Any four real numbers are on all fours with analogy

TL;DR

This paper develops a mathematical foundation for analogy among real numbers using generalized means, showing that any four increasing positive real numbers form a unique analogy and can be reduced to an arithmetic analogy.

Contribution

It introduces a unifying framework for analogies based on generalized means, extending the concept to complex numbers and providing foundational insights.

Findings

Any four increasing positive real numbers form a unique analogy.

Analogies can be reduced to an arithmetic form.

Analogical equations have solutions in complex numbers.

Abstract

This work presents a formalization of analogy on numbers that relies on generalized means. It is motivated by recent advances in artificial intelligence and applications of machine learning, where the notion of analogy is used to infer results, create data and even as an assessment tool of object representations, or embeddings, that are basically collections of numbers (vectors, matrices, tensors). This extended analogy use asks for mathematical foundations and clear understanding of the notion of analogy between numbers. We propose a unifying view of analogies that relies on generalized means defined in terms of a power parameter. In particular, we show that any four increasing positive real numbers is an analogy in a unique suitable power. In addition, we show that any such analogy can be reduced to an equivalent arithmetic analogy and that any analogical equation has a solution for increasing numbers, which generalizes without restriction to complex numbers. These foundational results provide a better understanding of analogies in areas where representations are numerical.