iSoul In the beginning is reality.

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Newton’s laws and their duals

The following is based on Classical Mechanics by Kibble and Berkshire, 5th ed., Imperial College Press, 2004, with the dual version indented and changes italicized.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

The most fundamental assumptions of physics are probably those concerned with the concepts of length and duration. We assume that length and duration are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific instant of duration space, and that there are universal standards of length and duration (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal length scale (in the sense that two observers who have symbasalized their clocks will always agree about the base of any event), that the geometry of time is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all chronations and legerities.

Read more →

Interchange of length and duration spaces

In geometry, a spherical coordinate system specifies points by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle, which is the horizontal angle between the origin to the point of interest.

A spherical coordinate system is implicitly behind length space (space-time) (3+1), with the temporal directional ignored for scalar time. That is, every instant in duration space (3D time) is projected onto a temporal sphere centered on the origin instant. The scalar time is the radial distime of each instant.

Let the spherical coordinates of length space be (r, θ, φ) with r representing the radial distance, and θ and φ representing the zenith and azimuth angles, respectively. Let the spherical coordinates of duration space (3D time) be (t, χ, ψ) with t representing the radial distime, and χ and ψ representing the temporal zenith and azimuth angles, respectively.

To represent the fullness of length space and duration space requires six dimensions (3+3), three for length and three for duration: ((r, θ, φ); (t, χ, ψ)) or (r, θ, φ; t, χ, ψ). Then length-duration (space-time) (3+1) can be represented by the coordinates [r, θ, φ; t] and duration-length (time-space) (1+3) by the coordinates 〈r; t, χ, ψ〉.

If rectilinear coordinates are used for duration space, say (ξ, η, ζ), then the time, i.e., radial distime, t = √(ξ² + η² + ζ²). The corresponding length space concept, base, is the radial distance. If rectilinear coordinates are used for length space, say (x, y, z), then the base r = √(x² + y² + z²). For 2D applications such as mapping, polar coordinates would be used instead of spherical, in which case r = √(x² + y²) and t = √(ξ² + η²).

To convert (3+1) function to a (1+3) function requires expansion to its (3+3) function, inversion by interchange of length and duration, and then contraction to (1+3). In symbols with ↑ as expand, ↓ as contract, and ↔ as interchange: (3+1) ↑ (3+3) ↓ (1+3), or (r, θ, φ; t) ↑ (r, θ, φ; t, χ, ψ) ↓ (r; t, χ, ψ). In this way length and duration, time and base are interchanged.

In more detail, a parametric length space function is converted to a parametric time function: r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r) = [t, χ, ψ] = t.

Take for example the definition v = dr/dt. We have: v = dr/dt = [dr/dt, dθ/dt, dφ/dt] = [r(t), θ(t), φ(t)]) ↑ [(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ ((r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = [dt/dr, dχ/dr, dψ/dt] = dt/dr = u. The result is that length space and duration space are interchanged, with length space vectors becoming bases and times becoming duration space vectors.

Functions may be converted by differentiation, then inversion, then integration. Take for example, the function s(t) = s0 + v0t + ½at². Differentiating twice leads to s(t)´´ = a = dv/dt. Expanding to spherical coordinates, inverting, and contracting to the radial component results in t(s)´´ = du/ds = b. Integrating twice produces t(s) = t0 + u0s + ½bs², which has the same form as the original function.

Speaking of reality

Anti-realism has been popular among the elites for some time. This has led to anti-realist speech spreading to the mass media and general culture. It has also led to much confusion and foolishness. One wonders how it will end, but reality can be averted only so long.

As a start toward speaking of reality the following terms are offered. Note that “pseudo” has been added to anti-realist conceits. If not now, then at some point people will be speaking of reality and will need some terms such as these.

Many terms could be used for the sexual obsessions of the elites. There is pseudo-sex, which means the false couplings of same-sex duos; and the pseudo-sexed, which means the false identities of those who reject biological sex. There are male and female variations of these false couplings and identities as well.

Since these pseudo-couplings have been legalized, there are pseudo-marriages, pseudo-weddings, and pseudo-spouses, too. Some claim to be in transition between their sex and a pseudo-sex, as if there were a middle ground between true and false. The law of the excluded middle has no exceptions so we have pseudo-trans, which is a kind of pseudo-squared.

Politics is much infected with anti-realism as well. There are the pseudo-progressives, who want western civilization to return to something like its pre-Christian condition. Pseudo-liberals want less liberty for the people and less protection for the unborn. Pseudo-conservatives are trying to change things back to a non-existent past.

Even science has fallen for anti-realism. There is pseudo-time, sometimes called deep time, which is the invented world that supposedly existed before time began to be measured (so much for empiricism). This leads to pseudo-history, which is history supposedly turned into a natural science, or rather a pseudo-science. This includes many pseudo-events that no one ever observed and pseudo-dates that no one ever recorded.

To this may be added the attempts to turn reality upside-down with pseudo-heros and pseudo-villians, the pseudo-art, pseudo-music, and pseudo-literature that turn from reality, and the pseudo-religions and pseudo-scriptures that worship a pseudo-god.

One wonders if any area of culture has not been infected with anti-realism. While few will accept these new terms today, there will come a time when many will return to reality. This is written for them.

Republican representation

This post builds on previous ones, such as here.

In the year 507 B.C., the Athenian leader Cleisthenes introduced a system of political reforms that he called demokratia, or “rule by the people.” This system was comprised of three separate institutions: the ekklesia, a sovereign governing body that wrote laws and dictated foreign policy; the boule, a council of representatives from the ten Athenian tribes; and the dikasteria, the popular courts in which citizens argued cases before a group of lottery-selected jurors. (reference)

The ancient Roman Senate was composed of patricians, members of the ruling families, who wielded varying amounts of influence and power in the Roman monarchy, republic, and empire. This aristocratic body is the forerunner of upper chambers of legislatures in the modern age.

Modern parliaments are descendants of the ancient ekklesia in single-chamber democracies. A legislative body of democratically-elected representatives is sufficient for this kind of democracy. A broader democracy includes two legislative chambers, with the lower chamber representing the people and an upper chamber representing tribes, ruling families, or key subdivisions of the country, that is, the land.

Representation of the traditional tribal, familial, or territorial alliances is important since they are the gluten than holds society together. While political principles and traditions are important, they alone cannot keep a society from separating, since they have no inherent attachment to a people or a place. There must be something so that a group of people are invested in the good of the country.

Hence a legislative body is needed that is tied to something tribal, familial, or territorial. In order to go beyond mere tribal or familial alliances, the territories of the people must be represented. A legislative body whose representation is not based on population will also mute the influence of gerrymandering.  The democratic approach to representation is through election so legislative divisions by territory are represented by the people who live in each territory.

One could go further and require that the electorate consist of those who live on land they own in the territory — or those who own their residence in the territory. These people are invested in the place. A republic includes both territorial-based representation and population-based representation. Hence a republic needs two legislative bodies with two different kinds of representation.

Dual calendar systems

The unit for all calendars is the day, the diurnal cycle of daylight and night. A lunar calendar is based on the monthly (synodic) cycle of the Moon’s phases. A solar calendar is based on the annual cycle of the Sun’s height above the horizon. A lunar-solar (lunisolar) calendar is based on the lunar month modified in order to match the solar (or sidereal) year. The solar-lunar calendar is based on the year but includes months similar to the lunar cycle.

“The lunisolar calendar, in which months are lunar but years are solar—that is, are brought into line with the course of the Sun—was used in the early civilizations of the whole Middle East, except Egypt, and in Greece. The formula was probably invented in Mesopotamia in the 3rd millennium bce.” (Encyclopedia Britannica)

The lunar and lunar-solar (lunisolar) calendars are the oldest calendar systems, and are still used in some traditional societies and religions. The Hebrew (Jewish) and Islamic calendars are examples of the lunar-solar calendar systems. Solar and solar-lunar calendar systems came from Egypt, Greece, and Rome. The solar-lunar month departs from the lunar month but combines to equal a year.

The question is why the Moon forms the primary cycle in some calendars, whereas the Sun forms the primary cycle in other calendars. The reason may well be that some societies think in terms of 3D time, whereas other societies think in terms of 3D space. The difference is that in 3D space the Earth revolves around the Sun and the Moon revolves around the Earth, whereas in 3D time the Earth revolves around the Moon and the Sun revolves around the Earth. In the former case the solar cycle is primary, whereas in the latter case the lunar cycle is primary.

When European societies considered the Earth to be the center of all celestial motion, their calendars were already established. So the correspondence between calendar systems and the dominant perspectives (spatial or temporal) applies to the original development of calendars.

Contraries as duals

Contrariety is a property of pairs of propositions, but it also applies to pairs of terms or concepts. “Two general terms are contraries if and only if, by virtue of their meaning alone, they apply to possible cases on opposite ends of a scale. Both terms cannot apply to the same possible case, but neither may apply.” (Aristotelian Logic, Parry and Hacker, p. 216) Opposite ends of a scale are also called extremes, which are contrasted with means between the extremes.

Every pair of contraries forms a duality by inverting the scale of which they are opposites. For example, quantitative contraries such as rich and poor become poor and rich when the scale is inverted. Every measurement scale can be inverted so in this sense a measurement and its inverse are a contrary pair that forms a self-duality. Every ratio or function of two variables, f(x, y), can be interchanged and form a duality, f(y, x). For example, the equation v = Δst can be interchanged to become u = v-1 = Δts.

The scale may be qualitative, too. For example, the qualitative contraries up and down become down and up, respectively, by looking upside-down. The contraries left and right become right and left when looked at facing the other way. Extension and intension are opposites that may be inverted by interchanging them with each other. Compare the duality of top-down and bottom-up perspectives.

“A pair of terms is contradictory if and only if by virtue of their meaning alone each and every entity in the universe must be names by one or the other but not both.” (Aristotelian Logic, Parry and Hacker, p. 216) May the terms X and not-X be made into duals? That depends. If not-X is the contradictory of X and means everything other than X, that includes things that are non-dual. But in some cases, not-X means the opposite of X, so that contraries are indicated.

Science, unity and duality

It is a Christian concept (or at least a theistic concept) that the world we inhabit is a universe. The existence of the universe requires there to be a perspective that encompasses the whole of the world, which is the perspective of a transcendent divinity. The universe is thus the whole of creation.

It is said that natural science studies the universe, but natural science today does not recognize a transcendent being, and so cannot genuinely recognize the universe. What can natural science recognize as the world that it investigates?

Natural science recognizes law and chance, the regular and the stochastic, but what determines the mix of law and chance? There are three possibilities: (1) the mix of law and chance is determined by law, in which case science investigates a cosmos; (2) the mix of law and chance is determined by chance, in which case science investigates a chaos; or (3) the mix of law and chance is determined by another mix of law and chance, which, if this duality continues at every level, indicates a dualism of law and chance as two independent principles for science to investigate.

Natural science seeks unity, so option (3) is distasteful. Option (2) is distasteful for aesthetic reasons, as well as for its lack of meaning. Option (1) is the least distasteful, and the science community increasingly states that they investigate a cosmos, a world of order that we inhabit. But mere law and order seems fatalistic, and the reality of chance keeps rearing its head, which undermines (1).

This pattern of seeking unity and finding duality occurs in other ways, too. Space and time are duals, but can they be unified by space or time? Either space alone is real (and time is unreal), or time alone is real (and space is unreal), or there is a duality of space and time that cannot be unified. Again, the first option is the most popular, though it has the same weaknesses as above.

The most satisfying answer for these dualities is that science investigates a universe, a unity that can be fully grasped only transcendently, but may be glimpsed by us. This gives us confidence that there is a unity, even if we haven’t yet found how that unity is shown by observation and experimentation. It is a qualified unity, which is not troubled by duality, and does not seek to force unity on a diverse universe.

Length and duration

Let us begin with (1) the motion of a body between two events and (2) two ways of measuring the extent of that motion: length and duration (or time). The measurement of length and duration is coordinated so that both measures are of the same motion. Length and duration are measured by a rigid rod and a stopwatch, respectively. A smooth manifold of length is called space (or 3D space), and a smooth manifold of duration is called time (or 3D time).

The length and duration of a motion are commonly measured along the trajectory (or arc) of the motion. The length along the trajectory of motion is the arc length (or proper length or simply length). The duration along this trajectory is the arc time (or proper time or simply time).

Once the length and duration are in hand, the next step is to form their ratio. The ratio with arc time as the independent variable and arc length as the dependent variable is the speed. Which is to say, speed is the time rate of arc length change.

Note that the ratio could just as well be formed in the opposite way, with the arc length as the independent variable and the arc time as the dependent variable. This ratio is called the pace from its use in racing, in which an arc length is first set and then the racer’s arc time is measured. As another way to state this, pace is the space rate of arc time change.

The reference trajectory for measuring the length of a motion is the minimum length trajectory between two event points. The length along this trajectory is the distance between the two event points, which forms the metric of space. Distance is represented as a straight line on a length-scale map.

The reference trajectory for measuring the duration of a motion is the minimum time trajectory between two event instants. The duration along this trajectory is the distime between the two event instants, which forms the metric of time. On a map, two isochrons are separated by a constant distime. Distime is represented as a straight line on an time-scale map.

Motion has direction as well as extent, and direction may also be measured in two ways. Consider the motion of rotation, which can be measured as a proportion of a circle and as a proportion of a cycle. For example, in an analogue clock a minute hand that moves the length of a right angle correspondingly moves a duration of 15 minutes and vice versa. The direction of motion may be measured by either length or duration.

A motion measured with length direction and distance comprises a vector displacement. A motion measured with time direction and distime comprises a vector dischronment. The ratio with time as the independent variable and displacement as the dependent variable is called the velocity. The magnitude of the velocity vector is the speed. The ratio with distance as the independent variable and dischronment as the dependent variable may be called the legerity. The magnitude of the legerity vector is the pace.

There are three dimensions of motion, and correspondingly three dimensions of length and duration. The three dimensions of length comprise space. The three dimensions of duration comprise time. The three dimensions of time come as a surprise, since the distime is often a parameter for ordering events. But the scalar distime should not be confused with the vector dischronment, which has three dimensions of motion measured by duration.

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 (or base) in time flows from the future toward the past.