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 ‘straight’ is just another kind of ‘curved’ and once smooth space resembles Alpine Lace Swiss Cheese. Needless to say, things did not go well there for poor Archimedes.

Years later when it came time to name the experimental school, they decided to call it, The non-Archimedean Academy…but not because Archimedes had (briefly) been a 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 an earlier TWS (5/06/2025) titled 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. If he answered correctly every question he could answer (within the limits of linearity), he would still end up with an F (37%). In an effort to mount a 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, Archimedes identified 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 could 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 the world takes on properties that would baffle even the most imaginative fiction writers, for example:

• All triangles are isosceles, and the two longest sides are always equal

• Every point in a sphere can be considered its center

• Any two spheres are either disjoint or one is contained completely within the other.

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’ and ‘cause & effect’, backbones of our disenchanted, secular logos, simply do not apply.

In such a world, all events are sui generis; 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:

• Event X may be a subset of either Set A or Set B or of neither Set A nor Set B.

• There is a Set C of which Event X is a subset.

• Event X cannot be in both A and B unless A is a subset of B and/or B is a subset of A.

‘And/or’ because the Foundation Axiom of standard ZPC Set Theory does not apply to these sets. A set can be a subset of itself, in which case the relationships among A, B, and C can be reciprocal, i.e. equalities. If X is in A, X can only be in B if B = A. Otherwise, X will be in A and C or in B and C or in C only.   

We are used to an Alpha-Omega time line, but here the Alpha event can also be the Omega event. The primordial event contains all other events including itself. A hierarchy of sets replaces linear spacetime as the organizational principle of this non-Archimedean universe. So…a tip of the hat for sure to this brilliant mathematician…but also a sigh of relief that we don’t live in his linear world.

Image: René Magritte. The False Mirror. Paris 1929