Polish notation was indeed published in 1920. However, it relies on knowing the arity of each symbol. Put another way, it relies on symbols *having* definite arities. Thus if f has 2 arguments, it has 2, not 1. This does not suit languages like Haskell at all well. Worse, as usually presented, it doesn't handle higher order functions. Suppose we wrote flip - 3 4 Polish notation doesn't really have the notion of a subexpression yielding an operator. One could patch Polish notation by assigning each symbol a rank as well as an arity, but I'll leave the details out because (a) I'm not sure it works very well, and (b) I've never found Polish notation (forward or reverse) to be even close to readable. Apparently the first known use of the parenthesis symbols () in England is in 1494. Presumably mainland Europe was using them before, but not all that long before. This was for asides in normal text. According to http://jeff560.tripod.com/mathsym.html the first uses of various kinds of brackets in mathematics go back to the mid 16th century. We haven't really had operator precedence for more than a few hundred years, or operators, come to that. So yes, it _is_ true "that people a couple of hundreds of years ago [could] manage to express [mathematics] without any parenthes[e]s at all". What clever notation did they use? Words. Lots of words. Sometimes heavily abbreviated words. Parentheses are *better*.