Natural logic is diagrammatic reasoning about mental models

Diagrammatic reasoning, based on visualization and analogy, is the foundation for reasoning in ordinary language and the most esoteric theories of mathematics. Long before they write a formal proof, mathematicians develop their ideas with diagrams, visualize novel patterns, and discover creative analogies. For over two millennia, Euclid’s diagrammatic methods set the standard for mathematical rigor. But the abstract algebra of the 19th century led many mathematicians to claim that all formal reasoning must be algebraic. Yet C. S. Peirce and George Polya recognized that Euclid’s diagrammatic reasoning is a better match to human thought patterns than the algebraic rules and notations. A combination of Peirce’s graph logic, Polya’s heuristics, and Euclid’s diagrams is a better candidate for a natural logic than any algebraic formalism. Psychologically, it supports Peirce’s claim that his existential graphs (EGs) provide "a moving picture of the action of the mind in thought." Logically, EGs have a formal mapping to and from the ISO standard for Common Logic. Computationally, algorithms for Cognitive Memory (CM) and virtual reality (VR) can support cross-modal analogies between language-like and image-like representations.