iSoul In the beginning is reality

# Problems in mechanics, part 3

This post continues a series of problems (part 1 here, part 2 here) based on the website Free Solved Physics Problems, this time concentrating on dynamics problems for time-space corresponding to problems in space-time.

Note: A newton is how much force is required to make a mass of one kilogram accelerate at a rate of one metre per second squared (1 N = 1 kg ⋅ m / s2 ). An oldton is how much rush is required to make a vass of 1 kilogram-1 expedite at a rate of one second per metre squared (1 O = 1 kg-1 ⋅ s / m2).

Problem 6.

A boy of mass 40 kg wishes to play on pivoted seesaw with his dog of mass 15 kg. When the dog sits at 3 m from the pivot, where must the boy sit if the 6.5 m long board is to be balanced horizontally? Solution here.

# Observation and transportation

Impossible objects such as the Necker cube above are drawings that appear as two different objects, in this case either a box standing out toward the lower left or toward the upper right. It can be seen as one or the other but not both simultaneously.

3D space and 3D time are like this. One can see either 3D space or 3D time but not both simultaneously. One may develop a unified 6D geometry for both of them but to measure rates either space or time must be reduced to a scalar or 1D quantity.

It is the same with observation and transportation. One can view a motion from the perspective of an observer (whether one is moving or on the sidelines) or from the perspective of a traveler (whether one is traveling or on the sidelines).

The observer sees motion taking place in 3D space ordered by scalar time. The traveler sees motion taking place in 3D time ordered by scalar space, that is, the stations.

# We the people

The preamble to the U.S. Constitution reads:

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.

This founding document is written in the name of “we the people”. Who are “we the people”?

# Time-space introduction

Length measures space and duration measures time. Length is a scalar, which combined with direction describes 3D space. Duration is a scalar, which combined with direction describes 3D time.

In relativity this might be considered trivial; length and duration may be converted into each other by multiplying or dividing by the speed of light. However, the actual measurement will either use rods or clocks, and so be qualitatively different. The act of measurement determines the type of measure more than the units.

Space and time both have three-dimensional geometries. For most purposes one needs to focus on either 3D space or 3D time and ‘scalarize’ the other. Space is scalarized by replacing each point with its distance from a reference point (usually called the origin). Such a scalarized point of 3D space is called a station.

Time is scalarized by replacing each instant with its distime from a reference instant. Such a scalarized instant of 3D time is called a time (as expected).

Space-time means 3D space measured by distance with scalar time. Time-space means 3D time measured by distime with scalar station.

# Science and history again

To some extent the sciences of society and history can be pursued as if they were natural sciences. For example, groups of people exhibit some characteristics of natural objects, and so reflect physics to some extent.

On the other hand, the physics of social beings is different in a complementary way from the physics of natural bodies. That is because social beings have purposes and plans. These can be accommodated within natural science only by including formal and final causes to some extent.

But knowledge of society and history are different from knowledge of the physical world. Their focus is different and the result is more likely to be a narrative than a theory.

The natural sciences emphasize quantities and have an over-riding principle of qualitative parsimony, often called Occam’s Razor. The sciences of society and history have a complementary principle of quantitative parsimony. This is seen in the increasing distinctions and qualities of society and history that resist generalization and lead to greater particularization.

While it would be best to have a balanced methodology of qualitative and quantitative parsimony, it may work well to have a dialectic of methodologies between two schools or disciplines, one with qualitative and the other quantitative parsimony. Then they can critique each other and seek to converge at a common solution.

# Science and metaphysics again

The Scholastics developed a cosmology with the Earth at absolute rest in the center of moving concentric spheres. Ptolemy’s geocentric astronomy with its epicycles was thought to be consistent with the Scholastic cosmology. When geocentrism was challenged by the early scientists, the whole Scholastic cosmology was thought to be undermined.

The difference between science and metaphysics has been confused ever since.

For example, in his General Scholium to the Principia Newton wrote:

For whatever is not deduc’d from the phenomena, is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.

But then in his Scholium to the Definitions Newton goes on to give his opinions about absolute time, absolute space, and absolute motion. So much for staying away from metaphysics!

As the sciences progress they normally come to question their metaphysical assumptions, reduce the metaphysics, and become more formal and mathematical. Quantum mechanics as an example of science with minimal metaphysics. Some people think that is a problem but I see it as a success.

Philosopher of physics Tim Maudlin describes quantum mechanics this way:

Unlike Relativity, there is no agreement among physicists about how to understand quantum theory. Indeed, the very phrase “quantum theory” is a misnomer: there is no such theory. Rather there is a mathematical formalism and some (quite effective) rules of thumb about how to use the formalism to make certain sorts of predictions. … The philosopher of physics cares about the underlying reality and attends to the predictions only insofar as they can serve as evidence for which account of the underlying reality is correct.” p.xiii, Philosophy of Physics: Space and Time by Tim Maudlin

Like most in his field, Maudlin develops a metaphysics that doesn’t go beyond the bounds of naturalism. That’s his business but it’s not the business of science.

Other sciences are entwined with metaphysics. For example, the development of geology and biology as historical sciences conjured up first thousands, then millions, and then billions of years of time before recorded history. With no quantitative parsimony there was no cost for such magic.

The existence of time before the history of humanity is really about metaphysics, not science. What can be done in science is to show that the assumption of pre-historic time fits well – or not – with other science. Those with a methodology that includes quantitative parsimony (with or without qualitative parsimony) have an incentive to avoid such large metaphysical quantities.

# Catholic and evangelical

The capitalized term Catholic refers to Roman Catholics, led by the Pope. The capitalized term Evangelical refers to revivalist Protestants, especially in English-speaking countries.

Lutherans are both catholic (uncapitalized) and evangelical (uncapitalized). This is sometimes called evangelical catholic, though it could as well be called catholic evangelical.

What is an uncapitalized catholic Christian?

The word ‘catholic’ means ‘universal’. A catholic Christian is a universal Christian. That is, one who identifies with whole Christian body throughout the history of the world. They retain traditions and doctrines that have had wide currency in the universal church, such as the liturgical calendar and the doctrine of the Real Presence in the Eucharist.

What is an uncapitalized evangelical Christian?

The word ‘evangelical’ means related to the gospel, the good news of grace through faith in Jesus Christ. An evangelical Christian is one who emphasizes the proclamation and sharing of the gospel, the authority of the Bible, and centrality of Christ.

# Formal and material space and time

Science makes no metaphysical claims but it is not unusual for scientists to make metaphysical claims, sometimes even in their scientific publications. That has confused the relationship between science and metaphysics. The philosophies of scientific realism and naturalism have further confused the relationship between science and metaphysics.

As a Christian I must say that if scientists make metaphysical claims, then their metaphysics should be consistent with Christian metaphysics. If scientists object to that, they should refrain from making metaphysical claims.

Science needs to begin without metaphysics. Mathematics has no metaphysics. So science should begin with mathematics. That is, mathematics should be the framework on which science is built.

Thus a science of space and time begins with a mathematical formalism. This formalism should be distinguished from the empirical units employed to measure space and time. As far as I know, that has not been done, so people have confused the measure of length with the form of space and the measure of duration with the form of time.

Isaac Newton separated his metaphysical claims about space and time into what he called scholia in his Principia. He should have just adopted a mathematical formalism and left out any metaphysical claims.

In order to distinguish the formal and material space and time, I have revised the Parallel Glossary for Classical Physics, see link above.

# George Washington’s warnings

Peter Lillback his article “The United ‘Statists’ of America?” in the book Statism: The Shadows of Another Night, edited by Charlie Rodriguez (2015) lists the following warnings given by George Washington in his 1789 address to Congress (with Lillback’s wording appended):

1. “I pretend to no unusual foresight into futurity, and therefore cannot undertake to decide, with certainty, what may be its ultimate fate.” Washington was not a prophet and could not make a final prediction about the ultimate fate of the Constitution.

2. “If a promised good should terminate in an unexpected evil, it would not be a solitary example of disappointment in this mutable state of existence.” In our uncertain world good things have often ended up as disappointing evils and this could happen with our Constitution too.

3. “If the blessings of Heaven showered thick around us should be spilled on the ground or converted to curses, through the fault of those for whom they were intended, it would not be the first instance of folly or perverseness in short-sighted mortals.” If we lose our Constitution’s blessings of liberty, it would not be the first time that human foolishness has squandered the blessings of heaven.

4. “The blessed Religion revealed in the word of God will remain an eternal and awful monument to prove that the best Institutions may be abused by human depravity; and that they may even, in some instances be made subservient to the vilest of purposes.” The word of God’s revelation of the Christian religion provides an eternal example of the fact that the best human organizations can be used for evil ends. (Washington is here referring to the events surrounding the crucifixion of Jesus Christ.)

5. “Should, hereafter, those who are entrusted with the management of this government, incited by the lust of power and prompted by the Supineness or venality of their Constituents, overleap the known barriers of this Constitution and violate the unalienable rights of humanity:” America’s future power-hungry leaders could get away with a disregard of the Constitution’s limitations and harm our unalienable rights because the voters have become lazy or selfish.

# Propositional logic calculation

George Boole is known for introducing a logical calculus for propositions in the mid-19th century. Although others before him such as Leibniz worked on logical calculi, Boole developed the first systematic one. Later C. S. Peirce and Gottlob Frege developed calculi that took into account the difference between universal and existential propositions. Since then many logical calculi have been developed, such as the Calculus of Indications previously noted here.

However, these calculi are not necessarily easy to calculate with. For that it is best to use something close to the familiar arithmetic and algebra. Here are two examples:

MIN-MAX LOGIC

The Boolean operations are negation (NOT, ¬, ~), conjunction (AND, ∧), and disjunction (OR, ∨), with the constants 0 (contradiction) and 1 (tautology). These correspond to the set operations complement (c, ´ ), intersection (∩), and union (∪) with constants ∅ (null set) and U or Ω (universal set).

Boolean logic may be represented by the following min and max operations:

 ¬a = 1 – a a ∧ b = min(a, b) a ∨ b = max(a, b)

Other operations may be defined from these such as material implication, ab = ¬ab, which corresponds to the subset proposition ab, and is represented by max(1 – a, b).

FINITE FIELD LOGIC

Propositional logic may be represented by any functionally complete binary calculus such as the finite (Galois) field of order 2. The constants are 0 and 1 with 1 + 1 = 0. Since ordinary arithmetic is a field, this representation is mostly familiar:

 ¬a = a + 1 a ∧ b = a · b a ∨ b = a b + a + b

Then ab is represented by a b + a + 1.