What’s the Matter with Archimedes?

It’s his first day in his new middle school and already he’s in the nurse’s office complaining of vertigo. “I feel like I’m being sucked into a black hole,” he explains with his usual touch of melodrama.

Turns out, our Archimedes is a bit OCD; specifically, he has ‘a thing’ about lines, straight lines: he’s not at home without them! His frustrated parents decided it was time to try some immersion therapy, so they sent him to a school where the word ‘straight’ was only used in the context of sexual orientation. Everything else was just ‘curved’ and space, once smooth, resembled Alpine Lace Swiss Cheese. Needless to say, things did not go well there for poor Archimedes.

Years later when it came time to rename the experimental school, they decided to call it The non-Archimedean Academy…but not because Archimedes had (briefly) been an ‘unsuccessful’ student there! Yes, the local adventures of the ‘linear lad’, as they called him, were still fresh on everyone’s lips, but the real story behind the school’s naming is even more bizarre.

Read on, McDuff: In another article on this site, Square Pegs in Round Holes, we used an analogy from solid geometry to demonstrate how much the fetish of linearity restricts and distorts our access to information about the world we live in. Reasoning from our model, we estimated that our refusal to color outside the ‘box’ (a straight edged cube) is blinding us to c. 63% of what we otherwise might know. 

Young Archimedes never had a chance at his new, ‘well rounded’ school. If he answered correctly every question he could answer (within the limits of linearity), he would still end up with an F (37%).

In his defense, Archimedes pointed out, correctly, that Pythagoras (c. 500 BCE), Plato (c. 375 BCE), and Euclid (c. 300 BCE) suffered from the exact same Learning Disability; but to no avail.  Probably as a response to this childhood trauma, Archimedes decided to devote himself to mathematics…turns out, this straight F student was brilliant at it! Who knew? (But then, Thomas Aquinas’ teachers called him a ‘dumb ox’.) 

Among his many claims to fame, the term ‘Archimedean’ came to define a property common to many, but not all, algebras and geometries; it has since been named for him: The Archimedean Property.

The Archimedean Property links geometry to the set of Real Numbers (extended to include Complex Numbers). Systems that incorporate non-reals, like surreal and p-adic numbers, do not conform to The Archimedean Property. (Surreal numbers include infinite and infinitesimal quantities.)

In Square Pegs (above), we explored important differences between Euclidean and non-Euclidean geometries…but both are consistent with The Archimedean Property. We imagined a hypothetical hike from Boston to Detroit and we pointed out why you might want The Archimedean Property to apply: Why run the risk of suddenly rematerializing in another place, at another time, or even outside of spacetime altogether. The Archimedean Property ensured that your bee line from Boston to Detroit did not take you through Memphis…unless your GPS was on the fritz. 

Without The Archimedean Property, your journey might have been even more eventful and a pop-up appearance on Beale St. could not be ruled out…but the Peabody (hotel) ducks are the least of your problems. Without The Archimedean Property, geometry loses its intuitiveness. The world takes on properties that would baffle even the most imaginative science fiction writers, for example:

➢ All triangles are isosceles, and the two longest sides are always equal  ➢ Every point in a sphere is its center 

➢ Any two spheres are either disjoint or one is contained completely within the other: o There is no touching: you touch me, I eat you…or you eat me. Eat or be eaten…or keep your hands to yourself! ➢ If B and C are disjoint subsets of A, the volume of (B + C) can be greater than the volume of A. In fact… 

o The volume of either B or C may be greater than A

o The volume of both B and C may be greater than A o (B + C) may be infinitely greater than A

o B and C may have ‘real number’ volumes while A’s volume may be infinitesimal.

If this is the first time you’ve encountered these strange propositions, you’re probably feeling like I did the morning I woke up on Europa, freezing, gasping for breath, and generally disoriented. You’ve entered a world in which ‘space & time, cause & effect’, backbones of our disenchanted, secular logos, simply do not apply.

In such a world, all events are sui generis and causa sui. No event causes any other event…or impacts it…or even influences it. However, events can be embedded in other events in every imaginable configuration resulting in Gaudi worthy architecture, subject to the following ‘engineering’ limitation: 

If B and C are sets in a Universe U, then there is a Set A of which both B and C are subsets. B may be a subset of C or C may be a subset of B or B and C may be disjoint. It is this final scenario – B and C are disjoint subsets of A - that produces mind bogglingly unexpected consequences (above).  

The Axiom of Foundation (AF) from standard ZPC Set Theory does not apply to these sets. Bertrand Russell notwithstanding, any set can be an element of itself, in which case the relationship between ‘the set as whole’ and ‘the set as part’ is recursive.  Whatever is done to the whole is done to the part and whatever is done to the part is done to the whole.

In this scenario, Set A has 4 members: Set B, Set C, Set A itself, and ø. In addition, there may also be a region (X) of A that lies outside both B and C. It is not itself a set, but it has a quantitative value that influences how and how much A differs from (B + C).   We are used to thinking of events in terms of a directional timeline running from Event Alpha to Event Omega, but here the Alpha event can also be the Omega event. The timeline is a curve, or better yet, a Mobius Strip. Every snake bites its own tail. The primordial event contains all other events including itself.

A potentially infinite hierarchy of embedded sets replaces flat and finite spacetime as the organizational principle of the non-Archimedean universe.  The author (John of Patmos) of the Book of Revelation (aka the Apocalypse) captured this relationship millennia earlier: “I am the Alpha and the Omega - says the Lord God - the one who is, and who was, and who is to come…the one who lives…the first and the last, the beginning and the end.” (Chapters 1 and 22, ‘the beginning and the end’)

As noted above, every moment in history is ‘its center’, as is every point in space. You are the center of the World…as am I. So…a tip of the hat to Archimedes, a brilliant mathematician…but also a sigh of relief that we don’t live in his boring, straight-edged world. 

Image: René Magritte. The False Mirror. Paris 1929