The making of a mathematical notion by analogy: the case of Hamiltonicity of (locally finite) infinite graphs

This article presents a toy model and a case study on how a mathematical notion gains acceptance over competing alternatives. We argue that the main criteria is success, in the sense of: (a) the new notion fitting the “mathematical landscape” and (b) it empowering mathematicians to prove publishable results in the modern academic landscape. Both criteria go hand in hand. We identify one particular way that both things can be established, namely by creating counterparts of existing structures in another area of mathematics, i.e. by making sure that analogical results hold. Unlike in previous accounts of analogical reasoning, we hold that, sometimes, this process involves intentional creation of parallelisms between domains rather than mere discovery. We show this by discussing the case of Hamiltonicity results for infinite graphs. We argue that a prominent aim of this new notion is to shape the target domain so that knowledge can be transferred from the source domain. In our case study this notion enables knowledge transfer from finite combinatorics to infinite combinatorics in graph theory. We study how the first suggested notion for the counterpart of cycle, namely the notion of the double ray, was replaced by a topologically motivated approach to better fit the general mathematical landscape and thus aiding with knowledge transfer across fields.