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David Cowles

Mar 1, 2024

“I want to repeal all the fundamental laws of Arithmetic.”

In the age-old battle between primary school students and their ‘minders’, we come down squarely on the side of the little Cherubs. Teachers never tire of drilling the rules of Arithmetic into their reluctant pupils’ skulls…even though Zeno proved 2,500 years ago that Arithmetic cannot adequately model any universe that includes motion. A pretty big miss, wouldn’t you say?

Arithmetic may apply to any universe with ‘continuous structure’ (remember the real number line?) but we don’t live in such a world! As Democritus conjectured and Planck, et al. proved, our Universe is quantized. 

“Whoa!” you say, “Calculus solved that problem.” And so we all thought…until Bertrand Russell showed that tacking Calculus onto Arithmetic doesn’t help at all. Both Arithmetic and Calculus are ‘non-physical’. They are useful tools for calculating certain quantities, but they are structurally inconsistent with the real world

They treat the world as if rather than as is…which is fine, so far as it goes: Euclidean Geometry does the same thing. That’s okay for most purposes. We make simplifying assumptions all the time. We treat the Earth, for example, as if it were a perfect sphere – it isn’t. 

Zeno could not reconcile the continuous nature of real numbers with the discontinuous (quantum) nature of reality. Neither could Stephen Hawking!  But I have a somewhat different bone to pick. I want to repeal all the fundamental laws of Arithmetic. If I have my way, Jack and Jill will no longer have to hear about Commutative, Associative, or Distributive Properties, but I want to start my crusade with the Transitive Property (TP): 

If a > b and b > c, then a > c.

Seems pretty innocuous, doesn’t it? Unless you’ve seen an episode of Doctor Who. According to the BBC, our wide but finite world contains a certain average-sized phone booth (or ‘box’)… remember them? The best adjective would be ‘cramped’; but inside this box, known as the Tardis, space is infinite: 

World > Booth (exterior) > Booth (interior) > World. 

Clearly, TP does not hold in the ‘real world’, at least not as it is defined by the BBC. Turns out, it doesn’t hold anywhere, no matter who’s defining it – all of which makes the 5 years kids spend in elementary school (US) ‘problematic’ at best.

BTW, I pointed out this apparent anomaly to an avid eight-year-old Doctor Who fan. I expected, “Aha!” To my surprise, this bright kid just stared at me and shrugged. He was struggling to be polite. He did not see this as a contradiction. He was not yet TP-poisoned. He was still neuroplastic, and I am grateful to him for showing me the limitations of TP.

Axioms (like TP) are funny things. They are assumptions, not subject to logical proof or requiring empirical validation. An axiom is falsified when normally intelligent people no longer see it as ‘obvious and incontrovertible’. An eight-year-old showed me that a ‘normally intelligent’ person can doubt the universal validity of TP. 

I’m afraid if I say that the transitive property (TP) is the root of all evil, you will stop reading this post, so I won’t. However, I do have (t)issues with TP. (No snickering, under 12s!) TP’s the very model of a modern ‘meritocracy’; it makes ‘hierarchy’ a fetish. 

  • I’m better than you, but that’s OK because you’re better than someone else – right?

  • You were abused by your father because he was abused by his, but don’t worry, you’ll get even; you’ll abuse your own children someday too. Fair enough?

  • The Untouchables is not a reference to Eliot Ness but to India’s ancient caste system.    

Fortunately, we don’t live in a world like this! To whatever extent the world is like this, we alone have made it that way. This is not the state of nature we inherited from Genesis…or from Jean-Jacques Rousseau. But you know that! How often have you said, “What goes around comes around”, and so it does. But then you go right back to teaching TP to your eight-year-old. Shameful!

I propose we build a new mathematics, replacing TP with a new paradigm, the Circular Property (CP): a > b, b > x … x > a.  CP describes a Chain of any length. The initial term and the final term of the Chain must be the same, and that term cannot appear anywhere else in the Chain. Whenever any initial term reappears as a final term, a complete Chain (loop) has been formed.

Any term can be the initial term (or the final term) of a Chain - alphabetical order not respected. “Pick a card, any card!” It’s like that cool trick your dad tried to teach you when you were small. 

At the end of the day, every Chain forms a loop. Any term in one loop can be a term in multiple other, intersecting (tangent) loops. The fabric of the universe may be a tight weave of such loops: Penelope and the Norns! Weaving as Sacrament: the whole is represented by, and instantiated by, the part. Knitting is liturgy!

CP corrects TP’s hierarchical bias. Imagine the social impact if we all knew that every subject is the ultimate object of its acts. Being as boomerang! No more child abuse, that’s for sure! Little crime, few murders. 

On the economic front, no more Class War. Workers win when owners win, and owners win when workers win! Instead of late-night bargaining sessions in smoke-filled rooms across from chanting picket lines and cops, negotiating teams will meet on the company lawn and sing Kumbaya – police welcome to join in. 

You’re chuckling. It’s one heck of a vision, that’s for sure. “But it’s not real,” you say. “It’s not physical.” Except it is! It’s entirely physical: Every action entails an equal and opposite reaction  (Newton). Every action!

Today, bleeding-edge scientists are working with a state of matter they call Time Crystals. In these event chains, time is effectively suspended (‘crystallized’). Process occurs outside of time.  It mimics CP:


If this sort of order is so natural and so clearly advantageous, why hasn’t anyone ever tried to build a social system around the CP Principle?

Well, have you heard of the Lex Telonis, an eye for an eye? It’s an attempt to implement CP, but it requires a huge, unmanageable intermediary apparatus: a police force, a judiciary, and, of course, executioners willing to blind a man with a hot poker. What could possibly go wrong?

Ok, then, how about the Beatitudes: “Blessed are the merciful, for they shall obtain mercy.” Again, what goes around comes around. Jesus dispenses with the administrative burden of a secular state, but he injects the process with his own intermediary, God. 

Christianity accepted God as the Uber-Omega but added back an infrastructure: Eschatology, Church, Clergy, Sacraments, etc. One can imagine Jesus, channeling Prufrock, “That is not what I meant; that is not what I meant at all”… or not, but that’s what Ecclesiology is all about. 


David Cowles is the founder and editor-in-chief of Aletheia Today Magazine. He lives with his family in Massachusetts where he studies and writes about philosophy, science, theology, and scripture. He can be reached at


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