There is a pleasure in recognizing old things from a new point of view. Also there are problems for which the new point of view offers a distinct advantage.

*Richard Feynman* (1918–1988),

1965 Nobel laureate in Physics

Quantum theory, while extending atomism almost limitlessly, has at the same time plunged it into a crisis that is severer than most people are prepared to admit. On the whole the present crisis in modern basic science points to the necessity of revising its foundations down to very early layers.

Erwin Schrödinger (1887–1961)

*Nobel Prize, Physics*, 1933

in, *Nature and the Greeks and Science and Humanism
*Cambridge University Press, Canto edition, 1996, p. 18.

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 symbolized 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 endeavor, 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?

…the **Fundamental Theorem of Arithmetic*** deals with this concept, but may be apparently consistent *if and only if* it is based on its own limited terms of reference. The definitions of integers and primes I see as problematic and the terms of reference too ‘closed’.

### Section overview

In the following section we *apply* the new theory of scale structures to

*extend the meaning and significance of a key class of numbers*in math (commonly known as ‘the squares’) from a scale-structure consideration of the plane- empower and
*simplify*the measurement of*areas* - explore the concept of
*relative*units and how these can render (geometrically) transparent the once-abstract ‘cloud’ surrounding many (quite fundamental) algebraic equations - apply scale structure theory to the
*Cosine Rule* *extend triangle classification*, introducing*two new fundamental classes of triangle*based on an application of the new theory- discover
*corresponding Pythagoras-like theorems*that relate to these new classes of triangle, both finding and visually-proving these theorems (which I name the*Eutrigon*and*Co-eutrigon theorems*, through a geometry of resonant scale structures) - relate the geometry of these theorems to number theory (finding parametric equations that generate new classes of
*triples*or*triad-pairs*corresponding to the new triangle types) - draw parallels between the tones and semitones of the classical diatonic scale of western music to resonant geometric and fundamental shapes and to the concept of the octave
- challenge the current definitions of ‘rational’ and ‘irrational’ numbers in math by implication of the aforementioned findings including the principle of Universal interconnectedness and of scale structure theory itself. [At the very least, the terminology, ‘rational’
*vis a vis*‘irrational’ (as applied to numbers) is misleading, shortsighted, and has undermined and clouded our ability to comprehend the magnitude and variety of unity that exists, and of which we are part, within this Universe.]

* The well-known and established *Fundamental Theorem of Arithmetic* states that all positive integers greater than 1 are either primes *or* can be *uniquely factorized into a sequence of primes (or powers of primes)*.