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Tag Archives: Mathematics

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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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,


\|\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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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

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 function Δt equals

The same except with Δt and Δx, respectively:

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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 away. We cannot plumb the ultimate depths within, the deep well of the heart. At the centre of it all is a bottomless pit, the hell of eternal darkness.

concentric spheres

The geometric inverse is with respect to a circle or sphere:

circle with line segmentBy Krishnavedala

P’ is the inverse of P with respect to the circle. The inverse of the centre is the point at infinity.

The order of events in this geometry is their distance from the horizon, not the centre. The return to home is the end of events, the final event. The later the event, the better, since it is closer to the end, to home.

The destination is where we’ve come from and where we return. It is a round trip, a circuit, a cycle of life and change.

What we call the beginning is often the end
And to make an end is to make a beginning.
The end is where we start from. T. S. Eliot, Little Gidding

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 and the dual may be used together.

(2) Number and algebra

The concept of counting and number is as universal as language, though the full definition of number did not occur until the 19th century. Algebra came to the West from India and Arabia in the Middle Ages but its formal definition did not occur until the 19th century. Abstract algebra also began in the 19th century.

The basic rules of algebra are as follows: addition and multiplication are commutative and associative; multiplication distributes over addition; addition and multiplication have identities and inverses with one exception: there is no multiplicative inverse for zero.

An idea of infinity comes from taking the limit of a number as its value approaches zero: ∞ ∼ 1/x as x → 0. Infinity can be partially incorporated via limits.

Dual: reciprocal numbers

An additive dual can be defined by negating every number. A more interesting dual comes from taking the multiplicative dual of every number. This latter case can be called harmonic numbers with harmonic algebra in which operations take place in the denominator.

The reciprocal isomorphism relates every number x in series to its reciprocal dual by H(x) := 1/y. The dual of zero is ∞.

Reciprocal algebra is the multiplicative inverse of ordinary algebra. There is a sense in which harmonic algebra counts down rather than up. Zero in reciprocal numbers is like infinity in ordinary numbers. Larger reciprocal numbers correspond to smaller ordinary numbers. Smaller reciprocal numbers correspond to larger ordinary numbers.

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 theory and logic

A set is defined by its elements or members. Its properties may also be known or specified, but what is essential to a set is its members, not its properties. The notation for “x is an element of set S” is “x ∈ S”. A subset is a set whose members are all within another set: “s is a subset of S” is “s ⊆ S”. If subset s does not (or cannot) equal S, then it is a proper subset: “s ⊂ S”.

The null set (∅) is a unique set defined as having no members. That is paradoxical but not contradictory. A universal set (Ω) is defined as having all members within a particular universe. An unrestricted universal set is not defined because it would lead to contradictions.

The complement of a set (c) is the set of all elements within a particular universe that are not in the set. A union (∪) of sets is the set containing all members of the referenced sets. An intersection (∩) of sets is defined as the set whose members are contained in every referenced set.

Set theory has a well-known correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to union, and conjunction (AND, ∧) corresponds to intersection. Material implication (→) corresponds to “is a subset of”. Contradiction corresponds to the null set, and tautology corresponds to the universal set.

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Harmonic algebra

This revised post follows up on harmonic (or parallel) addition mentioned in a previous post here.

Harmonic algebra 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 new additive unit. That is,

xy := g−1(g(x) + g(y))

xy := g−1(g(x) ∙ g(y))

etc., where g(x) = 1/x with x ≠ 0.

Harmonic algebra is isomorphic to the ordinary numerator operations exchanged with their denominator counterparts. It is like counting down from infinity, in which an increment of one reduces the amount slightly.

Harmonic addition (also known as parallel addition) is defined as a power operation:

O_{-1}(a_{1},a_{2},...,a_{n})\equiv \left ( \sum_{i=1}^{n}a_{i}^{-1} \right )^{-1} ,\,\, a_{i} \neq 0

with the understanding that (1/0) → ∞ and (1/∞) → 0. An intermediate value may be zero but not a final value.

Simple harmonic addition is thus defined as:

x \oplus y = \left ( \frac{1}{x} + \frac{1}{y} \right )^{-1} \equiv \frac{xy}{x+y}; \,\, x, y \neq 0

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