Order of operations paradox

There's a controversy that sweeps through social media, causing mathematicians, order-of-operations-enthusiasts and the average Jane or Joe to quarrel until they're red in the face. It's the question what is 6 / 2 * 3? This seemingly simple question is best described as bait for discussion, which if you're trying to reach a mass-audience on social-media, is the desired result of any public group. Facebook rates a post based on the number of comments and likes, without paying attention to the quality of the post itself. This results in the most controversial topics getting the most attention, which are sometimes far from being the most worthy of discussion.

So why is 6 / 2 * 3 a difficult question to answer? The short answer is because there is no single right answer. Anyone who answers 9 or 1 will be strongly opposed by someone arguing for the opposing number, so when everybody is accused of being wrong, you'll inevitably have people attempting to muscle their way into being right. Some people will quote the widely known order-of-operations and their subsequent acronyms BEDMAS, BODMAS or PEDMAS, and may come to the conclusion that division MUST be done before multiplication. So does imposing this rule on the problem give a definitive answer to the question? Not at all. For instance, just as imposing the rule 'I before E except after C' doesn't explain why the word 'their' doesn't obey it, it seems 6 / 2 * 3 refuses to play by the rules too. The simple answer is either/or, which is usually the answer people least want to hear. So how can opposing answers be correct in mathematics? Doesn't maths exist to provide definite solutions?

Look at it another way: division IS multiplication. Dividing by a number greater than 1 has the same effect as multiplying by a number less than 1 (and greater than -1). In fact, we don't need division at all, we can simply multiply numbers by fractions or decimal numbers and arrive at the same destination. Division is redundant in mathematics, it's simply a way of being more descriptive with multiplication and therefore making the concept is easier to understand. So how can we possibly give first preference to division or multiplication when division IS multiplication?

Surely if everyone followed the rules of order-of-operations, we can all get along... right? What if I told you that in some circumstances we want to multiply first? You might ask me about the circumstances. This is the core of the problem, mathematics exists to solve real-world problems, to be used in specific circumstances and not just arbitrarily solved. Mathematics without a purpose is like a language without any speakers, why would we need the language if nobody is there to speak it?

Let's assign some purpose to the calculation 6 / 2 * 3. Let's make it mean something! Say we have a farmer, he has a total of 6 cows and he must distribute them into trucks that each have space for 2 rows of cows by 3 columns of cows, how many trucks does he need? Let's write the problem simply: 6 cows spread across 2 rows by 3 columns, or 6 / 2 * 3.

To begin we multiply the number of rows by columns. This tells us how many cows each truck can hold. 2 * 3 = 6. Each truck holds 6 cows.

Now we divide the total number of cows by the number of cows each truck can hold. 6 / 6 = 1. The farmer requires a total of 1 truck.

Now alternatively let's try it using the order-of-operations method. If we divide the number of cows by the number of rows and then multiplying it by the number of columns, we arrive at the answer 9... a fair number of trucks for transporting 6 cows, seems like a bit of a miscalculation. So we can't fall back on the idea that order-of-operations can solve every problem. We can compromise by providing brackets, eg. 6 / (2 * 3), but the underlying point is that maths is a language to explain real-world concepts in simple terms. Without context, the method of deriving the answer can be ambiguous.

We can reach clarity by asking ourselves what is the desired result. Do we want to know how many cows fit in a truck (2 * 3 = 6) and then how many trucks can fit 6 cows (6 / 6 = 1)? Or do we want to organise 6 cows into 2 rows (6 / 2 = 3), each row contains 3 cows, and then give each row of cows a full column to move around in (3 * 3 = 9), we'd need 9 columns for that, and because each truck only has 3 columns, we'd need 3 trucks. It is therefore obvious that math's job is to serve the situation. Should we remove the situation, math is free to wander in any number of directions, like cows in an open field.

By looking at the desired results rather than the pre-existing rules, we can reach clarity on what we hope to achieve. Take a tradition for example, a tradition can be controversial. Many people may have ideas on how a tradition is done properly. The debate can rage on endlessly, both sides present their understanding of the rules, or the order-of-operations of this tradition. But why involve ourselves in this debate over rules when the answer can be determined by the desired result. In controversy, the rules are often shrouded in doubt, but by using the desired result as a starting point and working backwards we can understand whether the rules are worthy of following. While debating the rules can be done with good intentions, we can stand firmly on the foundation of results.