Sociology undergraduate here, who is struggling to "see the wood for the trees", as the idiom goes. My two brief questions are the following: Is there anything unique within sociological theory, or is it just a spin-off of philosophy that lacks training on how to think? Additionally, is it the case that a philosophy degree can open doors into other fields, but sociology is more limiting to academic mobility?

This will be a very subjective response; others who have some acquaintance with the two fields will answer differently. Also I'm pretty out of date on sociology. One thing that has not changed much there, though, as far as I'm aware, is that like many social sciences, it is deeply split between two subfields, which differ so much from each other that they might as well be separate fields. On the one side, there is mathematical and quantitative sociology, which operates largely with rational-choice models, treated quasi-formally, and on the other side there is qualitative sociology, which is no less empirical, but relies more on participant observation and what Clifford Geertz called "thick description." I think the answers to your questions depend to some extent on which of these two you are primarily interested in. Or you might, as a third option, be one of those hopeless idealists like James Coleman, who thought the two mutually alienated sides belonged inseparably together and that the example...

I am wondering if it is worth my time continuing to read philosophy. I have read quite widely in the hope of "broadening my mind", but lately I have noticed that while reading new material, I seize with pleasure on the points that confirm what I already believe--I am a practicing Roman Catholic--and sideline those I disagree with. I assume it would be the same, if I were a Marxist, Buddhist, agnostic, or nihilist, reading principally in search of what confirms my beliefs. I can see the point of reading philosophy, if you don't already have beliefs or opinions or simply need to produce an academic essay, but why bother when you know what you know? Have you had a different experience?

If you want to participate in the philosophy game, even passively, you have to pretend you have an open mind. Most philosophers don't, of course, but they do their best to pretend. And I have to say that the value system of the community (even among academics!) does seem to apportion the highest degree of respect and admiration to those who give the best appearance of having an open mind. From a philosophical point of view, your complacent attitude toward your own observations about yourself ("lately I have noticed" etc. -- without the slightest twinge of any critical impulse toward yourself) puts you outside the pale. If you really think it's just too much trouble to change that comfortable attitude, you should give up any interest in philosophy. It is not worth your time, or anyone else's. But before you settle into your comfortable ignorance, you should perhaps review, once again -- with care and attention, though -- the earliest Socratic dialogues, including especially the Apology of...

Can we really define Philosophy?

Think about it. Nearly all controversies with an intellectual component are at least partly controversies about what concepts to use. The deepest controversies in nearly all disciplines aren't the substantive ones, where people disagree about some particular theory or fact, they are the conceptual ones in which people disagree about how to talk about the substantive stuff, what concepts to use. Philosophy has generally been a clearing house for all these conceptual controversies from all the other disciplines, and from life in general (this isn't a definition, it's a stab at an empirical generalization, though admittedly a somewhat idealized one, in which I leave out a lot of stuff others might like to include within "philosophy"). Philosophical responses to these controversies have been all over the map. Sometimes philosophers just pick up those same controversies and carry on just as the physicists or lawyers or whoever might have done, in the same terms as the physicists or lawyers. Other times...

It's often said that we cannot predict which scientific discoveries will turn out to have practical value, and so we should encourage scientific curiosity and investigation even in cases where the subject matter seems frivolous or esoteric. To take one famous example, G.H. Hardy thought that number theory was perfectly useless, but it is now indispensable to cryptography. Could the same be said of philosophy? Are there philosophical theories that have had unforeseen benefits? Or is it safe to conclude that at least some philosophical pursuits really are just "useless"?

As useless as art, literature, music, or just about anything you do as an end in itself rather than a means toward some other end in itself. Most important science wasn't done for the purpose of achieving "practical" results, but to satisfy some inner compulsion, of the kind that Plato describes very well in his cave story. That said, just about all scientific disciplines have emerged, one way or another, from philosophy (though philosophy itself, if you consider Plato as an important milestone, seems to have been inspired by mathematics), so it's certainly been of very considerable practical use in that sense.