The "mathematics" and "geometry" of Special Relativity are based, not only on the axioms of mathematics (such as those of Peano, or those enunciated by Zermelo and Fraenkel, later extended by von Neumann) and on the postulates and propositions of Euclidean geometry, but also on the postulate that light propagates in a vacuum at a speed which is constant for all observers, regardless of the speed of the observer relative to the source of light.

Thus for their formulation, the "mathematics" and "geometry" of Relativity require a postulate additional to the axioms, postulates and propositions from which the rest of mathematics and geometry (as we know them to be) are formulated.

However, logically speaking, no axiom, postulate or proposition in mathematics and geometry may contradict another; and neither may it — nor any theorem derived from it — contradict any other theorem. If this occurs, that particular axiom, postulate or proposition cannot validly be a part of mathematics and/or geometry.

We prove hereunder that the Galilean theorem of addition of velocities holds true even if the rate of passage of time and the lengths of measuring rods is not constant from one inertial reference frame to another — as is claimed by Special Relativity. Since the Special Relativistic "theorem" of addition of velocities contradicts the Galilean theorem, it is demonstrated that the postulate on which Special Relativistic "mathematics" and "geometry" are based — namely the postulate of the constancy of the speed of light — cannot be a part of mathematics and/or geometry as we know them.

On a Cartesian graph with perpendicular x and y axes (the z axis being eliminated for the sake of simplicity — though it may be added if so desired), consider a point-like object O from the viewpoint of observers in two different inertial reference frames F and F', with the frame F' moving rectilinearly and uniformly, parallel to the x axis in the positive direction at a non-zero velocity v" relative to the frame F; the object O moving at a non-zero velocity v relative to the frame F in the same direction as that in which the frame F' is moving relative to the frame F; and the object O moving at a non-zero velocity v' relative to the frame F' in the same direction as that in which the frame F' is moving relative to the frame F.

Let the following be defined as hereunder:

1. (x, y) º a pair of co-ordinates representing a given point in the frame F, where the y co-ordinate is identical to the y co-ordinate of the object O;
   
2. (x', y') º the pair of co-ordinates representing the point in the frame F' which coincides to the point (x, y) in the frame F at any arbitrary instant in time 0.00, according to a clock associated with the frame F;
   
3. t º any arbitrary time interval immediately after the time instant 0.00 referred to above, the time interval t being measured by a clock associated with the frame F;
   
4. d º the distance, as measured by a measuring rod associated with the frame F, travelled parallel to the x axis by the object O in the frame F relative to the point (x, y) referred to above during the time interval t referred to above;
   
5. t' º a time interval, as measured by a clock associated with in the frame F', corresponding to the time interval t referred to above;
   
6. d' º the distance travelled by the object O in the frame F' relative to the point (x', y') referred to above during the time interval t' referred to above, as measured by a measuring rod associated with the frame F';
   
7. t" º any arbitrary time interval immediately after the time instant 0.00 referred to above, the time interval t" being measured by a clock associated with the frame F;
   
8. d" º the distance, measured by a measuring rod associated with the frame F, travelled by the point (x, y) in the frame F relative to the point (x', y') the frame F' during the time interval t" referred to above.
   

Then:

1. v = d/t,
   
2. v' = d'/t',
   
3. v" = d"/t",
   

Now:

1. Since by definition t" is arbitrary, we shall let t" = t.
   
2. Then v" = d"/t" = d"/t. (This is the speed of frame F relative to frame F'.)
   
3. From IV, VIII and 1. above, it is clear that during the time interval t, as measured by a clock associated with the frame F, the object O moves a total distance of (d - d") relative to the point indicated by the pair of co-ordinates (x', y') in the frame F' — and it is to be very carefully noted that the time interval t, as well as the distances d and d", are all measured with the help of measuring instruments associated only with the frame F.
   
4. v' = d'/t' is also the velocity of the object O relative to the frame F' — which is to say, v' is the velocity of the object O relative to the point indicated by the pair of co-ordinates (x', y') in the frame F', which in turn means that v' is equal to the distance (d - d") travelled by the object O from the point (x', y'), divided by the time interval t required by the object O to travel the distance (d - d") from the point (x', y').
   
5. For even in Relativity, the velocity of any object relative to a given point in space is the distance travelled by the object divided by the time interval required by the object to travel that distance away from that point in space — provided both the distance travelled by the object and the time interval required to travel that distance are measured using measuring instruments associated with a single frame of reference.
   
6. Now we noted in 3. above that t, d and d" are all of them measured by measuring instruments associated only with the frame F.
   
7. Thus v' = (d - d")/t = (d/t - d"/t) = (d/t - d"/t") = (v - v"), or v = (v' + v") — which confirms the Galilean theorem of addition of velocities.
   
8. Now the Lorentz transformation equations and the geometry of Minkowski space-time are obtained using the Relativistic postulate of the constancy of the speed of light regardless of the speed of the source of the light or of its observer; and among the equations calculated using this postulate is the so-called Relativistic "theorem" of addition of velocities, viz., v = (v' + v")/(1 + v'v"/c²).
   
9. But the equation in 8. above contradicts the equation in 7. above.
   
10. Since the equation in 7. above has been mathematically and geometrically proven, and since in mathematics and geometry, no theorem may contradict any other, the equation in 8. above cannot be a mathematical or geometrical theorem— i.e., a formula which can be proven; and as a corollary, the additional postulate which is required to formulate the equation in 8. above — namely the postulate of the constancy of the speed of light — cannot be a valid postulate of mathematics and/or geometry as we know them.