Archimedes and Christianity

Don’t blame Archimedes (b. 280 BCE); he got a bum rap. He did not invent the Archimedean Property nor did he lend it his name. That came much later when someone noticed that Archimedes’ work entailed certain previously unidentified assumptions about the structure of the phenomenal world.

The Archimedean Property (AP) boils down to the assumption that real phenomena can be exclusively and exhaustively modeled within the domain of Real Numbers (extended to include Complex Numbers). Certainly sounds reasonable and it is…for most purposes.

Sure, it fails to capture certain features of the deep structure of the Real World but how often does that come up in polite conversation? You can live your life from birth through NLE and never give a thought to anything beyond Real Numbers…and in fact that’s exactly how people lived from the time of Plato until the 18^th century.

We most often discuss AP in the context of Geometry (and Topology). The Euclidean Geometry we learned as adolescents assumes AP. So do most non-Euclidean Geometries (e.g. geometries applicable to curved spaces). However, there is a class of esoteric geometries that reject the Archimedean Property. 

They seek to model phenomena that can’t be adequately approximated using Real Numbers alone. These phenomena require the introduction of ‘Unreal’ Numbers such as infinities and infinitesimals and ‘p-adic’ numbers (loosely related to logarithms for the few of you out there who don’t have a PhD in Pure Mathematics, yet).  

Without AP, if any two entities intersect (tangent or overlap), one must be a subset of the other. Nothing touches, nothing overlaps, unless it absorbs totally or is totally absorbed. It’s a readymade plot twist for the next Sci-Fi Silver Screen blockbuster. It’ll be a remake of Goldfinger, which itself retells the legend of King Midas. 

I touch you, I absorb you…or you absorb me. This gives new meaning to ‘Kill or Be Killed!’ It is a game of tag…for adults, but a game with apocalyptic consequences. Not sure you want to go down this path? No problem: you can try it before you buy it. The game could be simulated on a computer; it could even anchor a whole new genre of computer games, competing with Fortnite, Grand Theft Auto, and Tour of Duty. 

Of course, such ‘individual imperialism’ is not the only possible response to life in a world without AP. For example, we might decide to live our lives in hermetically sealed pods. Or we might learn to live together, harmoniously and symbiotically, without invading each other’s personal space…what a concept! 

Or we could just go with the flow. I could welcome the other into myself and I could graciously reciprocate by freely allowing myself to be uploaded into ‘another other’. Again, feel free to try before you buy, but know this: the process may not be reversible…so your life literally depends on the result!

‘Another other’? Makes sense. A absorbs B, another, then AB is absorbed by C, another other, and so on ‘up the food chain’. But what if we turned Jacob’s Ladder into an Ouroboros? What if we allowed A to absorb B…and B to absorb A, simultaneously. What if we allowed our two sets to be subsets of themselves and of each other? Now that would be some cool place to live, agreed? Too much for you? Well, stay tuned!

You are the product of a single fertilized sex cell. Today, your body consists of 30 trillion descendants of that cell, working more or less in harmony to ensure your (temporary) survival and to allow you to make all the contributions to human history your parents had planned for you. There will be no peace on earth or cure for cancer without you and there would be no you without the immediate and enduring cooperation of 30 trillion independent organisms (100 trillion over your entire lifetime). 

But there is something very special about these cells. Unlike simple life forms (e.g. bacteria), most of the cells in your body include a nucleus that houses copies of your primordial DNA. All animal and plant life consists primarily of nucleated cells. But while the first living organisms (prokaryotes) appeared on Earth 4 billion years ago, the first nucleated cells (eukaryotes) evolved just 2 billion years ago.

More astounding, every eukaryotic cell on Earth today is likely descended from a single 2 billion year old ancestor. (By comparison, Abraham was a dilatant!) So how did the primal eukaryote evolve? Two unicellular life forms, bacteria perhaps, gave each other reck (Anaximander, Buber) and entered into a covenant (Jeremiah et al.).

The process was described in 1986 by Nobel winning biologist, Paul Simon: “If you’ll be my bodyguard, I will be your long lost pal.” (You can call me Al) Perhaps Paul was drawing on an earlier description: “You will be my people, and I will be your God!” (Jeremiah 7: 23).

And so it happened! One cell agreed to give up its independence to become the nucleus of another cell in exchange for that cell’s protection (an early version of the feudal contract)…and together they conquered the Earth. Alexander, Augustus, eat your hearts out! 

We life forms are stubborn. It took 2 billion years for two of us to decide that cohabitation and cooperation might be better than conflict. And now we’re surprised that there’s no peace in the Middle East? It was a high stakes gamble, a giant leap of faith, but boy did it ever pay off! We are indebted to these two risk-taking pioneers for everything from the majestic Redwood Forests to the teeming Gulf Stream Waters, for Mother Theresa, and for you. ‘Thank you’ doesn’t cover it.

There is a powerful Theory of Everything (TOE) that fundamentally depends on non-Archimedean topology. No, it’s not some esoteric version of String Theory: it does not require a cosmos with 10…or 26 dimensions. Nor is it obscure. In fact, every reader of this article is, happily or not, quite familiar with it. It’s called Christianity. 

Let’s just state it outright: Christian cosmology is fundamentally non-Archimedean! Write that 100 times on Sister Mary Martha’s blackboard. But don’t stop there! Christianity not only rejects AP; it also rejects the Transitive Property (if a > b and b > c then a > c) and the Fundamental Properties of Arithmetic (Commutative, Associative, and Distributive). 

Basically, Christianity repeals 5 years of primary school education. Or not! There is a real domain in which all these traditional properties hold but that ‘local’ domain is embedded in a much broader ‘universal’ domain where they do not. 

Arithmetic is a subset of mathematics. The Archimedean domain is a sub-region of the broader non-Archimedean Universe. Just as the Newtonian Universe is embedded in the Einsteinian Universe, so physics is embedded in theology. 

The defining doctrine of orthodox Christianity is Incarnation. The 2^nd person of the Blessed Trinity, the logos through whom ‘all things were made’ has in fact become one of those things. (John 1: 2 – 5) Contrary to the Foundation Axiom of Set Theory, the set is a subset of itself. God is the all-encompassing plenum and an irreducible quantum within that plenum.

The Son of God is God. According to the Nicene Creed, Jesus Christ is ‘true God (plenum) and true man (quantum)’. The whole is contained in one of its parts. Cosmos as Fun House! The reciprocal reflections in the Hall of Mirrors extend to infinity. The very concept of scale, so crucial to our understanding of the world, vanishes. The Cosmos is a fractal; no matter where you look it looks the same. It is a pattern endlessly embedded in itself.

Embedded, not conflated! The part does not become the whole, nor the whole the part. Call it the Pauline Exclusion Principle. According to Incarnation’s companion doctrine of Trinity, the plenum and the quantum remain distinct personae of the one Godhead. Perhaps the best description of this relationship is provided by Paul in his First Letter to Corinthians:

“When everything is subjected to him (Son), then the Son himself will be subjected to the one (Father) who subjected everything to him (Son) so that God may be all in all.” (15: 28)

‘All in all’, Anaxagoras’ pan in panti, describes the fundamental architecture of our non-Archimedean Universe. And it is also the Foundational Principal (Incarnation) of Christianity.  

It is no wonder that we are rapidly becoming unchurched. We have made a fetish of science. (Science is good but fetishizing anything is not.) Unfortunately, many Christians have responded to the fetish by rejecting the science. But science is not the antithesis of Christianity; it is a subset of it.

Once we have overcome the notion that the Universe is Archimedean throughout and properly understood the place of cosmology within metaphysics, we will suddenly find that science is not the enemy of religion but its most powerful engine. Truly, this is the Dawning of the Age of Aquarius. “Come Holy Ghost!”     

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Christ Pantocrator, 6th century, Saint Catherine’s Monastery, Sinai. The icon mirrors our theme by depicting Christ as both infinite plenum and finite quantum, true God and true man, embodying the non-Archimedean logic at the heart of Christianity.