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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)



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 perspective 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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