TUNING digest 589

Brian Writes:

>...but belief is *not* the same thing as proof...

Actually, there are those of us who believe that proof and belief
are exactly the same thing.

Others don't believe in believing, and view "proof" as a somewhat
underhanded means of persuasion.

>To my
>knowledge, there is as yet no known algorithm
>by which the entire field of reals may be sieved
>and by which all transcendental numbers will
>always be found.

There may be no such thing as a transcendental number. A number
cannot be recognized as transcendental, or even irrational, by
the mere quoting of digits, because no matter how many digits are
enumerated, one can never be sure that specifying more digits
will not result in a repetition, and thus reduce the number to a
mere rational.

Therefore, we do not have a transcendental, number, but a
transcendental algorithm. No possible enumeration of a subset of
the digits can allow us to determine whether the number is even
rational or not, much less distinguish the transcendentals from
the irrationals. An analysis of the generating algorithm may
give an indication.

I think that for the purposes of this forum, there is no
practical difference between transcendental numbers, irrationals,
and large irreducible rationals.

Anything that goes beyond the resolution of the ear is off the
scale as far as tuning is concerned.


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