From the some of the questions I see submitted to this site, it seems that many people expect philosophers to affirm that their faith / superstitious beliefs have some positive value or grounding in reality. I cannot however think of many modern philosophers who would support such a belief system, so my question is: why do people feel that philosophy will be more supportive of faith-based belief systems than science?

An interesting question. Your observation seems accurate. I don't know why people (I assume you mean the non-philosophical public) feel as you describe. However, I can explain why philosophers might be more willing to take a 'faith-based belief system' seriously. A scientist is professionally incapable of taking faith seriously. Scientific method, at least as it is frequently understood, begins by excluding faith as a proper object for scientific enquiry. Philosophical method, however, does not do so. Faith can be philosophically investigated, evaluated, perhaps even in some way justified. To be sure, many philosophers, at the end of their enquiries, arrive at a rejection of faith, but that is not a point from which philosophy commences.

Maybe there is some contrast in stereotypes, with scientists seen as more down to earth and philosophers seen as more speculative, and this leads folk to think that philosophers are more likely to take ungrounded claims seriously.

Here is another possibiliity. The great skeptical tradition in philosophy is an epistemic leveler. In light of Descartes' sceptical arguments about the external world and Hume's sceptical arguments about induction, the warrant for scientific claims can seem no better than any other claims. This train of thought may of course lead one to put less weight on science; but it may also lead one to take other sorts of claims more seriously. (Like many philosophers, however, I prefer to try to show what it wrong with those sceptical arguments.)

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