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A flurry of discussion around Andrzej Odrzywolek’s “mathematical minimalism” work is reviving an old question: what counts as an elementary function? Odrzywolek’s arXiv note claims a single primitive—an “exp-minus-log” style function (eml/elm) plus the constant 1—can bootstrap basic arithmetic, derive constants like c0, and build familiar functions including roots and trigonometric/hyperbolic forms. On Hacker News, readers push back that such claims depend on a restrictive definition: in classical differential-algebraic settings, adjoining algebraic roots changes what’s “elementary.” Critics note EML-only constructions are tied to solvable groups, echoing results related to quintic unsolvability.
Mark Carney’s new arXiv paper (2605.01636), submitted May 2, 2026, proves an inexpressibility result for Exp-Minus-Log (EML), a minimalist formal system introduced by Odrzywołek. EML reduces elementary complex-number functions to a constant 1 and a single binary operator E(α,β)=exp(α)−log(β), and is described as equivalent to Chow’s EL numbers. Carney shows that every number expressible in EML is computable, linking the system’s expressive power directly to computability theory. He then proves that Chaitin’s Ω_U, a canonical non-computable real, cannot be expressed in EML. The 5-page result formalizes a clear boundary on what EML can represent, relevant to logic and theoretical computer science.
A Hacker News thread discusses a blog post arguing that not every elementary function can be built from the operations “exp-minus-log” (EML). Commenters highlight that the claim hinges on the chosen definition of “elementary”: the blog uses a restrictive definition, while classical differential-algebraic settings admit algebraic adjunctions (e.g., adding polynomial roots). Several users note the core point echoes known results—Odrzywolek’s observation and Arnold’s proof related to unsolvability of the general quintic—and that EML-based constructions yield a solvable group, so they cannot represent arbitrary polynomial roots. The debate centers on terminology, whether the result is new, and the scope of claimed implications.
Mathematician Andrzej Odrzywolek posted an arXiv paper arguing that a minimal basis can generate all elementary functions using only a single function, referred to as “eml,” plus the constant 1. The article highlights equations from the paper’s supplement that show how to “bootstrap” core arithmetic operations—addition, subtraction, multiplication, and division—from eml. The paper and supplement reportedly extend the construction to derive mathematical constants such as π and to build functions including square, square root, and standard circular and hyperbolic functions. The work matters for foundational mathematics and for designing minimal computational or symbolic math libraries, suggesting that a very small set of primitives may be sufficient to express a broad range of standard functions. No publication date or further technical details are provided in the excerpt.
A recent arXiv note by Andrzej Odrzywolek demonstrates that the entirety of elementary functions can be constructed from a single function (elm) and the constant 1. The paper and its supplement provide explicit equations showing how to derive addition, subtraction, multiplication, and division from elm, and explain how to recover constants (like π) and common functions (squares, square roots, circular and hyperbolic functions). This minimalist construction is notable for its theoretical elegance: it reduces the set of primitives needed to generate standard mathematical operations, with implications for formal systems, minimal function algebras, and potential applications in symbolic computation or compact function representations. The work connects to prior ideas on bootstrapping small libraries and minimal gate sets.