Soft question: What if BODMAS/PEDMAS wasn't used. Would maths still "work"? I just watched this video (click here) on YouTube, where this guy tries to show that we take the rules of algebra for granted. He does this by showing examples of him using BODMAS backwards, and claims that it doesn't work. This completely messed up my mind.
The response to the video seem mixed, with mathematicians claiming that it proves nothing, while several maths teachers remarking that it was an amazing demonstration. I sort of got the gist of what he was saying, but is BODMAS the only system that can be used to solve equations, or is it sinply a form of notation? I have several questions:
1) Does this prove that BODMAS/PEDMAS is the only system which will help us solve equations in algebra? (Is the order of operations something discovered or agreed upon)
2) Or will changing the order of operations simply change the meaning of the mathematical statement, for example 11 in base 2 vs 11 in base 10? (In the video he says $2+2=22$, but would the + just carry the meaning of $* 10+$?
 A: Many other rules are possible.  The most important aspect is that the writer and the reader agree.  For example: on which side of the road should we drive?  The left or the right will work provided that we agree.  
As Arthur says, as long as brackets / parentheses are regarded as the highest priority, the other rules could be pretty much anything.  One possibility is that there are no other rules and the parentheses are mandatory.  In other words, $2 + 3 \times 4$ would not be a valid expression.  You would need to write either: $(2 + 3) \times 4$ or $2 + (3 \times 4)$. 
Or, you could devise a very different system which does not even need parentheses.  This has been done, see Polish Notation or Reverse Polish Notation which was once popular on calculators.  
Yet more schemes could be easily devised.  
However, most of us have agreed on one system so if you want to switch then you might be rather lonely and you will need to carefully warns others of your system.  
Is the accepted system the best?  That is hard to answer precisely and I think that most would agree that it is not perfect.  It does have some advantages, for example polynomials can be represented quite compactly which would not be the case in some other schemes.  
