Do people invent equations, or do they discover them? Examples of the sorts of things I am thinking of are Newton's laws of motion, or Mandelbrot's sets.

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It helps to begin by distinguishing laws of nature from our hypotheses about them. Then the first question is whether there really are laws of nature out there. I'm one of those philosophers who believes there are, though just what it takes to be a law is hard to say. For example, is a law just an objective pattern of properties, or does a law have a special kind of necessity? But some philosophers would deny that even the patterns are fully out there, because they hold that the structure of properties is something scientists impose on the world: the world does not come pre-carved into natural kinds.

So the answer to your question is a little complicated. Even if the laws depend on our own scheme of classification, it would probably be misleading to say we invent them: it is not as if we can just make them any way we like. And even if the laws of nature are fully objective and out there independently of us, it is still up to us to think up the hypotheses that are supposed to describe them. In any case, a lot of hypotheses we come up with are mistaken: even Newton's laws of motion are not strictly true. The true laws of nature may be entirely independent of us, but there is a clear sense in which our hypotheses are our construction, even if they are our best shot at getting at the truth.

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