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Rationale for Hermes’s use of the letter U for identity (projection) functions

Determine the historical or conceptual rationale for Hans Hermes’s choice of the letter U to denote identity functions (i.e., projection functions) in his formulation of primitive recursive functions, and clarify whether this notation was motivated by specific conventions or principles compared to the more common use of P for projection functions.

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Background

In the course of presenting the definition of primitive recursive functions, the paper notes a difference in notation between sources. Hermes (1965) refers to projection functions as “identity functions” and denotes them with the letter U, whereas other references (and the present paper) use the term “projection functions” with the letter P.

The author explicitly states uncertainty regarding why Hermes chose the letter U, while acknowledging that later authors followed Hermes’s convention. This unresolved question concerns historical or methodological reasoning behind a notation choice rather than a technical issue, but it is explicitly marked as unclear in the text.

References

In , the projection functions are called ``identity functions", and the letter $U$ is used. It is not clear to me why this letter was chosen, but I do note that seems to have followed suit by using the same terminology and the lower case $u$.

— A Characterization of Turing Machines that Compute Primitive Recursive Functions (2510.18283 - Schwartz, 21 Oct 2025) in Subsection “Primitive Recursive Functions” (Section 2), paragraph discussing notation differences