iSoul In the beginning is reality

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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Interesting universe

This post continues a series on history and science, see here and here.

The development of the comparative method in linguistics led to the genealogy of languages in the 19th century. This diachronic approach was largely abandoned in the 20th century with the rise of synchronic theories. In short, linguistics pivoted from history to science.

Modern science is basically synchronic, that is, spatially broad within a narrow time period. This arises most commonly with empirical methods, which can range throughout the earth and beyond but focus on contemporary observations. While it is possible to focus on a different time period, the common procedure is simply to assume that the past is like the present. We might call this the boring universe postulate: nothing significantly new ever happens.

Such an anachronistic method is anathema in the discipline of history, that is, diachrony. One cannot assume the past is like the present without evidence from past sources. Moreover, significant events are pivotal for history, unlike science. It is difference, not similarity, that drives history.

History is basically unpredictable, no matter how much minor predictions can be made. The empirical sciences extract what can be predicted from empirical sources but leave the unpredictable out, relegated to noise or chance. It would be better for the sciences to leave the unpredictable to history than to sideline it as if it were unimportant.

Historical science or scientific history are oxymorons. They seem to mean a science of history. The search for a theory of history was a focus of the 19th century with uniformitarianism and Darwinism (and Marxism for some) providing the top candidates. But the project is misguided: it would mean history in a boring universe, which would be history without meaningful history.

The history of the universe or nature or life are within the domain of history. Science is able to assist but it is presumption to substitute a boring universe for the real one. The universe, nature, and life are too interesting and meaningful for that.

From natures to nature

This post follows on a previous post here.

How did we get from natures to nature? In a word, nominalism. The many natures of pre-modern science have been transformed into one nature or Nature, reified if not personified as a thing or force or being.

Nominalism is the teaching that universals or qualities or natures do not exist. Only particulars or quantities or individuals exist. And the result is that only one universal or quality or nature is acknowledged to exist, the somewhat mystical universal quality or nature of everything that underlies all the particulars and quantities and individuals.

We can see nominalism in physics and chemistry, with the rise of the atomic model of nature as composed of one kind of atoms, with only different quantities and configurations to differentiate them. We can see nominalism in biology, with the rise of the evolution model of nature as composed of one kind of life, with only different lines of descent to differentiate individuals. We can see nominalism in politics and economics, with the rise of the equalized person interchangeable with any other person.

By why should this one universal or quality or nature exist at all? Why not go all the way and deny any universal or quality or nature? Nominalism has no defense against such a move. And so we are seeing nominalism end in nihilism, the denial of nature altogether.

We are also seeing the rise of an opposite extreme: that every individual is a unique kind of person and that every individual life is a species. If there are no permanent kinds or species, then individuals are the only kinds. Every person has a right to a unique identity, unique treatment, and unique pronouns.

Transforming 3D space into 3D time

There is a symmetry between space and time. As one can transform an observation by rectilinear motion (translation), or by rotation, or by a timeline change, so one can transform 3D space into an equivalent 3D time. This is not a continuous change so don’t expect a simple equation. There are four things that must be done to transform 3D space into 3D time, that is, 3+1 spacetime into 1+3 timespace:

(1) The ordering of events should be switched between a timeline (1D time order) and a placeline (1D space order). So a measurement of time, such as the duration from a reference event, should be switched with a measurement of place, such as the distance from a reference event.

(2) Scalars should be inverted: speed ⇒ pace, mass ⇒ 1/mass = vass, energy ⇒ 1/energy = invergy, work ⇒ 1/work = invork, etc.

(3) Vectors that are ratios of base units or products of base units should switch their numerators and denominators such that (a) the denominator becomes a magnitude of the former numerator and (b) the numerator becomes the vector with units of the former denominator: velocity ⇒ legerity, momentum ⇒ fulmentum, etc. This is similar to an inversion since s/t ⇒ t/s = (1/s)/(1/t).

(4) Other units should be derived from these, with new rates relative to the timeline for 3D space and the placeline for 3D time: acceleration ⇒ expedience, force ⇒ rush, power ⇒ exertion, etc.

There should be no time vectors in 3D space and no space vectors in 3D time. The distance from a reference place and duration from a reference event should be the same for both, apart from a change of reference points. The laws of physics should be the same for observation or transportation in each frame.

Equality and hierarchy

The state of nature was invented by Thomas Hobbes to support his idea of a social contract that was (or would have been) entered into by free individuals. In the natural state people would have been totally free but also lacking in security and other goods of society. So they voluntarily entered into a social contract that reduced their rights in exchange for social goods.

This placing of individual rights before social duties is what Harvey Mansfield called the beginning of liberalism. It is an egalitarian liberalism, since everyone is in an equal state of nature and has an equal right to make (or break) a social contract.

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Timelines and placelines

Events may be ordered in various ways (see here). Events ordered by time form a timeline, which is:

1. a linear representation of important events in the order in which they occurred.
2. a schedule; timetable.

This may be generalized to the following definition:

A timeline is an ordering of events by time or duration.

For example, below is a timeline of a Project Mercury flight:

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From 3D space to 3D time

We observe the sun and the moon traversing the sky. We know that the moon is objectively orbiting the earth but the sun is not. Where then is the sun that is observed traversing the sky in daily and annual cycles? It is not in 3D space. It is in 3D time.

Binary stars orbit their common barycenter. If the sun and earth were the only celestial bodies, it might not be clear as to which was orbiting which. But since there are other planets orbiting the sun, the only objective view is that all the planets are orbiting the sun (more precisely, the barycenter of the solar system, which is in or near the sun).

Compare sun-centered (heliocentric) and earth-centered (geocentric) frames of reference (click to enlarge):


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Direction for expedience and rush

Expedience is the change in legerity per unit of length. Average expedience is (u2u1)/Δs. If the second legerity is faster than the first, the time interval is shorter and the legerity is lower in value, which makes the expedience negative. Since increased motion is the default for expedience (as it is for acceleration), the default direction for expedience is in the direction opposite the motion. Negative expedience, or inexpedience, is in the direction of motion.

Rush, which is vass times expedience, is also in the direction opposite the motion. So the rush of levity is in the opposite direction as well. That is why cyclic motion in 3D time has a centrifugal rush. And that is why there is levity, not gravity, in 3D time although the motion is the same.

Newton’s 3rd law shows that forces (and rushes) come in direction pairs. So it is a matter of convention which direction is the primary one. The force toward the spatial center is primary in 3D space. The rush from the temporal center is primary in 3D time. Both directions are valid but one is primary.