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Transcendental Nature of Pi

March 22, 1987

Your concern for the existence of all the digits of the constant pi can be easily answered: Pi, like all mathematical facts, is a concept, not a thing.

The formal written expression of some approximation to pi is a thing that represents the concept (but only approximately). Even if that approximation has been carried to 134 million digits.

The concept of an infinite series of decimal digits to represent pi exists, whether or not we ever get around to calculating them all. And a method exists for their calculation, to as many places as we please or have resources for.

Mathematics has long been dealing with the apparent inconsistencies between mathematics and the real world as we observe it. That is why pure mathematics, in its abstractness, is completely divorced from reality. A fact that led Bertrand Russell to observe that pure mathematics is the subject of which we do not know what we are talking about, nor whether what we are saying is true.

Thus we can feel comfortable with the mathematical truth that there is an infinite number of rational numbers between 0 and 1, although we shall never tabulate them all. One can prove that there is a one-to-one correspondence between all of those rational numbers and the digits to express pi, because both have a one-to-one correspondence to the ordinal numbers: first, second, third, . . . all of which we shall never commit to paper.

More troubling is the incompleteness of mathematics, dished up for us in 1931 by the brilliant Kurt Goedel. Goedel proved that mathematics cannot be simultaneously consistent and complete. Consistent in the sense that one can rigorously prove that mathematics is free of contradiction; complete in the sense that all truths of mathematics can be derived from a finite set of axioms.

Or as Goedel's Proof is usually stated: If mathematics is consistent, it is not complete. And therefore there exist mathematical truths that cannot ever be proven. But even this melancholy conclusion is not all bad. If mathematics is incomplete, then we have less reason to believe that the universe is totally deterministic, and we can hope that man, as a part of it, is not mechanically determined.

There are, then, questions for which there are no answers. Thank goodness, a lot of 'em!

JOHN F. FERGUSON

Sherman Oaks

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