iSoul In the beginning is reality

Metaphors for time and space

George Lakoff and Mark Johnson have a chapter on time (Chapter 10) in their book Philosophy in the Flesh (Basic Books, 1999) that makes several points:

All of our understandings of time are relative to other concepts such as motion, space, and events. (p.137) Most of our understanding of time is a metaphorical version of our understanding of motion in space. (p.139)

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More equations of motion

Expanding on a previous post here, this is a summary of the equations of motion for space-time and time-space. See also a pdf version in the Time-space Glossary option above.

s = displacement magnitude, t = time magnitude, v = velocity, v0 = initial velocity, a = acceleration, u = tempo, u0 = initial tempo, b = expedience, ω = angular velocity, ω0 = initial angular velocity, ψ = angular tempo, ψ0 = initial angular tempo, θ = spatial angle, ψ = temporal angle, S = circumference, T = period, Rs = spatial radius, and Rt = temporal radius.

Linear w/3D space Linear w/3D time Angular w/3D space Angular w/3D time
Average Rate v =  Δst u = Δts ω = Δθt ψ = Δϑs
Average 2nd Rate a = Δvt b = Δu/Δs α = Δωt β = Δψs
Instantaneous Rate Velocity
v = ds/dt
Tempo
u = dt/ds
Angular velocity
ω = dθ/dt = dt/dϑ
Angular tempo
ψ = dϑ/ds = ds/dθ
Instantaneous 2nd Rate Acceleration
a = dv/dt
Expedience
b = du/ds
Tangential acceleration
α = dω/dt
Tangential expedience
β = dψ/ds
Centripetal/Radial 2nd Rate Centripetal acceleration
acen = v2/Rs
Centripetal expedience
bcen = 1/(u2Rs)
Radial acceleration
arad = Rs ω2
Radial expedience
brad = Rt ψ2
Uniform Tangential Rate vtan = 2πRs/T utan = T/(2πRt) vtan = Rs ω tan = Rt ψ
Circumference/Arc Length Spatial circumference
S = 2πRs
Temporal circumference
T = 2πRt
Spatial arc length
θ = s/Rs
Temporal arc length
ϑ = t/Rt
Period T = 2πRs/v T = 2πRtu T = 2π/ω T = 2π/ψ
Radius Spatial radius
Rs = S/(2πv)
Temporal radius
Rt = T/(2πu)
Spatial radius
Rs = ds/dθ = s/θ = v/ω
Temporal radius
Rt = dt/dϑ = t/ϑ = ℓ/ψ
Position s t On a circle: s = Rs θ On a cycle: t = Rt ϑ
Displacement s = s0 + vt t = t0 + us θ = θ0 + ωt ϑ = ϑ0 + ψs
Second Equation of Motion s = s0 + v0t + ½at² t = t0 + u0s + ½bs² θ = θ0 + ω0t + ½αt2 ϑ = ϑ0 + ψ0t + ½βs2
First Equation of Motion v = v0 + at u = u0 + bs ω = ω­ + αt ψ = ψ0 + βs
Third Equation of Motion = v0² + 2a(s – s0) u² = u0² + 2b(t – t0) ω² = ω0² + 2α(θ – θ0) ψ² = ψ0² + 2β(ϑ – ϑ0)

Relating space and time

In a sense, every distance can be converted into a duration or vice versa: simply multiply duration by the modal speed or multiply the distance by the modal pace. For example, every time can be multiplied by the speed of light in a vacuum and so be replaced by a distance. This is usually done with the invariant interval: ds² = c d – d.

However, these distances are not really durations and these durations are not really distances. Such manipulations cannot change whether the variables were measured synchronously or asynchronously, which is an essential part of what the units represent. These conversions have their place but they do not show the symmetry of space and time.

What does show the symmetry of space and time is a switch of perspective so that what was the independent variable becomes the dependent variable and vice versa. This means every speed becomes to a pace and 3D space is mapped to 3D time, and vice versa.

It’s not the modal rate that makes space and time symmetric, though that does fit with the symmetry. It’s the arbitrary nature of the difference between space and time that make them symmetric. There is no inherent reason why 3D space makes more sense than 3D time or vice versa. They are symmetric.

Religious freedom in two senses

I last wrote about religious freedom here. The post concerns how to define religion for purposes of religious freedom.

Basically, there are two ways to define religion: (1) a narrow, traditional sense in which religion means one of the world religions, which are concerned with worship of God or gods and/or following a certain way of life; or (2) a broad sense in which religion means what each person defines as the greatest good or ultimate concern and the lifestyle choices that follow from that. In the latter sense everyone has a religion; even those who are atheistic or anti-religious make a religion out of that.

The First Amendment to the U.S. Constitution lays out the two sides of religious freedom: negative freedom (freedom from) and positive freedom (freedom to). “Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof…” The incorporation doctrine applies this to the States, too. Consider these two sides in relation to the two definitions of religion.

(1) In the narrow sense of religion, government shall not promote or demote one traditional religion over another, shall not provide material support for traditional religious activities, and shall not interfere with traditional religious organizations and activities.

That protects religious organizations and leaders but what about adherents who want to apply their religion to their daily activities, their business, or other nominally secular actions? For example, can a business operate in accordance with religious principles if they conflict with generally applicable laws? The Supreme Court has said, No. So the narrow definition of religion causes problems for individuals.

(2) In the broad sense of religion, government shall not promote or demote one view of the greatest good or ultimate concern over another, shall not provide material support for anything that someone might consider religious, and shall not interfere with any organization or activity that people claim has religious significance for them.

Clearly, the broad sense of religion causes problems for government. For example, there are people who say it’s against their religion to pay taxes. Does that mean they get off free? Does anyone have a veto over laws based on something they claim is part of their religion? In the broad sense of religion, it seems so.

The Supreme Court has reacted against this stricture on government and affirmed the legitimacy of any government action that is not aimed against any particular religion. In 1990 an exemption was sought so members of a tribal religion could ingest peyote despite a ban on this drug. The Court denied this in terms that seemed to equate freedom of religion with freedom of speech: say anything you want, have any religious opinions, but the law applies to everyone (Employment Division v. Smith).

The problem is that the second definition is too broad and the first definition is too narrow. The way forward is to adopt a broader version of the narrow definition or a narrow version of the broad definition. What might this mean?

For example, it could mean that someone can’t just say, My religion forbids me to pay taxes. They need to demonstrate that this is part of a religious tradition or doctrine that is a central part of their life. This is similar to the process for obtaining conscientious objector status with the selective service system (military draft). It’s not easy to obtain this status, but it can be done by those with a strong case to be members of a pacifistic religion.

On the other hand, it should take more for the government to justify a law than merely that it advances a secular purpose. Many secular purposes these days are against the religious beliefs and practices of many people. The government should be required to show a compelling public interest in a law, or else carve out exceptions for religious objectors.

As government has grown, religious freedom has been under pressure to contract. This needs to change, without giving everyone a veto over laws they don’t like.

Parallel equations of motion

Expanding on the previous post here, this is a summary of the equations of motion for space-time and time-space. See also the Time-space Glossary option above.

r = displacement magnitude, t = time magnitude, v = velocity, v0 = initial velocity, a = acceleration, u = legerity, u0 = initial legerity, b = expedience

Space-time (3+1) Time-space (1+3)
Assumption da / dt = 0 db / dr = 0
da / dr = 0 db / dt = 0
First equation of motion dv / dt = a du / dr = b
v = v0 + at v = v0 + br
Second equation of motion dr / dt = v = 1/u dt / dr = u = 1/v
r = r0 + v0t + ½at² t = t0 + u0r + ½br²
Third equation of motion dv / dr = a/v = aℓ d / dt = b/u = bv
= v0² + 2a(rr0) = u0² + 2b(tt0)
2a = (v0²) / (rr0) 2b = (u0²) / (tt0)
Relation dv / du = a/bv = au/b du / dv = b/au = bv/a

Corresponding equations of motion

The classic equations of motion are for 3D space + 1D time (3+1). Equations of motion for 1D space + 3D time (1+3) were presented here. How do these equations relate to each other? Can one convert directly from one form to the other form?

Consider 1D space + 1D time (1+1), which can be generalized to other dimensions. The symbols used below are: acceleration a, velocity v, position r, expedience b, legerity u, time t, and the speed of light, c. Position in space and time are proportionate: r = ct.  Speed and pace are inverses of each other: v = 1/u.

There are two ways in which equations of motion can correspond to one another: the first way is to be an equal expression of the same equation, and the second way is to be a corresponding equation from the other perspective, switching the spatial and temporal. The first way is represented by ‘=’ and the second way by ‘⇔’.

Here is the case with zero acceleration:

r(t) = vtr0 = vt + ct0t(r) = urt0 = r/v + r0/c,

r′(t) = v = 1/ut′(r) = u = 1/v,

r″(t) = 0 ⇔ t″(r) = 0.

Here is the case with constant acceleration:

r(t) = ½ at² + v0t + r0 = ½ t²/b + t/u0 + ct0t(r) = ½ br² + u0r + t0 = ½ r²/a + r/v + r0/c,

r′(t) = at + v0 = t/b + 1/u0t′(r) = br + u0 = r/a + 1/v,

r″(t) = at″(r) = b.

Other dimensions may be added by applying the above to each component.

Note that uniform speed and pace are reciprocals of each other but uniform acceleration and expedience magnitudes are not. If a body increases velocity uniformly, its acceleration is constant but not its expedience.

Glossary of time-space terms

I’ve compiled a glossary of new terms on the top menu of this blog, or see here. These terms were coined for the study of 1D space + 3D time. It will be updated as needed.

A parallel comparison of space-time and time-space terms was added here.

“Synchronizing” space

An isodistance map shows the contours of equal distances from a central point. These would be circles on a map if distance is measured “as the crow flies.” The shapes vary if distance depends on a road network:

But how do you tell if two distances are the same? Different observers have different distance measuring instruments. There must be a way to ensure consistency of the distance measuring instruments – e.g., rigid measuring rods.

The problem is analogous to the need to synchronize clocks for 1D time. The Einstein synchronization convention is the standard way to do this:

Position 1 has clock 1 and rod 1; position 2 has clock 2 and rod 2. A light signal is sent from position 1 at time τ1 (defined from a specified point in time) to position 2 and immediately back, e.g. by means of a mirror. Its arrival time back at position 1 is τ2. This synchronization convention sets clock 2 so that the time τ3 of signal arrival and reflection is defined as

τ3 = τ1 + ½ ( τ2 − τ1) = ½ ( τ1 + τ2).

Synchronization is also achieved by slowly transporting a third clock from position ρ1 to position ρ2.

Similarly, one needs to synmacronize measuring rods for 1D space. As synchronize is from transliterated Greek syn + chron + ize (“to occur at the same time”) so synmacronize is from syn + macron + ize (“to occur at the same length”). Follow the convention above and then:

Use the constant speed of light, c, to convert the times into corresponding distances, setting rod 2 so that the distance of signal arrival and reflection is defined to be

3 = 1 + ½ (21) = ½ (1 + 2), i.e.,

ρ3 = ρ1 + ½ (ρ2ρ1) = ½ (ρ1 + ρ2), where ρ1, ρ2, and ρ3 are the distances corresponding to τ1, τ2, and τ3.

Synmacronization can also be achieved by slowly transporting a third measuring rod from position 1 to position 2.

Both space and time require consistent measuring instruments.

Characteristic limits

I have written about the characteristic (modal) rate for a mode of travel. This rate provides a factor for converting spatial into temporal measures and vice versa. It is possible that the characteristic rate is independent of any particular rate but often it is a function of many rates, such as the minimum, maximum, or mean. These functions may have some similarities with independent rates but they should be distinguished as relative characteristic rates.

An absolute characteristic rate is absolutely independent of any and all particular rates in a mode. This rate may be a limit of the rates but it must not be dependent on them in any way. So an absolute characteristic rate may be the infimum (greatest lower bound) or supremum (least upper bound) of the rates but strictly speaking may not be their minimum or maximum.

The speed of light in a vacuum is widely considered the maximum speed for signals. What is important is not so much that it is the speed of light as that it is the speed that is the supremum for the speeds of all bodies with non-zero rest mass. It so happens that the photon travels at this speed, called c.

I wrote recently about a lower limit on speed equal to the inverse of the speed of light. This is better described as the infimum of the speeds of all bodies with non-zero rest mass. Is there a particle that travels at this speed, 1/c? It might be a “lygon” for twilight particle (from lygo, Greek root for twilight).

Judging politicians

From what most media outlets say and the way most people talk, it would seem that the most important thing politicians do is make speeches. Talk, talk, talk, day after day. Some words bring headlines, perhaps unwanted. Other words bring praise or condemnation. In the end, what is most remembered about politicians is their words, not their actions.

Sometimes words and actions go together, as when a public official announces a decision they have the authority to make. Other times actions follow words, as when a politician promises they will do something specific, usually after the election. But most of the time words are sufficiently vague and actions are sufficiently long in coming that it’s hard to tell whether or not promises were kept.

What really makes a difference is what politicians do, not what they say. So the way to judge politicians is by their actions, and their results. That means one should pay little attention to their words and much more attention to their actions. Why is this so hard for people to do?

For one thing, the media make it hard to do. It is much easier for the media to talk about a politician’s words rather than their actions, which need to be explained and may get into complex details. And so the media focuses on a politician’s latest off-hand remark rather than on what document they signed or directive they gave.

Where is the action-oriented media? Mostly the business press. Those in business require knowledge of actions that may affect them rather than the daily brouhaha of political blather. Otherwise, specialized media have updates for particular issues. Are you interested in environmental issues? Subscribe to an environmental newsletter. Religious freedom? Follow a Christian news source. Your town council? Go to some meetings yourself.

The bottom line is that most talk by politicians is hot air, which should be no surprise. So don’t pay attention to it. Follow their actions instead. Chercher l’action.