Please note: This is the third session of a nine-week study group. For an overview of the entire effort, see below.
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). People think that lying, cheating, stealing, even murder (and any sort of violence) is always local when, in fact, within our model of the universe it instantly effects us all, everywhere, literally throughout the universe. Most everybody thinks the universe is huge, beyond comprehension, yet, in fact, it can be experienced as a rather small and intimate place.
You make a difference. Everything you say, or do, or think makes a difference. The image on the right is a slice of Narcissus by Caravaggio  and it goes to a question that we all ask, “Who am I?” Then we should also ask, “What is the meaning and value of my life?”
Our commonsense view of the world and this universe has failed us. Has there ever been a time in our history when people were not killing each other? Why?
Within the past few centuries, 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 when he 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. Nima Arkani-Hamed is 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  — and he 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.
A Brief History. I 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 both an implicit and explicit 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 space-and-time. Though largely unconstructed, it assumed there was 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 , 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).  His discussion (John Baez) is in context with the Standard Model.  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  which has become so much of the bread-and-butter as well as the bleeding-edge of physics today. 
The scholars of National Institute for Standards and Technology (NIST) have over 300 fundamental mathematical constants. Their work in 2010 is most comprehensive. 
More recently another scholar, Simon Plouffe, defined over 11.3 billion  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. )
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 just might be, we’d take the next generation of numbers (all ratios) that Simon Plouffe  has generated.
That is a lot of mathematics to integrate into this study to say the least.
Have A Little Pi. 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/simple/#Wilczek An intro to the 2006 work of Tegmark, Aquirre, Rees, Wilczek.
• 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 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.
(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)  were all partially right; certainly the universe looks like it could be a super-Massively-Parallel– processing exascale computer.  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 the 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 . 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 Frenkel and Robert Langlands — perhaps somebody can help with that — that study cause me to reflect on the very first notation.
Holding 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, none have redefined the infinite. As a result, they go in circles.
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 “a perfection” rapidly unfolding. By the tenth notation there are 1,073,741,824 scaling vertices. It all begins with the first doubling from 1 to eight. These could also be called pointfree, 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 these pointfree geometries, there are necessarily node-to-node relations, edge-to-edge relations, face-to-face relations, and object-to-obect relations. Then, there will be groups of object to groups of objects. All are keys to understanding. Within the dynamic image (GIF) just above, the centers of every sphere, actively defining the sphere also begin to define the first triangles, then the first tetrahedrons.
1 – The first tetrahedron has four faces or triangles.
2 – Within the next generation (a doubling), there are four faces per triangle, 32 per tetrahedron, and a total 256 faces.
3 – The next face of each tetrahedron has twelve faces per triangle You can readily (count them here). You can readily see how complex it all becomes very quickly.
It is easy to imagine the first octahedrons emerging. It will require more study of the dynamics of simple geometries to see how the the first pentastars emerge.
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.
 Narcissus by Caravaggio: Look into the mirror; what do you see? Today’s cultures encourage narcissism.
 Nima Arkani-Hamed
 A compilation of articles and resources about quantum foam and John Wheeler
 Baez, How Many Fundamental Constants Are There? Our references to John Baez
 Standard Models for Particle Physics and for Cosmology (Wikipedia)
 Physics beyond the Standard Model (Wikipedia)
 Beyond the Standard Model, Symmetry Magazine, John Womersley, Fermilab https://www.symmetrymagazine.org/
 CODATA Recommended Values of the Fundamental Physical Constants: 2010
 Steven Weinberg’s 2002 article, Is the Universe a Computer? within the New York Review of Books.
 The On-Line Encyclopedia of Integer Sequences (OEIS) and its 11.3 billion mathematical constants
 Simon Plouffe is a Montreal-based mathematician.
 The Universe As A Computer: Alan Turing, Alonso Church, Konrad Zuse, John Von Neumann, Stanislas Ulam, Stephen Wolfram, Benoit Mandelbrot, John Conway, and Ed Fredkin
 Super-Massively-Parallel–processing exascale computer (MA-PA for short).
 Each of these mathematical sets and groups are cited along line 11 within the chart and will be part of the development of notations 1-to-67.
 Birfurcation theory and our understand of fractals will be a central part of all future work as will quantum fluctuations and quantum foam.
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