A comment I made to an essay by Peter Jackson in the FQXi Essay contest:
Consider the number of squared quantities in physical theories along with the connection to complex numbers. If we have a squared value, then the ‘real’ (un-squared) value can be either positive or negative. Since we cannot know which it is, from only the squared value, there are two paths by which the squared result can be obtained. We cannot know which ‘path was taken’, so long as we live in a ‘real’ world constructed with Real numbers. In this world, there are always two paths. This is not the case if we use all complex numbers, in which case the path can be known.
Integers, Rationals, Reals, Complex, Quaternions are all ‘bracketed’ levels of mathematical knowledge with specific mathematical contexts that present a multitude of mathematical spaces. Algebraic equations with integer coefficients, rational coefficients, real coefficients are examples. Also equations with integer exponents and rational exponents bring us into much more complex mathematical spaces. We can consider algebraic equations with real coefficients and integer exponents or rational exponents. What about such equations with complex coefficients and complex exponents? Can we even do this?
Are we using the correct spaces to understand our physical world? (I will suggest we are using one layer of your hierarchical relationships). Do we have the correct tools to make use of certain mathematical spaces? It would appear we do not, since we cannot represent a complex number in the same way as we represent real, rational, and integer numbers – as completed values without unknowns (complex values today always have the unknown ‘i’). If this is true (we don’t have the correct tools), then we would be forced into ‘heuristic necessity’ (convoluted processes using our existing tools) in order to address situations that require math tools that we do not have.
If we do not have the appropriate tools, then what might change if we did have the appropriate tools? A complete complex value might be capable of measuring quantities we currently do not consider measuring, since we are unable to measure them using real (Real) values. If we could represent complex values as complete, without unknowns, then we should be able to compare them (greater or lesser) and rank them along a continuum. The universe might be better represented (not ‘Finally’ represented, however) using a possible complex continuum. (The separation of the Continuum Hypothesis from current foundational mathematics should allow this without destroying current foundations).
Given your presentation of red-green spin as a spinning sphere that ‘flips’, what if this is reasonably accurate, however the spinning sphere is occurring in a geometric space with a physical dimension that we currently do not include in our models (not some tiny rolled-up invisible one)?
If our mathematical tools cannot adequately measure quantities along this dimension, then we would not have a reason to include it in our models (even though this could be leading to ‘heuristic necessities’). It could be directly in front of us and we would not see it, because we do not realize it as a measurable quantity.
One last conjecture: This dimension is scale. It is exponential in character, is not measurable by current ‘real numbers’ (decimals are limited to specific scales), and is unaccounted for in current models of space (and it is ‘right under our nose’). Our body, at each level of scale – organ, cellular, molecular, atomic – is always a 3-D volume. This 3-D volume is populated by different objects at each level. Connecting the levels at any 3-D location gives a 4-D line (a line mutually perpendicular to the other 3). Note that a complex numeric system will likely require a system of numeric representation that, beyond the addition/subtraction, multiplication/division, and exponentiation/logarithm incorporated into current ‘real numbers’ (eg. decimals), will add integration/differentiation into the operations inherent in constructing a complex value – thus requiring a complete revamp (major upgrade) of all computing systems.
Maybe this is a maths fantasy and maybe it is maths imagination, yet not maths alone as it is coupled with a significant change in physical perspective. It also follows your hierarchic boxes within boxes approach, as it simply expands current ideas and levels, not destroying existing ones.