Algebra OK Euclidean geometry not OK
- From: Clifford Nelson <cnelson9@xxxxxxxxxxxx>
- Date: Sat, 20 May 2006 00:56:08 -0700
Our formal education in Euclidean geometry was started later (too late?)
in our lives than Bucky¹s four year old start with semi-dried peas and
toothpicks in kindergrarten. Bucky Fuller said algebra¹s O.K., but,
Euclidean geometry is not O.K..
Here are some quotes to back that up from Synergetics.
I haven't found anyone who can understand these quotes. It's as if their
eyes glaze over when they speed read it. Every beginning geometry text
book would be changed if everyone understood it because it is almost
impossible to disagree with the gist of it.
986.030 Abstraction
986.031 The scientic generalized eternal principle of leverage can be
experientially demonstrated, and its rate of lifting-advantage-gain per
each additional modular increment of lifting-arm length can be
mathematically expressed to cover any and all special case temporal
realizations of the leverage principle. Biological species can be
likewise generalizingly dened. So in many ways humanity has been able to
sort out its experiences and identify various prominent sets and subsets
of interrelationship principles. The special- case ³oriole on the branch
of that tree over there,² the set of all the orioles, the class of all
birds, the class of all somethings, the class of all anythings__any one
of which anythings is known as X . . . that life¹s experiences lead to
the common discovery of readily recognized, differentiated, and
remembered generalizable sets of constantly manifest residual
interrelationship principles__swiftly persuaded mathematical thinkers to
adopt the symbolism of algebra, whose known and unknown components and
their relationships could be identied by conveniently chosen empty-set
symbols. The intellectuals call this abstraction.
986.032 Abstraction led to the discovery of a generalized family of
plus-andminus interrelationship phenomena, and these generalized
interrelationships came to be expressed as ratios and equations whose
intermultiplicative, divisible, additive, or subtractive results
could__or might__be experimentally (objectively) or experientially
(subjectively) veried in substantive special case interquantation
relationships.
986.040 Greek Geometry
986.041 It was a very different matter, however, when in supposed
scientic integrity mathematicians undertook to abstract the geometry of
structural phenomena. They began their geometrical science by employing
only three independent systems: one supposedly ³straight²-edged ruler,
one scribing tool, and one pair of adjustable-angle dividers.
986.042 Realistically unaware that they were on a spherical planet, the
Greek geometers were rst preoccupied with only plane geometry. These
Greek plane geometers failed to recognize and identify the equally
important individual integrity of the system upon whose invisibly
structured surface they were scribing. The Euclidean mathematicians had
a geocentric xation and were oblivious to any concept of our planet as
an includable item in their tool inventory. They were also either
ignorant of__or deliberately overlooked__the systematically associative
minimal complex of inter-self-stabilizing forces (vectors) operative in
structuring any system (let alone our planet) and of the corresponding
cosmic forces (vectors) acting locally upon a structural system. These
forces must be locally coped with to insure the local system¹s
structural integrity, which experientially demonstrable
force-interaction requirements are accomplishable only by scientic
intertriangulations of the force vectors. Their assumption that a square
or a cube could hold its own structural shape proves their oblivousness
to the force (vector) interpatternings of all structurally stable
systems in Universe. To them, structures were made only of stone
walls__and stone held its own shape.
986.043 The Ionian Greeks seem to have been self-deceived into accepting
as an absolute continuum the surface of what also seemed to them to be
absolutely solid items of their experience__whether as randomly
fractured, eroded, or ground-apart solids or as humanly carved or molded
symmetrical shapes. The Ionian Greeks did not challenge the self-evident
axiomatic solid integrity of their supercialcontinuum, surface-face-area
assumptions by such thoughts as those of the somewhat later, brilliantly
intuitive, scientic speculation of Democritus, which held that matter
might consist of a vast number of invisible minimum somethings__to which
he gave the name ³atoms.² All of the Euclidean geometry was based upon
axioms rather than upon experimentally redemonstrable principles of
physical behavior.
986.044 Webster¹s dictionary denes axiom (etymologically from the Greek
³to think worthy²) as (1) a maxim widely accepted on its intrinsic
merit, and (2) a proposition regarded as self-evident truth. The
dictionary denes maxim as (1) a general truth, fundamental principle, or
rule of conduct, and (2) a saying of a proverbial nature. Maxim and
maximum possibly integratingly evolved as ³the most important axiom.²
Max + axiom = maxim. The assumption of commonly honored, customarily
accredited axioms as the fundamental ³building-blocks² of Greek geometry
circumvented the ever- experimentally-redemonstrable qualifying
requirement of all serious scientic considerations.
986.045 The Ionian Greeks assumed as fundamental geometric components
their line- surrounded areas. These areas¹ surfaces could be rough,
smooth, or polished__just as the smooth surface of the water of the sea
could be roughened without losing its identity to them as ³the surface.²
Looking upon plane geometry as the progenitor of subsequently-to-
be-developed solid geometry, it seemed never to have occurred to the
Euclideans that the surface on which they scribed had shape integrity
only as a consequence of its being a component of a complex polyhedral
system, the system itself consisting of myriads of subvisible structural
systems, whose a priori structural integrity complex held constant the
shape of the geometrical gures they scribed upon__the polyhedral system,
for instance, the system planet Earth upon whose ground they scratched
their gures, or the stone block, or the piece of bark on which they
drew. Even Democritus¹s brilliant speculative thought of a minimum thing
smaller than our subdimensional but point-to-able speck was speculative
exploration a priori to any experimentally induced thinking of complex
dynamic interactions of a plurality of forces that constituted
structuring in its most primitive sense. Democritus did not think of the
atom as a kinetic complex of structural shaping interactions of energy
events operating at ultra-high-frequency in pure principle.
986.046 Cubical forms of wood and stone with approximately at faces and
corner angles seemed to the Euclidean-led Ionians to correspond
satisfactorily with what was apparently a at plane world to which trees
and humanly erected solid wooden posts and stone columns were obviously
perpendicular__ergo, logically parallel to one another. From these
only-axiomatically-based conclusions the Ionians developed their
arbitrarily shaped, nonstructural, geometrical abstractions and their
therefrom-assumed generalizations.
986.047 The Greeks¹ generalized geometry commenced with the planar
relationships and developed therefrom a ³solid² geometry by in effect
standing their planes on edge on each of the four sides of a square base
and capping this vertical assembly with a square plane. This structure
was then subdivided by three interperpendicularly coordinate lines__X,
Y, and Z__each with its corresponding sets of modularly interspaced and
interparalleled planes. Each of these three sets of interparallel and
interperpendicular planes was further subdivisible into modularly
interspaced and interparallel lines. Their sets of interparallel and
interperpendicular planar and linear modulations also inherently
produced areal squares and volumetric cubes as the fundamental,
seemingly simplest possible area-and-volume standards of uniform
mensuration whose dimensioning increments were based exclusively on the
uniform linear module of the coordinate system__whose comprehensive
interrelationship values remained constant__ergo, were seemingly
generalizable mathematically quite independently of any special case
experiential selection of special case lengths to be identied with the
linear modules.
986.048 The Euclidean Greeks assumed not only that the millions of
points and instant planes existed independently of one another, but that
the complex was always the product of endlessly multipliable
simplexes__to be furnished by an innite resource of additional
components. The persistence of the Greeks¹ original misconceptioning of
geometry has also so distorted the conditioning of the human
brain-reexing as to render it a complete 20th-century surprise that we
have a nite Universe: a nite but nonunitarily- and-nonsimultaneously
accomplished, eternally regenerative Scenario Universe. In respect to
such a scenario Universe multiplication is always accomplished only by
progressively complex, but always rational, subdivisioning of the
initially simplest structural system of Universe: the sizeless,
timeless, generalized tetrahedron. Universe, being nite, with energy
being neither created nor lost but only being nonsimultaneously
intertransformed, cannot itself be multiplied. Multiplication is
cosmically accommodated only by further subdivisioning.
986.049 If the Greeks had tried to do so, they would soon have
discovered that they could not join tetrahedra face-to-face to ll
allspace; whereas they could join cubes face- to-face to ll allspace.
Like all humans they were innately intent upon nding the
³Building-Block² of Universe. The cube seemed to the Greeks, the
Mesopotamians, and the Egyptians to be just what they needed to account
their experiences volumetrically. But if they had tried to do so, they
would have found that unit-dimensioned tetrahedra could be joined
corner-to-corner only within the most compact omnidirectional conne
permitted by the corner-to-corner rule, which would have disclosed the
constant interspace form of the octahedron, which complements the
tetrahedron to ll allspace; had they done so, the Ionians would have
anticipated the physicists¹ 1922 discovery of ³fundamental
complementarity² as well as the 1956 Nobel-winning physics discovery
that the complementarity does not consist of the mirror image of that
which it complements. But the Greeks did not do so, and they tied up
humanity¹s accounting with the cube which now, two thousand years later,
has humanity in a lethal bind of 99 percent scientic illiteracy.
Cliff Nelson
Dry your tears, there's more fun for your ears, "Forward Into The
Past" 2 PM to 5 PM, Sundays, California time, at: http://www.kspc.org/
Don't be a square or a blockhead; see:
http://bfi.org/node/574
.
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