iSoul In the beginning is reality.

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.

Augsburg Confession briefly

The Holy Bible is the final authority in all matters of faith and practice for Reformation Christians. The Augsburg Confession is the doctrinal confession of faith adopted by the Lutheran Church. It is part of the Book of Concord, which includes the three ancient ecumenical creeds: the Apostles’ Creed, the Nicene Creed, and the Athanasian Creed. It also contains the Apology of the Augsburg Confession, the Smalcald Articles, the Treatise on the Power and Primacy of the Pope, Luther’s Small and Large Catechisms, and the Formula of Concord. (reference)

Martin Luther wrote the Catechisms and the Smalcald Articles. Phillip Melanchthon wrote the Augsburg Confession, its Apology, and the Treatise. So, Melanchthon wrote more of the Lutheran doctrines than Martin Luther. Note that the other writings of Luther have no official status among Lutherans, although most of them make for sound reading.

The Augsburg Confession was written in a particular historical context in which the Lutheran movement attempted to explain and justify itself to the religious and civil authorities of the day, notably Emperor Charles V of the Holy Roman Empire, at the Diet of Augsburg on June 25, 1530. Although reconciliation did not happen, the Augsburg Confession provided the primary confession of the Lutheran movement.

The twenty-eight articles in the Augsburg Confession consist of twenty-one statements of doctrine and seven declarations about abuses, and demands for reforms. Although Melanchthon and his associates compiled the confession, Luther approved it as an accurate account of his doctrine.

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Motion as a parade

Have you ever seen a parade? Have you ever been in a parade?

parade

A parade is basically a linear (1D) motion. It begins at a point in space and time and ends at a point in space and time. It is planned to progress in a certain order.

The view from the side sees the parade pass by. The parade participants change in time but the location does not. The parade represents the diachronic perspective of motion in time.

The bird’s-eye view from above (as of a camera on a drone) sees the parade as a whole. The time keeps changing but the general location does not. The parade represents the synchronic perspective of motion in time.

The view from a parade participant sees the streetscape pass by. The parade watchers change in space but the chronation does not. The parade represents the diachoric perspective of motion in space.

The view from the plan for the parade sees the parade as a whole. The stance keeps changing but the chronology does not. The parade represents the synchoric perspective of motion in space.

Prayers and benedictions

Prayers from the New Testament (NET Bible)

Our Father in heaven, may your name be honored,
10 may your kingdom come,
may your will be done on earth as it is in heaven.
11 Give us today our daily bread,
12 and forgive us our debts, as we ourselves have forgiven our debtors.
13 And do not lead us into temptation, but deliver us from the evil one. Mt 6:9-13

22 Jesus said to them, “Have faith in God. 23 I tell you the truth, if someone says to this mountain, ‘Be lifted up and thrown into the sea,’ and does not doubt in his heart but believes that what he says will happen, it will be done for him. 24 For this reason I tell you, whatever you pray and ask for, believe that you have received it, and it will be yours. 25 Whenever you stand praying, if you have anything against anyone, forgive him, so that your Father in heaven will also forgive you your sins.” Mk 11:22-25

17 I pray that the God of our Lord Jesus Christ, the Father of glory, may give you spiritual wisdom and revelation in your growing knowledge of him, 18 —since the eyes of your heart have been enlightened—so that you may know what is the hope of his calling, what is the wealth of his glorious inheritance in the saints, 19 and what is the incomparable greatness of his power toward us who believe, as displayed in the exercise of his immense strength. 20 This power he exercised in Christ when he raised him from the dead and seated him at his right hand in the heavenly realms 21 far above every rule and authority and power and dominion and every name that is named, not only in this age but also in the one to come. 22 And God put all things under Christ’s feet, and he gave him to the church as head over all things. 23 Now the church is his body, the fullness of him who fills all in all. Eph 1:17-23

I kneel before the Father, 15 from whom every family in heaven and on the earth is named. 16 I pray that according to the wealth of his glory he may grant you to be strengthened with power through his Spirit in the inner person, 17 that Christ may dwell in your hearts through faith, so that, because you have been rooted and grounded in love, 18 you may be able to comprehend with all the saints what is the breadth and length and height and depth, 19 and thus to know the love of Christ that surpasses knowledge, so that you may be filled up to all the fullness of God. Eph 3:14b-19

And I pray this, that your love may abound even more and more in knowledge and every kind of insight 10 so that you can decide what is best, and thus be sincere and blameless for the day of Christ, 11 filled with the fruit of righteousness that comes through Jesus Christ to the glory and praise of God. Phil 1:9-11

Do not be anxious about anything. Instead, in every situation, through prayer and petition with thanksgiving, tell your requests to God. And the peace of God that surpasses all understanding will guard your hearts and minds in Christ Jesus. Phil 4:6-7

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Independent uniform motion

This continues posts here and here.

The extent of a motion is measured by a reference motion, just as a length is measured by a reference length. The reference motion used to measure other motions is a uniform motion. Galileo’s definition of uniform motion is the following:

By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal. [Galileo’s Two New Sciences, Third Day]

Because equality is a symmetric relation, this could also be expressed as follows:

A steady or uniform motion is one in which the travel times of the moving particle during any equal intervals of space, are themselves equal.

Another way of saying this is that for a uniform motion the intervals of space and the corresponding intervals of time are proportional. That is, a uniform rate of motion is constant.

There are two measures of the extent of a motion, length and duration. Applied to the reference motion, these two measures produce two scalars, a measure of length called stance, and a measure of duration called time. Given a reference starting point, the length since the start point is called the stance, and the duration since the start instant is called the time. Because of the proportionality of uniform motion, given knowledge of stance or time along with the uniform rate, one may deduce the other measure.

The reference motion must also be an independent motion, not dependent on other motions, and it must continue indefinitely, so that any other motion would be at some point simultaneous or simulstanceous with the reference motion. Because of this, any motion may be a function of the reference motion.

This independent, uniform reference motion is commonly represented by a clock, which registers uniform motion continually. Even if the clock’s motion is a uniform circular motion, it represents a uniform linear motion as the numbers increase linearly. The reference motion may equally well be a metreloge, which is a uniform motion that registers length continually. As a clock may be a uniform angular motion whose angles register durations, a metreloge may be a uniform angular motion whose arcs register lengths.

Measures of the target motion may then be considered as a function of one of the reference measures, which acts as a parameter of the target motion. Parametric differential equations and geometry may then be used to represent the course of a motion.

Vital records of society

Vital records are the official documents of the birth, marriage, divorce, and death of members of a society. They are currently kept by an office of state government, but before the 19th or 20th century these documents were under the control of religious institutions, i.e., churches. For example, the state of Nebraska has birth and death records only since 1904, and marriage and divorce records only since 1909. Since marriage required a license, marriage certificates are often available since the 17th or 18th century in the older states.

State and national governments, however, have done poorly at this basic task. They have not recognized the personhood of children in the womb, they have made divorce “no-fault” and easy to get, and they have redefined marriage to eliminate the sex requirement. In some states, lawful death has been redefined to include assisted suicide.

With this in mind, it is time for religious institutions to take back the keeping of vital records. While legal requirements must still be met, religious requirements are different and need their own certificates and vital records. This is an opportunity to show the importance of recognizing the humanity of children in the womb by issuing a Life Certificate. It’s an opportunity to show the necessity for marriage to be kept for opposite sex couples only, and not same-sex or changed gender couples. It is an opportunity to set the requirements for divorce. And accurate death records would discourage the pretexts of assisted suicide.

Although religious institutions bear responsibility for this task, the operation may well be contracted out to a business. Because people move between the states and change church affiliation, it is best that a national database be established. This national database of vital records should be maintained with the highest integrity to safeguard how society treats the unborn, the institution of marriage, and the dying.

Simultaneity and simulstanceity

Max Jammer’s book Concepts of Simultaneity (Johns Hopkins UP, 2006) describes the significance, meaning, and history of simultaneity in physics. Here are a few excerpts from his Introduction:

… Einstein himself once admitted: “By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory.” p.3

That not only temporal but also spatial measurements depend on the notion of simultaneity follows from the simple fact that “the length of a moving line-segment is the distance between simultaneous positions of its endpoints,” as Hans Reichenbach … convincingly demonstrated. Having shown that “space measurements are reducible to time measurements” he concluded that “time is therefore logically prior to space.” p. 4-5

P. F. Browne rightly pointed out that all relativistic effects are ultimately “direct consequences of the relativity of simultaneity.” p.5

One might give the dual to the second statement as: That not only spatial but also temporal measurements depend on the notion of simulstanceity follows from the simple fact that “the duration of a moving line-segment is the time interval between simulstanceous chronations of its endpoints. Space is therefore logically prior to time.

In the next chapter, Terminological Preliminaries, Jammer clarifies the relevant concepts. It is ironic that he gives an early example of the metonym “of spatial terms to denote temporal relations that is frequently encountered both in ancient and in modern languages.” (p.9) Space has priority in language.

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Independent variable dimension

This continues the series of posts, see here.

Let’s begin with Galileo’s figure for uniform motion and uniform accelerated motion:

Falling projectile

Let the horizontal uniform motion be situated in a 2D x, y coordinate system:

2D uniform motion

The dependent uniform acceleration moves in an additional dimension, z, and so has 3D coordinates.

If the independent uniform motion is measured by time, then time has 2D coordinates. The coordinates are proportional to one another, and so may just as well be replaced by a scalar of the signed magnitude. The same can be done if the uniform motion has 3D coordinates. The scalar time is proportional to the common measure of time, with the appropriate rate of motion, it is the same as scalar time.

One can say the same if the independent uniform motion is measured by length, then length has 2D (or 3D) coordinates. This is nothing new, but because of the uniform motion, the coordinate lengths are proportional to one another, and so may just as well be replaced by a single scalar of the signed magnitude. In this way, length becomes a scalar, called here the stance.

Motion ordered by time or stance

This continues posts about Galileo such as here. Look again at this figure from Galileo:

Falling projectile

The horizontal line represents the independent variable, which is the horizontal component of a projectile with an initial speed that falls with uniform acceleration. The independent variable need not be horizontal, so let us represent it vertically with the vertical component of motion:

 

Four parallel lines

The first two lines represent the independent uniform motion but are measured differently, (1) by stance and (2) by time. The second and third lines show the correspondence between the independent variable (time) and the dependent variable (length). The first and fourth lines have the same relation: stance is the independent variable and duration is the dependent variable. Lines 1 and 2 are proportional, as are lines 3 and 4. As the units change, the functional relationship does not: the dependent variable is proportional to a square of the independent variable.

Time is the measure of duration of a uniform motion that is the reference motion, which means it is the independent variable. Similarly, stance is the measure of length of a uniform motion that is the reference motion, in which case it is the independent variable. The dependent variable for stance is duration in three dimensions, whereas for time the dependent variable is length in three dimensions.

Timeframes of reference

A kinematic frame of reference is a mathematical method to determine the position of points in abstract 3D space and scalar time. An inertial frame of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter is often envisioned as three mutually-perpendicular rigid rods attached at a common point, or a lattice of such rigid rods. In addition, there is envisioned a clock at every node of the lattice, which are all synchronized, which requires a method to synchronize them. The common point is called the origin point.

Such a frame of reference assigns coordinates in 3D space and 1D time to every event.

A lattice in all directions

A kinematic timeframe of reference is a mathematical method to determine the position of points in abstract scalar space and 3D time. An inertial timeframe of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter may be envisioned as three mutually-perpendicular rigid monorails attached at a common point. More fully, a timeframe of reference is a system of orthogonal rigid monorails with a regular succession of small, virtually frictionless monorail vehicles in uniform motion (think mag-lev). Such monorails record their location at every node. The start of monorails leaving the common point is the origin event.

Such a timeframe of reference assigns coordinates in 3D space and 3D time to every event.

monorails