The competition of alternative concepts can be quite sophisticated however. Let us illustrate this with three examples. The most illuminating example may well be Pythagoras’s theorem and its relation to the circle. This problem concerns mathematics, so that the discussion is less taxed by semantics and empirical matters - though there is of course the theory about empirical space. The second example of ‘falsification’ is surely in the realm of empirics. The third example concerns the distinction between determinism and volition.
Pythagoras and the circle
Regard a triangle with perpendicular sides a and b and hypotenuse c. There are two points of view:
1. Pythagoras proved [57] that the square of the hypotenuse equals the sum of squares of the perpendicular sides, i.e. that a2 + b2 = c2
2. For the circle, it is taken as the defining quality of the circle, and thus accepted without proof, that the points are at equal distances from the origin. In other words, a circle with radius c is defined as the collection of points (a, b) at a distance of c from the center. Thus a2 + b2 = c2 by definition.
The two points of view are presented in Figure 16. The definition of the circle can be taken for granted, since it is just a definition. On the other hand, it will be very useful to discuss the proof of the Pythagoras theorem, since then we see the need for a proof.
Let us take the square with sides z = a + b and surface z * z = z2 = (a + b)2. Within this square we can see four triangles with straight sides a and b and hypotenuse c, as has been done in Figure 16 in the square on the left.
In the square, another tilted square has been drawn, with sides c and thus a surface of c2. There are four surrounding triangles, each triangle has a surface of ½ a*b. The surface of the large square is equal to the surface of the tilted square and the four triangles.
Figure 16: Pythagoras and the circle