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Tag Archives: Mathematics

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.

Harmonic arithmetic

This post follows up on harmonic addition mentioned in the previous post here.

Harmonic arithmetic is an inverse arithmetic. 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.

Harmonic addition was defined as a power operation:

Simple harmonic addition is thus defined as:

x\oplus y=\left ( \frac{1}{x}+\frac{1}{y} \right )^{-1} = \frac{xy}{x+y}.

The harmonic additive unit is infinity instead of zero. Harmonic adding one to infinity equals one. Incrementing one leads to one-half, and in general each increment leads to a smaller number:

x\oplus 1=\left ( \frac{1}{x}+1 \right )^{-1} = \frac{x}{x+1}.

Harmonic subtraction is defined as:

x\ominus y =x \oplus (-y) = \left ( \frac{1}{x}-\frac{1}{y} \right )^{-1} = \frac{xy}{y-x}.

Note the reverse of x and y in the denominator. Harmonic multiplication is defined from multiple harmonic additions as:

x \odot y = \frac{x}{y}

which is surprisingly non-commutative. Harmonic division is defined then as:

x \oslash y=xy

which is surprisingly commutative. Harmonic exponentiation is defined from multiple harmonic multiplications as:

x\, (\wedge )\, y=x^{^{1-y}}.

The harmonic square is the inverse:

Regular and harmonic arithmetic are isomorphic with ordinary addition exchanged with harmonic multiplication and ordinary multiplication exchanged with harmonic division.

Harmonic arithmetic is like counting down from infinity, in which an increment of one reduces the amount slightly.

Means and operations

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

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

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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.

Laws of form

The remarkable book Laws of Form by George Spencer-Brown was published in 1969 and is almost forgotten today. The best expositors have been William Bricken with his boundary mathematics, Louis Kauffman with his knot theory, and Francisco Varela with his work on self-reference. Otherwise it has become something of an underground classic but otherwise forgotten. There are several reasons for the latter, including the exaggerated claims of the author and some enthusiasts. That said, I think it’s worth rehabilitating the Laws of Form (LoF) and rightly discerning its significance.

LoF is a work on diagrammatic reasoning in the tradition of Leibniz and CS Peirce. It is a calculus, complete with arithmetic and algebra, based on the act of making and indicating a distinction. Thus it is a work of mathematical realism, which begins to explain why it is not of interest to anti-realists. Its greatest accomplishment is the unified treatment of injunction and indication, of implication and negation via a single symbol, called a cross.

Here are the arithmetic axioms of the calculus of indications:

lof2

That’s it. The rotated “L” is the cross symbol. A cross next to another cross is equal to one cross; this is the Law of Calling. A cross inside another cross is equal to blank, that is, as if no cross had been written. This is the Law of Crossing, hence the name of the symbol, Cross.

This is a two-dimensional calculus, which gives it advantages that one-dimensional notation does not have. It also makes it hard to display typographically. The best alternative is simply to use parentheses or brackets:

( ) ( ) = ( ) and (( )) =   .

These arithmetic axioms can be used to derive two algebraic axioms:

((A) (B)) C = ((A C) (B C)) and ((A) A) =   .

From this a complete calculus can be constructed. It is isomorphic to Boolean algebra and other functionally-complete binary calculi, which is another reason LoF hasn’t stirred a lot of interest.

Things get more interesting as we review where this calculus comes from. Again this exhibits its realism; the standard approach for mathematics and symbolic logic is to begin with algebraic axioms or postulates without reference to any model or reality.

Let’s begin with a blank surface, say a blank page of paper. Now draw a distinction on this surface; that is, draw a closed curve or divide the page into two parts. Notice what has happened: part of the paper is distinguished from the rest of the paper by being to one side of the curve, say the inside. The curve separates the other side from the inside; call it the outside. But the original piece of paper is still there. We can still consider the whole piece of paper.

This process is symbolized by LoF as follows: what is outside the cross (or parentheses) can be seen inside the cross (or parentheses) if we change perspectives to the whole page. This is symbolized in a theorem:

(A) B = (A B) B.

So the distinction that is drawn is not between two contraries but within one space, represented by the whole page. It also shows the distinction can be undermined. This has been exploited to represent self-reference.

Much more could be said about LoF but that’s it for now.

Actual infinity

Before the 19th century it was commonly understood that only God (or perhaps the “gods”) were actually infinite.  If one spoke about the actual infinite, one was doing theology.  In mathematics infinity was considered a manner of speaking, which was clarified in the early 19th century with the careful definition of limits.

In the late 19th century Cantor’s infinite sets were seen as a challenge to this because they treated infinite sets as complete entities.  But there is still no need to consider this essentially different from the relative manner infinity is treated elsewhere in mathematics.

The idea that the universe may be eternal is ancient but there has never been a comprehensive treatment of what this would mean.  Theologians are still struggling to understand in what sense time could exist before the creation.  Nonmetric time seems to be the best solution.

The burden is on anyone who speaks of a physical infinite to explain in detail what they mean.  Otherwise, they’re just throwing words around.

August 2014

Means and Extremes

Means and extremes in classical mathematics have to do with proportions.

If A is to B as C is to D, we write A : B :: C : D.  This is ordered so that A is greater than or equal to B and C is greater than or equal to D.  A and D are called the extremes; B and C are called the means.

By elementary algebra the product of the extremes equals the product of the means:

A x D = B x C.

If B = C, then B is the mean proportional or geometric mean of A and D.  In that case B is the positive square root of A x D.

This provides a basic principle for centrism: the means are between the extremes in a principled manner.