
TL;DR
This paper develops an algebraic framework for understanding analogical proportions, enabling comparison of mathematical objects across domains, with implications for AI reasoning, learning, and creativity.
Contribution
It introduces a first-principles algebraic model of analogical proportions, embedding it into first-order logic and analyzing its mathematical properties.
Findings
Analogical proportions can be modeled using universal algebra.
The model is compatible with structure-preserving mappings.
It provides a foundation for AI reasoning and learning systems.
Abstract
Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, common sense reasoning, learning, language acquisition, and story telling. This paper introduces from first principles an abstract algebraic framework of analogical proportions of the form ` is to what is to ' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. It turns out that our notion of analogical proportions has appealing mathematical properties. As we construct our model from first principles using only elementary concepts of universal algebra, and since our model questions some basic properties of analogical proportions presupposed in the literature, to…
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Taxonomy
TopicsHistory and Theory of Mathematics
