In early 1927, Paul Dirac and Pascual Jordan, independently of one another, published their versions of a general formalism tying the various forms of the new quantum theory together and giving the theory's statistical interpretation in full generality. This formalism has come to be known as the Dirac-Jordan (statistical) transformation theory. A few months later, in response to these publications by Dirac and Jordan, John von Neumann published his Hilbert space formalism for quantum mechanics. The relation between the two formalisms can be captured in terms of a metaphor of arches and scaffolds that I have argued fits a number of instances of theory change in physics. What is unclear in this case is whether the story is best told with Hilbert space playing the role of the arch built on transformation theory as a scaffold to be dismantled once the arch could support itself, or with transformation theory playing the role of the arch and Hilbert space providing the scaffold built to prevent the arch from collapsing under the weight of its serious mathematical deficiencies. Either way, a narrative for this episode in the history of quantum mechanics based on the arches-and-scaffolds metaphor illustrates the promise of borrowing ideas from the approach to evolutionary biology known as evodevo for reconstructing genealogies of theories rather than species.