Friday, February 6, 2015

Foundation Paradox and the Last Digit

       Foundation Paradox and the Last Digit

The Foundation Paradox comes in various flavors, including the First Cause and the Munchausen Trilemma. To wit:

The First Cause:
Either: the chain of causation goes back infinitely, and there is no first cause;
Or: there is a first cause, itself uncaused;
Or; there is a first cause, caused by another cause, itself caused by the first cause, so causation loops.
And in none of these three cases is causation well-founded.

Munchausen Trilemma:
Any explanation is either:
infinite (it goes back forever);
incomplete (it starts with an unexplained explanation);
or circular (it loops).
And none of these are properly explanatory.

These two are clearly analogous; and they have many variants. (For instance, does the meaning of life have a meaning?)
The difficulty is often posed as theological, or scientific; but really the problem is mathematical. For consider the decimal expansion of a real number. Does it have a last digit?
Either: the decimal expansion is infinite, and there is no last digit;
Or: the decimal expansion terminates with a last digit and nothing but zeros thereafter;
Or: the decimal expansion loops in a cycle.
That is; a real number is either irrational, dyadic rational, or non-dyadic rational.

In general a sequence is either infinite or terminating or cyclic. This uncontroversial mathematical proposition implies theological and philosophical confusion.

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