“The habit has developed of assuming that a physical theory is necessarily sound if its mathematics are impeccable: the question of whether there is anything in nature corresponding to the impeccable mathematics is not regarded as a question; it is taken for granted.” Dingle 
Chambers Encyclopaedia – One kind of application of Special Relativity is to the fundamental concepts of physics, such as mass, energy, force etc. If the laws of conservation of mass and of linear momentum are to be valid in all inertial frames of uniform motion the mass of a particle is defined as m(1 − v²/c²)^{½} and its energy as mc²(1 − v²/c²)^{½}. Here m is the mass the particle would have in an inertial frame in which it was at rest (its proper mass) and v is its mass relative to the frame. It then follows that the laws of conservation of mass and energy are identical if we assume that the proper mass of a particle is itself equivalent to the amount of energy mc². These definitions of mass and energy imply that v must be less than c and it is for this reason that Special Relativity restricts itself to relative velocities not greater than c.
The two definitions have played a capital part in modern atomic physics. The equivalence of mass and energy helps to explain the phenomenon of radioactivity and the explosions of atomic bombs. It also accounts for the production of the vast quantities of radiation emitted by the stars without an apparent cooling of these bodies, the necessary energy being found in the conversion of a small fraction of their masses into radiant energy.
McCrea rather protests too much in his attempts to refute Dingle’s arguments, although he concedes that Dingle “has not made any mistakes in his algebra.” Instead, McCrea believes that Dingle “deals with objects to which the [SR] theory explicitly denies a meaning.” This, of course, begs the question: if the theory denies a meaning to a physical model, is the theory coherent? He claims that Dingle’s assertion is false because “he is not talking about the same thing, but two different things... equations (3) and (4) concern two different sets of events, and so they cannot contradict each other.” McCrea adamantly refuses to acknowledge the validity of coordinate transformation systems and his refusal to treat both events in a common system is contrary to both the letter and spirit of special relativity theory.
McCrea repeatedly asserts, in the summary in Special Relativity (2) and elsewhere, that “Because A is never at E_{1}, the phrase is meaningless and so Dingle’s (3) is meaningless. Correspondingly his (4) is meaningless.” That he should attempt to use “A is never at E_{1}” as valid criticism was regarded by Dingle as “indefensible.” Dingle had explicitly defined A as a clock distant from H. Clock A is never at E_{1}; it merely records the event.
To better illustrate the clock paradox Dingle developed his thesis further. A basic example of his argument is provided by considering only the x′axis and xaxis in two coordinate systems k and K. The coordinate system k is moving at uniform velocity v relative to K.
N  B  »»»»»»» v »»»»»»»  
xaxis of K, clock readings denoted by t, below 

A  H 
A and H are two regularly running clocks at a fixed distance from each other in coordinate system K. N and B are another pair of regularly running clocks at a fixed distance from each other in the moving coordinate system k. (The distances NB and AH are independent and arbitrary.)
The readings of N and B with respect to the coordinate system k are denoted by t′ and the readings of A and H, in the coordinate system K, by t. (For simplicity we regard K as stationary.) We now examine three successive events as k moves with uniform velocity v relative to K.
At event E_{0}, B is adjacent to A and both are observed to read zero. We also synchronize N and H to read zero at this moment.
t′ = 0  
N  B  
A  H  
t = 0 
At event E_{1}, B is adjacent to H; B is observed to read t′_{1} and H is observed to read t_{1}. Clock A is distant from clock H but reads t_{1} when H reads t_{1}.
t′ = t′_{1}  
N  B  
A  H  
t = t_{1} 
At event E_{2}, N is adjacent to A; N is observed to read t′_{2} and A is observed to read t_{2}. This is all independent of the theory – it is merely a description of a physical process. A theory is required when we wish to determine two independent things: (1) the values of t′_{1}, t′_{2} for given values of t_{1}, t_{2} or vice versa and (2) the relative values of A and B when these values apply. Clock B is distant from clock N, but reads t′_{2} when N reads t′_{2}.
t′ = t′_{2}  
N  B  
A  H  
t = t_{2} 
We now apply Einstein’s theory, supposing that A is fixed at the origin of the K coordinate system and B is fixed at the origin of the k system.
(i) t and t′ are related by the Lorentz transformation so that:–
Between events E_{0} – E_{1}  t′_{1} = at_{1}  (1) 
t_{1} in the K coordinate system is corrected by a (as A is offset) to read at_{1}
Between events E_{1} – E_{2}  t_{2} = at′_{2}  (2) 
t′_{2} in the k coordinate system is corrected by a (as B is offset) to read at′_{2}
and  a = (1 − v²/c²)^{½} 
(ii) The relative velocity a (the ratio of distance vs. time, sometimes denoted by µ) of k in relation to K is determined by choosing a pair of events and comparing the intervals between the readings of A and B at these events.
Einstein chose events E_{0} and E_{1}. At these events A reads 0 and t_{1} respectively and B reads 0 and t′_{1} respectively. The reason why A must be held to read t_{1} at E_{1} is that H reads t_{1} at this event, and by application of the Lorentz transformation the process by which A is set in relation to H synchronizes it with H.
Shortly however we reach a contradiction. Between events E_{0} and E_{1}, A advances by t′_{1} = t_{1} and B by t′_{1} = at_{1} by (1). Therefore:
rate of A  =  t_{1}  =  1  (3)  




rate of B  at_{1}  a 
But now choose events E_{0} and E_{2}. At these events A reads zero and t_{2} respectively and B reads zero and t′_{2} respectively. The reason why B must be held t′_{2} at E_{2} is that N reads t′_{2} at this event and the process by which B is set in relation to N synchronizes it with N.
However between events E_{0} and E_{2}, B advances by t_{2} = t′_{2} and A by t_{2} = at′_{2}, by (2). Then:
rate of A  =  at′_{2}  =  a  (4)  



rate of B  t′_{2} 
Equations (3) and (4) are contradictory; one result is the reciprocal of the other. Further, since v is nonzero, a is never 1 and it is obvious nonsense to derive (rate A)/(rate B) > 1 and (rate A)/(rate B) < 1 in the same system. Hence the theory from which they are derived must be invalid. Einstein in his theory gave only (3) and accepted it as the unique value of the rateratio; he did not verify the result by checking the interval between E_{0} and E_{2}. Had he done so he would have seen that his conclusion was untenable.
Comments are invited. paullb@totalserve.co.uk
Comment from Lev Verkvsky:
Special Relativity is completely erroneous since it is based on the wrong kind of [coordinate] transformations: the scale factor included in the original more general form of the Lorenz Tranformations, which incorporated with a scale factor (a Doppler effect), has been removed. Later Lorenz, and also Poincare and Einstein equated the scale factor to one without sufficient basis. The scope of the formulas was artificially narrowed, and they became incorrect. This led to a logical contradiction of the Theory, and to unsolvable paradoxes [including the clock paradox]. For more details, see "Memoir on the Theory of Relativity and Unified Field Theory" (published in Russian in 2000) and recently updated and published in English  as a PDF: