Thought Structures and their Morphisms

Structure-preserving maps (morphisms) are ubiquitous in mathematics. They are used as analogies and as tools for transforming problems into easier-to-solve formats. We argue that the usefulness of morphisms has a bigger scope. Mathemat- ics has unintentionally developed a grand theory of understanding and explanation since finding compatible operations across domains is a universal mechanism of any type of intelligence. Formally, morphisms are described by the equation

where · and ∗ are the two operations in the different domains. We can choose to transfer the inputs and carry out the computation by ∗ or compute with · first, and then transfer the result. If there is a morphic relation, then it does not matter where we perform the operations, as they are compatible.

An archetypical example is a cartographic map, but its geometric or topologi- cal features can be generalized. We define thought structures using directed graphs where an edge represents passing from one thought to another. The type of connec- tion does not matter much for this all-encompassing definition. It can be a random or guided association, a memory recall, or, in special cases, logical inference. Un- derstanding is then some (partial) morphism between thought structures or other systems. Thinking is compositional (as in category theory), and understanding is morphic (as in algebra). Compatible operations guarantee some useful similarity between the domains.

This argument is not entirely new. Any modeling relationship implies a morphic relation. Conceptual metaphors in cognitive linguistics are also based on the same mechanism. However, there are several misconceptions about the usefulness of morphisms: the seeming rigidity of the mathematical definition (the domains have to be fully defined) ruling out creativity, focusing on isomorphisms (1:1 scale maps) and not taking advantage of information reduction, and missing compositionality by using n-ary relations.

The morphic theory of understanding applies both to enhancing natural intel- ligence by deliberately creating and maintaining morphic relations for thinking, and to benchmarking artificial intelligence by looking for morphic representations. In this talk, we will demonstrate how the explicit formulation of morphisms im- prove explanations and describe these future applications. Further information and references can be found in the preprint at https://arxiv.org/abs/2411.06806.