iSoul Time has three dimensions

Category Archives: Knowing

epistemology, science, kinds of knowledge, methodology

From racing to relativity

There are three different contexts for 3D time, depending on whether stance is continuously increasing and, if so, whether there is a conversion factor between space and time:

(A) Stance is not continuously increasing. This is the situation of a race or sport in which game time has a definite beginning and ending. For example, in many sports the game lasts a specific time. In a race, the length of the course is set and the time for each contestant ends when they cross the finish line. The average pace of a contestant is their race time divided by the course length.

(B) Stance is continuously increasing and there is a conversion factor between space and time. This is the situation of the theory of relativity and some transportation settings in which the minimum pace is the conversion pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time is measured. It is like a race that has no finish line. Without an increase in time the pace is at a minimum (or the speed is a maximum). As the amount of time measured increases, the pace increases (or the speed decreases). Remember that a small amount of time per unit stance interval is a fast motion, whereas a large amount of time per unit stance interval is a slow motion.

In this way, the pace increases indefinitely. A pace of infinity would be at rest. A pace of zero is the minimum pace, which in relativity is the speed of light. That is, the speed counts down from the speed of light. This has been misinterpreted as a transformation with superluminal speeds, but they are counting down, not up, and so are subluminal.

For the dual Lorentz transformation

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ur) \; \textup{with}\; \lambda = 1/{\sqrt{1-cu}}.

The cu in λ is u/(1/c), which is the pace of the object divided by the pace of light. While cu is nominally equal to c/v, the denominator for pace is the stance, which is increasing at the conversion rate. As the time increases, the pace increases and the speed decreases from that of light toward zero. So, there is no square root of a negative number here.

(C) Stance is continuously increasing but there is no conversion factor between space and time. This is the situation of transportation in general. Space and time are not proportional, and the fastest route in space and time are in general different.

History and theology

What follows are excerpts from Ramsay MacMullen’s book Christianizing the Roman Empire, A.D. 100-400 (Yale, 1984). He begins with historiography pointers relevant to religious history.

My subject here is the growth of the church as seen from the outside, and the period is the one that saw the church become dominant, and Europe Christian. p.vii

My object is history. It might be, but it isn’t, theology. Accordingly, my view focuses naturally upon significance, the quality of weight that distinguishes historical phenomena from the (sometimes much more engrossing or at least more diverting) items of merely human interest that we see in the headlines of certain newspapers: ‘‘Mom Axes Babe” or the like. Significance, in its turn, indicates the degree to which many people, not just a few, are made to live their lives differently in respects that much engage their thoughts, not in respects they do not think about very carefully. … Significance must be compounded of both “many” and “much,” in a sort of multiplicand of the two elements. p.1

This is all elementary. Still, it needs to be said in order to explain the inclusion in my account of scenes not usually given much attention in books about church growth, scenes in which large numbers of persons are brought to a change in their religious allegiance, but namelessly—they are just ordinary folk of no account—and without great dramatic, further consequences in their manner of life. I think these scenes need to be included, along with Saint Augustine and a handful like him, because otherwise we would see only a church all head and no body, a phenomenon that affected only a few lives, a change without mass and therefore without historical significance. And that is the exact opposite of the truth. p.1

The process we are tracing, of the slow but gigantic growth of a community of believers, seems thus to have had at its heart a psycho-logical moment that might have been, though it was not always, quite uncomplicated; and that fact belongs by right, and not by later development, to the whole long process of ecclesiastical maturing. From the very beginning, Jesus’ disciples followed him instantly, without instruction; new adherents, by supernatural actions, were won to instantaneous belief, or trust (πιστις, “commonly mistranslated, ‘Your faith’ …,” with implications of doctrine, as has been pointed out). p.3-4

There is an obvious connection between simplicity of belief and rapidity of conversion: the simpler the set of ideas with their attendant feelings, the shorter must be the period of transition to the new. Which is not to say that much longer, complicated transitions may not also have had their abrupt moments, like Saint Augustine’ in the garden near Milan. The point is worth stressing because the more richly intellectual and dramatically interesting conversions naturally hold our attention best, and are most written about. p.4

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Invariant intervals

The spacetime interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, along with β = v/c and γ = 1/√(1 − β²):

\Delta x=\gamma (\Delta x'+\beta c\Delta t');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; c\Delta t=\gamma (c\Delta t'+\beta \Delta x').

The invariant interval is

(\Delta s)^2 = (c\Delta t)^2 -(\Delta x)^2 -(\Delta y)^2 -(\Delta z)^2

=\gamma ^2(c\Delta t'+\beta\Delta x')^2 -\gamma ^2(\Delta x'+ \beta c\Delta t')^2 -\Delta y'^2 -\Delta z'^2

Expand the squares and cancel the middle terms to get:

=\gamma ^2(c^2 \Delta t'^2(1-\beta ^2) -\Delta x'^2(1-\beta ^2)) -\Delta y'^2 -\Delta z'^2

=\gamma ^2(c^2 \Delta t'^2(1/\gamma ^2) -\Delta x'^2(1/\gamma ^2)) -\Delta y'^2 -\Delta z'^2

=(c\Delta t')^2 -(\Delta x')^2 -(\Delta y')^2 -(\Delta z')^2.

The spacetime interval is invariant over the dual Lorentz transformation (DLT). The following is a proof of this for the inverse DLT follows with temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity κ, along with ζ = u/κ and λ = 1/√(1 − ζ²):

\Delta x=\lambda (\Delta x'+ \zeta \kappa \Delta r');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; \kappa \Delta r=\lambda (\kappa \Delta r'+\zeta \Delta x').

The invariant interval is

(\Delta w)^2 = (\Delta x)^2 +(\Delta y)^2 +(\Delta z)^2 -(\kappa \Delta r)^2

=\lambda ^2(\Delta x'+ \zeta \kappa \Delta r')^2 +\Delta y'^2 +\Delta z'^2 -\lambda ^2(\kappa \Delta r'+\zeta\Delta x')^2

Expand the squares and cancel the middle terms to get:

=\lambda ^2(\Delta x'^2 (1-\zeta ^2) -\kappa ^2\Delta r'^2(1-\zeta ^2)) +\Delta y'^2 +\Delta z'^2

=\lambda ^2(\Delta x'^2 (1/\lambda ^2) -\kappa ^2\Delta r'^2(1/\lambda ^2)) +\Delta y'^2 +\Delta z'^2

=\Delta x'^2 +\Delta y'^2 +\Delta z'^2 -\kappa ^2\Delta r'^2.

Let Σ (Δri)² = (Δr)² and Σ (Δti)² = (Δt)². Then the two invariant intervals are

(\Delta s)^2 = (c\Delta t)^2 -(\Delta r)^2 \; \textup{and}\; (\Delta w)^2 = (\Delta t)^2 -(\kappa \Delta r)^2

These two invariant intervals are proportional if κ = 1/c:

(\Delta w)^2 =(\frac{\Delta s}{c})^2 = (\Delta t)^2 -(\frac{\Delta r}{c})^2 = (\Delta \tau)^2

In that case, the invariant interval for proper time is the same for both 3+1 and 1+3 dimensions. However, 3D proper time is limited to space-like intervals, while 3D proper distance is limited to time-like intervals. Since space-like intervals require superluminal speeds, they are considered impossible.

Symmetric transformations

What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.

For the (3+1) Lorentz transformation there are spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):

x'=\gamma (x - vt);\; y'=y;\; z'=z;\; t'=\gamma (t - vx/c^2).

The symmetric form is

x'=\gamma (x -\beta ct);\; y'=y;\; z'=z;\; ct'=\gamma (ct - \beta x).

The symmetric Lorentz transformation for vectors is

\mathbf{r'}_\perp = \mathbf{r}_\perp;\; \mathbf{r_\parallel }'=\gamma (\mathbf{r_\parallel } - \boldsymbol{\beta } ct);\; ct'=\gamma (ct - \boldsymbol{\beta } \cdot \mathbf{v}),\; \textup{with } \boldsymbol{\beta } =\mathbf{r_\parallel }/c.

For the (1+3) dual Lorentz transformation there are temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ux).

The dual symmetric form is

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; \frac{r'}{c^2}=\lambda (\frac{r}{c^2} - ux).

The dual symmetric Lorentz transformation for vectors is

\mathbf{t'}_\perp = \mathbf{t}_\perp;\; \mathbf{t_\parallel }'= \lambda (\mathbf{t_\parallel } - \boldsymbol{\zeta }\frac{r}{c});\; \frac{r'}{c}=\lambda (\frac{r}{c} - \boldsymbol{\zeta } \cdot \mathbf{u}),\; \textup{with } \boldsymbol{\zeta }=c \mathbf{t_\parallel }.

For the (3+1) Ignatowski transformation there are spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity V, with β = v/V:

x'=\frac{x-vt}{\sqrt{1-v^2/V^2}}; y'=y; z'=z; t'=\frac{t-vx/V^2}{\sqrt{1-v^2/V^2}}.

The symmetric form is

x'=\frac{x-\beta Vt}{\sqrt{1-\beta^2}}; y'=y; z'=z; Vt'=\frac{Vt-\beta x}{\sqrt{1-\beta^2}}.

For the (1+3) dual Ignatowski transformation there are temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity U, with ζ = u/U:

x'=\frac{x-ur/U^2}{\sqrt{1-u^2/U^2}}; y'=y; z'=z; r'=\frac{r-ux}{\sqrt{1-u^2/U^2}}.

The dual symmetric form is

Ux'=\frac{Ux-\zeta r}{\sqrt{1-\zeta^2}};\; y'=y;\; z'=z; \; r'=\frac{r-\zeta Ux}{\sqrt{1-\zeta^2}}.

The Galilean transformation has V → ∞:

x'=x-vt; \; y'=y;\; z'=z;\; t'=t.

The dual Galilean transformation has U → ∞:

x'=x-ur; \; y'=y;\; z'=z;\; r'=r.


Ignatowsky relativity

Vladimir Ignatowski (1875-1942) was a Russian physicist. “In 1910 he was to first who tried to derive the Lorentz transformation by group theory only using the relativity principle (postulate), and without the postulate of the constancy of the speed of light.” K M Browne gave a simplified derivation in the European Journal of Physics, 39 (2018) 025601, from which the key steps are presented below, followed by the corresponding steps for a dual transformation, switching space and time.

This is a derivation of the Ignatowsky transformation in which the axes x, y, and z are taken to represent space axes rx, ry, and rz with time t. The relativity postulate is taken to be: a valid relativistic transformation must be identical in all inertial frames.

Step 1. To find a valid transformation, we take the usual inertial reference frames S and S′ (the latter moving at velocity v in the +x direction relative to the former) for which intra-frame space is Euclidean but inter-frame space (measured from one frame to the other) may be non-Euclidean. Linear equations are necessary so that an event in one frame appears as a single event, without echoes, in the other. Initial conditions are x′ = x = 0 when time t′ = t = 0. We expect the generalised x equation to be the Euclidean equation x′ = xvt with an added multiplier, and if time is the fourth dimension, then the time equation will be similar but with two additional multipliers. The second of these, n, having the dimension of inverse velocity squared, is required to make the equation dimensionally correct. The y and z coordinates are not expected to be affected by x and t. The generalised transformation and its inverse are then

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Biological classes and ancestries

Taxonomy is the science of classification. Taxonomy applied to biology is a systematic approach to classifying organisms. It can be applied to all organisms at a particular time, throughout time, or within any context. Once a classification is determined, other questions arise such as whether there is an independent reason that organisms are in the same class together.

The basic question in all classifications is whether the objects to be classified fit within a class or belong to another class. The goal of a classification is to minimize the within-class differences and maximize the between-class differences. This is often done by defining a distance metric that quantifies the differences.

Carl Linnaeus is known as the father of modern taxonomy who formalized the binomial nomenclature and called the lowest classes species and genus (no doubt after Aristotle’s method of defining with species and genera). His original expectation was that these biological species were natural kinds that do not change over time. With the discovery that fossils came from dead organisms, it became clear that some of his species had changed over time.

The solution to this problem was to reclassify organisms both living and dead in a new classification system. But this was easier said than done since it took years for fossils to be examined. Meanwhile, people were anxious to know how all the diversity of species arose.

Charles Darwin’s hypothesis was soon adopted: species are temporary population groupings with universal common ancestry. If all species are temporary, there is only one fixed class: the class of all species. Others hypothesize  there are classes of species that are fixed and have separate ancestries, which supports design or special creation.

How can this dispute be resolved? Elliott Sober compares these two hypotheses in his book Evidence and Evolution. Sober argues for a likelihood approach to determining the better of two hypotheses. The law of likelihood states that evidence E favors hypothesis H1 over H2 if and only if the probability of E given H1 is greater than the probability of E given H2, or in symbols, P(E | H1) > P(E | H2). Note that this is a comparative approach; it only works when comparing two specific hypotheses.

In this case, the context is all species on the earth over all the history of life on earth. Hypothesis H1 states that there are multiple classes of species that span the history of life on earth, each class with separate ancestry. Hypothesis H2 states that there is only one class of species that span the history of life on earth, all with common ancestry.

Sober notes that Darwin routinely inferred common ancestry if there was some similarity between species. Sober calls this modus Darwin. It is better to have an overall metric of distance between species than rely on a few similarities. However, there is no generally accepted distance metric for species. In its absence, we can still make some inferences.

If there are many similarities between two species, that evidence is more likely given hypothesis H2 (common ancestry), though there is some likelihood given hypothesis H1. If there are discontinuities between two species, even if there are some similarities, that evidence is more likely given hypothesis H1 (separate ancestry).

Note that if someone proposes a possible sequence of events that explains a discontinuity given hypothesis H2, it is merely a possibility and lacks likelihood. But since hypothesis H1 includes partial common ancestry, it is likely with evidence of similarities as well as differences. The conclusion from this exercise is that separate ancestry is the superior hypothesis.

A problem arises when proponents of common ancestry insist there must first be an explanation of how these separate lines of ancestry originated. The best answer is that, just as abiogenesis is not part of the common ancestry hypothesis, so the origin of the separate classes of species is not part of the separate ancestry hypothesis.

Elemental inverse

Begin with elements. Elements are a very general concept: they may be either members of sets or distinctions of classes. As a set is defined by its members, so a class is defined by its distinctions. So, the elements of sets are members and the elements of classes are distinctions.

Sets may be divided into subsets or combined into supersets. Classes may be divided into subclasses or combined into superclasses. Distinctions may be between classes or within classes. Members may be within sets or without sets.

One might say that a class is just a set of distinctions, or one might also say that a set is just a class of members. But that would blur their differences.

Sets assume one knows members and is trying to combine them into the right sets. Classes assume one knows distinctions and is trying to divide them into the right classes. Aristotle assumed that classes could be known by defining them with the right distinctions. Empiricists assume that sets can be known by defining them with the right members.

Realists begin with classes. A tree is defined by its distinctions. Upon inductive investigation, trees may be grouped into types of tree. Upon deductive investigation, types of trees have certain properties.

Induction proceeds from classes to sets. Deduction proceeds from sets to classes. Sets and classes are like inverses of one another.

Both sets and classes are axiomatized by Boolean algebra with the axioms of identity, complementation, associativity, commutativity, and distributivity.

Physical history

At the highest level of classification, history may be divided into human history (better known simply as ‘history’) and physical history. The former is a large subject with many subdivisions, while the latter is usually turned over to the physical sciences. This is a pity since science and history are different disciplines (see posts here). What follows is a description of physical history as distinct from physical science.

History requires an agent of some kind. The environment is the proxy for an agent in evolutionary science. In physical history the agent is either humanity or one or more non-physical beings that connect to the physical world at its boundaries. The metaphysics of the latter are of no interest here, only their possibility. In other words, the physical universe has boundary conditions that are given; they are not a result of physical laws or processes.

But this sets up a potential conflict between a boundary condition which could have been the result of physical laws or process but was not. It would be simple to assume that all boundary conditions are such that they could not have been the result of physical laws or processes. But that assumes the limits of physical laws or processes are known, when they are to be determined rather than assumed.

Accordingly, the limits of physical laws and processes are themselves a matter of investigation. In other words, such limits are an open question. A good example of this is the argument for the existence of design in the physical world apart from human design. From human design we know something of what design is; if the physical world exhibits the features of human-like design but were not designed by humans, then a boundary condition has been found.

Otherwise, physical history is like human history. Physical particulars of the past are at the forefront, and universals of physical science are in the background. Whatever might be determined by physical science is acknowledged but the significant changes, the physical catastrophes and surprises, are granted a much greater rôle. There will no doubt be controversies between those who place much weight on key events versus those who look to the slow accumulation of little changes but such is usual for history.

Cycle of science

There is a well-known alternation of induction and deduction in science (click to enlarge):

induction-deduction cycleThe induction phase consists of data collection, data analysis, and model development. The deduction phase consists of taking the model, making hypothetical inferences, and following up with experiments that lead to new data collection. Then the cycle repeats.

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Isaiah Berlin on history and science

The following (long) excerpts are from Isaiah Berlin’s article “History and Theory: The Concept of Scientific History”, published in History and Theory 1 (1):1 (1960). Republished in Concepts and Categories: Philosophical Essays. NY: Viking Press, 1979. (online here).

HISTORY, according to Aristotle, is an account of what individual human beings have done and suffered. In a still wider sense, history is what historians do. Is history then a natural science, as, let us say, physics or biology or psychology are sciences? And if not, should it seek to be one? And if it fails to be one, what prevents it? Is this due to human error or impotence, or to the nature of the subject, or does the very problem rest on a confusion between the concept of history and that of natural science? These have been questions that have occupied the minds of both philosophers and philosophically minded historians at least since the beginning of the nineteenth century, when men became self-conscious about the purpose and logic of their intellectual activities. But two centuries before that, Descartes had already denied to history any claim to be a serious study. Those who accepted the validity of the Cartesian criterion of what constitutes rational method could (and did) ask how they could find the clear and simple elements of which historical judgements were composed, and into which they could be analysed: where were the definitions, the logical transformation rules, the rules of inference, the rigorously deduced conclusions? While the accumulation of this confused amalgam of memories and travellers’ tales, fables and chroniclers’ stories, moral reflections and gossip, might be a harmless pastime, it was beneath the dignity of serious men seeking what alone is worth seeking – the discovery of the truth in accordance with principles and rules which alone guarantee scientific validity.

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