Lately, I’ve been curious about the reasons why we can’t all agree on the soundness of certain logical arguments like we do mathematical derivations. Math is thought to be flawless because its content is always clear (i.e. it consists of well-defined elements like numbers and operations, etc.) and all derivations that stem from it are incontestable (i.e. mathematical proofs are not subject to opinion).
It has also been shown that propositions can be transcribed into formal logical notation (such as “If P then Q. P. Therefore Q.”) and rules of operation can be determined for the manipulation of this notation that mimics the rigor of mathematics. Why, then, do we not have a method for deciding whether philosophical arguments (which ideally should have a logical structure) are sound or not? Why does all philosophical discourse usually end in both parties agreeing to disagree? One would think that all such arguments can be transcribed into the formal notation and the matter can then be settled once and for all. So why do the same old topics come up over and over again? This never happens in mathematics. Why, if we have a formal method akin to that of mathematics, does it happen in philosophy?
My gut instinct tells me the primary culprit is our emotional attachment to the propositions we put into discourse. I also think it has to do with our drive to link our propositions, arguments, and conclusions to the real world, whereas in mathematics, all statements are understood to hold independently of the real world - that is, that mathematics finds its justifications internally.
What do others think?