v 6.20.00
20 Mar 2021
updated 20 Mar 2021

Pythagoras' Theory

No, not a typ% (or mirpsint) relating to the most famous maths theorem ever (though in fact that was also an alternative to Euclid's famous geometric Fifth Postulate, in which role it was more of a statement of mensurational faith).

I'm referring in fact to the central creed of the Pythagorean Brotherhood, that "All is Number" – though it's surprising infeasible to find how that was said in the original Greek, or much else about it. They were a wacky lot (for example, they refused to eat beans, as the shape of beans reminded them of testicles). A huge mythology has grown up around Pythagoras (and his followers) and much may be incorrectly attributed to them collectively.

But that doesn't matter – this assertion expresses perfectly my own conviction that not only does number express how the world (ie universe) works, it actually constitutes the very bodily essence of us and the substance of everything around us. We are number, the very cosmos itself is number.

Number, no added vitamins, just honest-to-goodness Number. I'll elaborate on that further down.


In another challenging statement, just as obscure if not more so, the first sentence of St John's Gospel famously proclaims "In the beginning was the Word, and the Word was with God, and the Word was God."

And since first hearing it read in church as a boy, I've always thought it was entirely implausible – how could words, let alone just one Word, be enabled to create the world?

Though, more recently, I've ruminated (as per Michael Coren) that the Archangel Michael might have summoned the Demiurges (one for Space and Time, the other for Matter), to say "I've just been to see Him, and He gave me the Word to have another go, but we all know what happened last time, we don't want a rerun of that, do we, thank you very much, narmean?"

Space and Time shuffled his feet, muttering apologies, and asked if he could try a triffic new release of GenRel? Matter, not to be outdone, expressed contrition for his part in the previous debacle, and asked if the very latest version of QuantMech would meet with His approval?

"OK, OK," said Michael wearily, "but make By-Our-Lady sure they're fully compatible this time. Or else," he added, "you wouldn't Adam and Eve wot a narsty wicked temper He's got if things go wrong again."

λόγος vs λέξις

The implication of "Word" as examined above is of course seriously unfair, and in digging a little deeper via respectable internet sources I've at last realised the real meaning of it – or at least an interpretation which makes sense.

John the Evangelist wrote his Gospel in Aramaic, the language of Christ himself, and it was later translated into Greek – and embellished with additional material – for the purpose of spreading his message further afield. But for present purposes let's focus on the translation from Greek to English.

From what word in Greek would the word "Word" in English be translated? In my role as T C Mits, I'd say "Logos". And that's indeed what seems to have been the case when the King James version was in progress.

But the word "logos" had a good many other meanings and usages in the highly sophisticated cultural life of the Graeco-Judaic world of Biblical times. One such meaning, but a very minor way, was indeed "word" – in the sense of a conventional sign, verbal or written, for something or another.

And the word used specifically for "word" in that sense was "lexis". But it was "logos" not "lexis" that the translators were faced with.

One of the most important sentences in the whole canon of Christian theology has been bungled, so it would seem to TCM. Its meaning has been lost in translation, into English at least. Of course theologians are fully aware of the intended meaning, but how many of the rest of us are? And all too many of us, natural ultracrepidarians, having found fault with the Sandal of Ambiguity proceed to rubbish the Toga of Truth. A large claim to make, but all too true in my own case.

So what are, in fact, these alternative meanings of "logos"?

  • Originally, in everyday Ancient Greek usage, it could mean "ground", "plea", "opinion", "expectation", "word", "speech", "account", "reason", "proportion", and "discourse".
  • But in their prolific schools of philosophy, the most compelling usages were those of Heraclitus, who used the term for a principle of order and knowledge, and Aristotle, who used it to refer to reasoned discourse.
  • Aristotle's usage has of course come down to us as Logic, the rules of systematic thought and discourse (verbal or symbolic).
  • Though Heraclitus' usage is just what we're looking for here - a template and process by which a concept could be defined and developed.
  • But then John (or his proxy translator into Greek) came along and b*ggered it all up by identifying Christ as being the personification of Logos, and indeed identified with it. Had he forgotten that rather resonant divine rebuke to Job back in the Old Testament, "Where wast thou when I was laying the foundations of the earth?"

I go with Heraclitus, and with his interpretation I can indeed accept that "In the beginning was the Logos, and the Logos was with Creator Mundi, and the Logos was Creator Mundi", and leave the nature of the Creator in the realm of perpetual mystery.

On a more human scale, Heraclitus' usage is nicely illustrated by the topic of logarithms, which struck terror into the hearts of most of us back in the mid 1950's. Progressive thinking has abolished them, as "calculators can do all that arithmetical drudgery", but of course they reassert themselves in the hard sciences. "Logarithm" is a portmanteau word, combining "Logos" = process, and "Arithmos" = number, signifying the power of a process to multiply and divide numbers of all magnitudes.

In everyday usage, however, the ambiguity remains – we would all understand "a logophile frequently consults his lexicon" as meaning "a word-lover often uses his dictionary" or that a lexicographer is a compiler of dictionaries.

I'm reminded of one of my father's favourite declamations

There's a lexicon written by Liddell and Scott
And some of it's good and some of it's not
The bit that is good was written by Liddell
And the bit that isn't was written by Scott

Liddell, the Dean of Christ Church Oxford, was the father of 'Alice in Wonderland', and the principal author of a classical Greek lexicon pre-eminent to this day.

Dip Tinking

As mentioned in the section Faith, Hope and Clarity above, it is now over 20 years since I was persuaded to attend an Alpha course being presented at an evangelical church in Reading. And curiously coincidental that Nicky Gumbel, the progenitor of the course, was based at Holy Trinity Brompton, the church in Kensington that I attended as an earnest seeker after truth in my early teens. Indeed he's now the vicar there.

The course was very well-meaning, but entirely unpersuasive – it begged all the questions that it purported to answer, and therefore failed to resolve them. But it did awake me from my dogmatic slumber, and I started to think once again about, in particular,

1] How can an immaterial Creator be capable of thought?

2] How is he able to create a material cosmos?

Though of course there are lots of other niggles, secondary stuff (Origin of Life, Sin, Virgin Birth, Pain and Suffering, Miracles, Atonement, Crucifixion, Resurrection, Trinitarianism, Life after Death, the usual things that have bothered us all over the last couple of millennia, and over which such quantities of ink and blood have been spilt to no effect).

It could be debated as to whether theology is a subset of physics, or physics is a subset of theology. Werner Heisenberg is (controversially) said to have remarked "The first gulp from the glass of natural sciences will turn you into an atheist, but at the bottom of the glass God is waiting for you." But I firmly believe that Questions [1] and [2] are strictly a matter of physics – and that the God to whom Heisenberg referred could equally well have been Einstein's "Der Alte", totally impersonal and utterly indifferent to the existence of Homo SapiensA.

Creating the universe is definitely a Boy Job, and so I'll refer to Creator Mundi, whereas creating life on at least this planet, and quite possibly elsewhere, is undeniably Skirt Work, and so I'll refer to Creatrix Vitae. Should it turn out that they're one and the same, I'll refer to the Creatron, in compliance with current standards of political correctness.

These amateurish lucubrations have no claim to originality or profundity, and simply represent the best efforts of an elderly recluse, neurones in free fall, trying for a rationale of the seismic shifts in religious and cosmological opinion over the last three generations. Professionals can curl their lip, though they're not likely to be reading this in the first place, it's "Sauve qui peut" for beliefs these days and this is my private philosophical lifeboat.


Question [1] How can an immaterial Creator be capable of thought?

The human power of thought (or merely rearrangement of our prejudices, as it was once described) is inextricably dependent on our brain, a biochemical by-product of organic life. So it's beyond our powers of imagination for us to conceive of how else it could be possible (apart from its limited simulation by electronic artificial intelligence software) whether on earth or anywhere else.

In his novel The Black Cloud, the cosmologist Fred Hoyle did go part-way in this direction, but an intelligent cloud of interstellar hydrogen is still a long way from a being outside space and time altogether.

But just because we can't imagine something doesn't mean it can't exist. On the other hand, we mustn't fall into the trap of supposing that if we can imagine it then it must exist, as was the basis of an attempted proof of God's existence popularised by mediaeval Roman Catholic scholastics such as Thomas Aquinas. After all, where are the unicorns?

But without some agency (a deus ex machina as the ancient Greek dramatists liked to invoke), capable of devising the most extraordinarily intricate and fruitful laws of physics, how can our universe be explained? As Paul Dirac remarked, "God is a mathematician of a very high order and He used very advanced mathematics in constructing the universe." And presumably very advanced mathematics can only originate from an entity capable of thought.

Like Stephen Fry in the QI programme, who would sepulchrally intone "Nobody knows" when an insoluble issue was under discussion, that's really all anyone can say.


Question [2] How is he able to create a material cosmos?

Richard Feynman once remarked

"If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied."

In 1990, the IBM logo and corporate motto were spelled out in individual xenon atoms on a single-crystal nickel substrate, using an STM (Scanning Tunnelling Microscope)

The notion of atoms was first conceived by Leucippus, popularised by Democritus, propounded by Newton, elaborated by Boscovich, extended by (Daniel) Bernoulli, Waterston, Maxwell et al to explain the nature of heat, and by Dalton to explain the laws of chemical combination. Complete respectability was finally conferred by Einstein in his explanation of Brownian motion.

The word "atom", implying indivisible, is actually a misnomer of course. More appropriate labels would have been "particle" (meaning a small part), or "corpuscle" (meaning a small body) or "molecule" (meaning a small mass), which feature the Latin suffix "culus" as diminutive. However, they have been put to work as particles meaning subatomic entities, corpuscles meaning Newtonian precursors of photons, and molecules meaning chemical combinations of atoms. Nothing has been wasted!

In the late 19th and early 20th century it was realised that, far from being indivisible, atoms actually comprised a number of different kinds of even smaller particles. And some of these smaller particles actually comprised even smaller particles!

Becquerel: alpha particle (helium nucleus), beta particle (electron), gamma particle (or ray)
J J Thompson: electron
Rutherford: nucleus (diminutive of nux = nut)
Mosley: proton
Chadwick: neutron

In particular, of course, nuclei consist of protons and neutrons in a bewildering number of combinations, and atoms primarily consist of nuclei and electrons.

Furthermore, subatomic particles obey laws of motion (quantum mechanics) remarkably different from the Newtonian laws established centuries earlier for macroscopic objects. But these new laws applied to the interactions between atomic nuclei and their retinue of electrons dramatically revealed the inner logic of the Periodic Table, and the reasons for chemical combination.

From the 1930's onwards, however, it became experimentally evident from the study of cosmic rays on the one hand and the development of increasingly powerful ground-level (or even underground) technology on the other, that the particles thus far discovered were only the first fruit of a bewildering proliferation of new and entirely unsuspected varieties (mostly of very short lifespans). It was all getting very, very complicated.

And from Dirac onwards, the relativistic theory of particle interactions with one another, and with radiation, known as QED (quantum electrodynamics), was becoming unimaginably complicated – Rutherford's criterion of a good theory being that you could explain it to an intelligent barmaid was by now long abandoned.

New categories of intranuclear force – the "weak force" (responsible for radioactivity) and the "strong force" (responsible for holding the nucleus together) – it does sound as though their job descriptions are somewhat at odds with each other – have been identified.

Electricity and magnetism, unified by Maxwell a century and a half ago, have been merged with the weak force and rebranded as the "electroweak" force. The strong force has yet to be merged, retaining its own job title as QCD (quantum chromodynamics).

There seems to be a new mantra, that not only does every particle have a field, but in addition, every field has a particle, and so on down the spiral staircase. Click here to see the currently-accepted roster of relatively stable fundamental particles (fermions) and their force-carriers (bosons)

In the first month or three of my 3 years in the Department of Theoretical Chemistry at the School of Chemical Sciences, University of East Anglia during the late 1960's, I mastered the rudiments of the semiclassical theory of electronic spectroscopy and optical activity, from the scanty selection of sources then available – I couldn't afford Eyring Walter& Kimball, and there was no helpful internet then. Absolutely no guidance from ones supervisor, of course.

The algebra was quite intricate, but didn't need anything tougher than A-level maths, and the exercise gave me confidence (which was sorely tried in the years that followed). The point is, however, that any higher level of rigour would have defeated me at that stage. How could any new graduate student possibly get off the ground with modern QED?

And the other point is that I'm deeply suspicious of all the fancy maths modern QED involves. As when a computer program gets too complicated, it's a sign that the enterprise is wrongly structured and should be abandoned in favour of something less elaborate. There's a very neat anecdote about von Neumann, though, which suggests that sufficient brainpower can win the day even if the going gets tough.

I wouldn't be at surprised, were I still around, if current approaches to QED were eventually to be seen as analogues of Ptolemaic deferents and epicycles upon epicycles, to be discarded and replaced by some visionary new Copernicus.

But getting to the point at last, what are all the fundamental entities in the Standard Model, fermions and bosons, actually made of? A naïve question, but somebody's got to ask.

If an infinite regression is to be avoided, the answer at some point has to be that they're not actually made of anything. I very strongly suspect that the leptons and the quarks are entities devoid of all size and substance, mathematical points in fact, but imbued with the properties appropriate to their category. Some properties are permanent (such as electric charge and spin) while others (such as mass and momentum) vary from instant to instant and from particle to particle, of course, depending on particular circumstances.

And of course the fundamental equations governing the interactions and motions of these mathematical points would be precisely as experimentally established over the last century or two by physicists who regarded these entities as very, very small, of course, but certainly not infinitesimal.

And if indeed the universe did erupt from a singularity, as is widely believed, there is absolutely no need to fret about its having been of infinite density (with all the contradictions that would involve), as any number of aboriginal mathematical points can be packed into a singularity.

Composite particles such as protons, neutrons and mesons are certainly not mathematical points, and do have a certain size of course, as they possess an inner structure arising from the interactions between their constituent quarks. But I suggest that they are nevertheless still just as insubstantial, or immaterialB, whichever adjective one might prefer.

And atomic nuclei, atoms, molecules – and stars, planets, tables and chairs, you and I, entities of whatever size, structure and complexity – are all ultimately constituted from pure Number.

So the age-old philosophical stand-off between an immaterial Creator and a material Creation, simply dissolves – the Creation is just as immaterialB as the Creator. At the moment of Creation, a torrent of Number erupted and took control of everything that has happened subsequently, precisely as Pythagoras and his bean-dodgers proclaimed 2½ millennia ago.


Rivalling the claims of Number and Word, however, is Harmony (as expressed by Dryden, for example, in his ode below), though Pythagoras himself was also fascinated by the close relationship between number and the harmonic ratios of pitch to tone. St Cecilia is of course the patron saint of music, and endured the usual painful death traditionally required for ultimate sanctification.


John Dryden (1631-1700)

(First verse)

From harmony, from heavenly harmony,
This universal frame began:
When nature underneath a heap
Of jarring atoms lay,
And could not heave her head,
The tuneful voice was heard from high,
'Arise, ye more than dead!'
Then cold, and hot, and moist, and dry,
In order to their stations leap,
And Music's power obey.
From harmony, from heavenly harmony,
This universal frame began:
From harmony to harmony
Through all the compass of the notes it ran,
The diapason closing full in Man.


A One of the most influential books in my early teens was Idiot Man (translated from the original French, l'Homme Stupide), by Charles Richet, 1913 Nobel laureate in Medicine or Physiology, and well to the right of Attila the Hun as regards racial and ethical issues. He proposed, mostly persuasively, that Homo Sapiens be rebranded as Homo Stultus, in regard to mankind's indefatigable cruelty, stupidity and folly.

Somehow it eventually got mislaid, but here is his Introduction, which gives a flavour of his punchy assertive style.

classiques.uqac.ca/classiques/ richet_charles/ homme_stupide/ homme_stupide_prologue.html


Linné, essayant de classer en bon ordre les diverses formes vivantes qui peuplent notre planète, a appelé l'homme, lequel constitue évidemment une espèce animale distincte de toutes les autres: Homo sapiens, l'homme sage.

Mais un tel éloge est manifestement injustifié. Car l'homme accumule de si abondants exemples d'extraordinaire bêtise, qu'il faudrait, pour se conformer à la réalité des choses, le dénommer tout autrement, et dire Homo stultus, l'homme stupide.

Quand nous consentirons à employer une classification zoologique sérieuse, il faudra adopter ce terme.

Dans ce bref écrit, nous établirons, ou du moins nous tâcherons d'établir, que l'homme est inférieur à la plupart des espèces animales pour le bon sens et la sagesse. Il me paraît même que nous aurions le droit de le qualifier de homo stultissimus, l'homme stupidissime.

Cependant, pour être modéré, nous nous contenterons de lui donner, sans superlatif, l'épithète qui lui convient: Homo stultus, l'homme stupide, et nous donnerons les preuves de son immense et incurable stupidité.

L'auteur ne se fait aucune illusion sur le sort réservé à cet examen de conscience qui froissera, qui offensera les intellectuels aussi bien que le populaire et qui laissera à tous une impression douloureuse.

Oui! Nous le savons.

Donc, ô lecteur, qui que tu sois, intellectuel ou artisan, ce livre va troubler, fût-ce pour un instant, la bonne opinion que tu as de toi-même. Il ébranlera cette conviction intime, que tu es sage, prudent, raisonnable. C'est peu agréable de s'entendre dire qu'on est stupide, et e'est plus désagréable encore d'en recevoir la démonstration.

Mais il ne s'agit pas de présenter, à la manière de Watteau et de Florian, des bergers d'opéra. Les paysans de La Bruyère n'ont pas de houlettes enrubannées, et j'estime, avec le vieux maître, que toute vérité est bonne à dire, si amère et décourageante qu'elle soit.


The immaterialism in which I believe (and which many others may do too, I don't know), shouldn't be confused with that advocated by the renowned clergyman and philosopher George Berkeley. It's many a long decade since I used to read about such matters, but I think his standpoint to have been that our conscious mind is our only route to an apprehension of the world around us, and that our concepts of this world are merely a mental synthesis of the percepts that our senses transmit to us of it. Ergo, for us the world is no more than a notion, of no independent reality.

This immateriality must logically extend to our body, and most importantly our brain – the seat of reason and all our mental processes. Does this not saw off the philosophical branch on which we are sitting? I don't know, and am not going to let it bother me. Berkeley must have thought this through to his own satisfaction and that of posterity.

Of course, the Berkelian immateriality of the world around us doesn't insulate us from coming to harm. An immaterial person might tumble over the edge of an immaterial cliff and suffer immaterial, but terminal, injuries in consequence!

And an immaterial boot aimed at an immaterial stone would of course rebound. Samuel Johnson's famous attempt at a refutation of Berkeley's idea was of course a rare instance of nonsense from the formidable sage.


After we came out of the church, we stood talking for some time together of Bishop Berkeley's ingenious sophistry to prove the nonexistence of matter, and that every thing in the universe is merely ideal. I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it -- "I refute it thus."

Boswell: Life of Johnson


The name of this fallacy is derived from a famous incident in which Dr. Samuel Johnson claimed to disprove Bishop Berkeley's immaterialist philosophy (that there are no material objects, only minds and ideas in those minds) by kicking a large stone and asserting, "I refute it thus."

As a by-product of his immaterialism, Berkeley coined the famous phrase (more or less), 'Esse Percipi Est' – 'To be is to be perceived' (the battlecry of the Copenhagen Interpretation of quantum mechanics). If you're not being observed you don't actually exist (Einstein plaintively enquired as to whether this principle applied to the moon: "Do you really believe that the moon isn't there when nobody looks?").

The Catholic priest and theologian Ronald Knox used the Esse Percipi principle as a humorous proof of the existence of God. The young man in the first limerick is clearly an Oxford undergraduate.

There was a young man who said "God
Must find it exceedingly odd
To think that this tree
Can continue to be
When there's no-one around in the Quad."

"Dear Sir: Your astonishment's odd;
I am always about in the Quad.
And that's why the tree
Will continue to be
Since observed by, Yours faithfully, God."

Science: Good, Bad and Bogus, Martin Gardner, OUP 1983, p186

The problem of saying what a quantum particle looks like when no one is looking at it is something like the problem of what a mirror looks like when no one is looking at it. Here is how the late J A Lindon, of Addlestone, England, expressed his befuddlement in a poem he sent me which I here print for the first time:

Look and See

I thought I knew my physics, though my knowledge isn't deep,
But now a problem haunts me that is ruining my sleep;
All night I toss and turn about, I stare into the gloom:
What does a mirror look like when there's no one in the room?

It can't reflect its owner, if the owner isn't there,
But surely it must still reflect his table and his chair,
His picture on the wall behind, the window and the door,
The potted fern, the chandelier, the carpet on the floor?

For these are there, in lots of light, before the mirror's gleam;
They're bound to be reflected, if it's real and not a dream;
But everything reflected forms an image in the glass,
That seems to come and move along and vanish as you pass.

So how a mirror looks to you depends on your position,
And here the books are guilty of a serious omission;
Away with "how it looks to you," put "seeming" on the shelf:
What does a mirror look like when reflecting by itself'?

There stands a mirror by the wall, a candle-flame before it
Reflected with such clarity a worm could not ignore it;
I take my toothbrush and my towel, yet wonder more and more
How it will look to no one there when I have shut the door.

I myself am proud to be what professional philosophers sneeringly refer to as a Naïve Realist, it saves time and brain-power. And it does respond vigorously to the oft-quoted query, "If a tree falls in a forest, and no one is around to hear it, does it make a sound?"

We NRs point out that, on all occasions of falling timber, a shock-wave (an objective phenomenon) is produced, that is detected by the audiferous sense-organs of any passing squirrel, fox or human. And these each, separately and individually, experience the subjective epiphenomenon of sound, a purely personal mental effect, that is entirely superfluous to the reality of the shockwave, or the tree for that matter.

Whatever side one takes in this debate, the elephant in the room is that of precisely what is the nature of consciousness, whether vegetable, animal or human, and I'd also like to contribute my own three-ha'penceworth to this thorny issue.