| Wayne Roberts © 2003 - 2008
Introduction to a revision of the concepts of 'number' and 'units of area' from a scale-structure perspective
Richard Feynman (1918–1988),
Erwin Schrödinger (1887–1961)
The foundational and 'early layers' that await revision (and to which Schrödinger alludes in the quote above) surely refer to that most abstract concept encompassed within the very 'notion of number'—that of the unit itself.
Here is a Nobel Laureate in Physics who is warning of a '...present crisis in modern basic science...'; moreover '...a crisis that is severer than most people are prepared to admit'. These are strong, courageous and challenging words from one well-versed in the concept of atomism and the notion of the unit which it reflects.
What exactly is a unit? And, by implication, a number? And how might these abstract concepts be best represented or symbolised within the jurisdiction of a language of which they form the parts? Is an 'intuitive knowing' (axiom) called for? A 'given' which must forever remain 'incompletely definable' (here alluding to Gödel's Incompleteness Theorem) because the concept of the unit lies at the heart of definitions themselves, and therefore any definition of itself must surely, ipso facto, be tautologous? We must remember that all things in our Universe (including 'definitions') are relative as required by the Principle of Universal Relativity —nothing is 'cut off' or separated, else it remains unknowable, neither may it 'influence' or 'be influenced'.
Where to from here?
We will take Schrödinger's advice and revise these 'very early layers and foundations'. But rather than first define, we will first 'undefine' or re-open and unravel the origins of several key concepts that have survived like 'sealed jars' amid the cobwebs of darkened shelves in a disused 'storeroom of known facts', largely unchallenged by mainstream institutionalized thinking for two-and-a-half thousand years, and which have persisted as part of the edifice of basic science (indeed of mathematics itself).
We will proceed to revise these foundations from the perspective of scale structure theory as presented in this web-document and its companion book. Hopefully, others will follow who, building upon and refining this new platform, may extend knowledge (in some instances, perhaps dramatically so) in their own fields of endeavour, and ultimately (we who share this dream do hope) to the assistance, betterment and rescue of humankind and its future generations. Perhaps such words sound too ernest or grandiloquent to some, but ours is, I fear, fast becoming an age of 'watered-down' dreams, an age of cynicism and laissez-faire, or, in the current vernacular, 'Who cares?'—at times thus altogether-lacking in a vision or concern for the future...of lost ideals and 'a sense of wonder'.
Hold on to your dreams if they have the potential to help others now or in the future. Together we may well achieve them.
Are primes the 'atoms' of number?
In the following section we apply the new theory of scale structures to
* The well-known and established Fundamental Theorem of Arithmetic states that all positive integers greater than 1 are either primes or can be uniquely factorised into a sequence of primes (or powers of primes).