 | ||||
GEOMETRY - CERTAINTY - CONTINUUM - NUMBERS - PHILOSOPHY
Synopsis of the book
THE
ARCHÉ
Many authors, over the centuries, have written about mathematicians believing that an all encompassing, unique, positive, single principle or arché (pronounced "ar-kay") forms a foundation for our numbers and mathematics. The proof of a justifiable basis for the certainty found in our number systems has been long desired, but whether the basis is empirical or idealistic is highly controversial. Our attempts to solve this theoretical problem have met with extreme difficulty. We must also keep in mind the ability to effectively analyze speculative topics is not always a universal human trait.
The objective of this book is not about mathematics and it does not contain references to or comments about religion, politics, societies or economics. The purpose of this work is to encourage others to know of their own accord a provable concept of a unique structure of idealistic forms which are "independent of sense experience” and how these fully related forms serve as a foundation for the certainty and the reliability inherent in our number systems. There is evidence of this ancient concept, although fragmentary, documented by a few of the Pre-Socratic philosophers, mathematicians and other men of the sciences in the 4th, 5th and 6th centuries B.C. It is my opinion; knowledge of this was within reach and useful to the temple and pyramid builders of Asia, Egypt and Central America. Also, reference to it may be found in old Chinese literature.
This concept, concealed and lost for over 2,400 years, is derived from the thoughts of an interrelated and interwoven sequence of predictable facts that lie “beyond what is perceivable to the senses” and is very likely the major reason those who searched were unsuccessful. These odd or peculiar forms by their strict adherence to pre-established principles extend the characteristic of predictability to related successor members and are a prime example of the "perfection of order". When viewed as a single entity this related set of geometric forms is instrumental in the establishment and is the source of a number continuum or the equivalent of an orderly infinite arrangement of every conceivable rational number. This predetermined continuum includes all numbers, those infinitely greater than one, one and those infinitely smaller than one. The fact that “zero” is not a member of this entity can be logically justified. Inherent in the concept of these idealistic forms are characteristics applicable to our common geometrical forms such as triangles, squares, cubes and pyramids.
Our common geometric forms, numbers and algebraic equations are some of the symbols and tools used to communicate the “thought” of this concept of certainty and predictability to the reader but numbers do not serve as the basis of the Geometrical Continuum. Empiricism can lead us to a plateau of possibilities but it does not truly indentify which of those paths will lead us to the identity of the "ideal forms". It has been stated "the path to the knowledge of these forms is only one way".
Although this concept is “of thought”, only logical and distinctive reasoning is employed in describing the forms. Awareness of and an understanding of the concept does provide a single foundation for our geometry, number, algebra and mathematics and how our manmade tools partake of the ideal forms. Whatever degree of certainty, stability or predictability there is to be found in the reality of our material world of phenomena can be linked to the characteristics displayed by the relationships of these proper and immutable forms. The significance of this predetermined and eternal concept may be found everywhere, but in its entirety there can be no worldly likeness because it is “idealistic” or the perfection of order.
In general, the Geometric Continuum is the conception of a structure of distinctive forms which are so fully related to each other that each successor form embraces and combines the characteristics or attributes of their predecessor forms all derived from a single arché in an infinitely continuing pattern. It is a body of preordained forms without an extrinsic influence of any nature, whatsoever
.
This concept of idealistic and unique forms is being described in a logical, rational and positive manner and provides a unifying base for the many highly valued works of others so often considered controversial. The presentation being positive in its approach is entirely without criticism of others, past or present, or of their work. Without the documentation of their thoughts and opinions, this work may not have been produced. Philosophy relating to the sciences of the Pre-Socratics is the major source of the information used by the author.
Included in the text are statements describing the necessity for the reader to logically reason and encouraging him to empirically verify the validity of every step rather than accepting anything, including the author's comments, as a “belief”. “Knowing oneself” that something is true is not the same as believing in another. Belief belongs to the category of uncertainty. Since the concept is of thought and requires logical reasoning, then an understanding and acceptance on your part, the author serves only as a temporary and limited guide in your intellectual quest for the truth of this universal arché. Your task will include “to doubt” as suggested by René Descartes in his “Discourse on the Method of Properly Guiding the Reason in Search of the Truth in the Sciences” and to “Know thyself” as spoken by Socrates.
The manuscript does not advocate the abandonment of our present day methods or procedures but offer you the opportunity of having as one of your choices a logical and unifying concept with a unified and justifiable foundation for geometry and number. Lacking self-awareness or an individual understanding of this ideal and proper concept you are without the option of using it should you choose to search for and compare the truths about the thoughts of Plato's realm of the "Real" composed of immutable and idealistic forms and his realm of "Reality" with influential material things having only temporary existences. The "Real" can only be the thought of an eternal and permanent "Master Plan" in comparison to the infinite diversity possible within our changing material world and universe being "Reality".
Those interested in set theory, numbers or philosophy may find this book of interest and also provide educators with the tools to explain the foundation for the reliability and certainty we find in our number systems.
Alfred James Crocker, Author7446 Royal Timbers Lane
Waterville, Ohio 43566-9406
Date of last revision 12/22/2010
