Dirac's Way to Quantum Gravity (via "Numerology")

Speaking of Dirac’s “dangerous numerology” of dimensionless quantities, those of us interested in bits may want to consider this “dangerous numerology” (which I must credit to Frederick Parker-Rhodes’s Theory of Indistinguishables’ Combinatorial Hierarchy and the Bit-String Physics it spawned with which I’ve had some peripheral involvement):

2^3-1 = 7
2^7-1 = 127
2^127-1 ≈ 10^38

Note 10^38 is in this sequence of Mersenne primes.

But now get a load of this:

3+7+127 = 137

Which is approximately the dimensionless electromagnetic scaling constant hc/(2 π e^2).

OK, but one might wonder “What about the other sums: 3, 10 and 137+2^127-1?”

Well, Noyes, et all founded an organization called Alternative Natural Philosophy Association to explore these numbers and have come up with an argument for why they may be considered the other 3 coupling constants for the 4 forces of nature.

I won’t try to apologize for that thesis here… I just figured it was worth mentioning.

Of course Feynman didn’t have the number “10” afixed to his wall, as he did “137”, but Pythagoras did much more than that: Pythagorean acolytes had to go around mute for years doing little else but meditating on the number 10, which they called the “tetraktys”. Likewise we may dispense with “3” as virtually every foundational metaphysics, numerology and theology is all over that.

But I don’t think any ancient mystery schools paid much attention to 137 let alone 137+2^127-1.


Here is a review paper from 2007 that describes Dirac’s large number hypothesis and the many variations on it and additional apparent “coincidences” found by others over the years, experimental evidence and arguments for and against it, and the status of investigation of changes in the gravitational constant G over cosmological time, “Large Number Hypothesis: A Review”. Full text [PDF] is available at the link.

The coincidences involving the radius and mass of the proton seem odd to consider fundamental to me since the proton is not a fundamental particle and most of its mass comes from the binding energy of the massless gluons that hold its three constituent quarks together.

Another numerological observation that’s always intrigued me is the universe’s inordinate fondness for the number three:

  • Three extended dimensions of space
  • Three colour charges for quarks
  • Three generations of fundamental particles (e.g. electron, muon, tau)
  • Three fundamental non-gravitational forces, mediated by gauge bosons

Wouldn’t it be neat if all of these were related somehow?


A rationale for using baryonic matter (protons and neutrons) in dimensionless number calculations relating macro to microcosmic numbers is that virtually all gravitation (spacetime curvature) may be on account of their mass. If one may attribute significance to the curvature around a proton’s surface, for example, to its contribution to cosmological phenomena like gravity, then both the mass and radius may be rationalized without too much of a stretch.

Take for example this calculation:


The 4 digit coincidence with a prediction of the pure-information derivation by the ANPA folks here is intriguing.


The small discrepancy, in the case of gravitation, between the Combinatorial Heirarchy’s dimensionless integers and its interpretation as dimensionless coupling constants of the 4 forces of nature, might reflect the remainder of gravitational mass in the universe after the proton is taken into account. So to get a handle on of what particle(s) this residual might consist, I wrangled Mathematica into solving for the mass of a hypothetical (composite) particle that exists in a ratio of 1:1 with protons. Unsurprisingly, it came up with something almost as massive as the neutron (or proton). I subtracted it from the neutron mass, leaving a final residual mass (after accounting for the neutron component of it) for this composite particle of about 6MeV/c^2.

What this seems to say is that if the CH approach is correct and if the universe had an equal number of neutrons and protons then the remaining mass of the universe that needs to be accounted for is about 6Mev/c^2 per proton, or about 10x the mass of the electron (0.5 MeV/c^2) – but is less than half of the ionization energy of hydrogen. There’s a lot of ionized hydrogen out there.


A rank ordering of particles and their use in calculating the dimensionless gravitational coupling constant, sorted by how closely they are to the combinatorial hierarchy’s 4th entry (the third entry being 137 corresponding to electromagnetic coupling aka the fine structure constant):

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Here is a talk on the pre-history of ANPA and future directions spawned by the original questions regarding the Combinatorial Hierarchy.

Note three things in particular, beyond the intriguing “numerology” pointed out above regarding the proton and the gravitational coupling constant:

  1. Michael Manthey (manthey@acm.org) was (and remains), to the best of my knowledge, the person most qualified to pursue the computational implications of the Combinatorial Hierarchy. He ended up patenting a qualitatively different computing paradigm to the Turing machine which he believes is necessary to generate true self-organization hence AGI.*
  2. The person that introduced me to the Combinatorial Hierarchy, the late Tom Etter, was not only the editor of the ANPA West Journal, but attended the 1956 Dartmouth Summer Workshop, apparently arriving with Ray Solomonoff. (I didn’t know about Tom’s history here until recently as he never mentioned it while we worked together on the foundation of programming languages at HP’s “Internet Chapter 2” project: eSpeak.)
  3. The Combinatorial Hierarchy was conceived as an outgrowth of Cambridge University’s “Language Research Unit” that was studying the theory of language. (I suspect this was, in turn, an outgrowth of Bletchley Park.)

* Manthey’s most recent presentation (corrected slides here) on his approach to what he calls “spatial computing”, in terms of Geometric Algebra, as qualitatively distinct from Turing machine computation that operationally defines “awareness” thence operationally defines “consciousness” as “awareness of awareness”:

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I posted a question to the cstheory stackexchange challenging them to refute Manthey’s bold theory regarding a spatial notion of computation that qualitatively differs from Turing’s and that is necessary to adequately model quantum theory.

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Since the prior Mathematica notebook achieved the statistically rather significant result that the square of the ratio of the Planck mass to the proton mass was a very close match to the last entry in the CH4, I decided to find out if there might be any meaning to the physical dimension of mass^2 (as there is when one squares the Planck length). So I asked a pretty run-of-the-mill question about Planck units at physics.stackexchange and immediately someone weighed in with “Close” vote – no explanation. It would be one thing if I’d brought up “137” or the combinatorial hierarchy or other such “numerology”, but this is exactly the kind of question that one would expect of a reasonably inquisitive, reasonably perceptive, non-transgressive and in-earnest college freshman in physics.

You really have to what kind of person sits around waiting to do stuff like this.

Like pulling teeth to get a straight answer out of those guys at physics.stackexchange, but here’s what I came up with as an answer that, although I don’t grok it, at least has enough keywords that I might be able to educate myself about what mass^2 “means”:

The Planck units are defined in terms of generalized “areas” in their respective Hamiltonian phase space coordinates. Hamiltonians use generalized coordinates yielding generalized “areas”. The so-called Planck area is only one kind of “area”: spatial area in the sense of ordinary 3D space we inhabit. But other Planck units are also “areas”, but in a “phase space” the dimensions of which are, physical dimensions as in dimensional analysis.

So, yeah, there is something quite deep going on with the notion of mass^2 particularly with regard to metaphysics, if the combinatorial hierarchy’s “numerology” is to be taken seriously.

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137 = Kabbalah
Can’t make this stuff up.


“You might say the ‘hand of God’ wrote that number [137]” – Richard Feynman

You will notice that there is no mention of kabbalah =137 in Wikipedia.

The Times of Israel has this article that does talk about 137 = kabbalah:

At the risk of going completely mystical on ya’ll:

The Combinatorial Heirarchy’s first 3 numbers are:

3, 10, 137…

3 is, of course, all over the place in mystical traditions. But what’s this “10” business about? Well both Pythagorean and Kabbalah traditions place “10” at the center of their “esoteric” initiation. Pythagoreans required initiates to keep silent for years while contemplating the number 10 in the form of the Tetractys.

Kabbalah’s central model of the cosmos is the:

Sefirot (/sfɪˈroʊt, ˈsfɪroʊt/; Hebrew: סְפִירוֹת, romanized: Səfīrōt, Tiberian: Săp̄īrōṯ ),[1] meaning emanations , are the 10 attributes/emanations in Kabbalah,[2] through which Ein Sof (The Infinite) reveals itself and continuously creates both the physical realm and the chain of higher metaphysical realms (Seder hishtalshelus ).

I don’t know what Kabbalah’s cosmogony says about how 3 gives rise to 10, but…

Cosmogony concerns itself with the unfolding into primordial Creation and Perception of phenomena by primordial Being. As I understand Frederick Parker-Rhodes’s original motivation in formulating the Combinatorial Hierarchy, it was just such an approach to what might be called “The Inevitable Universe”:

The Inevitable Universe- Parker-Rhodes’ Peculiar Mixture of Ontology and Physics*
Stanford Linear Accelerator Center Stanford University, Stanford, California 94309
Submitted to Proceedings of the 11 th Annual International Meeting of the ALTERNATIVE NATURAL PHILOSOPHY ASSOCIATION
Dept. of History and Philosophy of Science
Cambridge University Cambridge, England
September 14-l 7, 1989

  1. 2^2-1 = 3 (Trinity)
  2. 2^2-1 + 2^(2^2-1)-1 + 0 = 10 (Tetractys/Kabbalah Sefirot)
  3. 2^2-1 + 2^(2^2-1)-1 + 0 + 2^(2^(2^2-1)-1)-1 = 137 (Gematria of the Hebrew name for Kabbalah/ElectroMagnetism)
  4. 2^2-1 + 2^(2^2-1)-1 + 0 + 2^(2^(2^2-1)-1)-1 + 2^(2^(2^(2^2-1)-1)-1)-1 = 170141183460469231731687303715884105864 (Phenomenal Universe/Gravitation/Dirac Dimensionless Large Number Hypothesis)

The way we get from 3 to 10 in the Combinatorial Hierarchy is the recursive application of one layer of creation (the first being 3) to generate binary combinations which may be distinguished from that layer (ie: apply XOR) as well as from each other. In this case we have 3 entities/qualities/properties/attributes that may be present or absent in generating the 2^3-1 new, combination yielding 7 additional entities for a total of 10. This happens again with 2^7-1 to produce 137 in total, corresponding to the Electromagnetic scale. Then things go completely whacko and explode with 2^127-1 and we get gravitation. This may correspond to the kabbalah’s notion of the world coming into being in such an explosion that must be, in some sense, “repaired”.

What does kabbalah’s cosmogony say, if anything, about which 3 sefirot would be considered the progenators of the rest of the 10 sefirot?

And we’re left with the “coincidental” Gematria of 137 for the Hebrew word for kabbalah which, at least in my mind, points to someone figuring out the Combinatorial Hierarchy who then influenced Gematria. But, if we are to believe the antiquity of the Old Testament’s references to 137 (Ishmael’s age at death, Levi’s age at death and Amram’s age at death (and, according to the Times of Israel article Abraham’s age when commanded to sacrifice Isaac – but I can’t find support for this)), it probably means that someone had figured it out at least that early, which would mean prior even to Pythagoras. In any event, this stuff seems encoded in the Old Testament, what ever the antiquity of those references.

PS: I have to testify to a rather “positive vibe” hitting me when studying this stuff of a quality I only rarely experience – like once a decade or two – in my life. This “positive vibe” last happened to me in 2000 one morning when I woke up and somewhat marveled at it since I had nothing to which I could attribute it. Then I checked hits on my website. The hits for the Bowery Award for Amateur Rocketry had gone through the roof – hundreds of times greater than ever. Wired magazine had published a story on the CATS prize that linked to it. Reflecting on the “magic” of this sort of thing is when we are dealing with the “ElectroMagnetics” of human society – particularly when we deal with mass media – there are potentials at work (perhaps Magnetic Vector Potentials). These potentials are a two-way street between the mind of God – as metacognitive source – and our own mind. These potentials may be put to “use” as “tools” to manipulate masses – bending them to our will – but where the positive vibes come from is their potential to reorient us toward God’s will. The phrase “moral compass” comes to mind – although I’ve seen that phrase used in ways that point to the stampede of the mob as the alignment – rather than a higher agency.


To check how confident one might be in attributing significance to the Biblical occurrence of age-at-death of “137” I found this table:

The only verses in which the age at death is given literally would appear to be those in which there are no entries in the “Before” and “After” columns – since those columns are used to compute the age at death.

This then provides a probability distribution in which, although “137” is the maximum frequency with 3 occurrences, it does have a runner-up with 2 occurrences of “110”. I suppose an additional statistical test might be run but from this it would appear that it is not unreasonable to take this as a “signal” of some sort given that one had a prior expectation that 137 would be the most frequent.

Just to be clear about what my maximum-likelihood hypothesis is at this point:

The sources for these ages appear in the imputed “Priestly” texts of the Babylonian exile period which correspond to a number of “Priestly appropriations” from other cultures including the Epic of Gilgamesh among others such as evidenced by the Cyrus Cylinder:

This makes me suspect Gematria was a product of these sources, resulting in the otherwise bizarre coincidences involving not only the lifetimes of 137 years but also the Gematria for the Hebrew word for “kabbalah”. Why would the word “kabbalah” figure so prominently in the assignment of numeric values to the Hebrew alphabet? For the same reason 137 was encoded in those ages-at-death:

There is an ancient knowledge about mathematical metaphysics that was “Priestly appropriated” but lost in the translation as it became “mystical”, “esoteric” and otherwise subject to noise in transmission between generations. This probably happened with the Pythagoreans as well with, perhaps, some attempt to recover it among the neoplatonists and those they influenced in various esoteric schools of JudeoChristianity.


Here’s the Combinatorial Hierarchy’s first 2 echelons illustrated as a directed graph of primary color combinations for the combination of attributes present in the second echelon’s 7 “sefirot”:


It will be difficult to generate the third eschelon’s 127 “sefirots” in addition to this as destinations of the directed graph from the 7 “sefirots” of the second eschelon.

Another CH “coincidence” (this one good old 137 aka CH3) pointed out to me by Pierre Noyes and David McGoveran – which I’m throwing into the physics.stackexchange mosh pit before I proceed to the analogous “coincidence” involving gravitational potential energy and black holes and CH4:

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Did quark theory take a wrong turn in the 1960s? Peter Rowlands seems to think so (in this video presentation given to the CH crowd in 2021):

Specifically, He claims the Higgs mechanism is founded on the wrong quark theory:

“If integral charges are fundamental, different colours of quark automatically produce the two hypercharge values, 1 and -1, which we require for both isospin states and which would be repeated in each generation.”

Rowlands’s paper “Nilpotent Quantum Mechanics: Analogs and Applications”.

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I just had to save this interaction with a Defender Of The Faith in a screenshot for posterity as soon as it happened because the last time I nailed one of these pseudonymous Bloblings to the reddit wall s/he/it deleted their account and all their messages:

Aside from the pseudonymous Blobling “hecd212” failing to perform what it claimed was a “very easy” feat of confabulatory mathematics, its performance was even more laughable if you consider that there is really only one formula for the dimensionless gravitational coupling constant (dimensionless meaning pure number with no physical units) that uses the grand unification (Planck) mass divided by the mass of a chosen subatomic particle. Although there may be a non-arbitrary dimensionless parameter like 2π if angles are involved, you really aren’t free to confabulate the formula, as it accused me of doing.

So it was nailed to the reddit wall twice: 1) Having to come up with a “formula” that would put the proton at the top (which would be obvious gibberish), and 2) Having to use the actual formula for alphaG – as I had. I suppose I should add 3) That it should have known #2 being a Defender Of The Faith and all.

So my only real “arbitrary” choice was the choice of which particle’s mass I was to use. But I really wasn’t even free to do that since I’d already been working with folks using the proton mass. But, to avoid the accusation of “numerology” in getting the proton mass alphaG to ~2^127 I decided to write a Mathematica script that ran an exhaustive search of massive particles to see which of the best-matched the “numerology”.

As far as the theoretic motivation, I’d already provided that in links both here and in that hilarious exchange, with references going back to the Cambridge Language Research Unit in the 1950s looking for an alternative approach to natural philosophy grounded in language theory.


I ‘like’ most of your posts, meaning to honor your effort and dedication to important questions about the nature of reality. I wish I could say I actually understood many of them! Though I play in the minor league, I am glad for the existence of the majors - yes sir! (I salute the major leaguers among us; you know who you are).


Me too! Which is why I said to the blobling “I seek one.” meaning understanding of the work others, who I have good reason to believe are better equipped, have done.


I’m really popular over there at physics.stackexchange.com! Within 10 minutes of posting this question its GONE!


Having been duly chastened by my betters (and apparently those of Dirac, Eddington and Feynman) at physics stackexchange, I’ll not bother them with a refinement of something Dirac noticed.

But first, you have to notice this:

If you take the reduced compton wavelength of the electron and put it through a mere 360 degrees shift, you’ll invert its spin (say, from 1 to -1) due to the electron’s spin being 1/2. If, however you take it all the way through 720 degrees spin, you’ll get back to its spin state (say, from 1 through -1 and back to 1). By “reduced” is meant the wavelength is divided by 2pi radians (360 degrees), so that if one multiplies it by 2, you get the equivalent of 720 degrees which brings you through a complete cycle.

So, now, here’s the thing Dirac didn’t notice:

The 5th Catalan Mersenne prime (2^127-1) times the complete cycle of the electron is equal to the light-age of the universe to within 1%:


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