I feel as though we're skipping between different definitions of pure.

Pure maths is the most abstract, it's not designed to have any practical application, it's just about proving things about purely abstract entities. Whether or not that has consequences (such as showing there's no formula to solve quintic equations) doesn't matter to a pure mathematician. So on those grounds, maths is definitely purer than most science; it is focused *only* on mathematics rather than letting other fields get involved.

That doesn't mean the thought processes and the disocveries of those thought processes don't have applications.

But as to whether or not mathematics is a pure science, I'd say no because it's not a science. It's a tool.

That's why pure maths and all these strictly abstract proofs have any attention paid to them; it's not because anyone particularly cares that p^{2}-1 for any prime p is a multiple of 24, say, they care about what the proof of that requires, whether the tools there can be used elsewhere, and what the proof teaches us about numbers in general. Maths is a toolbox. Applying those tools is science.

Which then raises the question of whether maths is the only set of tools scientists can use.

And this is where definitions get even messier. I don't think equations are necessary for science *or* maths, sure they help, but you can describe what's going on and make predictions without a single number. Equations get you precision, but at the end of the day they just formalise something you could put into words anyway.

Meanwhile, with maths, you get fields like graph theory where proofs can go by without a single equals sign, purely from walking through and describing possibilities. One could argue that isn't mathematics, but I'd disagree. Because of that I'd say maths is required, because it and pure logic alone have enough grey areas that I'm content just calling them synonyms. As such I'd say science does require maths, but it does not require equations. Equations are a goal, but they're not needed to describe what happens; their prime use is in seeing whether or not a hypothesis is accurate by whether the necessary equations can explain what is observed.

But you can't have an empirical field without the toolbox used in maths. At a basic level an experiment or observation is just an attempt at a proof by contradiction; does this statement line up with what we can find out to be true? If yes, then it's possible but not proven. If no, then it is disproven.