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Tag Archives: Mathematics

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: harmonic 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 and harmonic algebra because of its relation to the harmonic mean.

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

For harmonic algebra: see here.

Harmonic algebra is the multiplicative inverse of ordinary algebra. There is a sense in which harmonic algebra counts down rather than up. Zero in harmonic numbers is like infinity in ordinary numbers. Larger harmonic numbers correspond to smaller ordinary numbers. Smaller harmonic 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 addition mentioned in a previous post here.

Harmonic algebra is an inverse algebra. It 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.

Regular and harmonic algebra are isomorphic with ordinary addition exchanged with harmonic multiplication and ordinary multiplication exchanged with harmonic division. Harmonic algebra is like counting down from infinity, in which an increment of one reduces the amount slightly.

Harmonic 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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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 x = b, where a ≠ b. If we multiply these equations together, we get

x² = ab,

in which the solution is x = √ab, so that x is the geometric mean of a and b.

If we make the equations homogeneous first, then multiply them together, we get: 0 = x − a and 0 = x − b, so that

0 = (x − a) (x − b) = x² − (a + b) x + ab.

The solution of the combined equation is either x = a or x = b. To combine equations with AND, multiply homogeneous equations together.

Another way to combine equations is to add them together. In this case, we get

x + x = 2x = a + b, or x = (a + b)/2,

so that x is the arithmetic mean of a and b. Homogeneous equations added produce the same result: 0 = x − a + x − b = 2x − (a + b), so that x = (a + b)/2.

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 subsets or combined into supersets. Classes may be divided into subclasses or combined into superclasses. Distinctions may be between classes or within classes. Members may be within sets or without sets.

One might say that a class is just a set of distinctions, or one might also say that a set is just a class of members. But that would blur their differences.

Sets assume one knows members and is trying to combine them into the right sets. Classes assume one knows distinctions and is trying to divide them into the right classes. Aristotle assumed that classes could be known by defining them with the right distinctions. Empiricists assume that sets can be known by defining them with the right members.

Realists begin with classes. A tree is defined by its distinctions. Upon inductive investigation, trees may be grouped into types of tree. Upon deductive investigation, types of trees have certain properties.

Induction proceeds from classes to sets. Deduction proceeds from sets to classes. Sets and classes are like inverses of one another.

Both sets and classes are axiomatized by Boolean algebra with the axioms of identity, complementation, associativity, commutativity, and distributivity.

Means and operations

The power means are defined for a set of real numbers, a1, a2, …, an:

power mean

The best-known of these are the arithmetic, geometric, and harmonic means, with p = 1, p = –1, and  p → 0:

arithmetic mean

harmonic mean

geometric mean

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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 x = x(t); y = y(t); z = z(t); where the coordinates of the point (x, y, z) of the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t), y(t), and z(t) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve1

Displacement Δr connecting points A and B on parametric curve r(t).

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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 the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t) and y(t) are assumed to be continuous with a sufficient number of continuous derivatives. The parametric representation of space curves is: x = x(t); y = y(t); z = z(t); t1tt2.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve

Displacement Δr connecting points P and Q on parametric curve r(t).

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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 and a point not lying on it such that …” Or scientists will say, “Occam’s razor is a principle of science” as if they can assert principles ex nihilo. How should these creations be interpreted?

Mathematicians write as if infinity were next door: “As x approaches infinity …” Scientists write as if the entire universe were in view: “The universal theory of gravitation states …” But universal theories turn out to have limitations. And the One who is actually infinite never appears in mathematics. So what do these locutions really mean?

Before the discovery or invention (which one?) of non-Euclidean geometry and its application to physics, it was common for people to think that Euclidean geometry described the space we live in. It is said that most mathematicians are Platonists, and believe that mathematical entities literally exist. Since the 19th century, the literal interpretation of science has been in ascendancy, in which nature is all that exists (i.e., scientific naturalism, see here).

Some say modern science was an unintended consequence of the Reformation’s rejection of levels of meaning in the Bible, which led to a more literal interpretation of God’s other book, the book of nature. The conclusion from all this is that mathematics and science need to be interpreted as much as religious or philosophical writings. What’s your interpretation?

The story of nothing

Mathematics is the study of nothing. We make something out of nothing, acting the creator in a world of nothing. Here’s the story:

In the beginning is nothing. Not totally nothing because we’re there. But a blank page, a clear slate, a tabula rasa.

We draw a distinction, a part of nothing. The indistinct blankness of nothing gains a something. We indicate the something. We indicate the original nothing. We develop a logic of nothing.

We draw a place of nothing, a point. We draw a line of points, then a plane, and a solid. We select an original nothing, an origin. We develop a geometry of nothing.

We draw a number of nothing, a zero. We add it, subtract it, and multiply it. We raise numbers to its power. We develop an arithmetic and algebra of nothing.

We reduce a number to nothing, an infinitesimal. We define a function of it. We take its tangent, its sum, and its mean. We develop a calculus of nothing.

We take the set of nothing, the null set. We intersect it, union it, complement it. We take the power set of it. We find its cardinality. We develop a set theory of nothing.

In the end we have nothing, nothing but mathematics.