iSoul Time has three dimensions

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Historians and scientists

Historians establish the facts of history, of what happened in the past. They do this with a variety of sources, some documentary, some physical, and whatever else they find is relevant. Key particulars are more significant than universals in establishing the facts of history. Historians may consider scientific theory in doing this, but they may also conclude that some things happened that don’t fit well with current scientific theory. Whether or not there was an earthquake in 1755 that destroyed Lisbon is a matter of history, not science.

Scientists are dependent on historians for the facts of history. Scientists do not get to establish the facts of history, nor the limits of what could have happened in the past. The latter restriction is difficult for scientists to observe. If historians establish facts that don’t fit well with current scientific theory, then scientists are likely to react defensively rather than revise their theories.

Biblical (or creation) scientists consider the Bible as the key to history, and limit science to that which is consistent with biblical chronicles. As with all scientists, they depend on historians for facts about the past but not all historians have a high view of the biblical record. Disagreements among historians lead to variations in science, since they are working with different facts about the past.

The different rôles of historians and scientists are often confused. Astronomy is a case in point. Astronomical historians may work with documents produced by those who could be considered scientists from the distant past. But the interpretation of ancient or medieval scientific documents is not part of science. Astronomical historians deal with the particulars of history, in which universals play only an indirect rôle.

Astronomical scientists deal with universals, as all scientists do, and make use of the facts of history along with recent observations. Scientists may advise historians but science is dependent on history for facts about the past, not the other way around.

Space and time as opposites

A theme of this blog is that space and time are dual concepts, which means they are two ways of understanding the same thing. But in what ways are space and time opposite concepts?

Space is oriented toward its origin, the place that motion begins. Time is oriented toward its destination, the time that motion ends. Both length and duration are measured from an “origin,” a reference point, which is a zero point for each, but zero speed leaves a body in space at the beginning, whereas zero pace puts the body in time at its destination.

Length in the denominator of speed is a measure of the progress from the origin to the current location in space, whereas time in the denominator of pace is a measure of the lag from the destination to the current location in time. A body at zero speed will remain at its origin and never reach its destination, whereas a body at zero pace will arrive at its destination in literally no time. A body with a small speed will take a long time to reach its destination, whereas a body with a small pace will reach its destination quickly.

Large quantities in space correspond to small quantities in time. Large quantities in time correspond to small quantities in space. A high speed is fast, and a small speed is slow. A small pace is fast, and a large pace is slow. Mass and vass are inverses, as are energy and lethargy.

The origin in space corresponds to the destination in time. Time in space flows from the past toward the future. Space (stance) in time flows from the future toward the past.

Arcloge

An arcloge (arc’-loje) is a continuous, independent measurement of length. That is, it measures an ever-increasing length, which is the stance, similar to how a clock shows the time. The term is a combination of arc (as in arc length) and loge (as in horologe, a clock).

What does an arcloge look like? Start with a sector, which is a geometric figure fixed to the center of a circle that sweeps out an angle and a curved edge:

Basic definitions

Independent variable is a quantity that is not dependent on another quantity, which is either (a) a quantity chosen before an experiment or race, or (b) an ever-increasing quantity. Dependent variable is a quantity that whose value is a function of another variable.

Space is (1) length; (2) a 3D differentiable manifold of length; (3) the order of events on a stance line; (4) the stance, the reading on an arcloge.

Time is (1) duration; (2) a 3D differentiable manifold of duration; (3) the order of events on a time line; (4) the time, the reading on a clock.

Length is measured by a rigid rod. 3D space is length measured in three directions of motion.

Duration is measured by a stopwatch, timer, or clock. 3D time is duration measured in three directions of motion.

Spacetime is the 6D manifold formed from 3D space and 3D time. Worldline is the path in spacetime traced out by an object in motion.

Displacement is the vector between two points (events) on a worldline. Distance is the magnitude of a displacement.

Dischronment is the vector between two points (events) on a worldline. Distime is the magnitude of a dischronment.

Reference frame (or frame) is an abstract coordinate system and set of reference points in 3D space that uniquely fix the coordinate system and standardize measurements. Rest frame of a body is the reference frame in which the body is moving at zero speed, which is the time conversion pace.

Reference timeframe (or timeframe) is an abstract coordinate system and set of reference points in 3D time that uniquely fix the coordinate system and standardize measurements. Freeflow frame of a body is the reference timeframe in which the body is moving at zero pace, which is the stance conversion speed.

Proper length is the length of a body measured by a rigid rod moving with it. Proper time is the time of a body measured by a clock moving with it.

Lorentz transformation is a set of equations that relate space and time coordinates of reference frames moving at a constant velocity relative to each other.

Dual Lorentz transformation is a set of equations that relate space and time coordinates of reference timeframes moving at a constant legerity relative to each other.

From racing to relativity

There are three different contexts for 3D time, depending on whether stance is continuously increasing and, if so, whether there is a conversion factor between space and time:

(A) Stance is not continuously increasing. This is the situation of a race or sport in which game time has a definite beginning and ending. For example, in many sports the game lasts a specific time. In a race, the length of the course is set and the time for each contestant ends when they cross the finish line. The average pace of a contestant is their race time divided by the course length.

(B) Stance is continuously increasing and there is a general conversion factor between space and time. This is the situation of the special theory of relativity and some transportation settings in which the conversion pace is the minimum pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time interval is measured. Without a time interval increase the pace is at a minimum (or the speed is a maximum). As the amount of time measured increases, the pace increases (or the speed decreases). Remember that a small amount of time per unit distance is a fast motion, whereas a large amount of time per unit distance is a slow motion.

In this way, the pace increases indefinitely. A pace of infinity would be at rest. A pace of zero is the minimum pace, which in relativity is the speed of light. That is, the speed counts down from the speed of light. This has been misinterpreted as a transformation with superluminal speeds, but because speed decreases as pace increases, object speeds are subluminal.

The dual Lorentz transformation (see here) is

$x'=\lambda&space;(x&space;-&space;ur);\;&space;y'=y;\;&space;z'=z;\;&space;r'=\lambda&space;(r&space;-&space;c^2ur)&space;\;&space;\textup{with}\;&space;\lambda&space;=&space;1/{\sqrt{1-cu}}$

with the understanding that c represents the inverse of the pace of light. The cu in λ is the pace of the object divided by the pace of light, with the stance increasing at the conversion rate. As the time of motion increases, the pace increases (and the speed decreases) from that of light toward the pace or speed of rest. So, the square root never becomes negative here.

(C) Stance is continuously increasing but there is no general conversion factor between space and time. This is the situation of general relativity and transportation in general. Conversion of space and time are local, not global, and the optimal route depends on whether space or time are optimized.

History and theology

What follows are excerpts from Ramsay MacMullen’s book Christianizing the Roman Empire, A.D. 100-400 (Yale, 1984). He begins with historiography pointers relevant to religious history.

My subject here is the growth of the church as seen from the outside, and the period is the one that saw the church become dominant, and Europe Christian. p.vii

My object is history. It might be, but it isn’t, theology. Accordingly, my view focuses naturally upon significance, the quality of weight that distinguishes historical phenomena from the (sometimes much more engrossing or at least more diverting) items of merely human interest that we see in the headlines of certain newspapers: ‘‘Mom Axes Babe” or the like. Significance, in its turn, indicates the degree to which many people, not just a few, are made to live their lives differently in respects that much engage their thoughts, not in respects they do not think about very carefully. … Significance must be compounded of both “many” and “much,” in a sort of multiplicand of the two elements. p.1

This is all elementary. Still, it needs to be said in order to explain the inclusion in my account of scenes not usually given much attention in books about church growth, scenes in which large numbers of persons are brought to a change in their religious allegiance, but namelessly—they are just ordinary folk of no account—and without great dramatic, further consequences in their manner of life. I think these scenes need to be included, along with Saint Augustine and a handful like him, because otherwise we would see only a church all head and no body, a phenomenon that affected only a few lives, a change without mass and therefore without historical significance. And that is the exact opposite of the truth. p.1

The process we are tracing, of the slow but gigantic growth of a community of believers, seems thus to have had at its heart a psycho-logical moment that might have been, though it was not always, quite uncomplicated; and that fact belongs by right, and not by later development, to the whole long process of ecclesiastical maturing. From the very beginning, Jesus’ disciples followed him instantly, without instruction; new adherents, by supernatural actions, were won to instantaneous belief, or trust (πιστις, “commonly mistranslated, ‘Your faith’ …,” with implications of doctrine, as has been pointed out). p.3-4

There is an obvious connection between simplicity of belief and rapidity of conversion: the simpler the set of ideas with their attendant feelings, the shorter must be the period of transition to the new. Which is not to say that much longer, complicated transitions may not also have had their abrupt moments, like Saint Augustine’ in the garden near Milan. The point is worth stressing because the more richly intellectual and dramatically interesting conversions naturally hold our attention best, and are most written about. p.4

Invariant intervals

The spacetime interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, along with β = v/c and γ = 1/√(1 − β²):

$\Delta&space;x=\gamma&space;(\Delta&space;x'+\beta&space;c\Delta&space;t');\;&space;\Delta&space;y=\Delta&space;y'\;&space;\Delta&space;z=\Delta&space;z';\;&space;c\Delta&space;t=\gamma&space;(c\Delta&space;t'+\beta&space;\Delta&space;x').$

The invariant interval is

$(\Delta&space;s)^2&space;=&space;(c\Delta&space;t)^2&space;-(\Delta&space;x)^2&space;-(\Delta&space;y)^2&space;-(\Delta&space;z)^2$

$=\gamma&space;^2(c\Delta&space;t'+\beta\Delta&space;x')^2&space;-\gamma&space;^2(\Delta&space;x'+&space;\beta&space;c\Delta&space;t')^2&space;-\Delta&space;y'^2&space;-\Delta&space;z'^2$

Expand the squares and cancel the middle terms to get:

$=\gamma&space;^2(c^2&space;\Delta&space;t'^2(1-\beta&space;^2)&space;-\Delta&space;x'^2(1-\beta&space;^2))&space;-\Delta&space;y'^2&space;-\Delta&space;z'^2$

$=\gamma&space;^2(c^2&space;\Delta&space;t'^2(1/\gamma&space;^2)&space;-\Delta&space;x'^2(1/\gamma&space;^2))&space;-\Delta&space;y'^2&space;-\Delta&space;z'^2$

$=(c\Delta&space;t')^2&space;-(\Delta&space;x')^2&space;-(\Delta&space;y')^2&space;-(\Delta&space;z')^2.$

The spacetime interval is invariant over the dual Lorentz transformation (DLT). The following is a proof of this for the inverse DLT follows with temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity κ, along with ζ = u/κ and λ = 1/√(1 − ζ²):

$\Delta&space;x=\lambda&space;(\Delta&space;x'+&space;\zeta&space;\kappa&space;\Delta&space;r');\;&space;\Delta&space;y=\Delta&space;y'\;&space;\Delta&space;z=\Delta&space;z';\;&space;\kappa&space;\Delta&space;r=\lambda&space;(\kappa&space;\Delta&space;r'+\zeta&space;\Delta&space;x').$

Symmetric transformations

What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.

For the (3+1) Lorentz transformation there are spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):

$x'=\gamma&space;(x&space;-&space;vt);\;&space;y'=y;\;&space;z'=z;\;&space;t'=\gamma&space;(t&space;-&space;vx/c^2).$

The symmetric form is

$x'=\gamma&space;(x&space;-\beta&space;ct);\;&space;y'=y;\;&space;z'=z;\;&space;ct'=\gamma&space;(ct&space;-&space;\beta&space;x).$

The symmetric Lorentz transformation for vectors is

$\mathbf{r'}_\perp&space;=&space;\mathbf{r}_\perp;\;&space;\mathbf{r_\parallel&space;}'=\gamma&space;(\mathbf{r_\parallel&space;}&space;-&space;\boldsymbol{\beta&space;}&space;ct);\;&space;ct'=\gamma&space;(ct&space;-&space;\boldsymbol{\beta&space;}&space;\cdot&space;\mathbf{v}),\;&space;\textup{with&space;}&space;\boldsymbol{\beta&space;}&space;=\mathbf{r_\parallel&space;}/c.$

For the (1+3) dual Lorentz transformation there are temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):

$x'=\lambda&space;(x&space;-&space;ur);\;&space;y'=y;\;&space;z'=z;\;&space;r'=\lambda&space;(r&space;-&space;c^2ux).$

The dual symmetric form is

$x'=\lambda&space;(x&space;-&space;ur);\;&space;y'=y;\;&space;z'=z;\;&space;\frac{r'}{c^2}=\lambda&space;(\frac{r}{c^2}&space;-&space;ux).$

The dual symmetric Lorentz transformation for vectors is

$\mathbf{t'}_\perp&space;=&space;\mathbf{t}_\perp;\;&space;\mathbf{t_\parallel&space;}'=&space;\lambda&space;(\mathbf{t_\parallel&space;}&space;-&space;\boldsymbol{\zeta&space;}\frac{r}{c});\;&space;\frac{r'}{c}=\lambda&space;(\frac{r}{c}&space;-&space;\boldsymbol{\zeta&space;}&space;\cdot&space;\mathbf{u}),\;&space;\textup{with&space;}&space;\boldsymbol{\zeta&space;}=c&space;\mathbf{t_\parallel&space;}.$

Ignatowsky relativity

Vladimir Ignatowski (1875-1942) was a Russian physicist. “In 1910 he was to first who tried to derive the Lorentz transformation by group theory only using the relativity principle (postulate), and without the postulate of the constancy of the speed of light.” K M Browne gave a simplified derivation in the European Journal of Physics, 39 (2018) 025601, from which the key steps are presented below, followed by the corresponding steps for a dual transformation, switching space and time.

This is a derivation of the Ignatowsky transformation in which the axes x, y, and z are taken to represent space axes rx, ry, and rz with time t. The relativity postulate is taken to be: a valid relativistic transformation must be identical in all inertial frames.

Step 1. To find a valid transformation, we take the usual inertial reference frames S and S′ (the latter moving at velocity v in the +x direction relative to the former) for which intra-frame space is Euclidean but inter-frame space (measured from one frame to the other) may be non-Euclidean. Linear equations are necessary so that an event in one frame appears as a single event, without echoes, in the other. Initial conditions are x′ = x = 0 when time t′ = t = 0. We expect the generalised x equation to be the Euclidean equation x′ = xvt with an added multiplier, and if time is the fourth dimension, then the time equation will be similar but with two additional multipliers. The second of these, n, having the dimension of inverse velocity squared, is required to make the equation dimensionally correct. The y and z coordinates are not expected to be affected by x and t. The generalised transformation and its inverse are then

Biological classes and ancestries

Taxonomy is the science of classification. Taxonomy applied to biology is a systematic approach to classifying organisms. It can be applied to all organisms at a particular time, throughout time, or within any context. Once a classification is determined, other questions arise such as whether there is an independent reason that organisms are in the same class together.

The basic question in all classifications is whether the objects to be classified fit within a class or belong to another class. The goal of a classification is to minimize the within-class differences and maximize the between-class differences. This is often done by defining a distance metric that quantifies the differences.

Carl Linnaeus is known as the father of modern taxonomy who formalized the binomial nomenclature and called the lowest classes species and genus (no doubt after Aristotle’s method of defining with species and genera). His original expectation was that these biological species were natural kinds that do not change over time. With the discovery that fossils came from dead organisms, it became clear that some of his species had changed over time.

The solution to this problem was to reclassify organisms both living and dead in a new classification system. But this was easier said than done since it took years for fossils to be examined. Meanwhile, people were anxious to know how all the diversity of species arose.

Charles Darwin’s hypothesis was soon adopted: species are temporary population groupings with universal common ancestry. If all species are temporary, there is only one fixed class: the class of all species. Others hypothesize  there are classes of species that are fixed and have separate ancestries, which supports design or special creation.

How can this dispute be resolved? Elliott Sober compares these two hypotheses in his book Evidence and Evolution. Sober argues for a likelihood approach to determining the better of two hypotheses. The law of likelihood states that evidence E favors hypothesis H1 over H2 if and only if the probability of E given H1 is greater than the probability of E given H2, or in symbols, P(E | H1) > P(E | H2). Note that this is a comparative approach; it only works when comparing two specific hypotheses.

In this case, the context is all species on the earth over all the history of life on earth. Hypothesis H1 states that there are multiple classes of species that span the history of life on earth, each class with separate ancestry. Hypothesis H2 states that there is only one class of species that span the history of life on earth, all with common ancestry.

Sober notes that Darwin routinely inferred common ancestry if there was some similarity between species. Sober calls this modus Darwin. It is better to have an overall metric of distance between species than rely on a few similarities. However, there is no generally accepted distance metric for species. In its absence, we can still make some inferences.

If there are many similarities between two species, that evidence is more likely given hypothesis H2 (common ancestry), though there is some likelihood given hypothesis H1. If there are discontinuities between two species, even if there are some similarities, that evidence is more likely given hypothesis H1 (separate ancestry).

Note that if someone proposes a possible sequence of events that explains a discontinuity given hypothesis H2, it is merely a possibility and lacks likelihood. But since hypothesis H1 includes partial common ancestry, it is likely with evidence of similarities as well as differences. The conclusion from this exercise is that separate ancestry is the superior hypothesis.

A problem arises when proponents of common ancestry insist there must first be an explanation of how these separate lines of ancestry originated. The best answer is that, just as abiogenesis is not part of the common ancestry hypothesis, so the origin of the separate classes of species is not part of the separate ancestry hypothesis.