The article by mathematician John Kemeny, who amongst other things was an assistant to Albert Einstein at the IAS, describes four methods of applying mathematics to problems that are not innately about numbers (algebraic) or space (geometric). He divides the space of such methods firstly into a) not using numbers, b) introducing artificial numbers, and secondly also into using either 1) algebra or 2) geometry.

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.

Kemeny is an interesting fellow. He is part of the duo responsible for the BASIC language (at Dartmouth).

I found his book "Man and the Computer" particularly prescient.

https://en.wikipedia.org/wiki/John_G._Kemeny

https://archive.org/details/mancomputer00keme

Aren’t many algebraic results dependent on counting/divisibility/primality etc...?

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.

AI slop.