                           20050527.00

     Ich finde es praktisch den pythagoreischen Satz als
Erklaerung nicht des genauen Messens der Seiten der Dreieke zu
deuten, sondern als Approximationen, als Annaeherungen.  Dadurch,
durch diese Erkenntnis wuerden Dedekinds Schnitte, insofern als
sie geometrisch veranlasst sind, ueberfluessig, und das Mysterium
der Unmessbarkeit, Incommensurability der Hypotenuse von 2, 3, 5,
6, 7, 8, 10, 11, 12, 13, 14, ... usw, wird voellig, gaenzlich ins
Numerische verlegt.

     The conclusion that it is not specifically the hypotenuse of
the right triangle which is incommensurate; but that all lengths
are incommensurate; and the hypotenuse is no more incommensurate
than the legs (limbs) of the right triangle.  The question is no
longer one of geommetry, no longer one of the measurement of
lengths.  The question is why is the square root of 2, 3, 5 etc,
is: a) not a whole number, b) not a fraction, c) not a
terminating decimal, d) not a repeating decimal?  Why does the
act of dividing not produce a natural number?  Is not the finding
of square root in effect the division of a number by an unknown
such that the divisor and the quotient are equal?  Is not the
finding of an nth root in effect the division of a number n times
by an unknown divisor such that the divisor and the ultimate
quotient are equal?

     If lengths are by their nature incommensurate, so are areas
and volumes similarly; and all the calculations of plane and
solid geometry turn into, reveal themselves as approximations.
The question becomes: why are there irrational numbers?  How is
numerical irrationality to be explained?  The question why are
roots irrational?  must be preceded by the question, why are most
fractions irrational?  Maybe the question should be, not why are
most roots irrational?  but why are some roots rational?  Maybe
the question should be, not why are most fractions irrational?
but why are some fractions rational?

     Possibly the Pythagorean theorem is an approximation
comparable to the approximation of Newton's law of gravitation?

     Strictly within number theory, the irrationality of square
roots acquires a totally different significance, and lends itself
to a reduction to the phenomenology of number.  i.e. one might
say: This is how numbers are; this is how numbers behave; this is
how the function of the human mind is reflected in multiplication
and division; this is how the structure of the human mind is
reflected in multiplication and division.

     Der erste Schluss eines solchen Umdenkens besagte, dass
_keine_ Laenge genau messbar ist, dass die Unmessbarkeit,
incommensurability, der Hypotenuse

     The rule of procedure therefore is: first eliminate the
approximations by accepting them, recognizing them as such.  stop
pretending that successive approximations will lead to a
cognitive limit.  accept the approximations for example, of
measurement, as unavoidavble, inescapable consequences of the
limitations of the human mind.  second: accept the natural
numbers, with Kronecker, as facts of nature; i.e. as phenomena
reflecting the correspondence of the human mind with nature.  as
phenomena reflecting the correspondence of the inside with the
outside.  as phenomena reflecting the fact that the human mind,
that human thought is _part_ of nature, and imbedded
(eingeschlossen) in it.  third: contemplate the correspondence
phenomena, and understand, interpret their limitations.  This
third step is clearly recursive; but the consequences and
implications of this recursion are to broad to fit into the
margin of this manuscript.

     I think I remember that the Pythagorean theorem can be, and
originally was, demonstrated geometrically, with ruler and
compass, without resorting to numbers and algebra.  The
discontinuity from which the irrationality arises might therefore
be attibuted to the numerization of length, i.e. ascribed to the
process of measuring, as process seemingly introducing precision,
but actually inherently imprecise.  Consider in this context that
the construction with compass and ruler, however intuitively
convincing, carries with it its own approximation, its own
imprecision; and whether the mechanical construction process
necessarily causes these approximations to cancel out, I am not
sure.

     One cannot make what is imprecise precise.  One compensates
for imprecision by statistical techniques, but these do not
eliminate the imprecision.

     ==========

     Another related, but different consideration.  If one
considers real numbers as a separate entity: Real numbers, as
distinct from rational numbers are generated by division. Finding
the nth root of a number is also a species of division.

     A number generated by division (Nummer) might be considered
a different number from a number generated by counting (Zahl).
Indeed, counting, telling, zaehlen all refer to the same mental
activity.  If one divided 1 by 3, one obtains a repeating
decimal.  If one divided 2 by 3, one obtains a repeating decimal.
If one divided 3 by 3, one obtains a repeating decimal also, but
without realizing it.  The computer program will generate as many
3's as there are significant figures when 1 is divided by 3.  The
computer program will generate as many 6's as there are
significant figures when 2 is divided by 3.  The computer program
will generate as many 0's as there are significant figures when 3
is divided by 3.  The 0 generated as a remainder when 3 is
divided by 3 is no different from the 6 generated as a remainder
when 2 is divided by 3 is no different from the 3 generated as a
remainder when 1 is divided by 3.  Indeed it may be a
misconception to think that natural (or rational) numbers are
susceptible to division at all.  What is susceptible to division
is only the irrational number.  Strictly one should write
1.0000.../3.0000...  Strictly division is NOT the inverse of
multiplication.

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