On one hand, we simply decided to use it that way. It is just a convention we agreed upon.

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On the other hand, when people were deciding on the convention, surely they had *some* reason, right?

I think they did, and that there is some underlying logic to it.

The mathematical operations have some sort of inherent order to them, basically about how 'strong' (in a sense) they are.

• You may have heard that multiplication is repeated addition. (e.g. 3x4 = 3+3+3+3 by definition).
• Exponentiation (taking a 'power') is repeated multiplication. (e.g. 3^4 is 3x3x3x3 by definition).
• So, in a sense, by being repeated versions of another operation, Exponents are stronger than Multiplication, and Multiplication is stronger than Addition (and Addition is stronger than Counting, for what it is worth).

We don't have to do operations in the order of most 'strong' to least 'strong', but it feels natural to do so, and so we formally choose and decide to do so.

So do exponents (aka powers) first, then multiplication, then addition.

But we've forgotten some operations, so let's add them in. Division is just as 'strong' as multiplication, because is the opposite, and can therefore undo exactly what multiplication causes. Subtraction is similarly the same strength as addition. So, in standard ways of listing the order of operations (like BODMAS/PEMDAS/etc), we obey that ordering.

I'll reiterate that despite this reasoning, it isn't lot as if we are logically forced to use this convention. It is just a fairly natural one to use.