Pythagorean Geometry Revisited

Pythagorean Geometry and Mathematics Inventions

The conventional notion that says that after three dimensions have been
generated and a solid has been measured, that a fourth dimension may be
applied to the solid, or to space, creating the new entity of time. That
doesn't make logical sense. The huge gaps in that thinking cannot be
patched up by means of long excursions through higher order mathematics.

The Ancient Greeks solved that dilemma around 500-400BC by means of
conceptualized the identifications of the concepts they were using. They
didn't try to force one type of concept onto another and to create
amalgams of unworkable ideas.

The Ancient Greek geometers well prior to Eudoxus, Plato, Aristotle, and
Euclid, for example, were able to conceptualize and graphically
represent, that is perceptibly represent, concepts of numbers for
integers, powers and roots. Pythagoras's geometrical system was able to
portray the inter-relationships of all, and also the commensurabilities
of some integers and all powers and roots. One part of his system was
that by means of his discovery of one new way to provide solutions to
triangles, he was also able to demonstrate his universal concepts of
these newly conceived number concepts of powers and roots of and to
generate particular numbers from them.

What he did was to divorce the system of powers and roots of numbers
from three dimensional measurements - and, also, to establish a concept
of such numbers that could generate the particular numbers based on
selected inputs.

He also divorced the method of three-dimensional measurements from the
geometric straight line, square, and cube. Prior to Pythagoras it was
known that the method of what we know to be the locus method of
generation of lines, planes, and solids, was correct only in that it was
an excellent device for teaching the geometric concepts. The problem
with that concept of the locus is that the entities generated were not
continuous of themselves - they were made of multitudes of
sub-components, either points or planes, for example. The new concept of
the continuous nature of of the principle of a line meant that there
were no points in it, and that it had no gaps. That was old thinking to
them, but the discovery of redefining geometrical magnitudes (or,
scientific concepts), and separating them from mathematical concepts
that could be demonstrated by means of measuring epistemological and
metaphysical existents, meant that a new world of mathematical and
geometric creations could be made. The cube was no longer defined by
measuring in three dimensions, or by lining up myriads of grains of
sand. The geometric existents were to exist conceptually with existing,
even, and continuous natures and generations. The concept of the line
was that it was an idea, and the principle of the idea was that it
continuously generated the integrated unit of the line entity. Similarly
with the plane and solid.

What they discovered is that the principle of these geometric, that is
epistemological, entities was that they were concepts, and that they had
to be understood as principles or causes (or units - that is, as Ayn
Rand would have said, as mental integrations.) At that point the point,
line, plane, and solid became separated conceptually from an enormous
number of possible concepts of number - notably, of integers, powers,
and roots. The problem of commensurability of these concepts of numbers
had existed prior to Pythagoras, and he separated and systematized the
numbers making it possible for the first time to generate any power or
root, generate particular and perceptible instances of it, and to solve
for the number.

The elaborate graphical system he created was also used to create the
architectural proportional systems of most of the era's beautiful great
temples including the Parthenon.

Except for angular and rotational measurements, solid measurements had
stopped with three dimensions. The great discovery of Pythagoras was
that he separated the system of the concept of numbers that are powers
and roots from plane and solid geometry. These days, some college morons
are still battling against the cube and wondering why the concept of
number, say, X^10, cannot be used to measure cubes. They are equally at
a loss when they are asked to compare that number to their own arbitrary
notions of time, for example.

They need to conceptualize the concepts of the point, line, square, and
cube, and to get rid of the notions of particular measurements and the
multitudinous particular loci as the basis for these unique concepts.

They need to do what Pythagoras did, and that was to also separate, that
is, to conceptualize, the concepts of numbers that are powers and roots
from three dimensional solids. They are different concepts, and are
related insofar as discoveries and demonstrations.

Pythagoras did not disallow the manifold possible particularizations of
the concepts of powers and roots. Quite the opposite - he provided a
system of graphically representing those numbers by means of
straightedge and compass, the method of drawing figures for the
solutions to triangles, some methods of applying the new concepts of
number to real world particular demonstrations and measurements, by
means of providing a system that could generate particular numbers in
the forms of architecture and designed objects, and by means of applying
the system of powers and roots to musical harmonies. In the latter he
only got so far as the application of conventional proportions to music,
but the higher order concepts of number are waiting in the wings for
theoretical development and particular musical demonstrations.

Objectivists would say that concepts are universal and objective, and
that they are not particular or arbitrary.

Geometry and mathematics students will see how universal concepts are
formed by induction, and that they may also be proved and demonstrated
in terms of particulars in the real world, and perceptibly, by means of
deductions from universals.

Ralph Hertle


BTW, this post was filed on alt.astronomy with three XPs, on 4/28/2004.
It was in reply to the post by Steve Nicholas from
, on the thread, " If the universe is shaped
like a cone why does the sky look round?" My reply to his post
disappeared from the thread and alt.astronomy. The post did go out to at
least one XP, and I forwarded a copy back to A.A. The post didn't,
however, go with the thread, and it should be a standalone post. I am
refiling the post on a new thread with minor changes and deleting the
forwarded post.

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