A recent mathematical paper claiming that a single operator can express all elementary functions is facing technical scrutiny. The critique, published by Robert Smith on www.stylewarning.com, argues that the 'exp-minus-log' function lacks the power to represent certain fundamental mathematical structures.
The debate centers on a paper by Andrzej Odrzywołek titled 'All Elementary Functions from a Single Operator.' The original work has gained traction online, with some observers calling the findings a 'breakthrough' or 'groundbreaking.' Odrzywołek suggests that the function E(x,y) := exp(x) - log(y), combined with basic constants, can construct everything from addition to pi.
However, Smith contends that Odrzywołek's definition of 'elementary' is too narrow. According to the report from www.stylewarning.com, Odrzywołek limits his scope to 36 specific symbols and ignores the broader mathematical definition used since the 19th century.
The limits of EML terms
In standard mathematics, elementary functions include arbitrary polynomial roots. Smith argues that 'EML terms'—expressions built from the exp-minus-log operator—cannot express these roots.
Using Khovanskii’s topological Galois theory, Smith demonstrates that the monodromy group of functions built from these terms remains solvable. This prevents them from reaching the complexity of the full class of elementary functions.
'The titular result does not hold in this setting,' Smith writes, noting that the EML function cannot serve as a continuous analog to universal logic gates like the NAND gate.
While Smith acknowledges the Odrzywołek paper is 'neat and thought-provoking,' he maintains that the claim of universality is mathematically inaccurate under standard definitions. The critique suggests that while the operator is expressive, it remains a subset of the broader mathematical landscape.
