Following up a conversation with John Amson at Rufflets Hotel, just outside St Andrews, I was led to a paper on combinatorial physics:

Ted Bastin, H. Pierre Noyes, John Amson, Clive W. Kilmister: On the physical interpretation and the mathematical structure of the combinatorial hierarchy, *International Journal of Theoretical Physics* **18** (1979), 445–488; doi: 10.1007/BF00670503

Here is the abstract. See what you make of it.

The combinatorial hierarchy model for basic particle processes is based on elementary entities; any representation they may have is discrete and two-valued. We call them Schnurs to suggest their most fundamental aspect as concatenating strings. Consider a definite small number of them. Consider an elementary creation act as a result of which two different Schnurs generate a new Schnur which is again different. We speak of this process as a “discrimination.” By this process and by this process alone can the complexity of the universe be explored. By concatenations of this process we create more complex entities which are themselves Schnurs at a new level of complexity. Everything plays a dual role in which something comes in from the outside to interact, and also serves as a synopsis or concatenation of such a process. We thus incorporate the observation metaphysic at the start, rejecting Bohr’s reduction to the haptic language of common sense and classical physics. Since discriminations occur sequentially, our model is consistent with a “fixed past-uncertain future” philosophy of physics. We demonstrate that this model generates four hierarchical levels of rapidly increasing complexity. Concrete interpretation of the four levels of the hierarchy (with cardinals 3,7,127,2^{127}-1∼10^{38}) associates the three levels which map up and down with the three absolute conservation laws (charge, baryon number, lepton number) and the spin dichotomy. The first level represents +, −, and ± unit charge. The second has the quantum numbers of a baryon-antibaryon pair and associated charged meson (e.g.,n^{-}n,p^{-}n,p^{-}p,n^{-}p,π^{+},π^{0},π^{-}). The third level associates this pair, now including four spin states as well as four charge states, with a neutral lepton-antilepton pair (e^{-}e or v^{-}v), each pair in four spin states (total, 64 states) – three charged spinless, three charged spin-1, and a neutral spin-1 mesons (15 states), and a neutral vector boson associated with the leptons; this gives 3+15+3×15=63 possible boson states, so a total correct count of 63+64=127 states. Something like SU_{2}×SU_{3} and other indications of quark quantum numbers can occur as substructures at the fourth (unstable) level. Breaking into the (Bose) hierarchy by structures with the quantum numbers of a fermion, if this is an electron, allows us to understand Parker-Rhodes’ calculation of *m*_{p}/*m*_{e} =1836.1515 in terms of our interpretation of the hierarchy. A slight extension gives us the usual static approximation to the binding energy of the hydrogen atom, α^{2}*m*_{e}*c*^{2}. We also show that the cosmological implications of the theory are in accord with current experience. We conclude that we have made a promising beginning in the physical interpretation of a theory which could eventually encompass all branches of physics.

My first reaction is something along these lines. Pythagoras is thought to have believed that “all is number”, and also to have believed in reincarnation. (Of course, we know nothing about what he really believed.) So perhaps these authors are channelling the spirit of Pythagoras.

Note also that *Schnur* is German for “string”, as claimed, but is also defined by the Urban Dictionary as “the ultimate insult that means absolutely nothing”. Interesting?

Actually reading the paper didn’t clear up all my doubts. The setting is sets of binary strings. A set of strings is called a *discriminately closed subset* if it consists of the non-zero elements in a subspace of the vector space of all strings of fixed length *n*. Such a subset has cardinality 2^{j}−1, where *j* is its dimension. Now a step in the *combinatorial hierarchy* involves finding a set of 2^{j}−1 matrices which are linearly independent and have the property that each of them fixes just one vector in the subset (I think this is right, but the wording in the paper is not completely clear to me). These matrices span a DCsS of dimension 2^{j}−1 in the space of all strings of length *n*^{2}.

Of course, the exponential function grows faster than the squaring function, so the hierarchy (starting from any given DCsS) is finite. Their most important example starts with *j* = *n* = 2 (two linearly independent vectors in {0,1}^{2} and their sum), and proceeds to *j* = 2^{2}−1 = 3, *n* = 2^{2} = 4; then *j* = 2^{3}−1 = 7, *n* = 4^{2} = 16; then *j* = 2^{7}−1 = 127, *n* = 16^{2} = 256; then *j* = 2^{127}−1 ∼ 10^{38}, *n* = 256^{2} = 65536 (but this is impossible, so I am not sure what it means to continue the hierarchy to this point).

They point out that the numbers 127 and 2^{127}−1 are close to, respectively, the fine structure constant and the ratio of strengths of electromagnetic to gravitational force between protons. If you use cumulative sums, you get 3, 10, 137 and a number which is again about 10^{38}, and of course 137 is even closer to the target. But I am not sure why you should do this, and in any case, the fine structure constant is measureably different from 137. The non-existence of 10^{38} linearly independent matrices of order 256 supposedly has something to do with “weak decay processes”.

The paper contains some constructions of the appropriate hierarchies. One could pose the mathematical question: do the required linearly independent non-singular matrices exist for any *n* and *j* for which 2^{j}−1 ≤ *n*^{2}?

By this stage I was floundering, so I gave up my careful reading. I noted at a certain point a calculation of the ratio of proton mass to electron mass giving a value 137π/((3/14)×(1+(2/7)+(2/7)^{2})×4/5), agreeing with the experimental value to eight significant figures. (There are three terms in the geometric series because the hierarchy falls over at step 4.) Of course, putting in a more accurate value for the fine structure constant would make the agreement less good: the authors do attempt to explain this.

(If you are a mathematician looking at the paper, the mathematics is put in an appendix starting on page 480, so you can avoid the physics.)

At another point in the conversation, John told me that he ran with the fox and hunted with the hounds. On the strength of this, I can perhaps attribute to him a certain degree of scepticism about all this.