iSoul In the beginning is reality

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Inverting motion curves

The mathematical problem is this: given a curve with distance coordinates that are parametric functions of time (duration), find the reparametrization of the curve with duration (time) coordinates that are parametric functions of distance. Symbolically, given the regular curve α(t) = (a1(t), …, an(t)), find β(s) = (b1(s), …, bn(s)) such that bi(s) = t(ai) for i = 1, …, n, where t is time (duration) and s is distance. (Greek letters are used for vectors, Roman letters for scalars.)

The solution is to invert each coordinate function and express them in terms of a common parameter. That is, set each ai(t) = s and solve for t to get t = ai-1(s) = bi(s) for the inverse coordinates in parametric form.

For example, consider a projectile fired from height h with velocity v at angle θ. The path of the projectile is represented by a parametric equation

α(t) = (a1(t), a2(t)) = (vt cos(θ), h + vt sin(θ) – ½gt²),

where g is the acceleration of gravity. Setting s = vt cos(θ) and s = h + vt sin(θ) – ½gt²), then solving for t results in the inverse coordinates, which are in two parts:

β(s) =(s/(v cos(θ)), (v sin(θ) + sqrt(2gh – 2gs + v² sin²(θ)))/g) going up, and

β(s) =(s/(v cos(θ)), (v sin(θ) – sqrt(2gh – 2gs + v² sin²(θ)))/g) coming down.

The spatial position vector α(t) corresponds to a temporal position vector β(s). As there are multiple dimensions of space, so there are multiple dimensions of time. But the time in multidimensional space is a scalar, and the space in multidimensional time is a scalar.

Essentials of Christian Thought, part 4

The previous post in this series is here.

The key to this middle way, if it is truly a middle way between extremes, is divine self-limitation—the idea that the God of the Bible is vulnerable because he makes himself so out of love. p.139

… the personal God of the Bible is revealed there as the one “principle of all things,” “both cause and reason” for everything else’s existence. [Emil] Brunner also rightly emphasized that for the Christian this is no “theory of the world,” no rational, speculative hypothesis, but revealed truth of the “one word of God.” p.142

Whether or not one takes the Genesis narratives of creation literally, their theological meaning is obvious to anyone who approaches them without bias against personal theism: The whole world, the universe, everything outside of God, was created by God “in the beginning.” p.143

And, yes, God has mind, intelligence, thought, purpose, but his essence is not “Mind” (Nuos) as Greek philosophy conceived it. p. 145

According to the biblical narrative, then, there are two basic categories of reality—God’s, which is supernatural and personal (but not human), eternal, independent, self-sufficient; and the world’s, which is dependent but good, filled with purpose and value and governed as well as sustained by God. p.145

The distinct, singular personhood of God, the reality of God as a being among beings, not an all-inclusive, unconditioned, absolute Being Itself, is a hallmark of the biblical portrayal of God. p.147

By the free act of creation, by creating something outside of himself with limited autonomy, the God of the Bible has become a being beside other beings and limited by them in a limited way. p.149

… the difference between God and humans is character, not personhood. p.149

As philosopher Plantinga explained, the scientific search for truth assumes nature is not all there is. If nature is all there is, then truth itself is a chimera and our human faculties for discovering and knowing it are unreliable. p.151

As already explained, according to the biblical view of God and the world, the world has a relative autonomy over against God—by God’s own design. Yet neither nature nor history are independent processes operating entirely under their own laws and powers. p. 151

Modern Christian thinkers such as Scottish philosopher Thomas Reid (1710–96), Horace Bushnell (1802–76), and C. S. Lewis, among many others, went out of their way to explode the myth that a miracles must be a divine interruption of nature—as if, in order to act in special ways, God must “break into” a world that operates like a machine alongside of, over against, and independently of God’s immanent, continuing creative activity. The biblical-Christian view of nature and history is the both are in some sens always already the activity of God. That is not to say that everything that happens in them is the direct, antecedent will of God; it is only to say that, from a biblical and Christian perspective, the very laws of nature are, in some sense, simply regularities of God’s general providential activity. And history is always being guided, directed, and governed by God—even when God’s human creatures, endowed with free will, rebel and act against God’s perfect will. According to a biblical-Christian worldview, God’s agency is always the principle and power underlying everything. p.152

That means, then, that a miracle is never a “breaking” of nature’s laws, a “violation” of nature, or a “disruption” of history’s story as if nature and history were normally operating under their own power and overcome by God “from the outside.” That is the myth about the supernatural and miracles imposed by modern naturalism. p.152

Rather, from a biblical-Christian perspective, a miracle is simply an event in which God acts through nature in an unusual way. p.152-3

The ultimate reality of the biblical narrative, God, is self-sufficient but also vulnerable. He is not dependent on anything outside himself and yet, at the same time, opens himself to influence by his own creatures. … God’s self-sufficiency is his freedom; his vulnerability is the product of his love. p.154

According to [Thomas F. Torrance], the Genesis creation narrative itself implies God’s entrance into time. p.157

Catholic Tresmontant affirmed that the God of the Bible, unlike the ultimate reality of Greek philosophy, is not an unchanging sameness but ever active life and action. p.157

For Cherbonnier, God’s immutability is simply his faithfulness, not his static being-ness without becoming or eternity without temporality. p.158

That is, the biblical story consistently correlates virtue and knowledge but not in the Greek sense of “to know the good is to do the good.” Rather, for the Bible and Christian thought generally, “doing the good,” by God’s grace and with faith, produces knowledge of ultimate reality as the ultimate good. p.162

But also, Brunner argued, the whole idea of an objective moral law, “right” and “wrong,” depends on ultimate reality being a personal God. p.162

For biblical-Christian thought, then, metaphysics and ethics are inseparable. p.163

Essentials of Christian Thought, part 3

This post continues from part 2, which is here. The following are more excerpts from Roger E. Olson’s The Essentials of Christian Thought.

For [Emil] Brunner, and for me, natural theology means only (1) that the biblical-Christian worldview better answers life’s ultimate questions than its competitors and alternatives, and (2) that eyes of faith for whom the Bible “absorbs the world” see the natural world as God’s good creation—”charged with the grandeur of God”—even if eyes of unbelief cannot see it as such. p.75

For biblical-Christian thought, in contrast with Greek philosophy, souls are created by God, they are not emanations, offshoots, of God’s own substance. p.81

Nearly all extra-biblical philosophies struggle with the [biblical] idea of a personal, related, vulnerable ultimate reality capable of being influences by what creatures do. p.84

Brunner believed God is revealed in nature and in the human spirit generally (general revelation). p.92

First, … nature and universal human experience, general revelation, yield only a “thatness” of God but not God’s “whoness,” personhood, and will. What humanity needs is to know God personally, not just God’s nature as ultimate reality. Second, according to Brunner, in complete agreement with most classical Protestant theology (and the Bible in Romans 1!), reason, or the use of reason, has been spoiled in humanity by sin. p.93

The reason the human person cannot use his own reason to arrive at a satisfying life philosophy or vision or reality is his own natural tendency to minimize evil—especially in himself. p.93

Brunner argued that “everyone who philosophizes does so from a definite starting point, upon which he, as this particular man, stands. The Christian philosophizes from that point at which God’s revelation sets him.” p.94

For Brunner, the God of biblical revelation is supernatural and personal but not human. p.95

God is both ontologically beyond and personally present. p.98

The point of this entire chapter is that there is a biblical, narrative-based metaphysic that contrasts with other metaphysical visions of ultimate reality, is not irrational, lies at the foundation of Christianity itself, and is being retrieved by Jewish and Christian scholars who are also separating it from extrabiblical philosophies that conflict with it. p. 99

Many scholars tend to define the difference between philosophy and theology as revelation—theology uses it and philosophy does not. There are, however, exceptions. “natural theology” is the rational exploration of the evidence of God in nature and universal human experience. “Philosophical theology” is philosophy that explores reasons for belief in God …. p.100

Brunner coined the term eristics for his own belief that, when set alongside alternative worldviews, Christian philosophy is superior. p.106

… the biblical narrative requires belief that God’s existence precedes the world’s not only temporally but ontologically. That is, the world is dependent on God, not vice versa. p.119

[Plantinga’s] conclusion, therefore, is that there is superficial conflict but deep concord between science and theistic belief, but superficial concord and deep conflict between science and naturalism. p.122

Humanism is simply any belief in the dignity and creativity of human persons, that human beings are unique and above nature, in some sense transcendent, capable of great culture achievements as well as terrible destruction. It places special value on humanity. … the real humanism is Christian humanism because of the biblical-Christian emphasis on humans as created in the image and likeness of God. p.123

functional naturalism—belief that although God exists and is person, he does not intervene in history or human lives, which are ruled by natural laws and explainable by science. p. 125

Classical Christian theism, born in the cauldron of philosophized Christianity in the second and third centuries in the Roman Empire, reached its zenith in Anselm and Aquinas. p.132

Gradually, Christian began to envision ultimate reality, God, along the lines of Platonic metaphysics—including the idea that God, being metaphysically complete and perfect in every way imaginable, cannot suffer or be affect by temporal events or creatures. The word for this was and is impassibility. p. 136

 

The next post in this series is here.

Logic as arithmetic

George Boole wrote on “the laws of thought,” now known as Boolean Algebra, and started the discipline known as Symbolic Logic. A different George, George Spencer Brown, wrote on “the laws of form,” which presented an arithmetic system underlying logic. Below are two symbolic logics equivalent to Boolean algebra that resemble ordinary arithmetic in some respects. To resemble arithmetic in other respects, use the Galois field of order 2, GF(2). Zero is taken as representing false, and one as true.

LOGIC OF SUBTRACTION

Subtraction

A – 0 = 1 – A = 1

A – 1 = A

Definitions

– A is defined as 0 – A (and so 0 is ”  “, ground, false)

A + B is defined as  A – (– B)

Tables

A 0 − A A − B 0 1 A + B 0 1
0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 1

Consequences

– (– A) = A

A − B = A ← B

A + B = A ∨ B

A + B = B + A

– is not distributive

 

DIVISION LOGIC

0 / A = A / 1 = 0

A / 0 = A

Definitions

/ A is defined as 1 / A (and so 1 is ”  “, ground, true)

A • B is defined as  A / (1 / B)

Consequences

1 / (1 / A) = A

A / B = – (A → B)

A • B = A ∧ B

A • B = B • A

/ is not distributive

Tables

A 1 / A A / B 0 1 A • B 0 1
0 1 0 0 0 0 0 0
1 0 1 1 0 1 0 1

 

Curves for space and time, continued

The following is a continuation and revision of the previous post, here.

Based on the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A pdf version in parallel is here.

Let a three-dimensional curve be expressed in parametric form as x = x(t); y = y(t); z = z(t); where the coordinates of the point (x, y, z) of the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t), y(t), and z(t) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve1

Displacement Δr connecting points A and B on parametric curve r(t).

Consider a segment (displacement) of a parametric curve r = r(t) between two points P(r(t)) and Q(r(tt)) as shown in the figure above. As point B approaches A or in other words Δt → 0, the length s becomes the differential arc length of the curve:

ds = |dr/dt| dt = | rt | dt = √(rtrt) dt,

where the superscript t denotes differentiation with respect to the arc time parameter t. The vector rt = dr/dt is called the tangent vector at point A.

Then the arc length, s, of a segment of the curve between points r(t0) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(rtrt) dt = ∫ √((dx/dt)2 + (dy/dt)2 + (dz/dt)2) dt.

The magnitude of the tangent vector is

| rt | = ds/dt = v.

Hence the unit tangent vector is

Ts = rt / | rt | = (dr/dt) / (ds/dt) = dr/dsrs,

where the superscript s denotes differentiation with respect to the arc length parameter, s.

If r(s) is an arc length parametrized curve, then rs(s) is a unit vector, and hence rsrs = 1. Differentiating this relation, we obtain rsrss = 0, which states that rss is orthogonal to the tangent vector, provided it is not a null vector. The unit vector

Ns = rss(s) / |rss(s)| = Tss(s)/|Tss(s)|,

which has the direction and sense of rss(s) is called the unit principal normal vector at s. The plane determined by the unit tangent and normal vectors Ts(s) and Ns(s) is called the osculating plane at s. The curvature is

κs ≡ 1/ρ = |rss(s)|,

and its reciprocal ρ is called the radius of curvature at s. It follows that

rss = Tss = κs Ns.

The vector ks = rss = Tss is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κs is nonnegative, thus the sense of the normal vector is the same as that of rss(s). For a three-dimensional curve, the curvature is

κs = |rt × rtt| / | rt |³.


Let a three-dimensional curve be expressed in parametric form as X = X(s); Y = Y(s); Z = Z(s); where the coordinates of the point (X, Y, Z) of the curve are expressed as functions of a parameter s (length) within a closed interval s1ss2. The functions X(s), Y(s), and Z(s) are assumed to be continuous with a sufficient number of continuous derivatives.

In vector notation the parametric curve can be specified by a vector-valued function w = w(s), where w represents the position vector (i.e., w(s) = (X(s), Y(s), Z(s)).

curve2

Distimement Δw connecting points C and D on parametric curve w(s).

Consider a segment (distimement) of a parametric curve w = w(s) between two points C(w(s)) and D(w(ss)) as shown in the figure above. As point D approaches C or in other words Δs → 0, the length t becomes the differential arc time of the curve:

dt = |dw/ds| ds = | ws | ds = √(wsws) ds,

where ws = dw/ds, which is called the tangent vector at point C. Then the arc time, t, of a segment of the curve between points w(s0) and w(s) can be obtained as follows:

t(s) = ∫ dt = ∫ √(wsws) ds = ∫ √((dX/ds)2 + (dY/ds)2 + (dZ/ds)2) ds.

The vector ws = dw/ds is called the tangent vector at point C. The magnitude of the tangent vector is

| ws | = dt/ds = u.

Hence the unit tangent vector is

Ttws / | ws | = (dw/ds) / (dt/ds) = dw/dtwt.

If w(t) is an arc length parametrized curve, then wt(t) is a unit vector, and hence wtwt = 1. Differentiating this relation, we obtain wtwtt = TtTtt = 0, which states that wtt is orthogonal to the tangent vector, provided it is not a null vector. The unit vector

Nt = wtt(t) / |wtt(t)| = Ttt(t)/|Ttt(t)|,

which has the direction and sense of wtt(t) is called the unit principal normal vector at t. The plane determined by the unit tangent and normal vectors Tt(t) and Nt(t) is called the osculating plane at t. The curvature is

κt ≡ 1/ρ = |wtt(t)| = |Ttt(t)|,

and its reciprocal ρ is called the radius of curvature at t. It follows that

wttTtt = κt Nt.

The vector kt = wttTtt is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κ is nonnegative, thus the sense of the normal vector is the same as that of wtt(t). For a three-dimensional curve, the curvature is

κt = |ws × wss| / | ws |³.


Here are some useful formulae of the derivatives of arc length, s, and the arc time, t:

v = st = ds/dt = | rt | = (rtrt)1/2 = 1/| ws | = 1/(wsws)1/2,

a = stt = dst/dt = (rtrtt) / (rtrt)1/2 = – (wswss) / (wsws)4/2,

sttt = dstt/dt = [(rtrt)(rtrttt + rttrtt) – (rtrtt)²] / (rtrt)3/2

= – [(wsws)(wswsss + wsswss) – 4(wswss)²] / (wsws)7/2,

u = ts = dt/ds = 1/| rt | = 1/(rtrt)1/2 = | ws | = (wsws)1/2,

b = tss = dts/ds = – (rtrtt) / (rtrt)4/2 = (wswss) / (wsws)1/2,

tsss = dtss/ds = – [(rtrt)(rtrttt + rttrtt) – 4(rtrtt)²] / (rtrt)7/2

= [(wsws)(wswsss + wsswss) – (wswss)²] / (wsws)3/2.

Curves for space and time

The following is slightly modified from the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT.

A plane curve can be expressed in parametric form as x = x(t); y = y(t); where the coordinates of the point (x, y) of the curve are expressed as functions of a parameter t (time) within a closed interval t1tt2. The functions x(t) and y(t) are assumed to be continuous with a sufficient number of continuous derivatives. The parametric representation of space curves is: x = x(t); y = y(t); z = z(t); t1tt2.

In vector notation the parametric curve can be specified by a vector-valued function r = r(t), where r represents the position vector (i.e., r(t) = (x(t), y(t), z(t)).

curve

Displacement Δr connecting points P and Q on parametric curve r(t).

Let us consider a segment (displacement) of a parametric curve r = r(t) between two points P(r(t)) and Q(r(tt)) as shown in the figure above. As point Q approaches P or in other words Δt → 0, the length s becomes the differential arc length of the curve:

ds = |dr/dt| dt = | r | dt = √(rr) dt.

Here the dot denotes differentiation with respect to the parameter t. Therefore the arc length of a segment of the curve between points r(t0) and r(t) can be obtained as follows:

s(t) = ∫ ds = ∫ √(rr) dt = ∫ √(x2(t) + y2(t) + z2(t)) dt.

The vector dr/dt is called the tangent vector at point P. The magnitude of the tangent vector is

| r | = ds/dt = v.

Hence the unit tangent vector becomes

T = r / | r | = (dr/dt) / (ds/dt) = dr/ds.

Here the prime ¹ denotes differentiation with respect to the arc length, s. We list some useful formulae of the derivatives of arc length s with respect to parameter t and vice versa:

v = s = ds/dt = | r | = (rr)1/2,

a = s•• = ds/dt = (rr) / (rr)1/2,

s = ds/dt = [(rr)(rr + rr) – (rr)²] / (rr)3/2,

u = t¹ = dt/ds = 1/| r | = 1/(r • r)1/2,

b = t¹¹ = d/ds = – (rr) / (rr)4/2,

t¹¹¹ = dt¹¹/ds = – [(rr)(rr + rr) – 4(rr)²] / (rr)7/2.

If r(s) is an arc length parametrized curve, then (s) is a unit vector, and hence = 1. Differentiating this relation, we obtain r¹¹ = 0, which states that r¹¹ is orthogonal to the tangent vector, provided it is not a null vector.

The unit vector

N = r¹¹(s)/|r¹¹(s)| = (s)/|(s)|,

which has the direction and sense of (s) is called the unit principal normal vector at s. The plane determined by the unit tangent and normal vectors T(s) and N(s) is called the osculating plane at s. The curvature is

κ ≡ 1/ρ = |r¹¹(s)|,

and its reciprocal ρ is called the radius of curvature at s. It follows that

r¹¹ = = κN.

The vector k = r¹¹ = is called the curvature vector, and measures the rate of change of the tangent along the curve. By definition κ is nonnegative, thus the sense of the normal vector is the same as that of r¹¹(s).

For a space curve, the curvature is

κ = |r × r| / |r|³.

Modernity and parsimony

I’ve written before about modernity here and parsimony here.

An age begins by repudiating something essential about the previous age. The middle ages started with repudiating the ancient gods and myths (cf. St. Augustine’s City of God). The modern age began with the Reformation, which repudiated the history of the Church and the pagan past of the Gentiles. It continued with scientists repudiating Scholasticism and Aristotle. And it came into its own by starting anew, whether in religion or science or politics.

If modernity starts with breaking free of the past, then what keeps it from flaming out into insignificance? The key for science was parsimony, commonly called simplicity. In contrast with the middle ages, which specialized in ad hoc explanations, the modern age adopted Occam’s razor, the law of parsimony, which privileged the fewest number of assumptions and kinds of entities.

Modernity took the law of parsimony to an extreme. It led to questioning, if not overthrowing, every tradition, every non-empirical entity, every metaphysics. The absolute minimum ontology was considered the best, which turned out to be the physical world.

Even the nature of physical things was questioned as unknowable, until the only nature left was the nature of the physical world. This nature became the idol of modernity, the one thing that could not be questioned. It became Nature, reified as something with a will of its own, something that led to human life, something that substituted for God.

As we break free of modernity, we can see its limitations and failures more and more. One is the bias of the law of parsimony: it meant qualitative parsimony but not quantitative parsimony. That is, only one or a few kinds of things could exist, but the number of them available for explanatory purposes was unlimited. This bias fit well with the use of mathematics as the language of science.

But mathematics is more than the study of quantity. It is also the study of space, structure, and change. And there is no good reason not to apply parsimony to all of them in finding the best explanation. Once we open up to the possibility of a balanced application of the law of parsimony, we can see some of the weaknesses of modern science.

Deep time was invented in the 18th century and exploited in the 19th and 20th centuries to explain the history of the Earth and the universe. What started with geology expanded to human history, biology, and cosmology.

It is all a matter of time scale. An event that would be unthinkable in a hundred years may be inevitable in a hundred million. Carl Sagan

Time is in fact the hero of the plot. … Given so much time, the “impossible” becomes possible, the possible probable, and the probable virtually certain. One has only to wait; time itself performs the miracles. George Wald

The flaw is simple: it’s too easy to “explain” anything. The violation of quantitative parsimony was the Achilles’ heel of modernity. The temptation to explain everything was too much to resist. And so, as with every age, modernity ended in failure. A great failure, but a failure nonetheless.

We can only hope that the current age will learn from the failure of modernity and seek a balanced parsimony.

Observers and travelers, continued

This post continues the topic of the previous post here. This is a post about two kinds of people. First a warning:

There may be said to be two classes of people in the world; those who constantly divide the people of the world into two classes, and those who do not. – Robert Benchley

Actually, this post is about two different roles that people take, though some people get stuck in one role or the other. Consider these pairs of complementary roles:

speakers and listeners, writers and readers, artists/performers and viewers, musicians and audiences, programmers/designers and users, producers and consumers, etc.

Scientists and engineers often have complementary roles: engineers making things that work in the world and scientists observing and seeking to understand the world. In the MBTI personality types, there are judgers and perceivers. Combine all these with travelers and observers, transmitters and receivers, of the previous post.

What is the basic distinction here? It’s between an active role and a passive role, between having a goal and a way to get there vs. letting things go and seeing what happens. In terms of Aristotle’s four causes, it’s between the final and formal causes vs. the mechanistic/efficient and material causes.

Aristotle give an example of a sculptor, in which they start with a final goal in mind and develop a plan, a design, a form. Then they take some material such as marble or clay and use tools to form it into something. An observer would only see the last two steps. They would have to infer the first two steps – or else stick to empiricism and ignore the first two steps.

In terms of studying motion, the distinction is between having a destination and moving there vs. starting somewhere and observing what motion there is. These two roles lead to the two approaches to space and time: 3+1 dimensions and 1+3 dimensions.

These roles are distinct even when they’re combined. For example, scientists do experiments, which requires an active role, but the purpose is to observe, which means to watch what happens.

In ethical terms, the distinction is between givers and takers, doers and hearers (James 1:22).

These roles are different enough so that communication may be a problem. They speak different dialects and some translation may be required for them to understand one another. Knowing about personality types provides a clue as to how to approach those who prefer to take a particular role.


A few favorites of this “different kinds of people” genre:

There are three kinds of people in the world: Those who know math and those who don’t.

There are 10 kinds of people in the world: Those who understand binary and those who don’t.

There are two types of people in this world: Those who can extrapolate from incomplete data

Observers and travelers

Let us distinguish between observer-receivers and traveler-transmitters. Although observers can travel and travelers can observe, insofar as one is observing, one is not traveling, and insofar as one is traveling, one is not observing. The main difference is this: traveler-transmitters have a destination but observer-receivers do not (or at least not as observers).

Compare the roles of the driver and the passengers in a vehicle: the driver is focused on the road and traveling to the destination, whereas the passengers are looking out the window and observing things in the landscape. These are two different roles.

Observer-receivers of motion naturally compare the motion observed with the elapsed time. But traveler-transmitters have a destination and naturally compare the travel motion with the elapsed distance, which measures progress toward the destination. Because of this, the frame of mind for observer-receivers is 3D space + 1D time, whereas it is 1D space + 3D time for traveler-transmitters.

Observers of the sky naturally think of celestial bodies as appearing when they are observed, as with celestial navigation. That is, they act as though the light observed arrives in their sight instantaneously.

Transmitters of light naturally expect that the light reaches its destination as they transmit it, as with visual communication. That is, they act as though the light transmitted arrives at its destination instantaneously.

This is consistent with having two conventions of the one-way speed of light (previously discussed here). To be consistent with the round-trip speed of light equaling the value, c, for all observers, that implies the following:

For observers: observed light is instantaneous but transmitted light travels at the speed c/2.

For transmitters: transmitted light is instantaneous but observed light travels at the speed c/2.

Although relativity theory is the scientific approach, for everyday life the above speeds make things simpler, and are fully legitimate.

Definition of vass

See also the related post on the Center of vass. Relativity has been addressed before, such as here.

Isaac Newton called mass “the quantity of matter”, which is still used sometimes, although Max Jammer points out how it has been criticized for centuries (see Concepts of Mass in Classical and Modern Physics, 1961). Other definitions arose in the 19th century. One is the ratio of force to acceleration, which assumes that force can be defined independently of mass.

Another approach is to define the equality of masses. For some such as Saint-Venant, “two bodies have equal masses if their velocity increments after impact are equal.” (ibid., p.91) For Ernst Mach equal masses “induce mutually equal and opposite accelerations.” (ibid., p.94)

Is there an independent definition of vass, the inverse of mass? One could modify these definitions of mass to define vass or equality of vasses:

Definition 1: Vass is the ratio of the surge to the prestination of a body.

Definition 2: Two bodies have equal vasses if their celerity increments after impact are equal.

Definition 3: Two bodies have equal vasses if they induce mutually equal and opposite prestinations.

In relativity theory, mass is dependent on velocity as follows:

m = γ m0,

with mass m, invariant mass m0, velocity v, speed of light c, and γ = (1 – v²/c²)–1/2.

It is easily verified that vass is dependent on celerity as follows:

ℓ = ℓ0 / γ,

with vass ℓ, invariant vass ℓ0, celerity u, pace of light ç, and γ = (1 – ç²/u²)–1/2.