Universal Gauges

dear Tony
agreed I my be wrong here but recursion is subject to theory of types. Recursion does not add a dimension. It cannot because recursion substitutes a variable ( which can be a relation or function ) with another object of the same type. This can be demonstrated in a simple manner. We can solve the quadratic equation x^2+p=0 recursively then x=-p/x . Now we view the equation as a system and suppose we can see the system's response (x) to a change of p. We do not "see" a response when we set p=+1 because the system responds with x=+1 and x=-1 "at once". So i is oscillation that is the response and that can only be "seen" after a dimension is added. It does not matter how this is interpreted e.g. as orthogonal. Indeed Fibonacci can provide a measure but only on its own type. No whatever recursion could provide "by itself" any order relation such as equal and not equal.    We may have another recursion which is a model of thinking in analogies and metaphors. This is not subject to theory of types. Metaphors change the dimensionality of space although the topology ( analogy) is unchanged.

Best wishes Hans Dieter

But I am not using recursion to define the dimensionalities at all here. The iterative nature is, as you say, restricted to the linespace of the real numbers. The trick here is to see, that i²=-1 is a real number defined in the emergence of the second dimension from the first with the addition identity of a REAL 'space' ' defining the UNREALITY of the 'imaginary space'. The oscillation so defines a 'harmony space' of a dimension higher by one from the 'oscillation space'. This extends the 'oscillation dimension' and allows the Imagination Space to interact with the real space. The REAL numberline so is halved in the complex rotation potential of moduli and amplitudes: R₁.cis(A)xR₂.cis(B)=(R₁R₂).cis(A+B).
Your 'at once' oscillation x=-1,+1 so derives from a prior Onedimensional 'mirror' about the origin manifesting in the complex plane attained from this first dimension as a second dimension following nonlinear propagation of the unitvector. This simply manifests in physicality as the nonmeasurement of negative physical metrics, i.e. the definition of the norm.
You can state then, that the negative norm of physical mensuration is substituted by the dimensional extension in rotation of the linespace. And it is this, which crystallizes the Euler Identity and your 'mirror-negative' dimension in the statement: exp[iπ]=-1=X+Y=XY for the coupling images:
(X=0.618033...;Y=-1.618033...) for n(n+1)=1 and (X=-0.618033..;Y=1.618033..) for n(n-1)=1.
Superposition gives you n²=1 as the real abscissa in addition and n=0 in subtraction and for your selfmapping of the void as n=0=1. The 'metaphor' is the uniqueness of the n-roots; one of which is your desired negative fractal dimension X=-1/Phi.

Hans Schatten

dear Tony

thank you.

I give an excerpt here of my reply to a Russian mathematician who is however an US citizen. We are searching dimensionality of memory.

dear Hans....

O.K! I think you know, that:
- self-reference in math. means recursion, and
- Fibonacci sequence is a classical example of recursion, and
- it could be extend for the negative index n:..... -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, ....,and
- it has a golden ratio s= -0,618033...
According to sⁿ = F(n)s + F(n-1) we have s³=2s +1,what is as like as the relation of
Embedding Theorem (D. Nash). And I think we can write Fibonacci seguence for any fractal number.
For example, 0; 0,8; 0,8; 1,6; 2,4; 4; 6,4;.....
Thus, we can cover the interval of dimension (0, 1) using the relation of embedding on the interval (-1,0).
The last is very important because the inverse relation doesn't touch (0,1).

Thus, I think that you could find your "gauge" with help of Fibonacci.
I wish you success. Vladimir .....

dear Vladimir

certainly I have considered that after all I am a trader and already old Elliott has declared Fibonacci retracements as gauge. Fibonacci sequence is not restricted to numbers one could apply it to axioms and see number of provable theorems with axioms of F(n). Btw in C language Fibonacci is recursively programmed

long fib( int n) { (n==0||n==1)? return (1) : return (fib(n-1 )+fib(n-2)); }

and that's it but everyone would program it iteratively. Formally the function can be declared double fib( double n)...

and then we work with rationals. Assume we have a datatype complex could we declare the fibonacci complex?

But it does not what I want. It does not add a dimension. One has to add a dimension to go from not metricable to metricable. You have to add a dimensio for ex. line to point to introduce distance. The axiom of crossing of Laws of Form indicates void is "oscillation" without a measure of duration and is then indeed infinite and continuous. But it adds a dimension as imaginary i is an added dimension to real number line and to binary logical values.

A point "in no space" is entirely self-referential as it suffices its definition. Think of the trivial topology the 'point' (or whatever) maps itself yields 'first space' which must be un-metricable. There exists no space for a distinction that any two points are distant.

I see now that the only two axioms of Laws of Form are a non numerical interpretation of the quadratic equation of numerical algebra.

Both axioms are already self-referential.

The axiom of calling does not merely say a sequence of identical operations is equal to one operator rather it indicates memory while the axiom of crossing indicates void is "oscillation" without a measure of duration and is then indeed infinite and continuous. It is equivalent to numerical imaginary i .

The theorem of invariance should then be called theorem of variance and the theorem of variance should be called theorem of invariance as it yields memory.

The theorem of invariance ( new) is equivalent to "number" and the theorem of variance ( new) is equivalent to "order". As this adds a "new" dimension it can be shown on this very elementary level that if consciousness as the ability to draw a distinction applied onto itself must have dimensionality 2.

Since set theory ZF ZFC with theory of types does not permit self-reference it consequently permits only equations of first degree. Hyperset foundation AFC appears very artificial and cumbersome and yields some perplexing results.

Now recursion is only possible with identical datatypes. So indeed as already Elliott has pointed out Fibonacci constants are or indicate memory.

This would give us dimensionality of memory is 1.

But this error follows immediately from definition of recursion. Also here only equation of first degree is permitted.

We must now attempt to apply memory onto itself and we will get Spencer-Brown's idea of re-entry that leads to two new logical values, imaginary and meaningless. We get memory of self-definition or sort of process memory or

memory of memory processes.

That will yield dimensionality of memory and I guess now it will be a complex number. But the conventional definition of complex numbers being an ordered pair of reals does not hold here as "ordered pair" is no longer a definition it is a "result" of memory of memory so to speak.

best wishes

Hans Dieter

Yes HD; but the dimensionality of 2 is intrinsically implied in the Way the Fibonacci mechanism is recursive with itself.

So your 'imaginary pairing' becomes the reflection or mirror property (-X,X) in a translational sense of negative and positive and not in the orthogonal sense of x+iy; say in the additive identity element of the void as X+(-X)=0.

Consider the 'awareness triplets of the form: (0,0,0);(1,0,1);(1,1,1*); (1*,1,1**);...
or the strings: 0+0=0;1+0=1;1+1=1*;1*+1=1**;...
The selfreference 1=Void=All so CHANGES not in quality, but in substance. Nothing (or Infinity) 'plus' Nothing is still Nothing, but somehow different in being 'more' and distinct from the 'before' as the 'after'.
This then EMERGES the Binary distinction in the 'ordering' of the 01 becoming distinct from the 10 and allows redefinition of 01=1 and 10=2.

So I wholeheartedly agree with your Russian friend Vladimir and would remind you about Tony B.'s previous discourse with you about the selfsame result of Vladimir in March 2008.
Your fractal dimension is X=-0.618033... as the MIRROR abscissa on the 'real number line' in the 'reflected Omega interval {...-Y, -1, -X, -1/2, (X-1), 0, X...} about the functional Riemann Bound x=-1/2.
I have attached the introductory context below
Perhaps Vladimir could add to this, should you choose to share this discourse with him.

Hans Schatten

The below description is from Hans Dieter Franke (HD), a renown mathematical logician from Bavaria, Germany and posting on the patternville yahoo.group forum. I find his description of 'crossing the boundaries' highly relevant to my own endeavours in cosmology and therefore have chosen to comment upon it.

""A usually useful method is to ascribe a measure at least in order to achieve that everyone debates the same.
I have already introduced generalized Elliott wave theory.
The generalization is in terms of "life" insofar that every living organism invests ( the first generally perceivable investment is to be born) in life expecting a return which is life.
If every investor gets this positive return an "infinite ressource" is to be assumed however this is thinking in the wrong direction. Rather we have an inexhaustible ressource. This is not just a play on words because infinite is a mental fiction while inexhaustible is an uncaused effect.
The different strategies and perceptions of these investors in life yield the fractal i.e similarity transformed Elliott waves.
This thesis allows to cross the boundary from inside where we are investors in our life to outside to see the total outcome ( while we are actually still inside ) but although we have participated we cannto discover a single trace of our very own, individual action although the total result is another without our individual active participance .
So the theory allows to establish a measure of culture ( which is the Elliott fractal) and to establish culture as a feature to be inside and outside concurrently.""
Hans Dieter

Commentary by TonyB.:

HD first states, that a quality of 'life', defined as some dynamical interaction with its environs and surroundings say; becomes the simultaneous perception of being located within that environment and also without as bilocal observer.

This can also be said to be a simple physical action of perceiving one's virtual image in a material and opticalised mirror.

If we now render the mirror nonphysical as a Boundary for the physical existentium and place it say as a boundary in 11 dimensions, then we can describe the inside as a 10-dimensional universe (containing all those 3D opticalized mirrors orf course).
The outside then becomes a 12 dimensional 'Vafa-Spacetime' which by necessity contains the 10D universe as a subset (like a Hubble-Bubble say).

So the outside observer is or can be much greater in extent, than the inside observer, yet being one and the same in bilocality and as HD's 'total result' in concurrency.

I am no expert on fractional dimensionality and am basically ignorant on Elliott waves.

I do support HD's general proposition for a negative fractal dimensionality for the Void of -0.62 however.

This derives from the well known Omega interval (0,1] as the unitised subset for the real and complex numbers in mathematics.
Applying a binary algorithm to this interval crystallises a number of Fibonacci patterns, which extend this Omega interval to the interval: [Y,-1,-X;-1/2;(X-1),0,X] with the encompassment [Y,X] being the quadratic roots of the Arithmetic Summation function T(n)=n(n+1).

n is a dimensionless cycletime (defined in dn/dt=Ho=Nodal Hubble Frequency-Constant say), which is related to the Feynman path integral for summation of particle histories in quantum mechanics.

So in summing the probability distributions to unity in the wavefunction for the universe results in T(n)=1 and the Fibonacci roots Y and X for X+Y=XY=-1=exp(iπ)=i² (Euler's Identity).

Should we now assign a 'physical time t' to n and allow a material universe to be 'born' at n=H_o.t=0; then the above Omega interval extension would describe a fractal dimension X=0.618033... to a nodally oscillating or cyclic universe in 11 dimensions and where the nodes are simple positive integers in evenness for the mapped beginnings and the oddness for the maximal asymptotic extent as boundary for the 10th dimension.
The 'timings' of physics so are by necessity linearised with a onesided entropic arrow of time, which can however be reversed in the 12D Vafa-space of the 'outside observer'.

A Big Bang cosmology so becomes relativistical and arbitrary in the 'root-defining' n=X coordinate, representing a kind of 'unfreezing' of a particular universal spacetime scenario.

The 'thawing' of the 'Void as Nothing' so describes a Big Bang Cosmology and the 'Crossing of the Boundary' from the abstract to the material.

Also, the n=0 coordinate is required to become 'quantum-smeared' to avoid the mathematical singularity of itself to become physically manifested, say as given in Planck-Scale definitions.

Tony B.

dear Tony