People don't view philosophy that way because philosophy never gets far from words. Philosophy and math started to fork at the point where people realized that substituting words into logical schema and then using logical transformations to create new sentences doesn't work reliably, and in fact doesn't really aid thought at all except by suggesting possibilities. At that point, logic became a philosophic failure, a dead end. A curiosity at best. There's still kind of a conceptual overlap; a philosopher might find it stimulating to think about the concept of N-value truth systems. However, he wouldn't think that mathematical theorems about such systems have any strict philosophical consequences. They would just be spurs for thought.
This was not the case at all where I did my undergrad, judging by the OpenCourseWare it doesn't appear to be the case at MIT and I would assume this is the same at any of the other major universities. Arguments were consistently and constantly described using symbolic logic and derivations.
I don't understand the animosity you have for philosophical logic. I've always felt it was an interesting compliment to what I did in computer science and it made me appreciate discrete math concepts to a greater degree. The fundamentals building blocks are the same, just a different application.
I don't have any animosity for philosophical logic, but I'm irked by the impression that mathematical logic's relevance to philosophy is as a tool of thought, i.e., that being a philosopher has some essential connection to being a practitioner of mathematical logic. Mathematical logic plays a similar role in philosophy that UML and design patterns play in programming. UML was hoped and perhaps originally intended (I don't honestly know) to be a powerful tool that could be used in CASE tools to accomplish programming in an easier and more reliable way than the clearly insufficient way that program now. It turned out that problems could be solved that way, but only after so much difficult "traditional" preliminary work that nothing was saved. By and large, UML is only used for documentation of patterns and high-level documentation of systems. You can undoubtedly program without using UML and patterns as tools, and people disagree about whether they are even useful.
In a similar way, it was hoped that a sufficient understanding of logic would enable one to reach philosophical conclusions more easily, with complete reliability. This sounds ridiculously naive to us, but we have the advantage of hindsight. At one time it was thought that with the investment of sufficient philosophical thought, logical rules of thought could be distilled and used to settle real problems. Unfortunately, it turns out that every application of logical rules requires so much examination of the validity of the application that little advantage is derived from logic beyond its ability to suggest and describe possible lines of thought, in the same way that UML and design patterns suggest and communicate but cannot replace the work of programming nor ensure the validity of resulting programs. Symbolic logic cannot be used to validate philosophical thought. To take an example of a deeply "logical" work, if you translated Spinoza's Ethics into symbolic logic, you would see nothing except some very trivial stuff and probably numerous errors. The profundity of Spinoza's work has little to do with logic in the mathematical sense. Once philosophers realized that, mathematical logic quickly became an ex-wife they had dinner with once a month. Never entirely forgotten, always thought of with affection, an important part of one's development, not without interest, but not looked to as a source of growth and vitality. You can certainly be a keen and insightful philosopher (though an ignorant one) without any understanding of mathematical logic, and a deep study of mathematical logic is rather useless for philosophy.
Now we muse about the "unreasonable effectiveness of mathematics in the natural sciences." At one time we hoped pure logic would help us decide questions about God and morality, and now we have learned to be surprised that it is useful for anything at all! Mathematical logic is an active field of mathematics, and philosophers have many problems to ponder, but there is no philosopher reading set theory abstracts looking for support in a dispute about aesthetics or ontology. Logic has become a branch of mathematics partly because it is necessary for mathematics and can be studied as mathematics, but also because no field outside of mathematics requires the solution of any complex problems in mathematical logic.
Even Godel's Incompleteness Theorem, while containing rich philosophical implications, did not actually apply to any philosophical ideas outside of math or logic because philosophers had long since given up using mathematical rules to "prove" or "derive" philosophical results. Philosophers were, as always, using language to express, justify, and criticize philosophical work. The use of logic as a tool for rigorously validating philosophical arguments, which was the original basis for the connection between philosophy and logic, is still a pipe dream.
I don't mean to denigrate either mathematics or anything outside mathematics by this. I have a math degree, but I'm the first person to admit that mathematical logic tells you nothing reliable about the relationship between "Every X is a Y, and Z is an X," and "Z is a Y." A proposition in symbolic logic can be derived mechanically according to its rules, but philosophy is uncertain and requires judgment. Mathematics is a useful tool for the natural sciences; in philosophy, mathematical logical is occasionally a stimulating subject for thought, but not a significant tool.
>The use of logic as a tool for rigorously validating philosophical arguments, which was the original basis for the connection between philosophy and logic, is still a pipe dream.
That's not strictly true. The Arab and Scholastic philosophers constructed a philosophical world-view, largely based on Aristotelian logic, which internally consistent fairly comprehensive. What resulted in its downfall, was not the failure of logic, but the desire for absolute truth. All forms of logic, whether mathematical or syllogistic, require argument from premises, which means that ultimately one must start with first premises that cannot be proved and must merely be agreed upon (making truth a matter of consensus.) Descartes tried to rectify this situation by assuming away all assumptions and proving first premises in a vacuum. He didn't get far (made too many logical fallacies) but he started a fad that overtook the philosophical realm and continues to this day.
My point being, that the problem wasn't the language (despite what the deconstructuralists might say,) which, if common and well defined presents no barrier, but changing ends in philosophical discourse.
