The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

Suppose that we adopt the rigid spherical ball picture, requiring that when two of the spheres collide they simply rebound perfectly elastically. That is to say, they separate again without any loss of energy (or of total momentum), as if they were perfect billiard balls. We also have to specify exactly how the forces are to act between one ball and another. For simplicity, we can assume that the force that each ball exerts on each other ball is along the line joining their centres, and its magnitude is a specified function of the length of this line. (For Newtonian gravity this assumption automatically holds true, by a remarkable theorem due to Newton; and for other laws of force it can be imposed as a consistent requirement.) Provided that the balls collide only in pairs, and no triple or higher-order collisions occur, then everything is well defined, and the outcome depends in a continuous way on the initial state (i.e. sufficiently tiny changes in the initial state lead to only small changes in the outcome). The behaviour of glancing collisions is continuous with the behaviour when the balls just miss one another. There is, however, a problem with what to do with triple or higher-order collisions. For example, if three balls A, B, and C come together at once, it makes a difference whether we consider A and B to come together just first, and C then to collide with B immediately afterwards, or whether we consider A and C to come together first, and B then to collide with...