For geometry not using numbers, he shows how graph theory can be applied to the problem of social balance as defined by psychologist Fritz Heider. This example is based on work by Dorwin Cartwright and Frank Harary.
For algebra not using numbers, he chooses the theory of group actions, and applies it to a way of preventing incestuous relationships that was used in some cultures, which works by assigning each child a group that they are exclusively allowed to marry in. This example is based on work by André Weil and Robert R. Bush.
For geometry using numbers, he uses an adjancency matrix to show how you can find out how many ways there are to send a message from one person to another in a network.
For algebra using numbers, he defines axioms for a distance function for rankings with ties, which can be shown to be unique (probably up to some isomorphy), and which can be used to derive a consensus ranking from a set of rankings. This appears to be the central piece of the article, as that is an example that he developed himself together with J.L. Snell and which was yet to be published.
Numbers are such a fundamental structure. I disagree with the premise that you can do mathematics without numbers. You can do some basic formal derivations, but you can’t go very far. You can’t even do purely geometric arguments without the concept of addition.
It’s not.
Mathematics is fundamentally about relations. Even numbers are just a type of relation (see Peano numbers).
It gives us a formal and well-studied way to find, describe, and reason about relation.
A logician / formalist would argue that mathematics is principally (entirely?) about proving derivations from axioms - theorems. A game of logic with finite strings of symbols drawn from a finite alphabet.
An intuitionist might argue that there is something more behind this, and we are describing some deeper truth with this symbolic logic.
My wife would have probably gone postal (angry-mad) if I had tried to form an improper relationship with her. It turns out that I needed a concept of woman, girlfriend and man, boyfriend and then navigate the complexities involved to invoke a wedding to turn the dis-joint sets of {woman} and {man} to form the set of {married couple}. It also turns out that a ring can invoke a wedding on its own but in many cases, it also requires way more complexity.
You might start off with much a simpler case, with an entity called a number. How you define that thing is up to you.
I might hazard that maths is about entities and relationships. If you don't have have a notion of "thingie" you can't make it "relate" to another "thingie"
It's turtles all the way down and cows are spherical.