iSoul In the beginning is reality

Biblical theism vs. classical theism

Biblical theism and classical theism have much in common, particularly the position that God is different in kind from all of creation. But there is an implicit principle of classical theism that I would put this way: “God only does what only God can do.” For example, because only God is transcendent, it is consistent with this principle that God creates from nothing.

“An architect of the universe would have to be a very clever being, but he would not have to be God…” Maurice Holloway, S. J., An Introduction to Natural Theology, pp. 146-47 (quoted here). However, there’s more than the existence of God at issue; there’s also the existence of mankind as a created kind, rather than a taxon only different in degree from other taxa.

Classical theists assert that there is only one causal act in God by which he causes ex nihilo whatever exists apart from himself. That is, God does not take something already existing and make it into something else. Why not? Because that would be doing something that a creature could possibly do.

They aver that if God designs creation, then he is doing something that others can do in some measure, which would be beneath God, as if God were merely a demiurge. Thus this view of God deprecates any divine association with design.

It’s like saying, “A human is different in kind from an ant, so since ants can crawl around, humans would never crawl around because that would be different in degree, not in kind.”

Wrong. To be different in kind does not entail being different in every respect.

God could take something already existing and make it into something else. Whether or not he has done so is another matter. The biblical theist insists that God has done so because that is what the Bible reveals.

God not only creates something from nothing but he also designs something from something previously existing. Genesis chapter 1 not only states, “And God said, “Let there be light,” and there was light.” (verse 3). It also states, “And God separated the light from the darkness.” (verse 4)

Genesis chapter 2 states, “the Lord God formed the man of dust from the ground and breathed into his nostrils the breath of life, and the man became a living creature.” (verse 7) And then, “the Lord God planted a garden in Eden, in the east, and there he put the man whom he had formed.” (verse 8) And further, “the rib that the Lord God had taken from the man he made into a woman and brought her to the man.” (verse 22).

An unprejudiced reading of these chapters shows God creating from no prior existing thing (e.g., light) as well as from some prior existing thing (e.g., by reforming a part of Adam).

At this point the classical theist may well bring up primary and secondary causality. God could have caused things to exist from nothing (primary), and then those things could cause existing things to change (secondary). That no doubt happens, but is not necessarily the only thing that happens.

The classical theist needs to show that God only creates from nothing and in no case from something. Or show that God’s primary causality is only final causality, and all other causes are secondary causes.

But both final and formal causes are primary causes. Secondary causes are the efficient and material causes. God causes both the end and the form of creation. Since formal causation is an act of design, God is a designer as well as a creator.

Teleological causation is from nothing. Formal causation is from something – whether it is an end (a telos) or a beginning (a material).

Genesis shows God causing kinds of creatures to exist, not mere taxa that differ in degree only. That entails design, a forming of something from something already existing. God does that in the act of creation.

Amateur and independent science

An independent scientist (or gentleman scientist) is someone who pursues scientific research while being independent of a university or government-run research and development body. “Self-funded scientists practiced more commonly from the Renaissance until the late 19th century … before large-scale government and corporate funding was available.” (Wikipedia)

Independent scientists are amateurs in the sense that they are doing scientific research for the love of it (the word is from the French amateur, “one who loves”) rather than as an occupation. They may have an occupation in a related field such as teaching science but their scientific research is done on their own time. Or they may be professional scientists in a specialty other than their research.

I remember years ago hearing the great Hungarian mathematician Paul Erdős remark that an “amateur mathematician” had done work in number theory. He explained that the amateur was a professional mathematician but not a professional number theorist. That made the person an amateur number theorist. It is the same with professionals in any specialty outside their own.

Some great scientists were professors of mathematics, such as Galileo, who was a professor of mathematics at the University of Padua, and Isaac Newton, who held the Lucasian Chair of Mathematics at the University of Cambridge.

In the history of science many breakthroughs have been done by amateurs. Here are some great amateurs or independent scientists:

Albert Einstein – physics
Antonie van Leeuwenhoek – microbiology
Charles Darwin – biology
Gregor Mendel – genetics
Joseph Priestley – chemistry
William Herschel – astronomy

One could add others who were primarily inventors such as Thomas Edison and the Wright brothers, since science is often given credit for inventions.

On a related note, Robert A. Stebbins wrote Amateurs, Professionals, and Serious Leisure (McGill, 1992) and other works on productive uses of one’s free time.

Measures of motion

This post follows others such as the one here and here. A background document is here.

One-dimensional kinematics is like traveling in a vehicle, and on the dashboard are three instruments: (1) a clock, (2) an odometer, and (3) a speedometer. In principle the speedometer reading can be determined from the other instruments, so let’s focus on a clock and an odometer. The clock measures time, which will be used to measure travel time. The odometer measures distance, or length, which will be used to measure travel distance.

From this information one can establish a reference frame to describe space and time. The reference frame includes a point of reference, called the origin (or destination depending on the context), and a unit vector for each dimension, from which one can determine a position vector in space or time. A position vector is the sum of position coordinates times unit vectors. For example, r(t) = x(t) î, with position vector r, position coordinate x, and unit vector î.

The path or route that the vehicle takes is mathematically a curve or arc. The travel distance or length of a trip is its arc length along the path, which is measured by an odometer. The travel time of a trip is its arc time along the path, which is measured by a clock or stopwatch. If the path is a mathematical curve, it can be integrated to find the arc length or arc time:

s = s(t) = ∫ || r′(τ) || , from 0 to t, with arc length s(t) and position vector r(t); or

t = t(s) = ∫ || w′(σ) || , from 0 to s, with arc time t(s) and time position vector w(s).

An interval of time is the arc time from time t1 to time t2. Similarly, an interval of space is the arc length from position s1 to position s2. If time or space are compressed to 1D, then an interval equals the difference between the endpoints: Δt = t2t1 or Δs = s2s1.

The speed of a body is the ratio of the arc length to the arc time: Δst. The pace of a body is the ratio of the arc time to the arc length: Δts.

The displacement of a body during a time interval is defined as the vector change in the position (x) of the body, which for 1D is:

Δrr(t2) − r(t1) = (x(t2) − x(t1)) î ≡ Δx(t) î.

Similarly, the distimement of a body during a space interval is defined as the vector change in the time position (ξ) of the body, which for 1D is:

Δww(s2) − w(s1) = (ξ(s2) − ξ(s1)) î ≡ Δξ(s) î.

The x-component of the average velocity for a time interval Δt is defined as the displacement Δx divided by the time interval Δt:

vavg ≡ Δxt.

Similarly, the ξ-component of the average celerity for a space interval Δs is defined as the distimement Δξ divided by the space interval Δs:

uavg ≡ Δξs.

The x-component of instantaneous velocity at time t is given by the slope of the tangent line to the curve of position vs. time at time t:

vx(t) = dx/dt.

The instantaneous velocity vector is then: v(t) = vx(t) î. Or more generally: v(t) = Σj vj(t) îj.

Similarly, the ξ-component of instantaneous celerity at position s is given by the slope of the tangent line to the curve of time vs. position at position s:

uξ(s) = ds/.

The instantaneous velocity vector is then: u(s) = uξ(s) î. Or more generally: u(t) = Σj uj(s) îj.

The x-component of the instantaneous acceleration at time t is the slope of the tangent line at time t of the graph of the x-component of the velocity as a function of the time: a(t) ≡ dv/dt. The instantaneous acceleration vector at time t is then a(t) ≡ a(t) î.

Similarly, the ξ-component of the instantaneous prestination at position s is the slope of the tangent line at position s of the graph of the ξ-component of the celerity as a function of the position: b(s) ≡ du/ds. The instantaneous prestination vector at position s is then b(s) ≡ b(s) î.

Space, time, arc length, and arc time

Let there be a displacement vector r that is a parametric function of arc time t so that r = r(t). Then define s as the arc length of r so that

s = s(t) = ∫ || r′(τ) || ,

where the integral is from 0 to t. Let us further assume that s is bijective so that the inverse function is t.

Let there also be a distimement vector w that is a parametric function of arc length s so that w = w(s). Then t is the arc time of w if

t = t(s) = ∫ || w′(σ) || ,

where the integral is from 0 to s. Now the arc time derivative of r, that is, the derivative of s with respect to t is

s′ = ds/dt = || r′(t) ||.

And the arc length derivative of w, that is, the derivative of t with respect to s is

t′ = dt/ds = || w′(s) ||.

From the inverse function theorem we have that

t′ = dt/ds = 1/|| r′(t) || = 1/(ds/dt).

And also that

s′ = ds/dt = 1/|| w′(s) || = 1/(dt/ds).

Putting these together we find

s′ = ds/dt = || r′(t) || = 1/|| w′(s) || = 1/(dt/ds).

And also that

t′ = dt/ds = || w′(s) || = 1/|| r′(t) || = 1/(ds/dt).

We then have

s = s(t) = ∫ || r′(τ) || = ∫ 1/|| w′(σ) || dσ,

where the first integral is from 0 to t and the second integral is from 0 to s. And also that

t = t(s) = || w′(σ) || = 1/|| r′(τ) || ,

where the first integral is from 0 to s and the second integral is from 0 to t.

Because of the difficulty of inverting s(t), this shows a bypass is available. That is, from r(t) we find

t(s) = ∫ 1/|| r′(τ) || ,

and from w(s) we find

s(t) = ∫ 1/|| w′(σ) || .

The interpretation is that the space vector, r, is a position vector function of the arc time, t, and the time vector, w, is a time vector function of the arc length, s.

Conservatives and liberals

The terms “conservative” and “liberal” are used in a variety of contexts but what is the distinction? They have come down to us through 19th century politics, but that turns out not to help much since many conservatives today would consider themselves as “classical liberals”. One can use alternate terms such as traditionalist and progressive, but they have various associations of their own.

I would say that the basic distinction is this: conservatives are most concerned with saving something – souls or money or traditions – and liberals are most concerned with spending something – lives or money or resources. That is, conservatives focus on what is worth keeping and liberals focus on what is worth spending.

Religious conservatives want to save souls, to promote what it is that brings salvation, to keep people from being or becoming infidels or unbelievers. Religious liberals want to spend their lives helping people, making the world a better place, doing something that needs to be done.

Economic conservatives want to save money, to buy only necessities, to keep money safe for future needs. Economic liberals want to spend money, to give to the poor, to use money to improve the world now. In the past, this has meant that conservatives had more money than liberals but that is not necessarily true today. Contemporary culture is a spendthrift culture, where most people do not save money either because they have more than enough already or because they live for the present.

Environmental conservatives are “preservationists,” those who value nature for its own sake and want to save it from development. Environmental liberals are “conservationists,” those who want to spend natural resources optimally for the sake of humanity. This is the inverse of what political conservatives and liberals want to do regarding the environment.

Political conservatives want to keep traditions that have worked for generations, to maintain the solvency of governing institutions, to preserve culture and society. Political liberals want to spend resources on improving society, to change what is wrong with society, to remake everything in light of their vision for the world.

In short, conservatives see the glass as half-full, and liberals see it as half-empty. Liberals see what the have-nots need, and conservatives see what the haves could lose. In the past conservatives were considered more pessimistic – seeing what could go wrong – whereas liberals were more optimistic – seeing what could work for the better. But today liberals are almost paranoid about the future – warning of disaster if society doesn’t change radically – whereas many conservatives are content to stay the course with only modest changes.

I have written before, here, about an inversion that can take place between conservatives and liberals. If liberals succeed at changing society enough, then conservatives may long to change things back to where they were before, whereas liberals want to keep their gains. Then liberals will resist change and conservatives will promote a return to what was lost. So conservatives become liberals and liberals become conservatives.

We save in order to have something to spend, and we spend in order to have something to save. The wise counselor advocates balance between these two movements. That is the centrist approach.

Polar coordinates for time-space

This post follows the material on polar coordinates from MIT Open Courseware, here. Instead of the space position vector r, we’ll use the time position vector w, and replace (arc) time with arc length, s.

In polar coordinates, the time position of an eventicle A is determined by the value of the radial duration to the origin, w, and the angle that the radial line makes with an arbitrary fixed line, such as the tx axis (x axis with time metric). Thus the trajectory of an eventicle will be determined if we know w and θ as a function of s, i.e., w(s) and θ(s). The directions of increasing w and θ are defined by the orthogonal unit vectors ew and eθ.

The time position vector of an eventicle has a magnitude equal to the radial duration, and a direction determined by ew. Thus

w = w ew.

Since the vectors ew and eθ are clearly different from point to point, their variation will have to be considered when calculating the celerity and prestination. Over an infinitesimal interval of arc length ds, the coordinates of time point A will change from (w, θ) to (w + dw, θ + ).

We note that the vectors ew and eθ do not change when the coordinate w changes. Thus dew/dw and eθ/dw = 0. On the other hand, when θ changes to (θ + ), the vectors ew and eθ are rotated by an angle .

A mathematical approach to obtaining the derivatives of the unit vectors is to express ew and eθ in terms of their Cartesian components along i and j. We have that

ew =  cos θ i + sin θ j

eθ = – sin θ i + cos θ j.

Therefore, when we differentiate we obtain

dew/dw = 0,   dew/ = – sin θ i + cos θ j = eθ

deθ/dw = 0,   deθ/ = – cos θ i – sin θ j = –ew.

Celerity vector

We can now differentiate w = w ew with respect to arc length and write

u = w′ = wew + w ew′,

where the prime indicates differentiation with respect to arc length. Or, using the above expression for eθ, we have

u = w′ = wew + eθ.

Here, uw = w′ is the radial celerity component, and uθ = ′ is the circumferential celerity component. We also have that $u=\sqrt{u_{w}^{2}+u_{\theta&space;}^{2}}.$ The radial component is the rate at which w changes magnitude, or stretches, and the circumferential component is the rate at which w changes direction, or swings.

Differentiating again with respect to arc length, we obtain the prestination

b = u′ = wew + wew′ + w′θ′eθ + wθ″eθ + wθ′eθ′.

Using the previous relations, we obtain

b = u′ = (w″wθ′²) ew + (wθ″ + 2w′θ′) eθ,

where bw = (w″wθ′²) is the radial prestination component, and bθ = (wθ″ + 2w′θ′) is the circumferential prestination component. Also, we have that $b=\sqrt{b_{w}^{2}+b_{\theta&space;}^{2}}.$

Change of basis

In many practical situations, it will be necessary to transform the vectors expressed in polar coordinates to Cartesian coordinates and vice versa. Since we are dealing with free vectors, we can translate the polar reference frame for a given point (w, θ) to the origin, and apply a standard change of basis procedure. This will give, for a generic vector A,

$\binom{A_{w}}{A_{\theta&space;}}=\bigl(\begin{smallmatrix}&space;cos\,&space;\theta&space;&&space;sin\,&space;\theta&space;\\&space;-sin\,&space;\theta&space;&&space;cos\,&space;\theta&space;\end{smallmatrix}\bigr)&space;\binom{A_{x}}{A_{y}}\:&space;\:&space;\:&space;\textup{and}\:&space;\:&space;\:&space;\binom{A_{x}}{A_{y}}=\bigl(\begin{smallmatrix}&space;cos\,&space;\theta&space;&&space;-sin\,&space;\theta&space;\\&space;sin\,&space;\theta&space;&&space;cos\,&space;\theta&space;\end{smallmatrix}\bigr)&space;\binom{A_{w}}{A_{\theta&space;}}.$

Equations of motion

In two dimensional polar coordinates, the surge and prestination vectors are Γw = Γwew + Γθeθ and b = bwew + bθeθ. Thus, in component form, with vass n, we have

Γw = nbw = n (w″wθ′²)

Γθ = nbθ = n (wθ″ + 2w′θ′).

Intrinsic coordinates for time-space

This post follows the introduction to intrinsic coordinates given here, and changes it for time-space (1D space + 3D time). So rather than the 3D space position vector r, we’ll use the 3D time position vector, w, and we’ll switch the arc time, t, with the arc length, s.

We follow the motion of a point using a time position vector w(s) whose position along a known trajectory in time is given by the scalar function t(s) where t(s) is the arc time along the curve. We obtain the celerity, u, from the space rate of change of the vector w(s) following the particle:

$\mathbf{u}=\frac{d\mathbf{w&space;}}{ds}=\frac{d\mathbf{w&space;}}{dt}\frac{dt}{ds}=&space;\dot{t}\frac{d\mathbf{w&space;}}{dt}.$

We identify the scalar $\dot{t}$ as the magnitude of the celerity u and dw/dt as the unit vector tangent to the curve at the point t(s). Therefore we have

u = u et,

where w(s) is the time position vector, $u=\dot{t}$ is the pace, et is the unit tangent vector to the trajectory, and t is the arc time coordinate along the trajectory.

The unit tangent vector can be written as et = dw/dt. The prestination vector, b, is the derivative of the celerity vector with respect to arc length. Since dw/dt depends only on t, using the chain rule we can write

$\mathbf{b}=\frac{d\mathbf{u}}{ds}=\ddot{t}\frac{d\mathbf{w&space;}}{dt}+\dot{t}\frac{d}{ds}\left&space;(&space;\frac{d\mathbf{w&space;}}{dt}&space;\right&space;)=\ddot{t}\frac{d\mathbf{w&space;}}{dt}+\dot{t}^{2}\frac{d^{2}\mathbf{w&space;}}{dt^{2}}=\dot{u}\frac{d\mathbf{w&space;}}{dt}+&space;\dot{u}^{2}\frac{d^{2}\mathbf{w&space;}}{dt^{^{2}}}.$

The second derivative d²w/dt² is another property of the arc time. We shall see that it is related to the radius of curvature.

Taking the space derivative of u, an alternate expression can be written in terms of the unit vector et as

$\mathbf{b}=\frac{du}{ds}\mathbf{e}_{t}+u\frac{d\mathbf{e}_{t}}{ds}.$

The vector et is the local unit tangent vector to the curve which changes from point to point in time. Consequently, the space derivative of et will, in general, be nonzero. The space derivative of et can be written as

$\frac{d\mathbf{e}_{t}}{ds}=\frac{d\mathbf{e}_{t}}{dt}\frac{dt}{ds}=\frac{d\mathbf{e}_{t}}{dt}u.$

In order to calculate the derivative of et, we note that, since the magnitude of et is constant and equal to one, the only changes that et can have are due to rotation, or swinging. When we move from t to t + dt, the tangent vector changes from et to et + det. The change in direction can be related to the angle (the differential angle).

The direction of det, which is perpendicular to et, is called the normal direction. On the other hand, the magnitude of det will be equal to the length of et (which is one), times . Thus, if en is a unit normal vector in the direction of det, we can write

det = en.

Dividing by dt yields

$\frac{d\mathbf{e}_{t}}{dt}=\frac{d\beta&space;}{dt}\mathbf{e}_{n}=\kappa&space;\mathbf{e}_{n}=\frac{1}{\rho&space;}\mathbf{e}_{n}.$

Here κ = /dt is a local property of the curve, called the time curvature, and ρ = 1/κ is called the radius of time curvature.

From the above we have that

$\frac{d\mathbf{e}_{t}}{ds}=\frac{d\beta&space;}{dt}u\mathbf{e}_{n}=\dot{\beta&space;}\mathbf{e}_{n}=\frac{u}{\rho&space;}\mathbf{e}_{n}.$

Finally, we have that the prestination can be written as

$\mathbf{b}=\frac{du}{ds}\mathbf{e}_{t}+\frac{u^{2}}{\rho&space;}\mathbf{e}_{n}=&space;b_{t}\mathbf{e}_{t}+b_{n}\mathbf{e}_{n}.$

Here $b_{t}=\dot{u}$ is the tangential component of the prestination, and bn = u²/ρ is the normal component of the prestination. Since bn is the component of the prestination pointing towards the center of curvature, it is sometimes referred to as the centripetal prestination. When bt is nonzero, the celerity vector changes magnitude, or stretches. When bn is nonzero, the celerity vector changes direction, or swings. The modulus of the total prestination can be calculated as

$b=\sqrt{b_{t}^{2}+b_{n}^{2}}.$

Relationship between t, u, and b

The quantities t, u, and bt are related in the same manner as the quantities t, u, and b for rectilinear motion. In particular we have that $u=\dot{t},\,&space;b_{t}=\dot{u},\,&space;and\:&space;b_{t}\,&space;dt=u\,du.$ This means that if we have a way of knowing bt, we may be able to integrate the tangential component of the motion independently.

The vectors et and en, and their respective coordinates t (tangent) and n (normal), define two orthogonal directions. The plane defined by these two directions is called the osculating plane. This plane changes from point to point and can be thought of as the plane that locally contains the trajectory (Note that the tangent is the current direction of the celerity, and the normal is the direction into which the celerity is changing).

In order to define a right-handed set of axes we need to introduce an additional unit vector which is orthogonal to et and en. This vector is called the binormal, and is defined as eb = et × en.

At any point in the trajectory, the time position vector, the celerity, and prestination can be referred to these axes. In particular, the celerity and prestination take very simple forms:

u = u et

$\mathbf{b}=\dot{u}\,&space;\mathbf{e}_{t}+\frac{u^{2}}{\rho&space;}\mathbf{e}_{n}$

The difficulty of working with this reference frame stems from the fact that the orientation of the axis depends on the trajectory itself. The time position vector, w, needs to be found by integrating the relation dw/ds = u as follows:

$\mathbf{w&space;}=\mathbf{w&space;}_{0}+\int_{0}^{s}\mathbf{u}\,&space;ds,$

where w0 = w(0) is given by the initial condition.

We note that, by construction, the component of the prestination along the binormal is always zero. When the trajectory is planar, the binormal stays constant (orthogonal to the plane of motion). However, when the trajectory is a time curve, the binormal changes with t. It will be shown below that the derivative of the binormal is always along the direction of the normal. The rate of change of the binormal with t is called the time torsion, σ. Thus

$\frac{d\mathbf{e}_{b}}{dt}=-\sigma&space;\,&space;\mathbf{e}_{n}\:&space;\:&space;or\:&space;\:&space;\frac{d\mathbf{e}_{b}}{ds}=-\sigma&space;u\,&space;\mathbf{e}_{n}.$

We see that whenever the torsion is zero, the trajectory is planar, and whenever the curvature is zero, the trajectory is linear.

Radius of curvature and torsion for a trajectory

In some situations the trajectory will be known as a curve of the form y = f(x). The radius of curvature in this case can be computed according to the expression,

$\rho&space;=\frac{[1+(dy/dx)^{2})]^{3/2}}{|d^{2}y/dx^{2}|}.$

Since y = f(x) defines a planar curve, the torsion σ is zero. On the other hand, if the trajectory is known in parametric form as a curve of the form w(s), where s can be arc length or any other parameter, then the radius of curvature ρ and the torsion σ can be computed as

$\rho&space;=\frac{(\dot{\mathbf{w}&space;}\cdot&space;\dot{\mathbf{w}&space;})^{3/2}}{\sqrt{(\dot{\mathbf{w}}\cdot\dot&space;{\mathbf{w}})(\ddot{\mathbf{w}}\cdot\ddot{\mathbf{w}})-(\dot{\mathbf{w}}\cdot\ddot{\mathbf{w}})^{2}}},$

$\textup{where}\:&space;\dot{\mathbf{w&space;}}=d\mathbf{w&space;}&space;/ds,\,&space;\textup{and}\:&space;\ddot{\mathbf{w&space;}}=d^{^{2}}\mathbf{w}&space;/ds^{^{2}}\:&space;\textup{and}$

$\mathbf{w&space;}=\frac{(\dot{\mathbf{w&space;}}\times\ddot{\mathbf{w&space;}})\cdot\dddot{\mathbf{w&space;}}}{|\dot{\mathbf{w&space;}}\times\ddot{\mathbf{w&space;}}|^{2}}$

$\textup{where}\:&space;\dddot{\mathbf{w&space;}}=d^{3}\mathbf{w&space;}/ds^{3}.$

Equations of motion in intrinsic coordinates

Newton’s second law is a vector equation, Γ = nb (with surge Γ and vass n; cf. F = ma), which can now be written in intrinsic coordinates. In tangent, normal, and binormal components, tnb, we write Γ = Γt et + Γn en and b = bt et + bn en. We observe that the positive direction of the normal coordinate is that pointing toward the center of curvature. Thus, in component form, we have

$\mathit{\Gamma}_{t}=n\,&space;b_{t}=n\dot{u}=n\ddot{t}$

$\mathit{\Gamma}_{n}=n\,&space;b_{n}=n\frac{u^{2}}{\rho&space;}.$

Note that, by definition, the component of the prestination along the binormal direction, eb, is always zero, and consequently the binormal component of the surge must also be zero.

Inverse causes

I’ve written about Aristotle’s four causes before (such as here and here). This also continues the discussion of observers and travelers, here.

Forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector (the device at the end of a robotic arm) from specified values for the joint parameters. Forward kinematics is also used computer games and animation. Inverse kinematics makes use of the kinematics equations to determine the joint parameters that provide a desired position for each of the robot’s end-effectors.

In other words, forward kinematics is for finding out what motion happens given particular inputs, whereas inverse kinematics is for determining how to move to a desired position. In terms of the four Aristotelian causes or explanatory factors, forward kinematics is concerned with the efficient and material causes, and inverse kinematics is concerned with the final and formal causes.

The surprising thing is that these two kinds of causes (higher and lower) are inverses of one another.

 Causes Higher Final Formal Lower Efficient / Mechanism Material

From the lower perspective one begins with some material. From the higher perspective one begins with the objective. From the lower perspective forces and laws make things happen. From the higher perspective following plans gets the job done.

One can see rôles parallel to the causes:

 Rôles Traveler Set the destination Plan the trip Observer Observe the motion See the material

And in robotics (or animation):

 Kinematics Inverse Pick the end position Plan the motions Forward Make the motions Pick the device

One could say that forward kinematics is for scientists and inverse kinematics is for engineers since the latter incorporate objectives and designs in their work but the former are focused on observation only. To go beyond observation scientists would have to open up to formal and final causes.

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