Any four real numbers are on all fours with analogy
Yves Lepage, Miguel Couceiro
TL;DR
The paper introduces a unifying framework for numerical analogy based on generalized means $m_p$, showing that for any four increasing positive real numbers $a<b<c<d$ there exists a unique $p\in\mathbb{R}$ such that $m_p(a,d)=m_p(b,c)$. This connects classical arithmetic and geometric analogies as special cases (e.g., $p=1$ and $p=0$) and demonstrates that any $p$-analogy can be reduced to an equivalent arithmetic form, with robust results including existence and uniqueness of $p$ and solvability of analogical equations over $\mathbb{C}$. The authors develop reductions to unit interval form, canonical forms, and cross-power relations, and provide practical computation via dichotomic search, along with extensions to negative numbers, non-real powers, and Boolean cases. The work offers a mathematically grounded lens for analyzing numerical relationships relevant to embeddings, knowledge graphs, and downstream AI tasks, potentially informing representation learning and analogy-based reasoning. Overall, the framework yields a rigorous, flexible foundation for understanding and manipulating numerical analogies across real and complex domains.
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.