|The Threshold referred to is the interface, the critical level of transition between the pre-physical and the physical-material realms of being,
and between the two primal polar opposite forces of the cosmos, levity and gravity, which manifest as expansion and contraction.
A familiar example of such a threshold in the physical realm
would be the boiling point and freezing point of water,
and at a more subtle level, the transitional phase between waking and sleeping consciousness.
A subsidiary polarity would be, for example,
the positive and negative poles of a battery.
In Threshold Mathematics the values of pi and the Golden Ratio
are found to symbolize this transitional quality.
Threshold Mathematics is not a new mathematical system. It is:
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.”
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