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.
I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers.
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.
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.