Functional definitions

Knowledge is knowing that a tomato is a fruit, wisdom is not putting it in a fruit salad.
— Miles Kington

I think one of the breakthrough moments in learning mathematics is when you realise that all of the definitions are just made up. Back when I was in high school and all the way into university, I was taught things like "a negative number times a negative number is a positive number", "anything except zero to the power of zero is one", "you can't divide by zero", "there's no square root of negative numbers", "actually we lied about that one". Well, actually, all of it was lies. None of those things are that way, they were just defined that way by some mathematician or other.

Many common terms for seeds and fruit do not correspond to the botanical classifications.
— Wikipedia article on Fruit

In the interest of balance, wouldn't it be prudent to include a section on the myriad criticisms of fruit. This page is so onesided.
— Wikipedia talk page on Fruit

And definitions, both inside and outside of mathematics, seem to follow this pattern quite frequently. Some people tend to claim that their set of definitions is absolutely and objectively correct. Indeed, they'd call them facts instead of definitions so as not to offer any implication of being arbitrary. On the other hand, you get people saying definitions are fundamentally arbitrary. What you call a fruit may as well be a vegetable, or a dinosaur, or whatever. What we call good could be called evil by someone else and it would make just as much sense. That viewpoint, however, leaves a lot to be desired.

You can know the name of that bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird. You’ll only know about humans in different places, and what they call the bird. So let’s look at the bird and see what it’s doing – that’s what counts.
— Richard Feynman (quoting his father)

Definitions are invented, yes, but not arbitrarily. We define things in certain ways because of the consequences of those definitions. For example, you can make your own branch of mathematics where multiplying negative numbers by negative numbers yields negative numbers. Nobody's stopping you! But other things would need to change too. Would your branch of mathematics end up consistent with these changes? Maybe! Would it end up useful? Probably not.

So too with other definitions. We can define a tomato as a fruit, vegetable, or dinosaur, but those definitions have consequences. If you're thinking about a tomato as something to eat, you're in for an unpleasant surprise if you expect it to have a similar taste or role in cooking as other fruit. On the other hand, if you want to grow a tomato, or analyse its reproductive cycle or relationship to other plants, you'll find thinking about it as a fruit saves a lot of time. However, regardless of your goals, it's unlikely that defining a tomato as a dinosaur will help you.

I think the only sensible way to think about definitions is functionally. That is, if I define a thing in this way, what does it buy me? What can I do with this newly defined thing that I could not do before, and what options does defining it in this way remove? Does it help me think more clearly about this thing or other things in the same category?

If, upon reflection, your definition doesn't do anything, you're probably better off without it.