Friday, April 30, 2010

Orders of Infinity

Originally posted 14 April 2010 on my temporary blogspace...

Eighteen months ago I posted a blog entry describing a Mass-Parity-Distance-invariant Universe. I got that idea while reading Roger Penrose's The Road to Reality during a month of vacation time away from online and other distractions. I couldn't see anything in the book that contradicted the idea. I knew String Theory is the most popular theory on how to unite the four forces, explain the Universe, and all that, but whenever I read up about it I sensed a certain inelegance about it all. Because I knew Penrose felt the same way, I was keen to plow through his book when I had the time. I haven't read up on physics much since writing that blog, but it's hard not to think about the maths when I'm out for a walk and daydreaming, and so I've had some further ideas for the maths...

Physics recap

I explained how by using a Mass-Parity-Distance (MPD) symmetry, similar to the well-known Charge-Parity-Time (CPT) symmetry, we would have a Universe where equal amounts of positive and negative energy fly off in opposite directions at the big bang, each side self-attracting but mutually repelling. At the Big Bang, the left-handed gravitons and left-handed neutrinos fly off in one direction, thus equating positive mass with normal matter in the large, while their right-handed counterparts would fly off in the other direction, equating negative mass with anti-matter. Matter and antimatter created from positive energy in our side of the Universe would both have positive mass, but a virtual particle-antiparticle pair in a vacuum would have an overall energy of zero, one of the pair having positive energy, the other, negative.

I also explained how such an MPD-symmetric Universe could explain dark energy, though I suspect for it to have its observed strength and timing, the observable Universe would be a very tiny proportion of the actual Universe. Just as our sun is one of about 100 billion in the Milky Way, and our galaxy one of about 100 billion in the observable Universe, so our observable Universe could also be a 100-billionth of the actual Universe. Homogeneity and isotropy would be a more local effect to this scenario. To picture it all using the common 2D-curved-space picture for general relativity, the positive matter would be on top of the sheet sinking downwards, but the negative matter would be under the sheet to indicate negative distances, floating upwards to indicate the negative mass. The positive and negative matter would both self-gravitate, but repel the other.

Cantor's hierarchy of infinities

I suggested the dual CPT and MPD symmetries suggest a mathematical model where the Universe is made up of two complex planes rather than four real lines. A complex plane is different to a 2-real-D plane in not having reflectional symmetry about the real line: two complex planes would have two such assymmetries, perhaps explaining the "parity" in both the CPT and MPD symmetries. But to see why our Universe would be two complex planes related in some way instead of some other structure, we'd need to understand what would make such a structure special. Why two complex planes and not three? And why use complex planes instead of real lines? What might make such a structure special could be the same thing that makes a single complex plane special. I suggest we look at its position in Cantor's hierarchy of infinities, as that seems more foundational than the other branches of mathematics, and move on up from there...

We know the zeroith order of infinity (a.k.a. finiteness) can define a logic system, by using two or more ordered finite values, i.e. false and true for boolean logic, more for other logic systems. The first order of infinity (a.k.a. aleph-0) can define an arithmetic system. The simplest is the natural numbers, created by starting at number 1 and applying induction. Extensions to this such as the integers and rational numbers are also aleph-0. Godel showed this arithmetic system cannot be both consistent and complete.

Things get more interesting at the second order of infinity. A next higher order of infinity is the power set of a lower one. The real numbers, introducing the square root of 2, extend the integers. We can prove they're at some higher order than the first order (integers, etc), but can't prove at what order they are. We therefore speculate they're at the second order (a.k.a. aleph-1), calling this the Continuum Hypothesis. Other number systems with the property of continuity (e.g. complex numbers, n-D manifolds) would then also be aleph-1, but complex numbers, which introduce the square root of -1, are "algebraically complete", not requiring any further extensions.

And what of aleph-2, the third order of infinity? The set of all curves (including fractal ones) is known to be at some order of infinity above aleph-1, hypothesized to be aleph-2. When looking at the curves, maybe it's best to consider the most complex curves first, such as 1D-lines with a (Hausdorff-) dimension of 2. The most famous of these are the Mandelbrot and Julia sets, which both happen to be defined on the complex plane. When we look at computer simulations of them, we see many concentric closed curves of various colors, reflecting the different integral-valued accuracies of calculation. If we could mathematically define a real-valued accuracy of calculation, would we still see concentric fractal curves? I suspect so, and that they would merge into a continuously-varying fractal structure with (Hausdorff-) dimension of 3, on the 2-dimensional complex plane. All of the Julia sets for the standard Mandelbrot look like they have this same concentricity property, though only some of them seem to have (Hausdorff-) dimension 3, the rest (on inspection) seeming to have dimension of less than 2. One is even a perfect circle, with only (Hausdorff-) dimension 1. And of course we could consider the nonstandard Mandelbrot views for all these Julia sets.

The Julibrot Set

Let's use these two fractals in a candidate mathematical model for our Universe. The Julibrot Set is defined as the topologically-4D union of all the Julia sets of the Mandelbrot set. If we consider real-valued accuracies of calculation for the entire Julibrot, we have a (Hausdorff-) dimensionality of somewhere between 4 and 6, embedded in the 4 topological dimensions. Now I suspect there could be a theorem concerning the (Hausdorff-) dimension in such a structure: I'll speculate it's 5 and see where that leads. If such a 2-complex-plane structure explains the 4D-spacetime of our Universe, then there's an extra non-expansive dimension supplied by considering the fractal curves at different positive-real-valued degrees of accuracy. This would explain the phenomenon of mass in our 4D-spacetime, along with certain rules of distribution within, which could be the law of Einsteinian gravity.

But where would this positive real number come from? In my other blog entry, I suggested we could relate each complex plane to the other probabilistically! Each position in a Julia set would correspond only probabilistically to the positions in the associated nonstandard Mandelbrots. This would explain why the said mass in our Universe doesn't conform fully deterministically to the large-scale gravity-based rules of distribution, but has degrees of freedom as ultimately enabled by the law of quantum physics, as powered by Planck's constant, being the measure of probability relating the two complex planes together. Such a structure consisting of 2 complex planes related probabilistically therefore would be the minimumly complexed structure that can squeeze in a fifth real-valued dimension, the one we know as Mass, into the four expansive dimensions.

What would be the order of infinity for this Julibrot-based Universe? There could be a theorem saying this structure is necessary and sufficient to contain all curves, and is therefore at the third order of infinity, aleph-2. Our Universe could then be the minimumly-complexed structure that can exist at aleph-2. Furthermore, just as the complex numbers are algebraically complete at aleph-1-infinity, so also our 4D-spacetime with its inbuilt phenomenon of Mass, obeying rules both of gravitation and of quantum physics, could also be complete in some way at aleph-2-infinity.

The directed dimension and inbuilt polarity

Is this probabilistically-defined Julibrot set the best model for our Universe? Any recursively-applied polynomial equation seems to give the basic Mandelbrotly-edged shape on the computer, so presumably they all have high enough Hausdorff dimension to be a candidate model. But only simple Julibrot Sets are reflectionally or rotationally symmetric in 3 dimensions, but not in 4, giving a dimension that looks like Time.

By looking at the Julibrot's constituent Mandelbrot and Julia sets on the computer, including their colored accuracy levels, we see that the Mandelbrot-real dimension is the assymmetric dimension, i.e. the Time dimension. As a bonus, the Mandelbrot-imaginary dimension is the only one that's reflectionally symmetric, therefore the one along which positive and negative matter flew apart, i.e. the Axial Space dimension. The Julia sets at each point below the standard Mandelbrot Time axis (where y=0 on the plane) are similar to the corresponding one above, except for being reflected through their Julia-real axis. This gives the appearance of the Mass rotating in opposite directions in each half of the structure, perhaps suggesting the opposite handedness of gravitons and neutrinos in each half of the mass distribution in our Universe. But in each half of the Mandelbrot Mass distribution, when we look toward the Time axis, the Mass appears to rotate in the same direction, perhaps suggesting why space appears to have an inbuilt polarity.

The Julibrot generated by the standard parameter plane doesn't bear an obvious visual resemblance to the MPD-symmetric Universe I've been describing, but by using a non-standard parameter plane for the Mandelbrot, we can shape it more like our Universe, such as the 1/μ-plane for a Big Crunch universe, or the 1/(μ + 0.25) plane for a Heat Death one, as pictured on this page.

We can also see suggestions of increasing entropy of Mass by looking at the Mandelbrot set. Because of the second law of thermodynamics, the entropy of the Universe at the Big Bang was at its minimum. In a Big Crunch Universe, the likely death scenario in both the positive matter and negative matter sides of the Universe is as two black holes spinning around each other for trillions of years before eventually smashing together, a high entropy end-state. We can consider the directedness of the dimension of Time as being caused by the increasing entropy of the Mass inside the Universe along the Time dimension. The standard Mandelbrot set on various parameter planes (i.e. the Time and Axial Space dimensions) seem to have one "creation" point, where the (Hausdorff-) dimensionality is 1 just at that point, situated on the Time axis, suggesting the low-entropy Big Bang.

So we could say that by increasing the mapping uncertainty between the Mandelbrot and Julia sets of complex planes (i.e. Planck's constant) up from zero, we create a fifth, non-expansive, dimension called Mass, which causes the Mandelbrot-real dimension to be directed and become Time, and the Mandelbrot-imaginary dimension to become a spatial axis around which the Julia planes rotate.

Electromagnetism

In this probabilistic Julibrot model I've described, Charge is seemingly absent. We first considered this type of model because of the CPT and MPD dual symmetries suggesting two complex planes, but have only derived out the entity of MPD-symmetric Mass following certain gravitational and quantum rules. We can consider the forces without an infinite range (the strong and weak forces) to be more minor details for filling in later, but can't consider electromagnetism that way.

We can put electromagnetism into this model by introducing an additional relationship into the structure, besides the probabilistic one between the two complex planes. The basic apparent difference between MPD-symmetric gravity and CPT-symmetric electromagnetism is that with gravity, like masses attract while unlike ones repel, whereas with electromagnetism, like charges repel while unlike ones attract. The logical effect of this is that gravity's masses are real numbers, enabling aggregation of mass, while charges must be discrete. These look like big differences, but there's a simple structural property we can introduce into the model which makes gravity and electromagnetism be exactly the same force!

Someone once asked me how we know the Universe isn't like at the end of the Men in Black: 1 movie, where it's all just a speck of dust on another creature's back. I've since realized that unless the Universe looks exactly the same at its largest scale as it does at its smallest, then it's not really a self-contained system. The generally-agreed smallest scale of the Universe is a particle and anti-particle splitting apart, then perhaps coming back together. In an MPD-symmetric Universe, the largest scale is positive matter and negative matter splitting apart in opposite directions along one dimension of space, then, in some cosmological theories, coming back together again. If the very first particles to split apart at the Big Bang were some form of particle/graviton coupling, splitting away from their opposite forms, then the negative mass will be mainly antimatter, just as the positive mass we observe around us is mainly normal matter. And a virtual particle and anti-particle splitting apart and coming back together would have matching positive mass and negative mass.

So, the only difference between gravity and electromagnetism in such a MPD-symmetric neutrino/graviton-origined Universe is that gravity is what we see when we're on the inside looking outwards, and electromagnetism is what we see when we're on the outside looking inwards. On the inside looking outwards, there's only one instance to look at, but on the outside looking in, we see many instances. From the inside looking out, it looks like MPD-symmetric positive and negative Mass obeying the laws of gravity, but from the outside looking in, it looks like CPT-symmetric positive and negative Charge obeying electromagnetic laws.

Black holes could fit naturally into this model. Some people speculate they're inherently similar to particles, others say they're the outside of other Universes. And when we look at a computer simulation of the Mandelbrot set, we see many near-similar instances of the large-scale Mandelbrot at smaller and smaller scales, ditto with the Julia sets, and perhaps this mathematical fact suggests in some way this physical fact about the Universe, that the Universe must always look the same at its largest scale as it does at its smallest. And in fact, this mathematical fact suggests perhaps we don't need to introduce this physical fact as an additional relationship into our structure, but perhaps it falls naturally out of it.

At the Big Bang

What might the Universe look like under this model in the first instant after the Big Bang? Because the neutrino is the particle defining matter/antimatter, I'll call the first particle to split from its antiparticle at the Big Bang a proto-neutrino. When the first protoneutrino/graviton coupling split apart from its opposite at the Big Bang, but before either the protoneutrino or the protoneutrino decayed into more particles, each half of the Universe looked like an expanding outer balloon with one decaying inner balloon inside it. Both outer and inner balloons had the same inherent structure: from the viewpoint of the space between them, the outer balloon appeared to obey the laws of MPD-symmetric Einsteinian gravity, and the inner balloon appeared to obey the laws of quantum electrodynamics. As the protoneutrino decayed into further different particles, the outermost scale and the innermost (quantum) scale always kept the same mathematical structure.

The protoneutrino further decayed into other particles, and in our own positive matter side of the Universe, in our own observable portion of the Universe, the fermions are presently the our well-known leptons, quarks, weak-force bosons, and whatever dark matter particles there are. In unobservable portions of the positive matter side of the Universe, perhaps the fermions are different. Perhaps the masses of particles change depending on when and where they are in the Universe. Perhaps other dimensionless constants, such as the fine structure constant, also change. In the negative matter side of the Universe, perhaps the first few quantum rolls of the dice caused the anti-protoneutrino to decay differently, resulting in a totally different life story there, but the overall mass distribution would be the same as in the positive matter side of the Universe, and there would also be a direct identity lineage from the protoneutrino to our present-day neutrino, ditto in the negative matter Universe. But despite what other changes there are in the structure of the Mass, the smallest scale would always have the same structure as the largest scale, and any changes to the structure of the Mass, such as values of particle masses or the fine structure constant, would actually be caused by this self-similarity requirement.

Higher orders of infinity?

So we've seen a possible model of our physical Universe by regarding Mathematics not as something that simply describes the Universe, but as something which the Universe is at some order of infinity. I suspect if my conjectures above are correct, then the core theorem deriving the axioms of General and Special relativity and quantum electrodynamics from an uncertainty-based Julibrot set would be as significant at aleph-2-infinity as the Cauchy-Riemann theorem is at aleph-1-infinity.

If our Universe is what exists at the third order of infinity, then what might constitute the next higher order of infinity? Using Cantor's theorem, by considering the power set of our Universe, we might say it's the set of everything that could have happened, could yet happen, and would yet have happened in our Universe, except for the number of dimensions. But how would we place our own specific Universe of actual happenings in this picture, with its unique quantum reductions into actualities? One person might say the power set is at a higher order of infinity than our own specific Universe of actual happenings, because of Cantor's theorem. But someone else might say the Universe of happenings is at a higher order, being the real, intentioned actualization, while the possibilities are at a lower order, being just the canvas for the actualization, so to speak. It sounds like an argument between atheists and theists, and perhaps never able to be proven either way.

So possibly our own consciousnesses can't really comprehend very high up the ladders of infinity. At the first order, we can't logically define a consistent and complete system. At the next order, we must hypothesize its actuality as The Continuum in our own Universe. And at another order or two higher, where our own consciousnesses dwell, we can't logically prove which of the two orders is the higher and which is the lower. If we could look higher up the hierarchy of infinities to the infinite level, then the hierarchy itself would be able to be counted by an induction-based counting scheme, and so become self-referenced, and even itself an inconsistent and incomplete arithmetic system, which it isn't. In fact, because different instances of our human consciousnesses can't agree which of the third and fourth orders of infinity is higher than the other, we can't even make the known physical representations of the orders of infinity into a propositional logic system, not knowing which to call True.

Moving each order of infinity up the ladder seems to require utilizing some new well-known mathematical concept. Moving from finiteness to aleph-0 requires Induction, and from aleph-0 to aleph-1 requires Continuity. If an MPD-invariant Universe is at aleph-2, then moving there requires Probability. This would be the lowest order of infinity which contains the entities of Time and Mass, as distinct from Space. And such entities, Time and Mass, are required for the mathematics of computational complexity, Time as a resource and Mass for building Turing machines. Perhaps the next order of infinity, aleph-3, requires the concept of Computation, which could explain the phenomena of consciousness. Such Computation is also required for calculating fractal curves in aleph-2 spacetime.

Computational complexity

Perhaps computational complexity will play an important role in a theory of the Universe. There's now many known complexity classes in the zoo. If we can slot the theory of computational complexity directly onto a mathematical structure which defines a space-distinct Time, which behaves according to the laws of our Universe, then many of these complexity classes, such as PSPACE and P(TIME), may meld into one when we factor in the effects of Special and General Relativity, such as time dilation and space contraction. PSPACE problems require more computation "power" than P(TIME) problems, but if space can become time due to high acceleration or a nearby strong gravitational field, perhaps ultimately they're really the same complexity class.

Similarly, the distinction between the P(TIME) and NP(TIME) complexity classes may not exist at Planck scales because of the quantum nondeterminism. Perhaps the electric field generated by human brain neural structure makes use of such quantum nondeterminism to produce the effect of consciousness. Perhaps both large-scale (relativistic) and small-scale (quantum) effects together reduce the many complexity classes down to a mere few. They seem to fall into four broad groups: logarithmic, polynomial, exponential, and recursive.

Do these complexity groups each match up somehow to the various orders of infinity I've described? Do aleph-0-infinite structures like the integers relate somehow to logarithmic-space computation? Do aleph-1-infinite structures like the complex numbers relate to polynomial-resource computation? Is our Universe an aleph-2-infinite structure? Is it an MPD-invariant "canvas" for our Universe of actual happenings, related somehow to exponential-resource computation? Are the quantum reductions that form the actual happenings in our Universe, including our own consciousnesses, an aleph-3-infinite structure? Related somehow to recursively-enumerable computation? And what could possibly lie beyond that?


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