What is "science"? What makes one endeavor scientific and another endeavor non-scientific?
It's tempting to say that science is what scientists do. It's a tempting answer, because different kinds of scientists do a lot of different things. The way biologists do biology is very different from the way physicists do physics; if they are very different, why is it legitimate to label both as "science" but label mathematics or philosophy as "not-science" just because mathematics and philosophy are very different from physics and biology. It's also tempting because we can say that science is just another social category; to talk about science abstractly risks the fallacy of reification. In much the same sense being "American" is just what people in the United States happen to do; to talk abstractly about some essential "Americanism" that would be true regardless of what any Americans actually did would be nonsensical reification.
A lot of people, myself included, think that, unlike Americanism, there really is something essential about science, something essential that would be true even if everyone who called themselves "scientists" didn't do it. Moreover, it's important to think and talk about the essential nature of science because science, done "correctly," has immense social value; done incorrectly, science is useless or harmful. Furthermore, what actual professional scientists actually do is a vastly more rigorous version of a general mode of thought, a mode that everyone should employ, at least about some of their ideas and behaviors.
Of course, there are no other modes of thought, modes that are not science, that also have value; I do not advocate "scientism" in the sense that science is the only valuable mode of thought.
If there really is something essential about science, then philosophers are better-situated than scientists to talk about it. Paradoxically, if there is something essential about science, then it is possible that scientists do things scientifically out of habit or tradition, without reflection or specific intention. They could be doing things scientifically, but doing those things not because they know those things are scientific, but just out of habit. It might also be true that because science, as actually practiced, is socially situated, scientists do specific things that might be irrelevant or contrary to what is essentially scientific because of their social situation. It could take an outsider to discern which of those things really comprise the essence of science, and which are just arbitrary habits or extraneous social behavior. For example, I've worked with a lot of scientists. Many of them have no idea what statistical tests do or how statisticians actually interpret them; they just click a button in Stata or Excel, and if the p value is less than 0.05, they publish. The statistics are essentially scientific; the choice of 0.05 (or any other specific number) as the threshold of significance is not essential to science.
Again, I don't wish to deprecate the social aspects of science: scientists are part of society, and they have to fit their practices to the larger social context. Still, if there is such a thing as what is essentially science, it is still important to discern what is scientific and what is social.
Philosophers are also well-situated to talk about the essential nature of science because when doing actual philosophy (rather than "philosopholgy," the study of the somewhat arbitrarily constructed canon of philosophy) are the least susceptible to authority and privilege. A philosophical work stands or falls on its argument, not on the reputation or privilege of its author. Robert Nozick's Anarchy, State, and Utopia, for example, is "authoritative" on a version of Libertarianism not because he holds any particular power to define it, but because he makes good arguments (well, as good as can be expected) and has persuaded a lot of people that his version of Libertarianism makes sense.
In this series, I will be talking about the philosophy of science as a philosopher. I claim no special privilege to define the essential nature of science, or even to establish that there is such a thing as an essential nature of science: my work, like all philosophy, stands or falls on the persuasiveness of my arguments.