There is never any justification for violence, not even the threat of violence.

We are all so finite, so fragile but filled with ego. We all need to be informed by the infinite!

by Bruce Camber, February 2018

— a work-in-progress —

We all so little understand the infinite. Its obtuseness has made for crazy religions, crazy science, and crazier people (along with their really crazy governments). We think lying, cheating, stealing, even murder (and any sort of violence) is local when it instantly effects us all, everywhere, literally throughout the universe.  Most everybody thinks the universe is huge, beyond comprehension, yet, in fact, it is rather a small and an intimate place.

You make a difference. Everything you say, or do, or think makes a difference. So please, please allow me a few minutes to attempt to convince you that is worth exploring further.

Our commonsense view of the world and this universe has failed us.

Most of us unknowingly adopted Isaac Newton’s  concepts about space and time from 1687. That is when he published his opus magnum, the Principia, and threw the world’s people the penultimate screwball – absolute space and time.

Absolutes are by nature abstract. We need more common ground to guide us. Those two seemingly innocent concept have become so baffling in light of current scientific research, some of the really-really smart ones are calling for space-and-time to be thrown out altogether. And by doing so, you can sure we will all become even more confused.

No kidding. One of the leading proponents of this “no space and no time” movement — his office just happens to be down the hall from where Einstein had his in Princeton, New Jersey [1] — is pushing hard and is ever so confident.

Redefine the finite in light of the infinite

Now, I’ll gladly stand among the discounted people by suggesting that infinity is better defined by three rather commonly used concepts:
•   continuity (my working page for this concept)
•   symmetry (working page)
•   harmony  (working page).

This definition of infinity will necessarily permeate, define, and give rise to all that is finite.

To introduce this point of view is the purpose of these writings.

1xI had formulated this notion of infinity sometime in 1971. As a conceptual frame of reference, though pleasing, it was hardly scientific. It went nowhere. It was too general and not necessarily relevant to current scientific research. Of course, continuity was the cornerstone of logic and science, but so was discontinuity. That net-net is a null. Symmetries were well-defined, but then there were quantum asymmetries. Harmony was illusive and seemed more experiential than experimental.

Something changed for me in December 2011.

When set within the chart, Big Board-little universe (December 2011) whereby “everything, everywhere for all time” was indexed, and with its simple geometries and even more simple doublings from the Planck Length, continuity-symmetry-and-harmony seemed to be more formidable.

First, all the numbers in these charts do not defy simple logic. Though a challenge, the chart maps a quiet expansion, a perspective created by this analysis of six sets out of 202 sets of numbers.  Second, doubling is a natural process, readily and graphically demonstrated, and it all begins with dimensionless numbers.  It doesn’t get any more simple than that. Third, in this model there is an implicit value component to space-time and matter-energy. And, that’s substantial.

In 1975, I began my doctoral studies at Boston University thinking that the essence of the infinite was continuity, symmetry and harmony, not absolute time. Though largely unconstructed, it assumed a special perfection that could only manifest in a momentary way within the finite. Though creating a seismic gap between my professors and me, I held firm yet eventually had to leave that work, unfinished, in 1980.

Can infinity be defined in terms of continuity, symmetry and harmony?

Continuity creates order.

It seems that I have always understood the infinite to be a perfection of some kind.  So, the key question becomes, “What has perfect continuity?” My answer: non-ending, non-repeating numbers. Enigmatic, these numbers are always unique and always the same. Think of pi and its 22 trillion digits (as of 2017).  Pi deserves all the attention that it gets.  I’ll argue that it is the most important of all the dimensionless constants; thinking about John Wheeler’s quantum foam [2], my guess is that it is the initial and a primary gateway between the finite and infinite.

One of our scholars asks, How many of these dimensionless fundamental constants are there? This depends on your opinion on some new developments, but my best guess is 26” (ninth paragraph). [3] His discussion is in context with the Standard Model. [4] Of course now, everybody is looking for the unifying “something” to simplify this model and it has engendered formal studies that are currently summed up as the Physics beyond the Standard Model. [5] It’s their bread-and-butter today, the mother’s milk of physics. [6]

The scholars of National Institute for Standards and Technology (NIST) have over 300 fundamental mathematical constants. Their work in 2010 is most comprehensive. [7]

More recently another scholar, Simon Plouffe, defined over 11.3 billion [8] mathematical constants (as of August 15, 2017). The results of his computation programs called the “Inverter” are now part of a website maintained within the On-Line Encyclopedia of Integer Sequences (OEIS). [9]

One might ask, “How many lines are there between the finite and infinite?” At the level of the Standard Model of Particle Physics, the simple number appears to be around 26. To begin to see the fullness of how the infinite encapsulates the finite, we’ll study the 300+ ratios from NIST. Then to begin to understand how exquisitely intimate it all is, we’d take the next generation of numbers (all ratios) that Simon Plouffe [9] has generated.

That is a lot of mathematics to integrate into this study to say the least.

If the actual interface between the finite and infinite is pi, the next challenge is to create a pathway between pi and all those numbers. We have used this little dynamic gif found within Wikipedia section about  cubic close-packing of equal spheres. It is one of several ways to go from the continuous to the discrete, from so-called quantum foam to tetrahedrons.  Prior discussions within this website have been rather limited:
•   https://81018.com/number particularly those numbers scholars thought were keys for understanding our universe.
•  https://81018.com/a0 and https://81018.com/a1 attempt to look at those numbers in light  first two notations within our chart of the universe.
•   https://81018.com/2017/10/16/eight/#Symmetry [11] where the image, a dynamic gif, shows the transition from circles and spheres to triangles and tetrahedrons.
•   https://81018.com/fabric/ is from my initial attempt to get inside Langlands  programs.  Though unsuccessful, it was the beginning of the notation-by-notation analysis.

Symmetries create relations.

Sphere to tetrahedron-octahedron couplet (very rough and even silly at times) 
Symmetries start with the transition from quantum foam to tetrahedrons, then it gets even more fun. Each face of the tetrahedron is a simple connection. Plate to plate, line to line, node to node, there is parity and direction that ostensibly becomes a simple machine, then a simple computer, and then multiple exascale computers. It would appear that Alan Turing, Alonso Church, Konrad Zuse, John Von Neumann, Stanislas Ulam, Stephen Wolfram, Benoit Mandelbrot, John Conway, and Ed Fredkin (and so many others) [12] were all partially right; certainly the universe looks like it could be a super-Massively-Parallel– processing exascale computer. [13] Perhaps we call it MA-​PA for short; just to beg the question, “Isn’t that a can of worms?” And, it’s okay because these worms may eventually lead us through a wormhole.

Also, the analogy is helpful because this node-to-node analysis has been going on for years under many different names by leading mathematicians and physicists: Loop Quantum Gravity, spin foams, string and M Theory, Causal set theory, non-commutative geometries, discrete models of space-time at the Planck scale (like lattice versions of loop quantum gravity), Cellular Networks and SU(2) gauge theory to make an ansatz. This list will be rank-ordered by complexity and there will be even more kinds of mathematics added over time.  All of these studies are part of line 11 within our chart between notation 1 and 67.

Many of these studies are relatively new for me; notwithstanding, I do not think any of them should be the domain of the specialists only. Conceptually, most of us should be able to engage their work. Of course, there will be many challenges along the way!

I can remember back in and around 1964 working with my dad and his friend with a 4’x4′  pegboard, where we did our first simple on/off switching experiments. Simple switches — off/on, yes/no,  0/1,  positive/negative — all apply.  Even though we graduated to a circuit board, there was always so much more to learn. In 1980, I was invited by two scholars to come to the Institut Henri Poincaré in Paris. Those discussions focused on Bell’s inequality and at no time did we think to engage Poincaré’s simple, off/on switching which he named birfurcation theory [14]. Now, pulling together as many loose ends as possible, I ask, “How does quantum foam generate its first relations whereby there is a bifurcation identity?”  How do simple switches —   positive/negative, off/on, yes/no, or 0/1 — apply? 

Though unable to get inside the work of Edward Frankel and Robert Langlands — perhaps somebody can help with that —  that study cause me to reflect on the very first notation.

Shoving aside all thoughts of a big bang, and starting with infinitesimal numbers for Planck Length and Planck Time and very small numbers for Planck Mass and Planck Charge, we observe how these numbers grow very large, very quickly. These are the first few steps close to the finite-infinite transformation.

Although Wheeler’s 1955 quantum foam has morphed in quantum spin states and loop quantum gravity (LQG) and its essentials are now studied by thousands in hundreds of different groups around the world [15], none have redefined the infinite. As a result, they going in circles. They’re stuck.

Quantum Foam is a soft combination of words. And in this regard, even the words, titanium balls, are soft as well. The spheres within that dynamic gif, with the first five notations, are defined by pure math — all the mathematics that define the space/time and matter/energy at the Planck scale. There is nothing foamy about these spheres. These are the base units of all space and time and all mass and charge.

Also, never stopping, this exquisite piling on —  the dynamic gif just above barely captures that potential — is perfection rapidly unfolding.  By the tenth notation there are 1,073,741,824 scaling vertices. It all begins with the first doubling with eight point free dimensionless vertices. The second has 64, and then 512 at the third, 4096 at the fourth, 32,768 at the fifth. These first five doublings capture the deep essence about our finite-infinite relation. Within pointfree geometries, those node-to-node relations are keys to understanding. The face to face relations are also defined within that dynamic gif. Here the first triangles become the first tetrahedrons which become the first pentastars which eventually become the first octahedrons:
•  The first tetrahedron has four faces or triangles.
•  The next generation has four faces per triangle or 16 in total.
•  The next face has twelve (you can readily count them here) yet 48 if all four faces are considered (and an argument should be made for it).
•   The next face has 36 and an argument should be made for 144.

Within four notations there are no less than 144 possible interfaces between the finite and infinite. Each node has a role that is currently be examined in depth by everyone of the  defined, then in trisection must also be counted.

Harmony as the essence of dynamics.

Though I have started to do some analysis of harmony, there is a long way to go. It is so key to the finite-infinite relation, every bit of that work will be included here and then extended into the notation-by-notation analysis. Every notation is linked by a 1:2 ratio. There are 19 primes from 0-to-67. I believe each of these primes introduce a new mathematical system and the best candidates will be a natural progression of mathematics represented as represented within Loop Quantum Gravity (LQG), spin foams, string and M Theory, causal set theory, non-commutative geometries, other discrete models of space-time at the Planck scale (like the lattice versions of loop quantum gravity), cellular networks and SU(2) gauge theory.

We will not have to learn each mathematical system. There are experts for that. We will have to have some help to rank-order, prioritize and begin to inter-relate all these systems as a whole.

More to come…  yes, still a work-in-progress!

Hilbert 1925 statement still stands firm, “… the meaning of the infinite, as that concept is used in mathematics, has never been completely clarified.” In physics, Max Tegmark, an MIT physicist, wanted to just retire the concept altogether.

References:

[1]  Nima Arkani-Hamed
[2]  A compilation of articles and resources about quantum foam
[3]  See: How Many Fundamental Constants Are There?
[4]  Standard Models for Particle Physics and for Cosmology
[5]
[6]
[7]  CODATA Recommended Values of the Fundamental Physical Constants: 2010
[8]  Steven Weinberg’s 2002 article, Is the Universe a Computer? within the New York Review of Books.
[9]
[10]
[11]
[12]
[13]
[14]

A Computable Universe: Understanding and Exploring Nature as Computation

26 dimensionless constants for the Standard Model, over 300 by NIST, and 11.3 billion through computer wizardry…

References:
Wikipedia’s Philosophy of space and time