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Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Metric postulates for time geometry

Geometry was developed by the ancient Greeks in the language of length, but it is an abstraction that may be applied to anything that conforms to its definitions and axioms. Here we apply it to duration. We will use Brossard’s “Metric Postulates for Space Geometry” [American Mathematical Monthly, Vol. 74, No. 7, Aug.-Sep., 1967, pp. 777-788], which generalizes the plane metric geometry of Birkhoff to three dimensions. First, Brossard:

Primitive notions. Points are abstract undefined objects. Primitive terms are: point, distance, line, ray or half-line, half-bundle of rays, and angular measure. The set of all points will be denoted by the letter S and some subsets of S are called lines. The plane as well as the three-dimensional space shall not be taken as primitive terms but will be constructed.

Axioms on points, lines, and distance. The axioms on the points and on the lines are:

E1. There exist at least two points in S.

E2. A line contains at least two points.

E3. Through two distinct points there is one and only one line.

E4. There exist points not all on the same line.

A set of points is said to be collinear if this set is a subset of a line. Two sets are collinear if the union of these sets is collinear. The axioms on distance are:

D1. If A and B are points, then d(AB) is a nonnegative real number.

D2. For points A and B, d(AB) = 0 if and only if A = B.

D3. If A and B are points, then d(AB) = d(BA).

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Length and duration in time and space

Length and duration are defined by their measurement. Length is that which is measured by a rigid rod or its equivalent.

Length is “extension in space” (Dict. of Physics).

Duration is that which is measured by a clock or its equivalent.

Duration is “time measured by a clock or comparable mechanism” (Dict. of Physics).

Time and space and defined as concepts. Time is a local uniform motion that indicates the length or duration of local events by convention.

Time is “The dimension of the physical universe which, at a given place, orders the sequence of events.” (Dict. of Physics).

Space is a three-dimensional expanse whose extent is measured by length and duration.

“Space, a boundless, three-dimensional extent in which objects and events occur and have relative position and direction.” (Encyclopedia Britannica)

The difference between length and duration is the relation between the observed and the observed, the measurand. The question is, which one is the reference quantity and which one is the measured quantity. If the observer is (or has) the reference, and the observed is the measurand, then the value measured is length. If the observer is the measurand, and the observed is (or has) the reference, then the value measured is duration.

Time is derived from (1) three orthogonal uniform motions, or (2) one orthogonal uniform motion in which the three orthogonal uniform motions are components, or (3) the distance from the origin of the one orthogonal uniform motion in (2). In both (1) and (2) time is derived from three-dimensional, whereas in (3) time is one-dimensional or a scalar.

Space is derived from (1) three orthogonal uniform motions, or (2) one orthogonal uniform motion in which the three orthogonal uniform motions are components, or (3) the distance from the origin of the one orthogonal uniform motion in (2). In both (1) and (2) space is derived from three-dimensional, whereas in (3) time is one-dimensional or a scalar.

The difference between time and space is the relation between the observer and observed, or reference and measurand. If the reference motion is at rest in the observer’s frame, then what is measured is the length of motion of an observed body. If the reference motion is at rest in the observed frame, then what is measured is the duration of motion of an observed body.

Two forms of time and space

The SI metric base unit of length is the metre. The SI base unit of duration is the second. Other units of length are the kilometre, millimetre, inch, foot, mile, etc. Other units of duration are the minute, hour, day (sidereal, solar, etc.), year, etc.

Duration time is also called time. Length time may be called stance. Length space is also called space. Duration space may be called chronotopy.

Time is an independent variable measured in units of length or duration. Space is a dependent geometry of positions measured in units of length or duration.

Time is indicated by a reference uniform motion, which in units of duration is called a clock and in units of length is called a metreloge. The reference rate of uniform motion is a convention that enables one to convert length to duration and vice versa. By appropriate choice of  units, this rate may equal one, in which case length and duration may be interchanged in calculations.

In uniform motion the spaces covered are proportional to the time elapsed. Independent space and time are two sides of the same coin.

The independence of time allows it to be selected independent of other variables, as in setting an appointment, a tempo for music, or the duration of a game. Its independence and its uniform motion allows time to change by some linear rule.

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

For this reason we also, from the day we heard about you, have not ceased praying for you and asking God to fill you with the knowledge of his will in all spiritual wisdom and understanding, 10 so that you may live worthily of the Lord and please him in all respects—bearing fruit in every good deed, growing in the knowledge of God, 11 being strengthened with all power according to his glorious might for the display of all patience and steadfastness, joyfully 12 giving thanks to the Father who has qualified you to share in the saints’ inheritance in the light. 13 He delivered us from the power of darkness and transferred us to the kingdom of the Son he loves, 14 in whom we have redemption, the forgiveness of sins. Col 1:9-14

11 And in this regard we pray for you always, that our God will make you worthy of his calling and fulfill by his power your every desire for goodness and every work of faith, 12 that the name of our Lord Jesus may be glorified in you, and you in him, according to the grace of our God and the Lord Jesus Christ. 2 Th 1:11-12

I always thank my God as I remember you in my prayers, because I hear of your faith in the Lord Jesus and your love for all the saints. I pray that the faith you share with us may deepen your understanding of every blessing that belongs to you in Christ. I have had great joy and encouragement because of your love, for the hearts of the saints have been refreshed through you, brother. Philemon 1:4-7

Dear friend, I pray that all may go well with you and that you may be in good health, just as it is well with your soul. 3 Jn 1:2

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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.

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.