iSoul In the beginning is reality

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.

Foundations of mechanics for 3D space or 3D time

The first edition of New Foundations for Classical Mechanics (1986) by David Hestenes included “Foundations of Mechanics” as Chapter 9. This was removed for lack of space in the second edition, but is available online as a pdf here. This space-time foundation may serve as a guide for the foundation of mechanics for either space-time or time-space. To do so requires introducing abstract terminology, notably:

position space → position geometry; time → event order; particle → point body; instant → point event; clock → event order indicator; simultaneity → correspondence; reference frame → frame.

The application of this abstract theory is to interpret the 3D position geometry with event order as either 3D position space with temporal event order (space-time) or 3D position time with spatial event order (time-space). It could also be applied to derivatives or integrals of these, e.g., a velocity space.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for 3D space or 3D time:


To begin with, we recognize two kinds of bodies, point bodies and bodies which are composed of point bodies. Given a body R called a frame, each point body has a geometrical property called its position with respect to R. We characterize this property indirectly by introducing the concept of 3D Position Geometry, or Relative Geometry, if you prefer. For each frame R, a position geometry P is defined by the following postulates:

  1. P is a 3-dimensional Euclidean geometry.
  2. The position (with respect to R) of any point body can be represented as a point in P.

The first postulate specifies the mathematical structure of a 3D position geometry while the second postulate supplies it with a physical interpretation. Thus, the postulates define a physical law, for the mathematical structure implies geometrical relations among the positions of distinct point bodies. Let us call it the Law of Geometric Order.

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Form and logic

I’ve written before about Laws of Form (the calculus of indications); see here and here.

In the beginning is an undifferentiated state, an unmarked space. The first distinction is the first differentiation, the advent of a mark, a cross, a form. The unmarked state is the urgrund of the form, its origin and basis. The marked state is the form of the mark, a cross into markedness. The form encapsulates the progression from undifferentiated unity to differentiated hierarchy.

Spencer-Brown applies the calculus of indications to propositional logic in his book Laws of Form (Julian Press, New York, 1972), Appendix 2. This requires an interpretation of the mark as either true or false. He acknowledges that the choice is arbitrary, and then takes the mark to represent truth. This corresponds to the common logical convention that a proposition is assumed to be false unless it is shown to be completely true.

True propositions can be conceived as arising islands of truth in a sea of falsehood; or they can be conceived as the sea itself, interrupted by islands of falsehood. They are opposite conventions, and there is no logical reason to prefer one or the other.

But they encode different conceptions of truth. Is truth rare or common? Is falsehood rare or common? Do we give a proposition the benefit of the doubt or accept it if it has some truth? Or do we reject all doubtful propositions or partially true propositions?

The form is pre-logical; it assumes no convention about truth values. The form also assumes no convention about classes. A logic of classes can adopt a convention that the mark is the universal class or the null class. It depends on whether classes are conceived as arising from a unmarked space of nullity or void, or from an unmarked space of everything or pleroma.

Laws of form easily fits universal forms of the logic of classes but existential sentences are problematic. Spencer-Brown observes this interpretative theorem:

An existential inference is valid only in as far as its algebraic structure can be seen as a universal inference.

This theorem holds for both conventions about classes, that is, whether existence is considered to stand out from a void or against a pleroma.

These dual conceptions lead to a distinction between sets, which are composed of members that may have certain attributes, and classes, which are composed of attributes that may have certain members. The null set is a set with no members. The universal set has all members within a given context. The universal class has no attributes. The null class has all attributes within a given context.

Beginning of the American revolution

The following chronology is based on the Timeline of the Revolutionary War and other sources.

French & Indian War 1754-1763 (part of the European war called the Seven Years War) – English victory was at the cost of a large debt. “It was that debt that caused the escalation of tensions leading to the Revolutionary War.”

Proclamation of 1763 – King George III’s proclamation that closed off the frontier to colonial expansion, which was resented by the colonists, who felt penned in on the East coast.

Sugar Act of 1764 (The American Revenue Act) – An act of Parliament that reduced the rate of tax on molasses and added other taxes, while Lord Grenville took measures that the duty be strictly enforced. This disrupted the colonial economy by reducing the markets to which the colonies could sell, and the amount of currency available to them for the purchase of British manufactured goods.

Currency Act of 1764 – An act of Parliament that prohibited the issue of bills of credit, which made it more difficult for colonists to pay off their debts to Great Britain.

Stamp Act of 1765 – An act of Parliament that imposed many taxes to pay Great Britain’s debt for the French & Indian War. It was their first serious attempt to assert governmental authority over the colonies.

Quartering Act of 1765 – An act of Parliament that required colonial governments to provide and pay for feeding and sheltering any troops stationed in their colony.

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