The sorcerer of 1710

During my career as a physicist and teacher of physics, I scarcely realised there was a scientific paradigm to change. I just got on with it without thinking much about what ‘it’ is, In that, I don’t think I was different from my colleagues. When, in my retirement, I read the long article on the 18^th century philosopher George Berkeley in the on-line Stanford Encyclopaedia of Philosophy, the following quotation awakened me to the fact that there are two scientific paradigms:

The sort of explanation proper to science, then, is not causal explanation, but reduction to regularity.

Causal explanation is a recognisable description of traditional science and paradigm I. It implies the existence of a large collection of named concepts like time, distance, speed and the speed of light.

Science based on paradigm II consists of the set of all data diagrams, each diagram being a simple line drawing with a few data points and a few numbers, the kind of thing that every scientist has in his or her notebooks. Data diagrams existed long before the arrival of man though. Apples fell from trees before Newton and pendulums swung before Galileo’s lantern. Data diagrams are found therefore in the past and the structure of the past provides their common framework and paradigm II.

The fundamental difference between the two paradigms is that paradigm I is created by scientists using their own concepts while the much simpler paradigm II exists already in each of our pasts and is there to be discovered (but not modified) by humans. Paradigm I science is scientists’ science. Paradigm II science is nature’s science.

According to paradigm II, an observer has a personal past in which the image of another object is seen by him to be as deep in that personal past as the object is far from him. Thus to see images very deep in his past an astronomer looks for objects which are very far from him. A personal past is instantly observable, as we see when we look at the night sky on a starry night. The night sky is part of one’s personal collection of data diagrams from very long ago.

Philosophers and scientists who have studied the writings of Berkeley still do not know what to make of him, but I think I do. He was both philosopher and scientist of course, but he was something far, far rarer than either of those. He was a paradigm changer, a sorcerer. People tend to read him with paradigm I in their mind and are baffled. Read him with paradigm II in mind and all is clear. I shall return to Berkeley and what happened in 1710 in a future article.

The two paradigms

Fig 1 summarises my conclusions on the paradigms.

Fig 1: The two paradigms of science.

Paradigm I is the paradigm of the 3-dimensional present. The world is described as a succession of short-lived 3-d presents strung along the axis of time. It consists therefore of one dimension of time and three dimensions of space and this 3+1 structure is its weak link. It is assumed that the directions in which we are designed to see are North-present, South-present, East-present, West-present, etc, but because sight is carried across the universe by light-waves at the relatively slow speed of 186000 miles a second, we only see the past.

Paradigm II is the paradigm of the personal 4-d past. In paradigm II, the directions in which we see are North-past, South-past, East-past etc. Those directions form a 4-d cone in the past and this conical structure is the paradigm’s strong point.

Einstein, the sorcerer’s apprentice

In 1905, paradigm I was in deep trouble. It had predicted that the speed of light would vary with the speed of the source of light, but after a careful search, no variation of the speed of light had been detected. (In paradigm II there is no speed of light so, of course, there are no variations either.) Einstein set out with the mission to save paradigm I by making its speed of light a constant. With that end in view, he devised a new law describing the way speeds combine, Einstein’s speed-addition law.

v₃=(v₁+v₂)/(1+(v₁v₂/c²))                 (1)

What Einstein did not know is that a similar equation arises in paradigm II, to describe how the sines of angles of divergence, S, combine.

S₃=(S₁+S₂)/(1+S₁S₂)                         (2)

Einstein introduced equation (1) into science with the aim of saving paradigm I but what he really did was to switch to paradigm II by equation (2), thereby converting his concept of relative speed into the sine of an angle and the speed of light into the sine of a right angle. After that, paradigm II was in control of his theory with its own diagrams, and Einstein was left trying to make sense of what was going on with irrelevant reference frames.

It is a good story and divides into two parts. Part A describes how paradigm II takes over Einstein’s theory. Part B is the story of how the hapless Einstein tries to present paradigm II as a modified version of paradigm I.

Part A

Equation (1) is Einstein’s answer to the following question. If v₁ is the relative speed of objects a and b, and v₂ the relative speed of objects b and c, all in the same line, what is the relative speed of objects a and c?

Equation (2) is the answer to a similar question. If S₁ is the sine of the angle of divergence of two lines a and b, diverging from a point, and S₂ the sine of the angle of divergence of lines b and c, diverging from the same point in the same plane, what is the sine of the angle of divergence of lines a and c?

The paradigm II diagram implicit in that question is fig 2.

 

Fig 2: The diagram implicit in equation (2)

Two paths, OX and OY, with gradients D and D^-1 diverge from a point O about an angle of divergence, α. From fig 2 one obtains D=tan(45+(α/2)) and then, by writing the tangent as sine divided by cosine and expanding the trigonometric functions one arrives at equations (3) and (4) where S=sin(α).

D²=(1+S)/(1-S)               (3)

and

S=(D-D^-1)/(D+D^-1)         (4)

D and S are simple numbers. If D=2, S=3/5 and if D=3, S=4/5. Conversely, if S=3/5, D²=4. There are no concepts of time, distance, speed or speed of light in fig 2. It is a data diagram of a kind that was created before the arrival of man by exploding seeds.

The broken arrows in fig 2 are along directions in which observers see and they show that observations from anywhere on OX show Y’s clock to be D times less advanced than X’s clock and observations from anywhere on OY show Y’s clock to have advanced D times more than X’s clock. By taking the readings of both clocks, two numbers are obtained and these serve as the coordinates of the point of observation. Fig 3 has sets of coordinates to illustrate that.

Fig 3: Coordinates of points along the diverging paths of fig 2.

Fig 4: A trio of diagrams like fig 2 related by equation (2).

When OM and MN are projected on to the y axis one obtains 1 and S and when ON is proected on to the x axis one obtains C=cos(α). From (4) one obtains S_3/2, S₂ and S₃ as 5/13, 3/5 and 4/5 and then by inserting these numbers into (2), one can confirm that (2) works for the trio in fig 3.

To obtain (2) one substitutes (3) into D₃=D_3/2D₂ to obtain (5)

(1+S₃)/(1-S₃)={(1+S₁)/(1-S₁))}{(1+S₂)/(1-S₂)}          (5)

and then, by taking the numerator-denominator/numerato+denominator of both sides of (5), one obtains (2). All the diagrams of paradigm II are knotted together by equations of this kind into a single logical complex.

Equation (3) is the answer to the question ‘How is the Doppler parameter, D, related to the sine of the angle of divergence, S, of the paths of two diverging objects?’ It is relevant therefore to all prosecutions for speeding based on evidence from Doppler cameras. That evidence has nothing to do with the concept of speed. By analysing data like that of fig 3, the evidence shows when the angle of divergence between the paths of a car and the camera has exceeded a permitted value.

Thus, straightforward chains of mathematical and geometrical logic lead from equation (2) to fig 2, its related equations, and observed data.

It was on to this integrated logical sea of paradigm II that Einstein launched himself on the frail craft of equation (1), equipped only in the way of navigational charts with reference frames from paradigm I.

Part B

It is surprisingly easy to make a paradigm II diagram look like a paradigm I reference frame. Fig 5 shows how.

Fig 5: Fig 5B is fig 5A with x units D times longer than y units.

Fig 5A is like fig 4 and shows three paradigm II diagrams. Fig 5B shows the same after units parallel to the x axes have been stretched to be D times longer than units parallel to the y axis. That does not change coordinates of course so fig 5B is still a set of paradigm II diagrams. To Einstein though, it was a reference frame with a peculiar defect.

The coordinates at N are not (d=v, t=1) as he expected, but (C/D, C). To put that right (as he saw it), he introduced a second coordinate, γ^-1, to enable him to ‘restore’ the coordinate at N to what he believed was the right number. Now though, he was in the hands of a disciplined logical system which did not permit ‘adjustments’ of that kind. He got a message from his paradigm II mathematics in the form of equation (6).

γ^-1=(1-(v/c)²)^1/2                (6)

Equation (6) told Einstein that he had one independent variable, not two, and that he should change his variable v/c to S=sin(α) and his variable γ^-1 to cos(α). (With paradigm I, a scientist is in control of his own theory and he decides how many variables it should have. With paradigm II, mathematical logic decides.)

Einstein did not change his variables as instructed so he failed to discover paradigm II, though he had obtained all its equations and had been told that its variables are D, S and C. He continued with his reference frames and time and distance paradigm and his v/c and γ^-1 variables even when, as in the twin paradox, they led to nonsense.

Einstein’s contrivances are the assumptions he made to reconcile his theory with paradigm I. Chief among these is that the speed of light is a constant. He had actually re-defined the speed of light, c, to be the sine of a right angle, 1, but he now awarded it the dimensions LT^-1 and declared it to be a speed. A velocity is a vector and a constant is a scalar but c is both. Why scientists accepted that (me included) is a mystery.

His second contrivance was his variable, γ^-1 which he attributed to ‘time dilation’. That was simply unnecessary because the coordinates in fig (6) are explained by paradigm II. Time dilation and length contraction have given rise to a complex folklore in which long trains are hidden in short tunnels and long poles accommodated in short barns. Those stories can now be put to rest.

The third contrivance is his explanations of redundancy equations. Einstein uses the symbols t, d, v, c, E, p, and m where paradigm II has none. All these symbols should therefore be changed to one or other of the numbers. S, C or 1. For example, Einstein’s equation E²=p²c²+m²c⁴ is really only 1=S²+C². People can stop being impressed by E=γmc². It only means 1=(S/S)1².

With Einstein’s theory of relativity, it is, as it is with the writings of George Berkeley. Both are best understood with paradigm II in mind.

That’s all for now. My next article will be about what happened in 1710.

Good bye and thank you for your company.