iSoul In the beginning is reality

Mechanics in multidimensional time

As (spatial) velocity and acceleration are vectors, so are their temporal analogues. This perspective makes sense because of the multidimensionality of time. There is an implicit recognition that time has directionality since time is commonly considered as any real number, and not a non-negative real number, which it would be if time were merely a magnitude. This “reverse time” is an example of time’s directionality — which however has nothing to do with reverse causal sequences.

The (spatial) length (especially of an object) is a magnitude that is used to represent physical space. Similarly, the duration (or length of time) is a magnitude that is used to represent physical time. We speak of the location or position in space of an object or a place. Similarly, we speak of the point in time or temporal position of an action or event.

A point in space or time is “that which has no part” (Euclid) whose location is represented by a position vector. A point itself is an abstraction that is zero dimensional but makes up all multidimensional abstract ‘spaces’ (which may represent space or time or whatever). If s is the distance of a point from a specified origin point in space, then its position may be represented by a position vector whose magnitude equals s. If t is the duration of a point in time from a specified origin point in time, then its position may be represented by a temporal position vector whose magnitude equals t.

The movement of a point through space in time may be represented by a vector function of temporal position t whose value is the spatial position at each temporal position t. The movement of a point through time in space may be represented by a vector function of spatial position s whose value is the temporal position at each spatial position s.

During the time interval (duration) Δt = t2 – t1, the position vector of an object changes from r1 = r(t1) to r2 = r(t2), with a displacement vector Δr = r2r1 (boldface represents vectors). The rate of change of the displacement vector is the average (time) velocity vector over the time interval, vavg = Δr / Δt. The rate of change of the average velocity vector is the average acceleration vector aavg = Δv / Δt.

Similarly, while traversing the space interval (length) Δs = s2 – s1, the position vector of an object changes from p1 = p(s1) to p2 = p(s2). The rate of change of the displacement vector is the average space velocity vector over the length of space, uavg = Δp / Δs. The rate of change of the average space velocity vector is the average space acceleration vector bavg = Δu / Δs.

Instantaneous velocity is considered to be measured over a differential of time (duration), dt. In that case the instantaneous (temporal) velocity is defined as v(t) = ds/dt and the instantaneous (temporal) acceleration as a(t) = dv/dt = d2s/dt2.

Similarly, the coincidental spatial velocity may be measured over a differential of space (length), ds. The coincidental spatial velocity is defined as u(s) = dt/ds and the coincidental spatial acceleration as b(s) = du/ds = d2t/ds2.

Measures of speed and velocity

The speed of an object is the ratio of distance (or length) traveled and the duration of travel. It is derived from the distance traveled during a given duration. It is expressed as the measured distance divided by the given duration, that is, distance relative to duration in units of distance over duration, e.g., m/s, km/hr, etc.

For example, the speeds of vehicles passing a fixed point along a roadway may be measured over a given duration by loop detectors and other fixed-location speed detection equipment. These are called spot speeds. The (arithmetic) average of such speeds is called the time mean speed since they are measured during a given period of time. Accordingly, each speed could be called a time speed.

But there is another, complementary way of determining speed. One can select a distance and measure the duration of travel while traversing that distance. Then the measured duration should be in the numerator to show the duration relative to distance, with units s/m, hr/km, etc. Unless the speed is constant, this is not the inverse of the time speed because the distances and durations will not match. It is called the pace, which means the change in time per change in position.

For example, probe vehicles may be in the traffic stream which measure their distance during a set period of time. Or these may be sampled using automatic vehicle location (AVL) data. The harmonic average of such speeds is called the space mean speed since it is measured over a given segment length. Accordingly, each speed could be called a space speed.

Why the harmonic average? Consider each space speed as an inverse speed: put the measured duration of travel in the numerator and the segment length in the denominator, so that the given segment length provides the units for this pace.

Now the average speed may be related to the average pace as follows: invert each speed to put the duration in the numerator and the length in the denominator, take their (arithmetic) average, and invert again to get the average speed. This is the harmonic mean of the space speeds.

Velocity is a vector of speed with the direction of movement. A time velocity may be defined as a velocity whose magnitude is a time speed, and a space velocity as a velocity whose magnitude is a space speed. If its magnitude is a pace, the components are duration divided by length, which is not velocity. It could be called legerity. Why might one use legerity instead of velocity? If the duration is measured for a given length, the legerity gives the appropriate measure: duration relative to length.

What does the direction of the legerity mean? Since it measures duration (relative to a given length), its direction is the temporal direction of movement. This shows again that the same three dimensions may be associated with time (duration) as well as space (length).

Homogeneity and isotropy of time

The homogeneity and isotropy of space are well-known. The homogeneity of time is partly known but is confused by an “arrow of time” concept that is not applicable to space and time. The isotropy of time is unknown (and usually denied) also because of confusion with an inapplicable “arrow of time” concept.

I previously wrote about the Multidimensionality of time. As space has three dimensions, so does time and they are the same three dimensions.

As space is homogeneous in each dimension, so is time. For example, it does not matter whether an experiment takes place “here” or 10 minutes north and 5 minutes east of “here” (if they are both inertial reference frames). The translational invariance of time is exactly like the translational invariance of space.

As space is isotropic, i.e., the same in all directions, so is time. For example, the duration measured by a clock is the same whether it is facing north, south, east, west, up, or down. And the duration is the same whether it is oriented horizontally, longitudinally, or transversely.

It is said that in classical mechanics time is reversible. This is a confused statement. What can be shown is that if a classical particle moves in one direction, its movement in the opposite (“reverse”) direction is also classical. Since both space and time are directional, that would equally well be true of space as of time but no-one says that space is reversible. It is best to leave questions of (ir)reversibility to thermodynamics, causality, etc.

Noether’s theorem shows that the homogeneity of space leads to the conservation of momentum, the homogeneity of time leads to the conservation of energy, and the isotropy of space leads to the conservation of angular momentum. I haven’t checked it yet but it is natural to expect that the isotropy of time leads to the conservation of rotational energy.

Multidimensionality of time

This post is another in a series on the duality of space and time. I have emphasized that the basis for space is length and the basis for time is duration. What, then, about direction? Does direction apply to both space and time? Yes, and in the same manner.

If someone says, “The hotel is 10 minutes away by car” how is that different from saying “The hotel is 5 miles away by car”? One provides a duration and the other a distance. Neither provides a direction. Both require movement to measure. They are exactly parallel.

If someone says, “The hotel is 10 minutes north by car” how is that different from saying “The hotel is 5 miles north by car”? One provides a duration and the other a length, each with a direction. Both require movement to measure. They are again exactly parallel.

Is the direction “north” part of space in one case but not in the other case? Then what does “north” mean in the sentence “The hotel is 10 minutes north by car”? It means that the direction “north” and the duration “10 minutes” are combined, just as we combined the direction “north” and the length “5 miles”. It would be arbitrary to say that direction applies to space (length) and not to time (duration).

So what is direction? It is something independent of length and duration, that is, it is independent of space and time but can be applied to either space or time. Direction is what makes the scalar “lengths” into a vector of directed lengths, often called a displacement. In the same way, direction is what makes the scalar “duration” into a vector of directed duration, which could be called a temporal displacement.

Is the concept angle only related to space? Look at the hand of a clock. Is it measuring an angle of space or of time? Both. We read a clock directly as time, a duration measured by revolving hands. But we recognize the spatial angles, too, and can use clock numbers to indicate space, as in “10 o’clock high” for a direction in space.

But if there are three dimensions for direction in space, does that mean there are three dimensions for direction in time? Yes, and they are the same three dimensions. For example, an isochrone map shows contour lines (isolines) for durations in two dimensions. It is like an isodistance map which shows travel lengths in two dimensions. The only difference between these maps is whether durations or lengths are shown; the two dimensions are the same.

So when we say that looking into outer space is looking back in time, that includes the three dimensions we see.

Space, time, and arrows

This post is a continuation on the duality of space and time. The basis of space is distance (or length) and the basis of time is duration. It must be emphasized that both distance and duration are scalars, i.e., they have magnitude but no direction. They are not one-dimensional because that would entail direction, represented by a positive and negative quantity. So scalars are non-negative real numbers (zero is a degenerate case).

Consider two sentences: “The Arcade building was a block long.” “They were stuck in traffic on Lake Shore Drive for 12 hours.” The first sentence expresses a distance and the second expresses a duration. Note that both sentences use the past tense. In the present tense the Arcade building in Chicago’s Pullman district doesn’t exist because it was demolished in 1926, and the mammoth traffic jam on Chicago’s Lake Shore Drive in February 2011 is over. This parallel shows that anyone who wants to say the past time is in the opposite direction from the present time could equally well say that past space is, too.

The problem is Arthur Eddington’s “arrow of time” which says time is one-way or asymmetric. But time in this sense has to do with tense, not duration, and has no application to space and time. Note that more recent work on “arrows of time” has focused on thermodynamics, causality, etc., and not on space and time.

What, then, is the meaning of a time line that goes from negative to zero to positive? If “now” is at the zero point, isn’t the negative part in the past? In fact, this is no different from a “space line” that goes from negative to zero to positive with “here” at the zero point. The location of a point in the past would be negative, but that does not lead us to say that space is one-way or asymmetric.

Putting “-t” into an equation of physics does not change the tense or make the present precede the past. It simply reverses the direction of the duration. If “+t” is to the right, then “-t” is to the left.

Arrow of tense

The arrow of time is a concept developed by Arthur Eddington in 1927. It is an arrow that points from the past through the present into the future. One problem with this concept is that multiple futures are possible; it would have to be a many-headed arrow. Another problem is that it could just as well be pointing from the future through the present to the past. The choice is arbitrary and may simply reflect a progressive bias.

One could as well speak of an arrow of place that points from there (where one was) to here (where one is) to there (where one is going). So both space and time have their arrows.

A deeper problem with the concept is that it’s really about tense in language. Different languages have different ways of indicating the time when an action or event occurs, or when a state or process holds. The past, present, and future tenses are one means of doing this. But there are other tenses such as the still sense, indicating that that a state is still the case. And some languages such as Chinese are tenseless.

So the arrow of time would be better called the arrow of tense and understood as a property of language. If the arrow of time is used at all, it should be paired with the arrow of place.

Duality of space and time

Several dualities of space and time are known, but there are thought to be exceptions for the dimensions of space and the arrow of time. It turns out these are not exceptions; space and time are fully dual. To understand this first note that movement is required for the measurement of time and space, and then compare the various meanings of the words time with the parallel meanings of space (or place):

(1) Time as duration, a period of time, is a length of time, analogous to a length of space. Duration and length are both scalar quantities.

(2) Time as points in time, instants of time, associated with specific actions or events. This is analogous to points in space, locations, which may also be associated with actions or events. Duration is a difference between two points in time as length is a difference between two points in space.

(3) Time as tense, a grammatical sense which expresses how an action or event relates to the present time, usually relative to the moment of speaking. This corresponds to language which expresses how an action or event is oriented toward the present location, usually relative to the place of speaking.

(4) Adverbs of time are relative to the speaker and include now, yesterday, tomorrow, later, etc. These correspond to adverbs of place, which are relative to the speaker and include here, there, down here, over there, etc.

(5) Time as the arrow of time is the forward flow from past times to the present time to future times. There is a corresponding flow from past places to the present place to future places which could be called the arrow of place. These are one-dimensional views of time and space which could be reversed by looking backwards.

(6) The speed of an object is a scalar measure of its rate of movement, expressed either as the travel length divided by the time taken (average speed) or the rate of change of position with respect to time at a particular point (instantaneous speed). To examine the relation of speed with time and space, consider highway traffic flow measurement which distinguishes two types of average speed:

The time-mean speed is the arithmetic mean of the vehicle speeds measured at one roadside location. The space-mean speed is the harmonic mean of speeds measured by the travel times collected between roadside locations (or on probe vehicles between two locations). Why the harmonic mean? Because the units are in the numerator, so it is a kind of inverse speed (the inverse of the duration of travel divided by the unit of travel length).

From this we may define the time speed as the travel length divided by the unit of travel duration. If measured at a point or “spot” it is called a spot speed, which is the instantaneous speed of a vehicle at a specified location. The space speed may be defined as the inverse of the duration of travel divided by the unit of travel length. For constant speeds, these values are equal but they are conceptually different.

The time speed has spatial direction but no time direction because the temporal denominator is a scalar. For the space speed the measurement of duration has temporal direction but no spatial direction because the spatial denominator is a scalar.

Rectilinear motion is along a straight line, with the distance from a point in that line varying with the time. Angular motion is the rotation of an object about a fixed point or fixed axis in a given time period.

Velocity is the rate at which an object changes its position. A time velocity may be defined as the vector of travel through space divided by the scalar unit of travel duration. The space velocity may be defined as the inverse of the vector of travel through time divided by the scalar unit of travel length.

(7) Direction is a vector of orientation or movement whose magnitude may be a length or a duration. A movement from here to there is also a movement from now to then which may be expressed as a vector. We tend to think of this in spatial terms but it may equally well be thought of in temporal terms.

There will be more to come on this topic but the bottom line is that length and duration are both scalars that may become dimensioned or tensed in an appropriate context. Space and time are dual.

Christianization of the world

In Mt 13:33 reports of Jesus: He told them another parable. “The kingdom of heaven is like leaven that a woman took and hid in three measures of flour, till it was all leavened.” Christianity is the leaven of the world. Put it into the world and gradually the whole world is leavened. This is the Christianization of the world.

Note that leaven works from the inside out, not the outside in. Christianization does not mean the world is given a coating of Christianity in hopes that it will penetrate further down. Rather, it means that the leaven of Christianity is put into the middle of the world and gradually works its way throughout.

When this is done, the outside at first may look as if nothing much has changed. But if the inside has changed, then sooner or later the outside and everything in between will change. That is what genuine Christianization means.

Christianity is a meta-religion: it “comes after” religion because it takes a religion and transforms it. The first religion Christianity was applied to was Judaism, as recorded in the New Testament. After that, the pagan religion of the gentiles in Europe was Christianized. Many of the customs associated with Christianity today come from Christianized Judaism and paganism. For example, Easter is a Christianized spring festival (the name comes from the Teutonic goddess of spring) and Christmas is a Christianized winter solstice festival.

Other religions have not been Christianized as much, but they could be. Music is one aspect which has been Christianized. There are Christian songs in every music tradition. Converts from any religion should be able to retain parts of their culture with a new focus and interpretation. Christianity is not about replacing the cultures of the world but about redeeming them.

The kingdom of God is the Christianization of the world. Where the kingship of Christ is, there is the kingdom of God. The world in all its diversity can be redeemed — and preserved — through Christ.


Political liberty still lacking

We in the United States like to consider our nation “free” and even the leader of the “free world”. Compared with totalitarian and authoritarian regimes of the past and present, yes, we are free. But that should not be the standard of comparison. The standard should be full political liberty, and there we are still lacking.

What are we missing? Let me list some:

(1) We lack a full vote. In an election for public office with more than two candidates, we can only vote for one candidate. After we have voted for one candidate, we cannot vote for any other candidate. So we have a partial vote, and this can lead to candidates with the largest plurality but without a majority being elected. This can lead to the least-preferred candidate in a three-way election being elected. All because people are only allowed to vote for one candidate.

Voting is an up-or-down decision but that decision is about each candidate for public office. People have a full vote only if they can vote for every candidate of their choice. This is called approval voting because voters can vote for every candidate they approve. Essentially, they are putting candidates into two groups: those they approve and those they don’t. That is a full vote and it is missing today.

(2) We lack the right to freely form political parties. Instead, the two major parties have rigged the system to make it difficult to establish a political party and get on the ballot. And political parties are regulated by the State instead of having freedom to conduct their affairs as they wish. For example, political parties do not have the right to determine membership in their party. We have primary elections in which voters can choose any ballot, so members of one party can vote in the primary for another party and determine their candidate.

While basic accountability for political parties should be enacted, political parties should be free to conduct their affairs as they wish. Let the voters decide whether they like them or not.

(3) We lack the right to freely contribute to political candidates. There are limits on what individuals can give, what parties can give, what campaigns can give, and what unaffiliated organizations can give. This gives an advantage to incumbents and those who can get free media attention. It undermines a free and open political process — contrary to the claims of those who promote these limits.

All that is needed is that the public is informed in a timely manner as to where each candidate’s finances are coming from. Publishing contributions over a specified amount can easily be done today. Then we can let the public decide if they like what they see or not.

There are other ways in which our liberty is unjustly restricted. The Institute for Justice among others is working on them: economic liberty, educational liberty (school choice), First Amendment defense (freedom of religion and speech), and private property defense.

In some ways the United States is free, but in other ways we still need to fight for freedom. The struggle goes on!


From history to nature

Over the centuries the various sciences have developed from a focus on history to a focus on nature, that is from a temporal or diachronic focus to a spatial or synchronic one. Saussure saw this in linguistics and reoriented it from a focus on historical language change to language as a system. Both have their place but historic study finds few natures, i.e., invariants, whereas the study of natures discovers many invariants.

For example, astronomy and physics in ancient times focused on cycles and the “harmony of the spheres” but in modern times focuses on a four dimensional continuum. Chemistry has developed from an alchemical focus on transmutation to a modern focus on the periodic table and compounds. Biology still focuses on temporality with its concentration on origins and history; to further develop it will need to focus on the nature of biological kinds. Geology has a similar focus on temporality so it will need to focus more on the nature of geological features.

Both History and Nature have been used by atheists as substitutes for God — in the 18th century Newton’s system was seen as Nature in control, then in the 19th & 20th centuries Darwin’s evolution was seen as History in control. So both approaches can be carried to extremes and will be by some.

Biology — whether evolutionary or creationary — needs to move from defining species or created kinds in terms of descent from original organisms to defining them in terms of their nature, e.g., as either having something in common (an essence) or a some type of interconnectivity (a topological definition).