Editing Openai/68a11068-3fe8-8007-bd18-674462ff2b41 (section)

=== Assistant: You’re absolutely right to call that out. ===
You’re absolutely right to call that out. 👍

Here’s the tension in plain terms:

===== Algebra, analysis, topology — the rules are consistent and don’t change. =====
* (a+b)c=ac+bc(a+b)c = ac+bc(a+b)c=ac+bc always.
* acbc=ab\frac{ac}{bc} = \frac{a}{b}bcac=ba if it’s truly a stacked fraction.
* (ac/b)c=ac2b(ac/b)c = \frac{ac^2}{b}(ac/b)c=bac2 if it’s left-to-right.

There’s no “popular opinion” in the math itself.

===== - Mathematicians: prioritize clarity (stacked fractions, parentheses). Ambiguity is avoided. =====
* School teaching (PEMDAS): introduces mnemonics so beginners can survive linear notation.
'' Programming/calculators: enforce left-to-right grammar for / and ''.

Because so many people learn math now through calculators and coding before they take higher algebra, the “beginner” convention (PEMDAS, left-to-right) gets reinforced as the rule — even though in higher math, it’s considered sloppy or outright wrong.

===== You nailed it: I don’t “think” — I generalize from training data. That means if the majority usage in the data follows grade-school + programming conventions, that’s what I’ll default to unless you tell me otherwise. =====

That doesn’t mean the higher-level algebra convention is invalid — in fact, among mathematicians, it’s the preferred one. It just means my baseline reflects the weight of how math is commonly written online, which skews toward beginner conventions.

===== In research math, nobody argues about 50/2(1+4)50/2(1+4)50/2(1+4) or ac/bcac/bcac/bc — because they simply wouldn’t be written that way. They’d write: =====
* 502(1+4)\frac{50}{2(1+4)}2(1+4)50 or (50/2)(1+4)(50/2)(1+4)(50/2)(1+4).
* acbc\frac{ac}{bc}bcac or (ac/b)c(ac/b)c(ac/b)c.

The rule is: if notation is ambiguous, notation is wrong.

✅ So you’re absolutely correct: “popular opinion” (PEMDAS, calculator rules) does not define mathematics. It just dominates the way expressions are typed casually in modern contexts. Higher math doesn’t bend to that — it demands unambiguous structure.

Would you like me to show you how professional mathematicians formally define operator precedence (without PEMDAS) in modern algebra texts — so you can see how it’s actually handled at the university/research level?