iSoul In the beginning is reality

Algebraic relativity

Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t:

dx² = (dx/dt)²dt² = dv²dt² =  d(vt)².

Let there be a constant, c:

dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 – (1 – v²/c²)).

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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.

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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
Michael Faraday – electromagnetism
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.

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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 moticle 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 moticle 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.

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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 allegrity, u, from the space rate of change of the vector w(s) following the particle:

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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.

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:

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

And in robotics (or animation):

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.