What Happened to Dingle?
On Thu, 20 Jul 2000, Harold Ensle wrote:
> H.Dingle really saw what was going on. He also had a simple proof
> that showed that SR was impossible. For some reason SRists can't
> seem to understand it.
L Hoffman wrote:
>Would you care to summarize Dingle's proof? I haven't seen it yet.
You almost certainly HAVE seen it already. It's the same "proof"
that every anti-relativity crackpot puts forward, in one form or
another. In a nutshell, Dingle considers two systems of inertial
coordinates x,t and x't' with a relative velocity of v, and then very
elaborately and clumsily constructs the partial derivative of t' with
respect to t, and the partial derivative of t with respect to t'. He
notes that these partials are equal, and declares this to be logically
inconsistent. Sad.
In a more familiar context, Dingle's "proof" can be applied to show
that Euclidean geometry is "impossible". Consider two Cartesian
coordinate systems xy and x'y' with a common origin, but rotated
by an angle of q with respect to each other. Given the xy coordinates
of any point, we can compute the x'y' coordinates by means of the
equations
x' = cos(q) x + sin(q) y y' = -sin(q) x + cos(q) y
Conversely, we can solve these equations for x and y in terms of
x' and y' to give the inverse transformation
x = cos(q) x' - sin(q) y' y = sin(q) x' + cos(q) y'
This is just elementary linear algebra. Now, if we hold y constant
and vary x, how much does x' vary? In other words, what is the
partial derivative of x' with respect to x? Letting Dx'/Dx denote
this partial, it's obvious that Dx'/Dx = cos(q). Now we ask a
different question, namely, if we hold y' constant and vary x',
how much does x vary? This is equivalent to asking for the partial
of x' with respect to x, and of course we have Dx/Dx' = cos(q).
Dingle's confusion is that (like some clueless freshman calculus
students) he imagines Dx'/Dx and Dx/Dx' are algebraic reciprocals
of each other, which would imply that 1/[Dx'/Dx] = Dx/Dx', and
therefore cos(q) = 1/cos(q), which is impossible for any q other
than 0. Must we conclude that Euclidean geometry (and linear
algebra) is inconsistent?! No, because Dingle's argument is
obviously specious. Partial derivatives can't be algebraically
inverted.
The application of Dingle's argument to the Lorentz transformation
is exactly the same. Two inertial coordinate systems xt and x't'
with a mutual relative velocity v are related according to the
equations
t' = (1/g) t - (v/g) x x' = (1/g) x - (v/g) t
where g = sqrt(1-v^2). The inverse transformation is
t = (1/g) t' + (v/g) x' x = (1/g) x' + (v/g) t'
In this case we have the two partial derivatives Dt'/Dt = 1/g and
Dt/Dt' = 1/g. Dingle erroneously assumes that 1/[Dt'/Dt] = Dt/Dt',
and so arrives at 1/g = g, which is impossible for any v other
than 0. Again the fallacy is the assumption that partial derivatives
can be algebraicaly inverted.
Of course, we CAN invert total derivatives, so let's see what happens
if we take the absolute differentials (for any constant v) of the
time transformation equations. We have
dt' = (1/g) dt - (v/g) dx
dt = (1/g) dt' + (v/g) dx'
Dividing the first equation by dt and the second by dt' gives
dt'/dt = 1/g - (v/g) dx/dt
(1)
dt/dt' = 1/g + (v/g) dx'/dt'
Considering the first of these equations, notice that in terms of
the unprimed reference frame we have dx/dt = 0 (by definition),
and so we have dt'/dt = 1/g. However, in terms of the primed
coordinates we have dx/dt = v, which gives
dt'/dt = (1/g)(1-v^2) = g
This shows how we arrive at either one of the partials, depending
on which direction in spacetime we are considering. Likewise for
the second transformation equation we can consider the cases when
dx'/dt' = 0 or -v, giving the results 1/g and g respectively.
Also, since equations (1) are reciprocals of each other, we can
multiply them together to give unity, i.e.,
[dt'/dt][dt/dt'] = [1 - v dx/dt][1 + v dx'/dt']/g^2 = 1
Notice that dx/dt is the velocity of one worldline with respect to
some arbitrary reference frame, and dx'/dt' is the velocity of the
other worldline with respect to that same reference frame. Let us
denote these velocities by u and w respectively. Recalling
that g^2 = 1-v^2, the above equation becomes
[1-vu][1+vw] = 1-v^2
Solving for u gives the familiar formula
w + v
u = -------
1 + wv
which is the relativistic speed composition formula. Needless to
say, there is nothing inconsistent or self-contradictory here.
Dingle was simply making an elementary error. Pictorially, his
claim was that the segment ratios a/b and c/d in the figure below
must be equal, whereas special relativity claims they are reciprocal.
A N H B
\ / \ /
\ / \ /
\/ \/
/\ /\
/ \b c/ \
/ \ / \
a/ \/ \d
/ . '/\' . \
/ . ' / \ ' . \
./' / \ '\.
t'=0 . ' / / \ \ ' . t=0
' / / \ \ ' .
The more interesting question is what happenned to Dingle. For
most of his life he wrote approvingly about relativity, beginning
with an article in 1922. He even wrote a text book on special
relativity (1940) in which he carefully explained the relativity
of simultaneity, and so on. But then in his later years (apparently
beginning around 1959) he embarked on a passionate anti-relativity
crusade, and exhibited a complete inability to grasp the relativity
of simultaneity. This culminated in his 1972 book entitled "Science
at the Crossroads". The obvious question is, how could he explain
a concept for years, and then later demonstrate a complete failure
to grasp that same concept?
I think the answer can be found through a careful reading of
Dingle's essay for the Encyclopedia Britannica on the philosophical
consequences of relativity (written during his pro-relativity
days). This article, in retrospect, shows that his acceptance
of special relativity (he never claimed to understand general
relativity) was based firmly on the notion that there is no
external objective reality. He believed that the essential
significance of relativity was that, in his words,
...the idea of something existing objectively, which physical
measurements revealed, had to be given up... The philosopher
must henceforth interpret physics in terms of operations and
their results alone, leaving external existences out of
account... Physics was thus thrown back on the unadorned
description of itself as the discovery of relations between
the results of chosen operations of measurements.
It's clear that Dingle accepted relativity (prior to old age) as
simply a collection of brute facts that need not yield any coherent
picture of an objective external reality. His view was similar to
the modern acceptance of quantum mechanics with the "measurement
problem" unresolved, i.e., we can't think of a realistic model
of an external reality that always yields the results of our
measurements; we can only describe the patterns in those results
as abstract brute facts that must be accepted. In other words,
Dingle's attitude (in his early days) was that "one does not
understand relativity, one merely gets used to it". This of
course is a paraphrase of a famous remark concerning quantum
mechanics, but the point is that Dingle was entirely mistaken
in applying this "shut-up-and-calculate" approach to relativity,
because in fact relativity (unlike quantum mechanics) is an
entirely classical theory, and is firmly based on a perfectly
coherent model of objective external reality. In retrospect we
can see that the young Dingle never grasped this model - indeed
his whole philosophy of science (in those years) was that
relativity had rendered all such models unviable.
If he had applied this line of reasoning to quantum mechanics, he
would have been in the mainstream of scientific thought, which
still today has been unable to reconcile the full range of
demonstrated quantum phenomena with any classically realistic
objective model. However, Dingle was very mistaken in applying
this line of reasoning to special relativity, which is a purely
classical theory with a perfectly sound objective model (Minkowski
spacetime). Looking back at Dingle's early explanations of the
twins paradox, we can see that he had NEVER grasped this simple
model. Instead he had simply told himself (like someone thinking
about Schrodinger's cat) that when we make certain measurements
we get certain results, despite the fact that he himself did not
understand it. He accepted this because he believed that NO ONE
understood it, and in fact he believed that that was the whole
point of relativity, i.e., that we must now believe things
that cannot be "understood" in the classical sense of being
manifestations of an external objective reality.
Then in his later years he rejected this approach (as did most of
the formerly enthusiastic circle of logical positivists), and decided
that we cannot reasonably dispense with the idea of an objective
reality. (Ironically, this was also Einstein's mature view.) The
problem was that once he made this change, it exposed the fact that
he had never grasped the simple objective model of special relativity.
In fact, he had spent much of his life trying to convince himself and
others that no such model was possible, and that this impossibility
was the whole message of relativity. He could not, in his old age,
accept the idea that in fact relativity had a perfectly simple
objective model, and that his views on this subject, to which he
had devoted much of his life, had always been fundamentally flawed
and misguided. To contemplate such a possibility is not within the
capacity of old men (as is demonstrated daily in internet newsgroups).
On a related point, I note that Professor McCrea (with whom Dingle
had his famous "flame war" in the pages of Nature in 1962) just
passed away in April of last year (1999) at the age of 94. Obviously
McCrea was on the right side of the debate, but in reading his
rebuttals of Dingle's position I think they weren't as clear and as
direct as they could have been. Of course, any false premise can be
refuted in infinitely many ways, and there is always the temptation
to pile refutations on top of each other, even though the effect of
this is often to blunt rather than sharpen the refutation.
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