0.

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With and between independent variables

This post continues the previous post here on independent and dependent variables.

The selection of a physical independent variable (or variables) applies to a context such as an experiment. Within that context all other variables are, at least potentially, dependent on the independent variable(s) selected. Functions with the physical independent variable as a functional independent variable have a common independent variable and form a kind of closed system.

Nevertheless, an independent variable in one context may be the same as a dependent variable in another context. The functions in both of these contexts may be related. That is, there may be relations between the functions of one independent variable in one context and the functions of another independent variable in another context.

~For example, speed (the time speed) is defined as the change in distance per unit of time with time as the independent variable. This appears to be equal to the reciprocal of the pace, that is, the change in distance per unit of time with distance as the independent variable. This is called the space speed (sometimes called the inverse speed).

~However, these two speeds are not the same. One difference is how the speeds are aggregated or averaged: time speed requires arithmetic addition and arithmetic averaging but space speed requires harmonic addition and harmonic averaging. The arithmetic average of speeds v1 and v2 is (v1 + v2)/2. The harmonic average of space speeds v1 and v2 is 2/(1/v1 + 1/v2).

A quasi-variable is a dependent variable in the denominator as if it were an independent variable. If it is a first-order variable, one can take the reciprocal to put it into its proper dependent position, the numerator. Second and higher order variables need to use their inter-functional relation.

One should always use mathematical variables and functions that mirror the physical variables and functions. A physically independent variable should always appear as the argument in a mathematical function. A physically dependent variable should always appear as a function, even if the functional relation is uncertain.

To continue the example above, the space speed should be stated mathematically as dx/dt(x) so that distance is shown as an independent variable and time is shown as a dependent variable. Harmonic addition (see harmonic algebra) is for adding quotients whose denominator is a function of the numerator.

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 any genuine science, a ‘demonstrative’ kind of enquiry. That is to say, experiential knowledge of the facts of motion was superseded by rational knowledge of the causes of those facts, this being accomplished by deductions from fundamental principles, or ‘common notions’, and definitions which were accepted as true. These facts of motion were understood as expressions of common experience rather than as generalisations based upon experiments. This was because the results of the experiments that could be performed were sufficiently uncertain and ambiguous to prevent reliable generalisation; discrepancies between conclusions derived from principles, and experimental results, could be tolerated. The appropriate model of a demonstrative science was Euclidean geometry, where the credibility of a theorem about, say, triangles depends not on how well it fits what we can measure but on its derivability from the basic axioms and definitions of the geometry. p. 23

For Galileo and his contemporaries there was a good reason why demonstration, or proof from first principles, rather than experiment, was required to establish general truths about motion. Any science—scientia—must yield knowledge of what Aristotle had called ‘reasoned facts’, i.e. truths which are both universal and necessary, and such knowledge—philosophical knowledge—can only be arrived at by demonstration. p. 24

there was a long-standing disagreement about the role that mathematics could play in natural philosophy, even though mathematics was able to give certain knowledge. p. 24

In some contexts, notably astronomy and geometry, the more elaborate and intellectually demanding methods of mathematics were often useful and appropriate, but in such contexts it seemed clear that those methods were applicable in so far as what was needed were re-descriptions which could help people formulate accurate predictions. ‘Hypotheses’ which successfully ‘saved the phenomena’, in the sense that they could be used as starting points for derivations of accurate predictions, could meet this need. p. 25

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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 which, when they are present in perceptible phenomena, constitute their aesthetic excellence. p.264

Those of our senses through which we are able to perceive some things as beautiful are therefore involved in an intimate collaboration with mathematical reason. p.264

Since beauty is the manifestation to the senses of that which reason understands as perfect in form, the senses to which beauty is undetectable lack sensitivity, which sight and hearing possess, to those distinctions which, from a rational point of view, are the most significant. p.265

the mathematical sciences have a single objective, the analysis and understanding of the formal basis of beauty p.266

The conception of mathematical science which Ptolemy has presented is that of a capacity that does not merely analyse sets of quantitative relations, but homes in on those that are of special significance, and discovers the principles on which their significance rests. p.268

Composition order

Written compositions organized by temporal order are narratives. Items such as descriptions of people, places, or objects are organized as they occur to the narrator, for example, as the narrator takes apart an object or walks through a building or meets various people. This is a common method of composition but there are others.

Spatial order is another method of composition. Items such as descriptions of people, places, or objects are organized by their physical or spatial positions or relationships, for example, starting at the top and proceeding downward. Explanations of a geopolitical matter might proceed in geographic order.

Travel can be described temporally or spatially. An itinerary is usually arranged temporally but telling about it afterwards might be more interesting if arranged spatially. There are other principles of organization such as climactic order (order of importance) and topical order.

In science the independent variable determines the type of organization. If the independent variable is time, the organization is temporal. If the independent variable is space or distance, the organization is spatial. The stance in spatial organization corresponds to the time in temporal organization.

The values of the independent variable are the index to the order of the composition. If the independent variable is time, then the times indicate the steps in the order. If the independent variable is space or distance, then the stances indicate the steps in the order. Once the step is indicated, the composition may be the same: whether it’s Tuesday, so the tour is in Paris or it’s Paris, so the tour is on Tuesday makes no difference.

History and science balanced

As I’ve noted before (here etc.) history and science have different aims and methods. Mixing them just confuses both of them. There is no genuine “historical science” or “scientific history”. History narrates particulars among unique events. Science theorizes universals among repeatable events. In physics time is homogeneous: an experiment is the same whether conducted today or 100 years in the past or future. That is not true in history. Time is not homogeneous there.

History and science can and should balance one another. The more science expands its universals, the more history can point out particulars that are overlooked or are important in a particular context. The more history focuses on unique particulars, the more science can point out the significance of universals.

The homogeneous and inhomogeneous aspects of time can both be known only by balancing history and science. One could say something similar about all universals and particulars. The universal and particular aspects of reality can both be known only by balancing history and science.