Not all elementary functions can be expressed with exp-minus-log

Scope of “elementary functions”

• Major point of contention is the definition of “elementary functions.”
• One camp uses the narrow, analysis-style list: polynomials, exp/log, trig/inverses, arithmetic, composition.
• Another uses the broader, Liouville/differential-algebra notion: includes all algebraic functions and roots of arbitrary polynomials (e.g., Bring radical).
• Several commenters note that in modern algebra/Computer Algebra contexts, “elementary” typically includes solutions of polynomial equations; others find this usage unintuitive or “non‑elementary.”

Quintic equations and EML’s limits

• The blog post argues: functions generated by exp-minus-log (EML) form a class whose monodromy groups are solvable.
• Quintic root–solving in general has non‑solvable Galois/monodromy groups, so a “largest real root of a general quintic” function cannot be expressed via EML.
• This is positioned as different from the classic Abel–Ruffini theorem, though related in spirit.
• Some view the blog’s criticism as mainly about terminology: if “elementary” includes arbitrary polynomial roots, EML is not universal for elementary functions.

Analogy to NAND and universality

• Original excitement framed EML as a continuous analogue of NAND / Toffoli / Fredkin: a single primitive generating “all” elementary functions.
• Critics say this analogy breaks:
  • NAND is complete for discrete Boolean functions.
  • EML is not complete for the broader “elementary” class that includes polynomial root functions.
• Debate over whether invoking unsolvable quintics here is akin to criticizing NAND for not solving the halting problem; some say that’s a poor analogy because polynomial roots are efficiently approximable.

Decidability and expressiveness

• Discussion on whether equality of two EML expressions is decidable.
• Some claim undecidability by analogy with known theorems (Richardson, Laczkovich); others note the details don’t straightforwardly carry over and that this remains unclear.
• Contrast drawn with NAND circuits, where equivalence of finite Boolean circuits is decidable.

Perceived significance and hype

• Several commenters see both the original EML result and the debunking blog as mathematically modest, even undergrad-level, but interesting as recreational math and for search techniques.
• Others criticize online reactions as overhyped, with claims about “up‑ending mathematics” seen as unwarranted.
• Some still value the work for providing concrete search machinery for single-operator representations and for exploring non-uniqueness of such operators.