iSoul Time has three dimensions

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Science and Hypothesis excerpts

What follows are excerpts from the book Science and Hypothesis by Henri Poincaré, translated (1905) from La Science et l’hypothèse (1902).

p.xxiii The latter [definitions or conventions] are to be met with especially in mathematics and in the sciences to which it is applied. From them, indeed, the sciences derive their rigour; such conventions are the result of the unrestricted activity of the mind, which in this domain recognises no obstacle. For here the mind may affirm because it lays down its own laws; but let us clearly understand that while these laws are imposed on our science, which otherwise could not exist, they are not imposed on Nature. Are they then arbitrary? No; for if they were, they would not be fertile. Experience leaves us our freedom of choice, but it guides us by helping us to discern the most convenient path to follow.

p.xxv Space is another framework which we impose on the world. Whence are the first principles of geometry derived? Are they imposed on us by logic? Lobatschewsky, by inventing non-Euclidean geometries, has shown that this is not the case. Is space revealed to us by our senses? No; for the space revealed to us by our senses is absolutely different from the space of geometry. Is geometry derived from experience? Careful discussion will give the answer—no! We therefore conclude that the principles of geometry are only conventions; but these conventions are not arbitrary, and if transported into another world (which I shall call the non-Euclidean world, and which I shall endeavour to describe), we shall find ourselves compelled to adopt more of them.

Read more →

Two measures of motion

By common experience, we know there are three dimensions of motion. That is, space, which is the space of motion, is three dimensional. To measure the extent of motion requires comparing one motion with another, of which there are two ways: length and duration. The length of a motion is measured by comparing it with symmacronous but not necessarily synchronous motion. The duration of a motion is measured by comparing it with synchronous but not necessarily symmacronous motion.

Length of motion considered by itself forms a length space, which is space with a metric of length. Duration of motion considered by itself forms a duration space, which is space with a metric of duration. Since there are three dimensions of motion, length space and duration space are both three dimensional metric spaces. By convention, both are Euclidean. The length metric is called distance. The duration metric may be called distime.

Each point in length space has a length position (LP) vector that begins with the length origin. Each point in duration space has a duration position (DP) vector that begins with the duration origin. The magnitude of a length position vector is called the stance. Every point in length space that is equidistant from the origin has the same stance. The magnitude of a duration position vector is called the time. Every point in duration space that is an equal distime from the origin has the same time.

Stance and time are vector magnitudes, with their direction ignored. Stance is a radius from the origin of length space. A unit of length is the absolute value difference between two stances, that is, between the radii of two length vectors with unit difference. Time is a radius from the origin of duration space. A unit of duration is the absolute value difference between two times, that is, between the radii of two duration vectors with unit difference.

The rate of motion measured by the length of motion per unit of duration is called speed. The rate of motion measured by the duration of motion per unit of length is called pace. Note that a faster speed is a larger ratio, whereas a faster pace is a smaller ratio. Also, the ratio of a slower speed to a faster speed is less than one but the ratio of a faster pace to a slower pace is less than one.

The vector rate of change in the length vector per unit of duration is called velocity. The vector rate of change in the duration vector per unit of length is called legerity. The vector rate of change in velocity per unit of duration is called acceleration. The vector rate of change in legerity per unit of length is called expedience.

The length position vector of a trajectory evolves as a function of the time. The duration position vector of a trajectory evolves as a function of the stance. These functions are inverses of one another.

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).

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 symmacronized their clocks will always agree about the stance 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 chronitions and legerities.

p.3-4 To specify positions and time, each observer may choose a zero of the time scale, an origin in space, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a frame of reference. The position and time of any event may they be specified with respect to this frame by the three Cartesian co-ordinates x, y, z and the time t. … The frames used by unaccelerated observers are called inertial frames.

p.3-4 To specify chronitions and stance, each observer may choose a zero of the stance scale, an origin in 3D time, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a time frame of reference. The chronition and stance of any event may they be specified with respect to this time frame by the three Cartesian co-ordinates ξ, η, ζ and the stance r. … The time frames used by inexpedienced observers are called facile time frames.

Read more →

Spherical coordinates more or less

“In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified 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 of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.” (Wikipedia)

A 3D time spherical coordinate system is implicitly behind space-time (3+1), with the two angles ignored for scalar time. That is, every instant in 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 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 3D time be (t, β, α) with t representing the radial distime, and β and α representing the temporal zenith and azimuth angles, respectively.

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

If rectilinear coordinates are used for 3D time, say (ξ, η, ζ), then the radial distime t equals √(ξ² + η² + ζ²). The corresponding spatial concept, stance, is the radial distance, which if rectilinear coordinates are used for 3D space, say (x, y, z), then the stance r equals √(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).

The result is that to convert an invertible (3+1) function to a (1+3) function requires expansion to the (3+3) function, inversion, and then contraction to (1+3). In symbols, (3+1) ⇑ (3+3) ⇓ (1+3), or (r, θ, φ; t) ⇑ (r, θ, φ; t, β, α) ⇓ (r; t, β, α). In this way space and time are interchanged.

In symbols, from a invertible parametric space function inverted to a parametric time function (with ⇑ as expand, ⇓ as contract, and ↔ as invert): r = [r, θ, φ] = r(t) = [r(t), θ(t), φ(t)] ⇑ [(t´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ ((r´, θ´, φ´), β´(r´, θ´, φ´), α´(r´, θ´, φ´)) ⇓ [t(r), β(r), α(r)] = t(r) = [t, β, α] = 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´, β´, α´), θ´(t´, β´, α´), φ´(t´, β´, α´)] ↔ ((r´, θ´, φ´), β´(r´, θ´, φ´), α´(r´, θ´, φ´)) ⇓ [t(r), β(r), α(r)] = [dt/dr, dβ/dr, dα/dt] = dt/dr = u. The result is that space and time are interchanged, with spatial vectors becoming radial distances and radial distimes becoming temporal vectors.

Functions that are not invertible may be inverted by differentiation, then integration. Take for example, the function s(t) = s0 + v0t + ½at². Differentiating twice leads to s(t)´´= a = dv/dt. Expanding, inverting, and contracting 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 had already be set. 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 duality 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. It gives us confidence that there is a unity, even if we haven’t yet found how that unity is shown by observation and experimentation. This 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.