iSoul In the beginning is reality

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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.

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

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

 

Essentials of Christian Thought, part 2

This post continues from part 1, here.

One characteristic of the book is that the “essentials” or “metaphysics” that Roger E. Olson elucidates are somewhat buried among the text dealing with the competing alternatives. What follows are excerpts that focus on the essentials of Christian/biblical thought itself.

A basic presupposition of this book is that the Bible does contain an implicit metaphysical vision of ultimate reality—the reality that is most important, final, highest, and behind everyday appearances. p.12

Ultimate reality is relational. p.13

Ultimate reality is personal, not impersonal, and humans reflect that ultimate reality in their created constitution—what they are. Here we will call that “Christian humanism.” p.17

Here metaphysics is simply another word for investigation into the nature of ultimate reality. p.19

… both Tresmontant and Cherbonnier argued very cogently that the biblical philosophy is holistic, not requiring supplementation by extrabiblical philosophies … and that the biblical philosophy is fundamentally contrary to Greek philosophies. p.22

in this postmodern age every philosophy is rooted in some story and tradition based on it, and that for the Christian “the Bible absorbs the world”—the biblical story, narrative, is the lens through which the Christian sees reality as God’s good creation (for example). p.23

belief in the supernatural (something above and free from nature and nature’s laws) is no more a matter of faith, “seeing as,” than belief in naturalism (that nature and its laws are all that are real). p.33

The biblical-Christian vision of reality is a “view from somewhere,” … that … better answers life’s ultimate questions than any competing worldview or metaphysical vision of reality. p.39-40

… Christian theology’s main task is not correlation with other, non-Christian worldviews or plausibility structures, but self-description of the Christian view of reality from within the Christian tradition-community inspired by the biblical story. p.41

… being Christian means, in part, seeing the world as the reality described, or presupposed, by the Bible. p.43

… [Hans Frei] argued that faithful Christians ought to take the Bible seriously as “realistic narrative.” In other words, the Bible ought not to be viewed either as history in the modern, literal sense (viz., a textbook of facts about history) or as myth (symbolic representation of universal human experience). Rather, a Christian should find the meaning of Scripture out outside it—whether in outer history or universal human experience—but inside of it. p.43

Frei’s point is simply that the meaning of the Bible is not outside of it. p.44

The Bible depicts ultimate reality—the highest, best, final, eternal reality upon which all else is dependent—as supernatural and personal but not human. Here supernatural simply means “beyond nature,” not bound to nature and nature’s laws, free over nature, not controlled by nature. Some people would prefer the word transcendent for all that … p.53

The Bible depicts ultimate reality as personal, which here means having intelligence, thought, iintentions, actions, and some degree of self-determination. It also means “relational”—being in relation to others, drawing one’s identity partly, at least, from relations with others. p.53

… the long history of philosophical metaphysics, from Plato in ancient Greece to Hegel in nineteenth-century German, has tended to depersonalize ultimate reality, to represent ultimate reality as impersonal, a power, force, or principle behind appearances. p.56

… the ultimate reality of the Bible, Yahweh, God the Lord, is personal in the primary, supreme sense, the pattern of true personhood, which human beings are personal in the secondary sense, copies of the pattern of true personhood. p.57

In Athens Paul articulated concisely what later Christian thinkers came to refer to as God’s transcendence and immanence—that God is both present within creation and exalted above creation as its source and sustainer who needs nothing. p.62

Summing up, the biblical view of ultimate reality is that it is not an it but a he. According to the biblical narrative … ultimate, final, eternal, all-powerful, all-determining reality is a personal being both beyond the natural world and dynamically present within it. This metaphysical vision has variously been labeled “personalistic theism” and “biblical theistic personalism.” At the heart of ultimate reality, the one unifying source behind and withing everything, is an intelligence, free agency, and independent will marked by loving-kindness and justice. p.63

The next post in this series is here.

Essentials of Christian Thought, part 1

The Essentials of Christian Thought: Seeing Reality Through the Biblical Story by Roger E. Olson was published by Zondervan in 2017. It’s 256 pages long in seven chapters with as many “Interludes” but no bibliography or index. The author gives a video introduction here.

The intended audience for the book is those who accept the Bible as a guide “to the nature of ultimate reality” (p.11). Its purpose is to describe that ultimate nature (or metaphysics) according to the Bible. Much of the book is spent delineating differences between the biblical metaphysics and that of others. The author leans heavily on four authors (in order of the number of references):

Edmond La Beaume Cherbonnier (1918 – 2017), “an American scholar in the field of religious studies. He served as Professor of Religion at Trinity College, Connecticut”. Wikipedia

“Is There a Biblical Metaphysic?”, Theology Today, 15(4), January 1959, pp. 454–69.
Hardness of Heart, Doubleday, 1955.
“Biblical Metaphysic and Christian Philosophy”, Theology Today, 9(3), October 1952.
“The Logic of Biblical Anthropomorphism,” in Harvard Theological Review 55(3), 1962, 187-206.

Claude Tresmontant (1925 – 1997), “taught medieval philosophy and philosophy of science at the Sorbonne.” Wikipedia

A Study of Hebrew Thought, tr. by Michael Francis Gibson, Descle, 1960.
Christian Metaphysics, Sheed and Ward, 1965.
The Origins of Christian Philosophy, Hawthorn Books, 1963.

Emil Brunner (1889 –1966), “a highly influential Swiss theologian who, along with Karl Barth, is associated with Neo-Orthodoxy or the dialectical theology movement.” Theopedia

The Philosophy of Religion from the Standpoint of Protestant Theology, tr. by Bertram Lee Woolf, James Clarke, 1958.
Revelation and Reason: The Christian Doctrine of Faith and Knowledge, tr. by Olive Wyon, SCMP, 1946.
“Nature and Grace” in Natural Theology, tr. by Peter Fraenkel, Geoffrey Bles, 1946.

Abraham Joshua Heschel (1907 – 1972) “was a Polish-born American rabbi and one of the leading Jewish theologians and Jewish philosophers of the 20th century.” Wikipedia

Man is Not Alone, Farrar, Straus and Giroux, 1976.

The next post is here.

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.