'What is Scientific Philosophy?'

"Scientific philosophy" as I will be using it here is an eighteenth century invention by now-forgotten philosophers (McLaurin, 's Gravesande) or not read as philosophers anymore (Euler) (and then opposed by now-canonical philosophers like Berkeley, Hume, Kant, and folk that are fun to read like Mandeville and Diderot) that, after the split between philosophy and science, was re-introduced into philosophy by people like Russell, and echoed by Carnap, and Reichenbach. Scientific philosophy has six characteristics:

1. Empirical ‘success’ trumps other (rational/methodological) claims. Given that scientific philosophers sometimes retreat to the idea that philosophy is an a priori discipline, the 'empirical' (in 1) is often re-packaged as, say, inference to the best explanation in light of a variety of enduring 'scientific virtues' (i.e., simplicity, scope, predictive power, fruitfulness, exactness, etc.) 
2. (a) Physics is the foundational science and/but it (b) has no need for ultimate foundations. While 2(a) may seem obvious (see, e.g., Ladymann & Ross) due to its universal scope, its foundational nature was contested well into the nineteenth century. One could imagine, say, the science of information taking over as the foundational science in the future.
3. Within scientific philosophy reason limits itself in various ways: in doing so (a) it avoid the fallacy of systematicity because it does not try to say everything about everything; (b) it embraces the intellectual division of labor (from 3(b)); it avoids the fallacy of (metaphysical) foundationalism because it has no need to try to to secure its practice in un-shakeable, first principles (see 2(b)). So, it is no surprise that Russell rejected the principle of sufficient reason or Bradley's regress argument. 
4. Scientific philosophy is a self-directed, autonomous practice; once one has mastered certain rigorous tools, one moves from one given experiment/solution (etc.) to the next problem. Given the emphasis on rigor, it is no surprise that:
5. Scientific philosophy is often opposed to a licentious or unintelligible alternative(s) associated with past failures, sometimes even moral. (Exhibit a.) It, thus, embraces commitments to transparency (and clarity). 
6. Scientific philosophy offers submission to the facts (recall) and is disciplined (recall) by way of a careful, painful, modest and most importantly open-ended progressive method. This entails that any scientific philosopher will enter a pre-existing, moving research trajectory and can expect to die before any destination is ever reached.

It follows from this list that scientific philosophy is in a certain sense doctrine neutral.  As long as one allows that the claims of physics are open to revision, scientific philosophy is welcoming to a whole range of views about reality. This is one reason why much recent analytical metaphysics can legitimately present itself as a species of scientific philosophy. Of course, at any given time the burden of proof is on those that challenge scientific orthodoxy.

It also follows that one should not conflate formal and mathematical methods in philosophy with scientific philosophy, even if it is certainly true that formal and mathematical methods often play a non-trivial role in scientific philosophy (due to the significance of exactness). The recently popular experimental philosophy (Xphi) is as much a species of scientific philosophy as, say, formal philosophy. As Hannes Leitgeb recently emphasized, mathematical and experimental methods need not be opposed.

Scientific philosophy also always risks two, Kuhnian dangers: (i) embrace of mythic history in order to facilitate (and shorten) the student's entry into mastery of the rigorous tool-kit and problem-space; and, thus (ii) the possibility of so-called Kuhn-loss (see especially Hasok Chang). One interesting version of a possible Kuhn-loss in scientific philosophy occurs when in order, say, to formalize an original problem (let's call it 'Z'), Z is transformed into something akin to it (say, in order to make it tractable), let's call it 'Z*', and special features of Z that are not visible in Z* are systematically overlooked or black-boxed in practice. Clarity may sometimes conceal. A related, third danger is that (iii) once one progresses, progress becomes to be valued for its own sake or (more tacitly) at the expense of other ends. Of course, in practice such risks may well be outweighed by many benefits (see for an excellent discussion Leitgeb's paper); I am especially partial to the topic neutrality of many tools in scientific philosophy and the relatively low barriers to entry of scientific philosophy.

Finally, there are two justificatory challenges for the scientific philosopher: (i) it cannot really ground itself; it can seduce and inspire, but due to the self-limiting nature of reason within it, it cannot ground itself morally (and probably epistemically); (ii) it cannot really explain why some given problems matter more philosophically than others, expect by an appeal to the scientific virtues. Thus, in practice, scientific philosophy is often either married to an engineering-ethic, when the constrains are given, or a kind of soft-utilitarianism in which good consequences, as such, are embraced. (Of course, formal methods can also be used in defense of other substantive moral visions!)

*I thank Irene van de Beld for recent discussion of these issues. My errors should not be attributed to her.