Threshold Mathematics

Threshold thinking offers a newly emerging perspective on our world and ourselves. It draws on many sources, ancient and contemporary, and requires some considerable revising of old habits and assumptions in thinking.

It is applicable to all disciplines, secular and spiritual. Preparatory work is in progress for publication of papers on Threshold Psychology, Science, Healing, Economics, Politics, Spirituality.
The Threshold referred to is the interface, the critical level of transition or transformation, between the two primal polar opposite forces of the cosmos, expansion and contraction. These are also known under various other names, and manifest in secondary polarities such as the positive and negative poles of a battery. Other such thresholds, as examples, would be the 'boiling point' of liquids, and the transition phase between waking and sleeping consciousness. In Threshold Mathematics the values of pi and the Golden Ratio are found to symbolize this transitional quality.

Threshold Mathematics

Threshold Mathematics is not a new mathematical system. It is:

  1. a psychological perspective on mathematics, a looking in from outside, a kind of meta-mathematics;
  2. an inclusive approach which incorporates both a broad overview and a focussed insight into the heart/core of the subject under consideration;
  3. a philosophical view which reveals some fundamental limitations of current orthodox mathematics.

One professor of mathematics, in response to this new perspective, quoted Archimedes� famous claim, �Give me somewhere to stand, and I�ll move the world."

Quotes re the Primes (pdf)

Threshold Mathematics (pdf)

The Prime Numbers and Riemann Hypothesis Enigmas: A Concise Threshold Mathematics Resolution (pdf)

The Prime Numbers and the Riemann Hypothesis: Minimal version (pdf)

A Commentary on Threshold Mathematics (excerpt)

I see [Threshold Mathematics] as prepared seriously to question the foundational assumptions of both classical and modern mathematics. [It] recognises that these assumptions entail the paradoxical treatment of ‘space’ or ‘Aether’ as an external nothingness or ‘absence’, which is given no numerical value, yet isolates discrete units of ‘matter’. Numerical and geometric systems based on such treatment are incapable of representing natural flow, because they cannot bridge the ‘gap’ from discontinuity to continuity, i.e. they can only begin and end at discrete points that have nothing before or after them.

[Threshold Mathematics] illustrates the difficulty of making discrete assumptions with the 'problem of the primes' whose pattern of occurrence in classical unidirectional or progressive terms can only be understood in terms of a non-classical mathematical approach, whereby numerical identity is given bidirectionally by dynamic balancing (harmonisation) of inner with outer worlds. Such balancing involves a relational interface or ‘Threshold’, which places 'One' and all the primes not as discrete, space-excluding unit masses along a horizontal line, but as space-embodying, dynamically sustainable 'resonant cavities' or ‘harmonics’.

Dr Alan Rayner, Reader, University of Bath