What do people mean when they say they don't believe in the rules of math? For example, I recently received a message on Reddit from someone who specifically stated he doesn't believe in the order of operations, and that it is some conspiracy and that I shouldn't teach it at all. I don't understand what belief has to do with it? Isn't arithmetic objective? Isn't all applied mathematics objective? Can someone just dismiss a fundamental rule of math? I am so confused now. How on earth would anything ever work if there were no rules to remove ambiguity from mathematical expressions? Help? (Yes - I looked at MANY posts about order of operations. I haven't seen anybody or any question/answer address this notion of "believing" in math as though it were a matter of opinion.) Confused and frustrated. Please help.
 A: Order of operations is more a question about notation and communication.
In formal mathematics, you might require parentheses everywhere, so that you never write $1+2\cdot 3,$ or even $1+2+3,$ but rather $1+(2\cdot 3)$ and $(1+2)+3$.
The reason we decide to add order of operations to our informal notation is to make our mathematics cleaner and easier to read. You end up with a lot of parentheses if you don't have an unambiguous order of operations. We pick a standard order of operations, rather than everybody picking their own order, because the goal is clearer communication.
In particular, since order of operations is merely a question of language, it's a bit like a person saying, "I don't believe in English," or "I don't believe in Arabic numerals." Order of operations is a standard to make communication easier and cleaner. 
We've picked an order of operations. It is neither true nor false, it is just either "this is how we are talking," or "this is not how we are talking."
There have been programming languages, such as APL, that use the left-to-right order of operations. So then $1+2\cdot 3=(1+2)\cdot 3$. 
And then there are languages where we don't use "infix" notation at all - where we write $+(1,\cdot(2,3))$ for $1+(2\cdot 3)$.
A: The order of operations, and other things, are not provable, but they are set and agreed upon conventions made by mathematicians. These such conventions avoid confusion in certain situations. 
Let's say you have the expression $3\cdot3+4$. If the order of operations didn't exist, Person A could evaluate the expression as $3\cdot3+4=9+4=13$, while Person B could essentially write $3\cdot3+4=3\cdot7=21$. Obviously, $13\neq 21$, so one expression can't have two values. So mathematicians decided on the order in which multiple operations are carried out (the familiar PEMDAS).
If someone wanted to rebel against a rule of math, they could, but everyone else would be following the rules, and he/she will be the only person who wants to dismiss a rule. So math rules are subjective, because mathematicians decided on them. So too bad for the guy who sent the message.
However, note that some rules are just obvious that they exist (like the Commutative Property of Multiplication). These rules are rigid, and could not be broken in the real number system. (See comment below)
(If they had organized the order of operations differently, we would be doing things a different way! Imagine doing addition before exponentation!)
A: A conspiracy "theorist" is essentially paranoid and delusional and cannot be reasoned with. Anything you say is dismissed as propaganda from the world-wide conspiracy against him, while he feels his importance in having discovered a great truth. He has the advantage, and the security, of never being wrong about anything. But he also has a constant resentment about the lack of acknowledgement, from others, of his importance. 
Recommended: A Budget Of Trisectors by Underwood Dudley..... I would call this book the tale of a mathematician's close encounters with cranks of the mathematical kind. 
