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A trivial refutation of one of Dingle's Fumbles
Too simple to bother, but what the heck...

- See also this nice little chat at the Wikipedia -

In this appendix to "Science At the Crossroads" on page 230, Herbert Dingle writes:

(start quote)
Thus, between events E0 and E1, A advances by t1 and B by t'1 = a t1 by (1). Therefore
 
           
...
Thus, between events E0 and E2, B advances by t'2 and A by t2 = a t'2 by (2). Therefore
 
           

Equations (3) and (4) are contradictory: hence the theory requiring them must be false.
(end quote)

Dingle should have written as follows:

(start correction)
Thus, between events E0 and E1, A, which is not present at both events, advances by t1 and B, which is present at both events, by t'1 = a t1 by (1). Therefore
 
           
...
Thus, between events E0 and E2, B, which is not present at both events, advances by t'2 and A, which is present at both events, by t2 = a t'2 by (2). Therefore
 
           

Equations (3) and (4) are consistent and say that any event's coordinate time is always larger than its proper time:
hence there is no reason to say that the theory requiring them must be false.
(end correction)

Click to play with the figure



Compare with the following simple analogy. This is how Dingle should have found a trivial contradiction in the laws of geometry and perspective.

\color{OliveGreen}\text{X}\color{Black}\text{覧覧覧覧覧覧覧}\color{Blue}\text{Y}     X and Y are twin brothers
\color{Black}\text{A}\color{Black}\text{覧覧覧覧覧覧覧}\color{Black}\text{B}     A and B are facing one another with some distance between them
\color{Red}\text{P}\color{Black}\text{覧覧覧覧覧覧覧}\color{Brown}\text{Q}     P and Q are twin sisters

So X and P are standing nearby next to A, and Y and Q are standing nearby next to B.

A looks through a gap between her fingers at the twin brothers X and Y, and she notices that X's gap (nearby A) is twice as large as Y's (nearby B). Therefore for A it is true that
\frac{\color{OliveGreen}\text{gap near A}}{\color{Blue}\text{gap near B}} = 2 \qquad \text{(3)}
B looks through a gap between his fingers at the twin sisters P and Q, and he notices that P's gap (nearby A) is half as large as Q's (nearby B). Therefore for B it is true that
\frac{\color{Red}\text{gap near A}}{\color{Brown}\text{gap near B}} = \frac{1}{2} \qquad \text{(4)}
Clearly, equations (3) and (4) are contradictory: hence the theory of perpective behind them must be false.

Or is it?

Of course it isn't. The equations are just poorly expressed, and should be expressed as follows:

For A, looking at twin brothers X and Y, it is true that
\frac{\color{OliveGreen}\text{X-gap near A seen by A}}{\color{Blue}\text{Y-gap near B seen by A}} = \frac{\color{OliveGreen}\text{local gap}}{\color{Blue}\text{remote gap}} = 2 \qquad \text{(3)}
For B, looking at twin sisters P and Q, it is true that
\frac{\color{Brown}\text{Q-gap near B seen by B}}{\color{Red}\text{P-gap near A seen by B}} = \frac{\color{Brown}\text{local gap}}{\color{Red}\text{remote gap}} = 2 \qquad \text{(4)}
Clearly, equations (3) and (4) are consistent: hence there is no reason to think that the theory of perspective behind them must be false.

The trick to make this happen: proper understanding, proper labelling, proper expressing. Trivial.

Hit this to mail me.
(-: Dirk Van de moortel ;-)

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