What's really wrong with 6÷2(1+2)? (Part (14÷7(5-3)+2)÷3)

In Part (14÷7(5-3)+2)÷3 (read it first if you haven't already), I introduced a problem: what does 6÷2(1+2) equal? The post ended with the rather strange, and seemingly exaggerated, assertion that "6÷2(1+2)" is not actually math at all. Here, in part (14÷7(5-3)+2)÷3, the mystery will be resolved!

This word is edible

Now, saying that "6÷2(1+2)" isn't actually math might come as a bit of a shock... until you realize that the word "pizza" isn't (usually) edible. That is, just as "pizza" is used to describe a food, even though the word itself isn't actually a food, "6÷2(1+2)" is just a way of communicating a mathematical idea - the symbols themselves aren't math. What we have here is a written language, a set of symbols that by themselves mean very little, but offer meaning based on how they're arranged, and on what we as a society have decided the arrangements should represent. In both English and this math language, you can make nonsense like "uesohfiwjaoiesjd" and "++4×-÷2=-12÷..×.4++3-", short pieces of information like "butter" or "23", longer expressions like "a stack of pancakes" or "94+2", or sentences that actually make a claim, like "Juice is dry" or "17-4=2." In English, you might have different conventions ("color" or "colour") or ambiguous situations ("I caught a butterfly with a net." "Wow, I've never seen a butterfly carrying a net before!") or statements that sound strange but are actually ok ("The horse raced around the barn fell"), so it's not surprising to see things like that happen in math, right? So is the problem just that we're using a written language, which will never be as precise as we want?

Nope. It is actually possible to produce a written language that's consistent and unambiguous; the problem is deeper than that. Here's the difference between "pizza" and "6÷2(1+2)" (besides the fact that one has tomato sauce, and the other is pizza). The word "pizza," in a sense, completely captures the nature of what it describes. Of course, I can add other description words if I want to narrow down my type of pizza of course, and I could give synonyms or definitions, or translate it into another language, if I wanted to say the same thing without actually using the word "pizza." But in the end, if you want to describe that flattened dough with toppings, you can't really get any closer than "pizza" (or the same word in some other language), because the word "pizza" was defined to mean exactly that. Because of its definition, it manages to communicate the essence of what it tries to describe. But I claim that the mathematical expressions we all know and love don't.

Before I can get to that though, I need to talk about another mathematical idea. Though students often don't learn about this until late middle school or even high school, it's actually a concept far more basic than things like adding and subtracting: the function. Simply put, a function is any rule that gives an output based on an input. So there's

If this picture makes sense to you,
you may skip the next couple paragraphs.

• a function that takes any number as input, and gives back its triple as output; 
• a function that takes any person as input, and gives the number of Facebook friends that person has as output; 
• a function that takes a word as input and gives its definition as output; 
• a function that takes a colour as input and gives its complementary colour as output; 
• a function that takes a word or phrase as input and gives the first Google search result of that word or phrase as output... 

as you can see, functions are really basic, really fundamental. Any time you want to connect two pieces of information, you can think of it as a function. The usual way to write down a function is to give its name, then the input in brackets afterwards. So if we call the Facebook friend function "FBfriends," then FBfriends(Jonathan Love) = 1176. If we call the tripling function "f," then f(4) = 12. Functions can also take in more than one piece of information; so for example, you can describe

• a function that takes two people and gives the number of mutual friends on Facebook;
• a function that takes two cities and gives the distance between them;
• a function that takes a person and a year, and gives the amount of time between that year and the year the person was born;
• a function that takes three mountains and gives the height of the tallest one;
• a function that takes twenty countries and gives the average population density of all of them...

As an example of one of these, if we call the function that measures distance between cities "Heretothere," then Heretothere(Moscow, New Delhi) = 4,348 km. Another thing you can do with functions is put the result of one function inside another. For example, I could take the distance from Moscow to New Delhi (Heretothere(Moscow, New Delhi) = 4,348 km) and triple it (f(4,348 km) = 13,044 km). This can also be written as  f(Heretothere(Moscow, New Delhi)) = 13,044 km.

It turns out that all the operations you remember from elementary school are actually functions. Given two numbers, you can add them (we'll call this function "add"), subtract them ("sub"), multiply ("mult"), or divide ("div")*. So for example, add(2, 3) = 5, div(24, 3) = 8, sub(19, 2) = 17. And there are others like powers and roots but I won't get into those for now.

*For various reasons, it actually makes more sense a lot of the time to use a different set of functions. Keep the adding and multiplying, but then have two functions that only take one input: an additive inverse function "ainv" and a multiplicative inverse function "minv," so that ainv(x) = -x, and minv(x) = 1/x. But that's not necessary to my point, so I'll stick with what most people are used to for now.

Now, so far, all we have is a different way of writing the same thing: add(2, 3) and 2+3 have the exact same meaning, as do sub(2, 5) and 2-5, or mult(6, 4) and 6×4. And it seems like all I've done is make things longer and more complicated. But let's see what happens if we go a bit further...

Sorry man, can't squeeze in another input.

How would you write 12+33+72 using functions? The natural reaction might be to just write it as add(12, 33, 72). But wait! The function "add" only takes two inputs, you can't just shove another one in there. If that sounds like just a stupid rule, how would you explain sub(10, 4, 3)? What gets subtracted from what? Or even more confusing, What would Heretothere(Moscow, New Delhi, Paris) be? How would you try to explain the "distance between three cities?" There are ways you could make it work - the shortest path connecting all three, or adding up each of the individual distances, for example - but then you're making a new function. The fact is, if you have a function that takes two inputs, it must take two inputs.

If you look again at 12+33+72, there are two + signs, so you're actually adding twice. Add two of the numbers, and then add the result to the third number. But you'll notice that there are two ways to do that - you can add 12 and 33 first to get 45, and add 72 to that, OR add 33 and 72 to get 105, and add 12 to that. Or, writing out the functions, you can have either add(add(12, 33), 72), or add(12, add(33, 72)), but you have to choose one or the other. So which is it? Find the answer in Part (14÷7(5-3)+2)÷3.