Why coin tossing is random

Let's imagine flicking a coin into the air with the thumb.   We can suppose that the exposed face of the coin on landing is determined by the number of half-turns, n, that the coin makes in flight---if the coin lands on a soft surface we can ignore its motion after  it has touched down.  n will be proportional to the time t that the coin is in the air and its initial angular velocity w.  A little Newtonian dynamics shows that t is proportional to the coin's initial vertical velocity u imparted by the flick.  So n is proportional to uw.  Hence the dependency of the toss on u and w is qualitatively given by the diagram below.

The u-w plane divides into alternating Head/Tail bands whose boundaries are hyperbolae uw=constant.  Now, if we could control u and w precisely we could fix the outcome of the toss---the motion is quite deterministic. But human physiology prevents us.  No matter how hard we try to be consistent we will end up imparting a u and w that lie randomly within some region such as the oval shown dotted.  If this is large enough to intersect with more than one of the H/T bands then the outcome will be indeterminate.

This kind of delicate dependency on initial conditions explains a great deal of the indeterminacy we find in the macroscopic world.   All that is required is that regions of indeterminacy in the space of initial conditions (such as the oval above) be crisscrossed by regions of distinct deterministic outcome (like the bands of H and T).

We can apply this to the philosophy of mind.

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