iSoul In the beginning is reality

Equations of motion in space-time and time-space

First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = t, position = r, initial position = r(t0) = r0, velocity = v, initial velocity = v(t0) = v0, v = |v| = speed, and acceleration = a.

First equation of motion

v = ∫ a dt = v0 + at

Second equation of motion

r = ∫ (v0 + at) dt = r0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (rr0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (rr0)).


Next, here is a derivation of the time-space equations of motion, in which prestination is constant. Let position (displacement) = r, time = t, initial time = t(r0) = t0, celerity = u, initial celerity = u(r0) = u0, u = |u| = pace, and prestination = b.

First equation of motion

u = ∫ b dr = u0 + bt

Second equation of motion

t = ∫ (u0 + br) dr = t0 + u0r + ½br²

Third equation of motion

From u² = uu = (u0 + br) ∙ (u0 + br) = u0² + 2r(bu0) + b²r², and

(2b) ∙ (tt0) = (2b) ∙ (u0r + ½br²) = 2r(bu0) + b²r² = u² ‒ u0², it follows that

u² = u0² + 2(b ∙ (tt0)).

Christianity and science

A good summary of the myth of a long-running conflict between Christianity and science is in Timothy Larsen’s “War is Over, If You Want It” (September 2008). This warfare myth was invented in the 19th century by people such as TH Huxley who either should have known better or were purposely stirring up animosity. It is composed of individual myths that “support” it, such as the myth that Christians thought the earth was flat in the Middle Ages or the myth that Christians opposed the use of anesthesia during childbirth in the 19th century.

Larsen references Frank M. Turner’s “Contesting Cultural Authority” (Cambridge, 1993), as someone who “persuasively argued that the notion of a conflict between theology and science was generated as part of a campaign of professionalization by would-be scientists.” (p.150) It’s almost forgotten today, but the profession of a scientist didn’t exist until the late 19th century. Before that, science was developed by amateurs (including clerics) who had the leisure and interest. TH Huxley and others fought against such people because they stood in the way of a new class of professional scientists.

Although the warfare meme is vastly exaggerated, there are enough misunderstandings that the opposite idea of integration isn’t realistic. For example, it is said that many Christians quickly accepted Darwin’s theory of evolution in the 19th century and later. But what is overlooked is the fact that Christians misunderstood Darwin and substituted their own ideas of evolution by law or miracle.  Theistic evolution is common among Christians who either insert a law-bound version for Darwin’s undirected version or else invent undetectable miracles that make it God-directed.

Many have noted that modern science developed in a Christian matrix. If science jettisons its Christian roots, it loses a reason to expect an ordered universe that can be understood by human beings. It may either adopt a multiverse that just happens to have order in one universe or drift toward non-causal explanations in a chaotic universe.

Some scientists want to deepen the Christian roots of science rather than cut them off. They are mostly creationists or intelligent design proponents. Those who follow TH Huxley will have nothing of it. But some are willing to entertain new proposals. As the modern era comes to a close, we can expect that modern science will change into something else.

Derivation of Newton’s second law

It is often said that Newton’s laws are laws of nature, which can only be determined by observation. That’s true in the sense that the definitions required are based on inductive reasoning. However, once these definitions are in hand, it should be a deductive science. This is classical science, in the sense of Euclid, Archimedes, Plato, and Aristotle.

Here is a derivation of Newton’s second law in space-time, with distance, s, time, t, and mass, m:

Velocity, v := ds/dt.

Acceleration, a := dv/dt.

Mass flow rate, := dm/dt.

Weighted distance, ș := ms.

Momentum, p := dș/dt = d(ms)/dt = (mds + sdm)/dt = m(ds/dt) + s(dm/dt) = mv + sṁ.

If mass is constant, then p = mv.

Force, F  := dp/dt = d(mv + sṁ)/dt = d(mv)/dt + d(sṁ)/dt = (mdv/dt + vdm/dt) + (sd/dt + ds/dt) = ma + vṁ + sṃ + ṁvma + 2ṁv + sṃ, where = d/dt.

If mass is constant, then F = ma.

Here is a derivation of Newton’s second law in time-space, with vass, n = 1/m:

Celerity, u := dt/ds.

Prestination, b := du/ds.

Vass flow rate,  := dn/ds.

Weighted durationț := nt.

Celentum, q := dț/ds = d(nt)/ds = (ndt + tdn)/ds = n(dt/ds) + t(dn/ds) = nu + tṅ.

If vass is constant, then q = nu.

ElaphrenceΓ := dq/ds = d(nu + tṅ)/ds = d(nu)/ds + d(tṅ)/ds = (ndu/ds + udn/ds) + (td/ds + dt/ds) = nb + uṅ + tṇ + ṅu = nb + 2ṅu + tṇ, where  = d/ds.

If vass is constant, then Γ = nb.

Transformations with two speeds of light

I’ve discussed before how there may be two different one-way speeds of light since they are matters of convention. See here and here. This post is concerned with the transformation between two observers in such a case. I continue to work with the standard configuration, see here.

The standard transformation of reference frames begins with two frames in uniform relative motion along one axis (usually called x). Here we take the spatial axis to be the r-axis, which parallels the spatial axis of motion. Similarly, the temporal axis is taken to be the t-axis, which parallels the temporal axis of motion.

The notation here is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames. One can generalize the results here to other directions by rotation.

The two frames are differentiated by primed and unprimed letters. They coincide at time t = 0 and their relative speed is v. The one-way speed of light is c1 in one direction and c2 in the opposite direction such that the round-trip speed equals c, the standard speed of light in a vacuum. That is, the harmonic mean of c1 and c2 equals c.

The trajectory of a reference particle or wave that travels at the speed of light follows these equations in both frames:

r =  c1t or r/c1 = t and r′ = c2t′ or r′/c2 = t′.

Consider a point event such as a flash of light that is observed from each reference frame. How are its coordinates in each frame related?

The basic relations are: r′ = rtv = r (1 – v/c1) and t′ = t (1 – v/c1). Next a factor, γ, is included in the transformation equations:

r′ = γ (rvt) = γr (1 – v/c1), and

t′ = γt (1 – v/c1),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations use the other speed of light:

r = γ (r′ + vt′) = γr′ (1 + v/c2), and

t = γt′ (1 + v/c2).

Multiply each corresponding pair together to get:

rr′ = γ²rr′ (1 – v/c1)(1 + v/c2).

Dividing out rr′ yields:

1 = γ2 (1 – v/c1)(1 + v/c2).

Solving for γ leads to:

γ = [(1 – v/c1)(1 + v/c2)]–1/2, which applies if |v| < |c|.

This is the Lorentz transformation with two one-way speeds of light.

Clock race

This post continues previous ones contrasting ancient and modern space and time, such as here.
Nordic Sun Chariot in Bronze
The above bronze-age depiction of the Sun on a chariot shows a common image from antiquity: the Sun crossing the heavens daily. The path of the Sun was also described as traversing a celestial circle (or sphere) and going around a racecourse. These images show that the clock of the Sun was considered as covering a distance in space, in contrast with the modern concept of simple duration.

Psalm 19
1 The heavens declare the glory of God,
and the sky above proclaims his handiwork.
2 Day to day pours out speech,
and night to night reveals knowledge.
3 There is no speech, nor are there words,
whose voice is not heard.
4 Their voice goes out through all the earth,
and their words to the end of the world.
In them he has set a tent for the sun,
5 which comes out like a bridegroom leaving his chamber,
and, like a strong man, runs its course with joy.
6 Its rising is from the end of the heavens,
and its circuit to the end of them,
and there is nothing hidden from its heat.

These are more examples of the interchange of travel distance with travel time that occurred in the transition from ancient to modern thinking. We can undo this interchange and find an alternative way of conceiving motion.

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 = Δts ω = Δθt = v/R ψ = Δϑs = /R
Average Rate 2 a = Δvt b = Δs α = Δωt β = Δψs
Instantaneous Rate Velocity

v = ds/dt = 1/

Celerity

= dt/ds = 1/v

Angular velocity

ω = dθ/dt = dt/dϑ

Angular celerity

ψ = dϑ/ds = ds/dθ

Instantaneous Rate 2 Acceleration

a = dv/dt := 1/b

Prestination

b = d/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 = 2/Rt = /Rs

Radial acceleration

arad = Rs ω2

Radial prestination

brad = Rt ψ2

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

Rs = S/(2π) = Rtv

Temporal radius

Rt = T/(2π) = Rs

Spatial radius

Rs = ds/dθ = s/θ = v/ω

Temporal radius

Rt = dt/dϑ = t/ϑ = /ψ

Circumference

Arc Length

Circumference

S = 2πRs = 2πRtv

Circumference

S = 2πRt/ = 2πRs

Spatial arc length

θ = s/Rs

Temporal arc length

ϑ = t/Rt

Period T = 2πRs/v = 2πRt T = 2πRt = 2πRs T = 2π/ω S = 2π/ψ
Position Distance: s Duration: t Arc distance: s = Rs θ Arc duration: t = Rt ϑ
Displacement s = s0 + vt t = (s ‒ s0) θ = θ0 + ωt t = (θ θ0)ψR2
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 = (-0/a) +

√[(0/a)2 + 2(ss0)/a]

θ = θ0 + ω0t + ½αt2 ϑ = (-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 + ℓs s = (ϑ ϑ0)ωR2 ϑ = ϑ0 + ψs
First Equation of Time-Space 1/v = (1/v0) + (s/a) =0 + bs s =  (ψ ‒ ψ0)/β ψ = ψ0 + βs
Second Equation of Time-Space s = (-0/b) +

√[(0/b)2 + 2(tt0)/b]

t = t0 +0s + ½bs² θ = (-α/ω0) +

√[(α/ω0)2 + 2α(tt0)]

ϑ = ϑ0 + ψ0t + ½βs2
Third Equation of Time-Space t = t0 + (202)/2b ² =0² + 2b(t t0) ϑ = ϑ0 + (ψ2ψ02)/2β ψ² = ψ0² + 2β(ϑ ϑ0)

 

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.