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Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

Space-time exchange postulate

Rates of motion are almost always expressed as a ratio with respect to time. For example, the average speed of a body is the travel distance of the body divided by the travel time. This makes the independent variable time and distance the dependent variable.

However, there is no physical dependency of motion on time rather than distance. One could just as well express the average rate of motion as the travel time of the body divided by the travel distance. The ratios are equally valid.

This is a general result. There is a binary symmetry between space and time. Travel distance and travel time are interchangeable as far as the equations of physics are concerned. J. H. Field has expressed this as a postulate for space-time exchange (STE):

(I) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of four vectors. (A four-vector has three components of length and one of distime.)

He avoids the question of 3D time by limiting the STE to the direction of inertial motion. Here we generalize the STE postulate to include 3D time:

(II) The equations describing the laws of physics are invariant with respect to the exchange of space and time coordinates, or, more generally, to the exchange of the spatial and temporal components of six-vectors. (A six-vector has three components of length and three of time.)

Field found the STE to violate Galilean symmetry, but this is incorrect because time is three dimensional, and there is a co-Galilean transformation symmetric to the Galilean transformation.

The STE postulate affirms the complete symmetry of space and time, which is built on the symmetry of length and duration. As distance is the metric of space, a kind of length, so distime is the metric of time, a kind of duration. The metric of space or time may be used to organize events linearly, with equivalence classes defined for events at the same position in the order.

Rest in space and time

Rest means no motion, or at least no motion detected by an observer.

We know what rest in space means: staying in the same place. That is, rest means no change of position, no travel distance, no length of motion. At rest the numerator of the speed is zero.

Yet clocks tick on. The denominator of speed is not zero. So the speed of rest is zero, that is, a length of zero divided by a non-zero quantity of time. Speed v = Δxt = 0/Δt = 0.

What is rest in time? It means staying at the same time. That is, rest means no duration of motion, no travel time. At rest the numerator of the pace is zero.

In this case, is the length of motion zero, too? No. For pace length is the independent quantity. It doesn’t depend on the motion. It depends on the given length or unit of length. So the pace of rest is zero, that is, a time of zero divided by a non-zero length. Pace u = Δtx = 0/Δx = 0.

Yet a zero pace seems to say one gets a change of place with no lapse of time. What gives?

Length in the pace ratio is the independent variable. Whether length is conceived to be continually increasing, as if it were a clock, or just a quantity of length for comparison, it is independent of the motion measured. The numerator, the time, is what is measured and compared with a quantity of length to determine the pace.

It is similar with speed. Whether or not there is a clock ticking away, the denominator is a quantity of time compared with a quantity of length. All the clocks in the world could be broken, yet the denominator of speed, the change in time, would still be non-zero.

Relativity alone

In a paper titled Nothing but Relativity (Eur. J. Phys. 24 (2003) 315-319) Palash B. Pal derived a formula for transformations between observers that is based on the relativity postulate but not a speed of light postulate. In a paper titled Nothing but Relativity, Redux (Eur. J. Phys. 28 (2007) 1145-1150) Joel W. Gannett presented an alternate derivation with fewer implicit assumptions. Here we’ll use Pal’s approach to derive the time-space version.

Consider two inertial timeframes S and , where the second one moves with legerity u, along the t-axis, with respect to the first one. There are two other time axes. The coordinates and radial distance in the S-timeframe will be denoted by t and x, and in the timeframe will be denoted with a prime. The time-space transformation equations have the form:

= T(t, x, u) and = X(t, x, u),

and out task is to determine these functions. A few properties of these functions can readily be observed. First, the principle of relativity tells us that if we invert the legerity in these equations, we must obtain the same functional forms:

t = T(t´, x´, –u) and x = X(t´, x´, –u).

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Circular orbits

*** Revised from June 2017 ***

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius: R, for space, and Q, for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions from the perspective of the Sun toward each orbiting planet:

  1. Each planet orbits the Sun in a circular path with radius R in 3D space.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is a constant, v.

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Change and stability

Evolution or alteration means change over time. Sameness over time is called permanence or stability. The study of change or the lack of change over time is called history or diachrony.

Change happens. But sameness happens, too. One easily sees that sameness happens in the natural realm much more than change. That is not the result of chance but of law, which is why natural science is able to articulate laws and predict the future. The natural future is like the natural past.

Some changes are unpredictable individually but have predictable distributions or aggregates. Many sciences from statistical mechanics to quantum mechanics to genetics are stochastic in nature.

Similar to the coastline paradox, the amount of change depends on the length of the “ruler” used to measure change. If it is a small ruler, one measures minutiae, and more changes will be found. If it is a large ruler, one measures key features, and fewer changes will be found.

The conceit of evolutionary biology is that very low rates of unpredictable change over very long periods of time can result in all the biological diversity of today. It is an appeal to the imagination more than an appeal to knowledge. Without imagination, the argument becomes an assertion of mere possibility, rather than plausibility, probability, or necessity.

But if very low rates of unpredictable change can determine what happens, how much more can very high rates of predictable stability. One does not need to appeal to the imagination to see that stability is the rule, and exceptions only prove the rule.

Organisms are similar in some respects but not in other respects. If one focuses on minutiae, there are many differences. It is part of the conceit of evolutionary biology to overstate the importance of minor differences such as color and understate the importance of major differences such as body plan.

One might hope that biologists would be working toward finding the optimum characteristics to measure biological change. Alas, they are determined to find the smallest ruler and overstate change as much as possible.

I predict that a more mature biology will seek the optimum measure of change, and will accept that some characteristics are permanent features of a body type.

Galilean transformation expanded

The Galilean transformation is typically presented for motion in direction of the x-axis, with the other axes unchanged:

x´ = xvt, y´ = y, z´ = z, and t´ = t,

where v is the relative velocity of the observers. This is incompatible with the Lorentz transformation, but more than that, it is inconsistent with the two-way (round-trip) speed of light in a vacuum.

The Lorentz transformation can be made compatible with the round-trip speed of light if light is considered to travel instantaneously to its observer, which is usually the final leg. The speed of light for the other part of the round trip can be inferred so that their harmonic mean equals c, which is the most that is known (see One-way speed of light).

That is, if the speed of light in the Lorentz transformation is allowed to approach infinity, then the transformation will approach the Galilean transformation. Here the Galilean transformation arises as a limit of the Lorentz transformation by the speed of light approaching infinity, rather than usual the relative velocity approaching zero.

The Lorentz transformation for motion in the direction of the x-axis is:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (tvx/c²), with γ = (1 – v²/c²)–1/2,

where γ (gamma) is the Lorentz factor. As c → ∞, γ → 1 and t´ → t. This can only be the case for one part of the light trip, which we’re taking as the last part of the trip.

Why the last part? Because that’s what is observed, directly or reflected in a mirror. And in everyday conversation the place where something is observed to be is spoken of as where it is now. Even with the convention of a constant speed of light, one has to be very pedantic to keep correcting others and oneself by saying that where something is seen to be is in fact where it was in the past.

For a round trip, the speed of light for the part not directly observed can be inferred from the empirical result that the round trip speed equals the constant, c:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (t – 4vx/c²), with γ = (1 – 4v²/c²)–1/2.

The speed of light for the unobserved part is inferred from the necessity that the harmonic mean equals c:

(1/c1 + 1/c2) = 2/c,

where c1c/2 as c2 → ∞. This harmonic mean of speeds is the arithmetic mean of paces. What is actually measured is the pace of light from the independent length traversed in the dependent time.

Length and time parallels

This post continues the parallelism between length and time, and includes some new terms.

Length and time both have base units in SI metric: the meter (or better: metre to distinguish it from a device) and the second. They can both be associated with direction. Length in a direction is from or toward an event place. Time in a direction is from or toward an event time.

Multiple dimensions of length are called space. However, space can mean merely the space between two points. To designate 3D space, let’s use the Latin spatium (space). Analogously, let’s use the term tempium to designate 3D time (cf. Latin tempus, time).

Events ordered by time are in time order. Events ordered by length are in length order. Events ordered by importance could be said to be in magna order.

Things are persistent events. Things have length. Things have three dimensions of length. Events have duration. Events have three dimensions of duration. The extent of space between things is called distance. The extent of tempium between events could be called temstance.

Relative space is divided into here and there; “here I am, there I was, there I will be.” The present tense of space is here. The past or future tense of space is there. Here I am. Some places were traversed in the past. Some places will be traversed in the future.

Relative tempium is divided into now and then; “now I am, then I was, then I will be.” The present tense of tempium is now. The past or future tense of tempium is then. Now I am. Some times were traversed in the past. Some times will be traversed in the future.

Matter is a spatial substance. Figure is a temporal substance. Matter has mass, solidity. Figure has vass, lightness. Many sports move matter, such as a baseball pitcher throwing the ball. A figure skater traces out figures in space and time, that is, spatium and tempium.

Observers and participants

Observers detect objects and events with objects. These objects are essentially passive; they must be made to do things by force and work.

Participants are subjects among subjects, actively engaging in events and making them happen. Subjects participate and participants are subjects.

The perspectives of an observer and a participant are inverses of each other. They are different attitudes. An observer has an attitude of standing apart from the world. A participant has an attitude of being part of the world. The world is placed in different contexts because of the attitude of the contact.

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Evolution for everyone

The word evolution is related to the terms evolve and evolute, and originally meant an unrolling. It acquired a sense of development in the 19th century and was associated with progress, especially as promoted by Herbert Spencer. Charles Darwin used it in print only once since his theory was not a theory of progress. “But Victorian belief in progress prevailed (and the advantages of brevity), and Herbert Spencer and other biologists after Darwin popularized evolution.” (source)

Today the basic meaning of the word evolution is change over time. That is, evolution refers to a process that changes one form into another form over time; in short, transmutation. There are various proposed means or mechanisms of evolution but they are all asserted to produce change over time.

Thus the concept of evolution is the opposite of the idea that forms do not change over time. What makes it complex is that some forms may change over time but not others. But no one today seriously alleges that there is no significant change over time. In that sense, we are all evolutionists.

Then we need terms to distinguish the different kinds of evolutionary concepts. One could simply attach the names of their originators, but their concepts are modified over time so additional terms would be required. We need simple terms to designate the main types of evolution. Three-letter acronyms would help, too.

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Technology and science

It’s not uncommon to hear an argument like this: “If you use modern technology, you are buying into all of modern science.” But that’s like saying, “If you celebrate Christmas, you are agreeing with all Christian doctrines.” For example, many Japanese celebrate Christmas, but only 1% of the country is Christian. Similarly, all sorts of people use modern technology, from children to terrorists, who aren’t adopting modern science. So this argument is not true.

This is related to the argument that modern science deserves all the credit for modern technology. But that’s like saying all the credit for modern science should go to mathematics, since science uses mathematics. So this argument also not true.

Consider some great inventors: Cai Lun (paper), Johannes Gutenberg (movable type), Jethro Tull (seed drill and horse-drawn hoe), Abraham Darby (pig iron), John Harrison (marine chronometer), Alessandro Volta (electric battery), Samuel Morse (telegraph), Karl Benz (petrol-power automobile), Thomas Edison (electric light bulb, phonograph, motion picture camera), Alexander Bell (telephone), Nikola Tesla (fluorescent lighting, induction motor, AC electricity), Rudolf Diesel (diesel engine), Wright brothers (airplane), Alexander Fleming (penicillin), John Baird (television), and Enrico Fermi (nuclear reactor).

A few of these inventors are known as scientists (Volta, Tesla, Fleming, Fermi) but most are not. They had various backgrounds and much of their interest was in practical advances, not theoretical ones. Also, the practical use of technology requires advances in engineering, which is not the same as science. Engineers do much of the work implementing technology but get little credit.

Moreover, the development of technology arguably derives the most impetus from those in business and investment who provide the capital to market and improve the devices. Without them, inventions would remain like Da Vinci’s diagrams lying dormant for centuries.

The science community does often get (or take) credit for technology. And they have an incentive to, since they are a prestige-driven occupation. The amount of funding that goes to basic research is directly related to the prestige of scientists. And scientists in the universities are part of the prestige-driven model of funding and promoting higher education.

So no, someone using modern technology is not buying into all of modern science. Nor do scientists deserve all the credit for modern technology.