Why do we have the present order of operations, and how do hyperoperations fit in?

Something that's been bugging me for a fairly decent while is the order of operations - not so much using it, however, as to understanding where it comes from.

Typically we're introduced to it in the "PEMDAS" order in school (or "BEDMAS" or whatever acronym you were given):

• First: Parenthetical/bracketed expressions
• Second: Exponentiation (and its inverse, roots)
• Third: Multiplication (and its inverse, division)
• Fourth: Addition (and its inverse, subtraction)

...all typically done in a left-to-right order. But this seems arbitrary in a sense - why this particular ordering of steps, as opposed to any other particular ordering? Why should I calculate $$4+5\times 6$$ by finding $$5\times 6$$ first, as opposed to $$4+5$$ first?

Obviously, we adopted a standard in order to prevent ambiguity - but then, why this standard?

I could not find anything on the matter that felt really satisfying. For example, a lot of the previous discussion on MSE ultimately boiled down a few points:

• It was agreed upon by mathematicians. (Okay, but why did they agree on this?)
• It makes for more efficient notation once everyone is on the same page. (Yes, but would that not also hold for any ordering?)
• Furthering the previous, it leads to more efficient notation for polynomials. (But as noted in the comments, Polish/reverse Polish notation also can resolve a lot of that.)
• Our end goal is an order of operations that drops a lot of the parentheses, for readability's sake. (Understandable but then why is this convention the most efficient in that regard?)

However, recalling a video by the Taylor series on hyperoperations, I did notice a bit of a pattern (which I later found touched on in opening of the Wikipedia article on hyperoperations).

Consider characterizing each of our operations by what they basically are: exponentiation is repeated multiplication, which is repeated addition, which is basically repeated application of the successor function. So we can sort of call our operations this:

• Operation $$0$$ - Successor function (repeated addition of $$1$$ in effect)
• Operation $$1$$ - Addition (repeated succession - repetitions of operation $$0$$)
• Operation $$2$$ - Multiplication (repeated addition - repetitions of operation $$1$$)
• Operation $$3$$ - Exponentiation (repeated multiplication - repetitions of operation $$2$$)

The pattern becomes evident if we adopt this scheme:

• Operation $$n$$ is just repeated applications of operation $$n-1$$
• Equivalently, repeated application of operation $$n$$ yields operation $$n+1$$

So we ask - what is operation $$4$$? Of course, in this scheme, it's repeated exponentiation - that is to say, tetration! Rather fitting since "tetra-" is the prefix for $$4$$, but I digress.

Of course, then, this sequence of hyperoperations also uses pentation, hexation, and so on, as repeated applications of the previous.

All of this, then, begs a couple of questions:

• Is this "sequence of operations" the reason we define the order of operations as we do - that is, we do exponentiation before multiplication, because the former is a repetition of the latter? Or is it just a "happy consequence" of whatever the reason is?

• Where do these hyperoperations, e.g. tetration and pentation, fit into the scheme? If the previous is true, then this suggests that we would handle all bracketed expressions, then the highest $$n$$-ations, then the $$(n-1)$$-ations, and so on - that is, we would handle hexation, then pentation, then tetration, then exponentiation, and so on.

• I don't know the history, but it's also worth noting that $(a\cdot b)+c$ is not "immediately reexpressible" but $a\cdot (b+c)$ is. That might be part of it - the idea may be that "$a\cdot b + c$" should be "maximally simple" in some sense. – Noah Schweber Feb 28 at 1:25
• You are asking a question about the history of mathematics, so you should really do some research into that history for yourself before asking for mathematical explanations for historical decisions. Here is a fun link to start from blogs.scientificamerican.com/roots-of-unity/… which demonstrates how before algebra a univariate polynomial was read out as a sum of monomials (implying a bit of BODMAS, particularly when you typeset $1 + x^2$ as $1 + xx$ as was done in days gone by to reduce the cost of typesetting). – Rob Arthan Feb 28 at 1:54
• Read A History of Mathematical Notations, by Florian Cajori for answers to most of your questions. – Somos Feb 28 at 3:00
• You're right that the successor operation is exactly the same operation as addition of 1. However, whether you define addition in terms of the successor operation or in terms of addition of 1 matters. If you say that by definition, $\forall x \in \mathbb{N}\forall y \in \mathbb{N}x + (y + 1) = (x + y) + 1$ and $x + 0 = x$, then the definition of addition is circular. What is 0 + 1? It's (0 + 0) + 1. Now how do you compute what (0 + 0) + 1 is? We see that 0 + 0 = 0 so (0 + 0) + 1 = 0 + 1 which in turn by definition is (0 + 0) + 1 so you never really defined addition. If you instead define – Timothy Sep 17 at 0:09
• addition in terms of the successor operation, that resolves the problem. It can then be shown that the successor operation is equivalent to addition of 1. – Timothy Sep 17 at 0:13