iSoul In the beginning is reality.

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

History and science balanced

As I’ve noted before (here etc.) history and science have different aims and methods. Mixing them just confuses both of them. There is no genuine “historical science” or “scientific history”. History narrates particulars among unique events. Science theorizes universals among repeatable events. In physics time is homogeneous: an experiment is the same whether conducted today or 100 years in the past or future. That is not true in history. Time is not homogeneous there.

History and science can and should balance one another. The more science expands its universals, the more history can point out particulars that are overlooked or are important in a particular context. The more history focuses on unique particulars, the more science can point out the significance of universals.

The homogeneous and inhomogeneous aspects of time can both be known only by balancing history and science. One could say something similar about all universals and particulars. The universal and particular aspects of reality can both be known only by balancing history and science.

A theory of time

The Background: The hands of a clock are in motion relative to the observer’s rest frame yet they display the present time of the rest frame. The motion of the clock hands is identified with the time of the rest frame.

The speed of a body is its distance traversed per unit of time. The inverse of speed is called pace, which is the time of travel per unit of length. The vector version of speed is the displacement of a body per unit of time (velocity), but what is the vector version of pace? Call it lenticity, which would seem to be a kind of displacement in time per unit of length, but that implies there are three dimensions of time. The rest of this article defines time and shows that it has three dimensions, although two of its dimensions are usually latent.

The term time has many different meanings, but it is unavoidable because of the lack of alternatives. The main thing to remember is that time in physics is duration, Δt, an interval or length of time (cf. Newton). Let us call this the distime because it is analogous to distance in space. Since time is homogeneous, in physics it makes no difference what the particular date and time are. It’s the same with space: only the length, Δx, a distance, an interval or length of space that matters, not the particular coordinates.

The Theory: An observer is a device or person capable of making measurements relative to a frame of reference. An inertial observer is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer. An inertial frame is one that is not accelerating. A frame of reference here shall mean an inertial frame.

An observer P measures space coordinates relative to a frame K at rest relative to P. Call frame K a space frame for P. Space (3D space) is the R3 geometry of places and lengths in a space frame. A place point is a point in 3D space. The space origin is a reference place point. The location vector of a place point is the 3D space vector to it from the space origin. The coordinates of place points are called locations relative to the frame K of observer P and are measured in terms of length of space. Let the space axes in K be designated as x1, x2, and x3.

An observer P measures time coordinates relative to a frame L in uniform motion at a standard rate relative to P. Call frame L a time frame for P. Time (3D time) is the R3 geometry of times and durations in a time frame. A time point is a point in 3D time. The time origin is a reference time point. The chronation vector of a time point is the 3D time vector to it from the time origin. The coordinates of time points are called chronations relative to the frame L of observer P and are measured in terms of length of time (duration). Let the time axes of L be designated t1, t2, and t3.

Every observer has a space frame and a time frame, which together form a complete frame of reference. Let the direction of motion of the time frame relative to the space frame be the x1 axis, and let this axis be coincident with the t1 axis. In general, the coordinates of a point in space and time will then be ((x1, x2, x3); (t1, t2, t3)).

Since uniform motion is one-dimensional, only one coordinate from either the space frame or time frame is required as the independent denominator for rates of motion. For rates of velocity, acceleration, etc., only one time coordinate is needed; the other two time coordinates are zero, so time (duration) is a scalar. For rates of lenticity, retardation, etc., only one space coordinate is needed; the other two space coordinates are zero, so space (stance) is a scalar.

If a time frame is moving at velocity vc relative to its associated space frame, then the other time coordinates are zero: ((x1, x2, x3); (t1, 0, 0)). The time coordinate (t1, 0, 0) may be expressed by a scalar; call it t. The result is the space coordinates (x1, x2, x3) and a time scalar, t.

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Abstract and concrete movements

Abstraction in Western culture has increased over time, so much so that Hegel made this the engine of history: his dialectic is a progression from the concrete to the less concrete, the abstract to the more abstract. Certainly, the history of natural science shows this progression. Modern physics is more abstract than classical physics. Every science becomes more abstract over time.

Increased abstraction in society and politics requires larger collections of people. Equality with increased abstraction requires equality within larger groups of people. For example, pan-European equality is less abstract than equality within global equality. Increased abstraction requires loyalty to ever larger groups.

History does seem to progress toward greater abstraction. Tribal cultures gave way to city-states, then to nations, then to globalism. In the U.S., there has been a progression from an English culture to a European culture, to a Euro-Afro-Latin culture, to an increasingly global culture. Those who promote this movement are called “progressives”. Those who resist it or support caution about it are called “conservatives”.

In sub-cultures of the West and in some non-Western societies there are movements in the opposite direction, toward more concreteness. They are often called “regressive”, which assumes a prior progressive movement. They could simply be called “concretive” (or “introgressive”) since they prefer the more concrete to the more abstract.

Those who prefer more concrete or at least a less abstract culture are considered traditional, old-fashioned, or backwards. In order to engage their opponents, traditionalists need to justify their preference for the concrete in more abstract ways, which they may find difficult. But the concrete has its advantages as much as the abstract does.

One danger of greater abstraction is that one loses touch with concrete reality. After all, human beings are concretely embodied. Concrete food, shelter, and much more are necessary for human life. Traditional social and political structures have much experience and stability behind them and so “should not be changed for light and transient causes” (the U.S. Declaration of Independence). And the new global human who ignores the local culture where they happen to be is looking for misunderstanding and worse.

In fact, there is no global, pan-religious, pan-racial, pan-sexual, pan-economic, pan-linguistic culture. Is such a culture even possible? In this world, that is highly doubtful. People are both concrete and abstract, body and spirit.

Concrete and abstract movements both have their place. Cultures will lean more toward one than the other, but both are legitimate.

What is a clock?

What is a clock? it is a device that measures time, but what are the essentials of a clock? I submit these are the essentials of a clock:

(1) A clock requires a uniform motion. Because only the kinematics (not the dynamics) are significant, a uniform rotation is acceptable. But because the result will be represented as a line – a timeline or time axis – a linear uniform motion has a more direct connection with what is measured, so let us take the first essential as a uniform linear motion.

(2) In order for clocks to be measuring alike, it is necessary that there be a standard rate for all clocks. In addition, clocks should have a standardized beginning point, so that clocks are interchangeable.

(3) A clock requires a pointer which indicates the present time on a time scale as it moves at the standard uniform rate. This would be the hands and dial on a common analogue clock. On a linear clock it is a part whose position in motion is interpreted as the present value of time. The pointer and scale are essentials of a clock.

Furthermore, a clock must be interpreted as showing the present time of the observer’s rest frame.

Figure 1

All the essentials of a clock can be represented by a frame in standard uniform motion relative to the observer’s rest frame. In that case, a clock should be definable in terms of frames of reference: one rest frame and one frame in uniform motion relative to the observer’s rest frame, as in the following.

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Space, time, and dimension

The post continues the ones here, here, and here.

There are three dimensions of motion. The extent of motion in each dimension may be measured by either length or time (duration). There are three dimensions of length and three dimensions of time (duration) for a total of six dimensions.

But there is no six-dimensional metric. Why? Because a metric requires all dimensions to have the same units, which requires a ratio to convert one unit into the other unit. The denominator on a ratio is a one-dimensional quantity, which means either the length or time dimensions need to be reduced by two dimensions.

This ratio is a conversion factor that is either a speed, which multiplied by a time equals a length, or a pace, which multiplied by a length equals a time. In general a speed is the ratio Δdr²/|Δdt²| = Δdr²/(Δt1² + Δt2² + Δt3²)1/2, and a pace is the ratio Δdt²/|Δdr²| = Δdt²/(Δx² + Δy² + Δz²)1/2. The denominator is a distance or distime, which is a linear measure of length or time (duration).

The conversion factors required are the speed of light in a vacuum, c, or its inverse, the pace of light in a vacuum, k. The resulting four-dimensional metric is either c²dt² − dx² − dy² − dz² (with time reduced to one dimension) or dt1² − dt2² − dt3² − k²dr² (with space reduced to one dimension).

These metrics are often simplified by taking c = 1 and k = 1 so that symbolically they are the same. Their units are not the same, however.

Each metric may be further reduced by separating space and time, as in classical physics. Then the space metric is |Δdr²| = (Δx² + Δy² + Δz²)1/2 and the time metric is |Δdt²| = (Δt1² + Δt2² + Δt3²)1/2. In the classical (3+1) of three space dimensions and one time dimension, time is replaced by its metric, and in the classical (1+3) of one space dimension and three time dimensions, space is replaced by its metric.

Space and time as frames

An observer is a body capable of use as a measurement apparatus. An inertial observer is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer.

An observer makes measurements relative to a frame of reference. A frame of reference is a physical system relative to which motion and rest may be measured. An inertial frame is a frame in which Newton’s first law holds (a body either remains at rest or moves in uniform motion, unless acted upon by a force). A frame of reference here shall mean an inertial frame.

A rest frame of observer P is a frame at rest relative to P. A motion frame of observer P is a frame in uniform motion relative to P. Each observer has at least one rest frame and at least one motion frame associated with it. An observer’s rest frame is three-dimensional, but their motion frame is effectively one-dimensional, that is, only one dimension is needed.

Space is the geometry of places and lengths in R3. A place point (or placepoint) is a point in space. The space origin is a reference place point in space. The location of a place point is the space vector to it from the space origin. Trime (3D time) is the geometry of times and durations in R3. A time point (or timepoint) is a point in trime. The time origin is a reference time point in trime. The chronation of a time point is the trime vector to it from the time origin.

A frame of reference is unmarked if there are no units specified for its coordinates. A frame of reference is marked by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A space frame of observer P is a rest frame of P that is marked with units of length. A time frame of observer P is a motion frame of P that is marked with units of duration.

Speed, velocity, and acceleration require an independent motion frame. Pace, lenticity, and retardation require an independent rest frame. These independent frames are standardized as clocks or odologes so they are the same for all observers.

Let there be a frame K1 with axes a1, a2, and a3, that is a rest frame of observer P1, and let there be a motion frame K2 with axes 1, 2, and 3, that is a motion frame of P1 along the coincident a1-a´1 axis. See Figure 1.

Two frames

Figure 1

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Vectors and Functions in 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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Galilean decompositions of the Lorentz transformation

The background for this post is here.

For space with time (3+1):

The gamma transformation (matrix Γ) expresses the time dilation of clocks and length contraction of rods with a relative speed:

\begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} = \begin{pmatrix} \gamma t \\ x/\gamma \end{pmatrix} = \begin{pmatrix} t' \\ x' \end{pmatrix}

Use vector (t  x)T. The gamma transformation is conjugate to the Lorentz boost (matrix Λ) by the Galilean transformations (G, GT), i.e., GTΓG = Λ:

\begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{pmatrix}

or

\begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\gamma & 0 \\ 0 & \gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} = \begin{pmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{pmatrix}

This expands to

\begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\gamma & 0 \\ 0 & \gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\gamma & -\beta\gamma \\ 0 & \gamma \end{pmatrix} = \begin{pmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{pmatrix}

The matrix second from the right represents the Tangherlini transformation (or inertial synchronized Tangherlini transformation).

For time with space (1+3):

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Lorentz transformation derivations

What follows are four derivations of the Lorentz transformation from the complete Galilei (Galilean) transformations in space with time (3+1) and time with space (1+3). Their intersection is linear space and time (1+1), which is the focus of the derivations. The other dimensions may be reached by rotations in space or time.

I. Space with Time (3+1)

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance x and x′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the point P:

x = ct, and x′ = ct′.

A. Time velocity

Define velocity v as the time velocity vt = ds/dt. Consider the standard Galilean transformation of ct and x with a factor γ, which is to be determined and may depend on β, where β = v/c:

x′ = γ(x − vt) = γ(x − βct) = γx(1 − β).

The inverse transformation is the same except that the sign of β is reversed:

x = γ(x′ + vt′) = γ(x + βct) = γx′(1 + β).

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Home is the horizon

As there is an inverse or harmonic algebra, so there is an inverse geometry, an inverse space. 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