iSoul In the beginning is reality

Centripetal prestination

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius, R, and the angle of motion, θ, by using Rs and Rt, and θs and θt, respectively. Then S = 2πRs and T = 2πRt. Also, Δs = Rs Δθs, Δt = Rt Δθt, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

What is the acceleration that occurs in uniform circular motion? Christiaan Huygens was the first to answer this in 1658. Here is a simple derivation:

An object in uniform circular motion traverses a circle at constant speed, v. Its spatial position can be represented by a vector, Rs, which changes its angle but not its magnitude. The distance traversed in one cycle is S = 2πRs. The period or duration of one cycle is T = 2πRt = 2πRs/v.

The velocity vector of this object can also be represented by a vector that changes its angle but not its magnitude. The accumulated change in velocity is 2πv. The magnitude of the acceleration is the change in velocity divided by the duration:

a = 2πv / T = 2πv / (2πRs/v) = / Rs.

Another derivation uses a diagram such as this:

centripetal acceleration diagram

Substituting Rs for r and θs for θ, the derivation is as follows:

Δs = Rs Δθs

v = |Δs| / |Δt|, and

a = |a| = |Δv| / |Δt| = v θs| / |Δt| = v s| / (Rst|) = t| / (Rst|) = / Rs.

Additionally,

a = / Rs = / (Rtv) = v / Rt = Rs / Rt².


What is the prestination that occurs in uniform circular motion? A simple derivation follows the first method above:

An object in uniform circular motion traverses a circle at constant pace, u. Its time position can be represented by a vector, Rt, which changes its angle but not its magnitude. The period or duration of one cycle is T = 2πRt. The distance traversed by one cycle is S = 2πRs = 2πRt/u.

The celerity vector of this object can also be represented by a vector that changes its angle but not its magnitude. The accumulated change in celerity is 2πu. The magnitude of the prestination is the change in celerity divided by the distance:

b = 2πu / S = 2πu / (2πRt/u) = / Rt.

A second derivation uses a diagram similar to the one above with these substitutions: rRt, θ → θt, st, and vu. Then,

Δt = Rt Δθt,

u = |Δt| / |Δs|, and

b = |b| = |Δu| / |Δs| = uθt| / |Δs| = ut| / (Rts|) = u² |Δs| / (Rts|) = / Rt.

Additionally,

b = / Rt = / (Rsu) = u / Rs = Rt / Rs².

Motion equations revised

Previous posts on motion equations, for example here, showed a misleading parallelism. Although there is a formal parallelism as shown, it is more accurate to show the inverse equations. The parallel equations above and below have been revised accordingly.

Parallel Equations

  Linear w/3D space Linear w/3D time Angular w/3D space Angular w/3D time
Average Rate v = Δst u = Δts ω = Δθt = v/Rs ψ = Δφs = u/Rt
Average Rate 2 a = Δvt b = Δus α = Δωt β = Δψs
Instantaneous Rate Velocity

v = ds/dt = 1/u

Celerity

u = dt/ds = 1/v

Angular velocity

ω = dθ/dt = dt/dφ

Angular celerity

ψ = dφ/ds = ds/dθ

Instantaneous Rate 2 Acceleration

a = dv/dt := 1/b

Prestination

b = du/ds := 1/a

Tangential acceleration

α = dω/dt

Tangential prestination

β = dψ/ds

Centripetal

Radial Rate 2

Centripetal acceleration

acen = v2/Rs = v/Rt

Centripetal prestination

bcen = u2/Rt = u/Rs

Radial acceleration

arad = Rs ω2

Radial prestination

brad = Rt ψ2

Uniform Transverse Rate v = 2πRs/T u = 2πRt/S vtan = Rs ω utan = Rt ψ
Radius Spatial radius

Rs = S/(2π) = Rtv

Temporal radius

Rt = T/(2π) = Rsu

Spatial radius

Rs = ds/dθ = s/θ = v/ω

Temporal radius

Rt = dt/dφ = t/φ = u/ψ

Circumference

Arc Length

Circumference

S = 2πRs = 2πRtv

Circumference

S = 2πRt/u = 2πRs

Spatial arc length

θ = s/Rs

Temporal arc length

φ = t/Rt

Period T = 2πRs/v = 2πRt T = 2πRt = 2πRsu T = 2π/ω S = 2π/ψ
Position Distance: s Duration: t Arc distance: s = Rs θ Arc duration: t = Rt φ
Displacement s = s0 + vt t = (s ‒ s0)u θ = θ0 + ωt t = (θ θ0)ψRt2
First Equation of Space-Time v = v0 + at t = (vv0)/a ω = ω0 + αt t = (ωω0)/α
Second Equation of Space-Time s = s0 + v0t + ½at² t = (-u0/a) +

√[(u0/a)2 + 2(ss0)/a]

θ = θ0 + ω0t + ½αt2 φ = (-β/ψ0) +

√[(β/ψ0)2 + 2β(ss0)]

Third Equation of Space-Time = v0² + 2a(s s0) s = s0 + (v² ‒ v0²)/2a ω² = ω0² + 2α(θ θ0) θ = θ0 + (ω2ω02)/2α
Distimement s = (t ‒ t0)v t = t0 + us s = (φ φ0)ωRs2 φ = φ0 + ψs
First Equation of Time-Space 1/v = (1/v0) + (s/a) u = u0 + bs s =  (ψ ‒ ψ0)/β ψ = ψ0 + βs
Second Equation of Time-Space s = (-u0/b) +

√[( u0/b)2 + 2(tt0)/b]

t = t0 + u0s + ½bs² θ = (-α/ω0) +

√[(α/ω0)2 + 2α(tt0)]

φ = φ0 + ψ0t + ½βs2
Third Equation of Time-Space t = t0 + (u2u02)/2b u² = u0² + 2b(t t0) φ = φ0 + (ψ2ψ02)/2β ψ² = ψ0² + 2β(φ φ0)

 

Reality and conventions #4

This post continues a series of posts. The previous one is here.

Modern natural science attempts a systematic account of the causes of change in the physical world, and is willing to go against the appearance of the physical world if that will further its goals. This differs from the ancient Platonic attempt to “save the appearances” at all costs by placing appearances within an ad-hoc but meaningful system.

In one sense, philosophy is the helpmeet of science. It aids in the task of putting our conceptual household in order: tidying up arguments, discarding unjustified claims. But in another sense, philosophy peeks over the shoulder of science to a world that science in principle cannot countenance. As Professor Scruton put it elsewhere, “The search for meaning and the search for explanation are two different enterprises.” Science offers us an explanation of the world; it may start out as an attempt to explain appearances, “but it rapidly begins to replace them.” Philosophy seen as the search for meaning must in the end endorse the world of appearance. The New Criterion, vol. 12, no. 10

Saving the appearances famously led to tweaking Ptolemaic astronomy despite its inability to explain why celestial bodies should move in epicycles. The Newtonian system didn’t give ultimate explanations but at least it gave laws that applied on Earth and skyward.

Yet there is nothing “wrong” with saving appearances such as the motion of the Sun relative to the Earth. In that sense, geocentrism was never wrong despite generations of people being taught so. Whether saving the appearances or saving the system is a goal, both must accept some conventions that include things such as the celestial body of reference – or lack thereof.

One may legitimately pursue a phenomenal science that saves appearances by sacrificing some consistency in conventions. For example, the Moon is in orbit relative to the Earth and the Sun is in a different kind of orbit relative to the Earth. In order to save both of these appearances, one would have to use a gravitational dynamics for the Earth-Moon system and a levitational dynamics for the Earth-Sun system. Awkward, perhaps, but legitimate.

Reality and conventions #3

This post follows on the previous post here, as well as other posts such as here.

The one-way speed of light is a convention (see John A. Winnie, Philosophy of Science, v. 37, 1970). The two-way (round-trip) speed of light is known to be c, but the one-way speed may vary between c/2 and infinity, as long as the two-way speed equals c. This means that those who say the light from a star took X light-years to reach the Earth are speaking of a convention rather than an actual duration.

A convention cannot be “cashed in” to become reality. For example, one cannot adopt a convention that some pebble is worth a million dollars, and take it to the bank and expect them to exchange it for a million dollars. According to their convention, it is worthless. If both follow the same convention, they can make the exchange, but even then it is based on a convention, not on an intrinsic reality.

Similarly, the time for starlight to reach the Earth cannot be cashed in for time on Earth. If the one-way speed of light equals c, then some galaxies appear to be billions of light-years away. But this time is the result of a convention, not an actual duration. This time cannot be cashed in to be an actual duration on Earth. Conventional years are not actual years.

It’s well-known that all motion is relative. That means what bodies are in motion are relative to a frame of reference, and there is no preferred frame of reference. Ironically, Galileo Galilei, who is credited with discovering the relativity of motion, is also known for claiming that the Earth moves around an immovable Sun rather than the converse. Whether the Earth or the Sun moves is a convention relative to a frame of reference, not a reality that all should recognize. Whichever convention is adopted cannot be cashed in for a state of rest or motion.

Conventional science is science with standard conventions. Unconventional science is science with non-standard conventions. Both are legitimate forms of science. Their conclusions should be the same, even though their conventions are different.

Reality and conventions #2

This post continues the topic of the previous post here.

Every pair of contrary opposites may have one or more conventions associated with it. That is because there is a symmetry between the two that can be reversed. Note this is not the case with contradictory opposites: they are not symmetric. Note also that terms may be symmetric without the references of the terms being exactly symmetric.

I’ll start with the latter point. A common example is the terms for male and female. In some respects they are symmetric opposites but in other respects they are not. The language can mislead on this point. Males and females have some similarities, some contrary (or complementary) differences, as well as differences that are not contraries, just different. Some aspects of male-female relations are conventions but not every aspect is.

The deconstructionists associated binary opposites with power structures (not unlike Hegel). They would reverse the meaning in order to undermine them. That assumes pairs are complete contraries, which is not as common as they thought. Deconstructionism works mostly on texts, in which the language of contrary opposites is deconstructed. The conventions associated with contrary opposites can be reversed but not all binary opposites are genuine contraries.

Contradictory opposites such as good and evil or true and false are not symmetric, contrary to the language that is often used. Not-evil is not necessarily good and not-false is not necessarily true. What is a matter of goodness or truth are not mere conventions.

There is a reality independent of us (or of our minds) but some things are conventions that are dependent on us. Motion is real but all motion is relative so it is a convention as to what motion is relative to. Galileo and the Scholastic philosophers (and their supporters) were wrong to think of the Earth as either only at rest or only in motion. Whether or not the Earth moves is a convention.

Reality and conventions #1

This post relates to the previous post here, as well as posts on light conventions here and here.

There comes a point in science in which a convention needs to be adopted in order to avoid confusion and ensure consistency. The tendency, however, is to think that the convention adopted is real, that is, that reality uniquely matches the convention. But that is an illusion since a different convention can legitimately be adopted.

This happens more often that we might realize. I have not tried to catalog all the conventions of science but here are some:

  1. Units of measure. These are all conventions, and there are variations such as the inch-pound units.
  2. Statistical significance. A p-value of 0.05 is often used, but it is a convention, not statistics.
  3. Negative charge of the electron. The current and the flow of electrons are in opposite directions.
  4. “A rod is undergoing tension. Is this negative or positive? In steel and concrete studies, tension is positive. In soil studies, tension is negative.”

Some of these conventions are a matter of choosing a value as the standard, others involve selecting a positive and a negative type (direction, charge). The positive type would seem to be the main or default one, as with arithmetic, but this may not be the case.

The conventions on the one-way speed of light show that the question relates to the status of the observer. Is the observer always right? That leads to one convention, in which the incoming speed of light is instantaneous. Is there an average that is right? That leads to Einstein’s convention, in which light travels at the average of the two-way speed of light.

Scientifically the latter is more straightforward but the problem is that it entails that some observations need to be corrected. The former may be more awkward but it has the advantage that “the observer is always right.” This accords with a common-sense realism and empiricism.

Consider optical illusions. They are something that appears one way but under further investigation are another way, such as the horizontal lines in this cafe wall illusion that appear to be sloped:

cafe wall illusion

But what about refraction? When we see a stick in water, it appears to bend but when we put our hand in the water, there is no bend. Yet we do not call this an illusion. We call it refraction. That is, it is an optical phenomenon and the appearance of a bend is real.

So it is a matter of classification. Yet all classifications are a matter of convention. We cannot get away from convention. In that sense reality is ambiguous or (à la Heisenberg) uncertain.

Since Plato and Aristotle science has included an attempt to “save the phenomena”. Although they meant different things by this phrase, it does indicate the primacy of phenomena. After all, there is no science (except perhaps for mathematics) without appearances. If all appearances are illusory, then appearance is not something to be explained but to be explained away.

A common-sense realist takes appearance as reality, with the understanding that some reflection is needed to avoid mistakes.

The knowledge the realist is talking about is the lived and experienced unity of an intellect with an apprehended reality. This is why a realist philosophy has to do with the thing itself that is apprehended, and without which there would be no knowledge. (#5 in A Handbook for Beginning Realists by Étienne Gilson)

Properly apprehended, the world of appearances is the real world. The observer is always right.

Gravitation and levitation theories

A theory of levity might be hilarious but the basic sense of the word levity is lightness, the opposite of gravity’s heaviness. In Aristotle levity is like buoyancy, as some things in water float and others sink.

Aristotle commits himself to gravity and levity as two distinct qualities, both of them positive. Fire has levity alone, earth gravity alone; air and water have both levity and gravity in different degrees. Aristotle, Physics, Vol. I, Loeb Classical Library, p. lxiv.

Levity was conceived as working in the opposite direction of gravity, but otherwise they were alike. The concept of levity was finally dropped in early modern science.

When Newton devised his theory of gravitation to explain the motion of celestial bodies, he said that the force of gravity was directed radially toward the body with greater mass. This was understandable since it is a theory of gravitation and since the greater mass would be associated with greater significance. But this is a convention.

One could just as well devise a theory of levitation with an “unforce” that is the same as Newton’s force of gravity except that it is directed radially toward the body with lower mass. Both gravitation and levitation are radial but levitation is about lightness, not heaviness. There is nothing illegitimate about such a theory, though it may seem strange to those raised on gravitational dynamics alone. After all, a satellite in geosynchronous orbit is hovering over the Earth, that is, levitating.

In time-space a different concept of levitation is available (see here). Levity may be measured by gorce, vass, and prestination, in which larger values indicate a smaller gravity. It would be natural to associate the direction of the gorce of levity toward the body with greater vass, which means less mass since vass is the inverse of mass. This means that the “center” of motion is the body in orbit or falling. It’s all a matter of perspective.

There is a larger issue here, which we will address in the next post.

Simple harmonic motion

This post is related to the one on circular orbits. I’ll continue to follow the exposition in Elements of Newtonian Mechanics by J.M. Knudsen and P.G. Hjorth (Spriner, 1995), this time starting with page 33. As before, the point is to derive the theory for time-space that is symmetric with space-time. Although the parallel theory is shown for 1D space + 1D time, it may be expanded to 1D space + 3D time.

A harmonic oscillator is like the spring below with a bob (weight):

Simple Harmonic MotionThe force is an example of Hooke’s law. Let’s first look at the horizontal case:

Fx = –kx,

where k is a constant determined from the properties of the spring. From Newton’s second law:

m d²x/dt² = –kx,

where m is the mass of the bob, x is the displacement, and t is the time. The solution to this differential equation can be written in the form:

x(t) = As cos(ωt + θs),

where ω = √(k/m), A is the spatial amplitude, and θ is the spatial phase angle. The period T must satisfy

cos(ωt + θ) = cos(ω(t + T) + θ).

Since the cosine is periodic with the period 2π, we have

ωT = 2π,

or

T = 2π/ω = 2π √(m/k).

Thus the period is independent of the amplitude. Note that

d²x/dt² = –ω²x.

Now consider a bob (weight) on a vertical spring under gravity. The total force acting on the bob is:

F = –ky + mg,

where k is the spring constant. The equilibrium position is y0 such that:

y0mg/k.

The equation of motion is

m d²y/dt² = –ky + mg,

which can be written in the form

d²y/dt² + (k/m)(y – y0) = 0.

The solution to this differential equation is

y – y0As cos(ωt + θ).

This shows that the bob will oscillate around the equilibrium position with the period (distimement extent), T, which is the extent of the distimement:

T = 2π √(m/k).


Now let’s look at simple harmonic motion from the perspective of time-space. Hooke’s law in the horizontal case is this:

Γt = –k′t,

where k′ is a constant determined from the properties of the spring. From Newton’s second law:

n d²t/dx² = –k′t,

where n is the vass of the bob, t is the time, and x is the displacement. The solution to this differential equation can be written in the form:

t(x) = At cos(ψx + φt),

where ψ = √(k/m), At is the temporal amplitude, and φt is the temporal phase angle. The displacement extent S must satisfy

cos(ψx + φ) = cos(ψ(x + S) + φ).

Since the cosine is periodic with the period 2π, we have

ψS = 2π,

or

S = 2π/ψ = 2π √(n/k′).

Thus the extent S is independent of the temporal amplitude. Note that

d²t/dx² = –ψ²t.

Now consider a bob (weight) on a vertical spring under gravity. The total gorce acting on the vass is:

Γ = –k′t + nh,

where k′ is a spring constant and h is the gravitational prestination. The equilibrium time position is t0 such that:

t0nh/k′.

The equation of motion is

n d²t/dy² = –k′t + nh,

which can be written in the form

d²t/dy² + (k′/n)(t – t0) = 0.

The solution to this differential equation is

t – t0At cos(ψy + φt).

This shows that the bob will oscillate around the equilibrium time position with the displacement extent, S:

S = 2π √(n/k′).

Centrism and extremism

I’ve written on my understanding of centrism here and here.

The essence of centrism is an acceptance of a limit for everything. This means there are limits in all directions. The image of this is a closed convex curve with a center in the middle of the region enclosed.

Without limits, there is no center. A center is always within limits. If there is any direction without a limit, the curve is not closed and there is no center.

Non-centrists are extremists in at least one way. They reject a limit in at least one direction. They are not only not in the center, but they reject the existence of a common center.

The slogan “No enemies on the Left” is a left-wing motto that goes back at least to the 1930s. It reflects an attitude that in the direction of leftist politics, there is no limit. Because it lacks a limit in at least one direction, it is extremist in at least one direction.

Most political groups promote some cause or idea that takes precedence over all other causes or ideas. They may hold these in a limited way, but unless they have ways of limiting the range of their support, they will tend to go further and further in that direction. They are or will become extremists.

1D space + 3D time again

I’ve written about 1D space + 3D time before, such as here and here.

Time is measured in terms of the annual calendar and the daily clock. Clocks are based on the apparent motion of the Sun and other celestial bodies across the sky, divided into 24 hours. The calendar is based on the annual cycle of the Sun’s declination (the angular distance north or south of the celestial equator), divided into 12 months and 365 (or 366) days. Thus time is measured in terms of the Earth’s orbital and rotational periods.

Length may be measured similarly. The original definition of the metre (meter) in 1793 was one ten-millionth of the distance from the equator to the North Pole. What is required to measure 1D space is a natural periodic motion whose travel distance may be accumulated indefinitely. What corresponds to the clock could be the motion of the night shadow across the face of the Earth at the equator:

So the evening and the morning were the [Nth] day. Genesis 1

The circumference of the Earth is 40,075,000 m. This is the distance moved by the edge of the day-night boundary in 24 hours. Then one hour is equivalent to 1,669,792 m and one second is equivalent to 464 m. The shadow motion may be accumulated indefinitely to make 1D length, similar to 1D time.

Whichever dimension is limited to 1D is the measure of a motion from the perspective of the object in motion, whether this is measured by the distance or duration. Since the 1D variable is the independent variable, this is selected as a standard motion available (in principle) to everyone. So the 1D variable becomes a universal length or time against which to measure motions in 3D of time or space, respectively.