Re:Why read old philosophy?
by foldr on Wednesday January 11, 2017 @05:39AM
(#13373024)
>More generally, philosophers rarely phrase their logic in a formal language (although they do on occasion), while mathematics has been done only in a formal logic since the advent of proper axiomatic foundations in the early 20th century.
I don't see any distinction between the two disciplines here. Philosophers make use of formal logic when it makes sense to do so (lots of discussion of the ontological argument makes uses of formal logic, for example [1]). Mathematicians follow exactly the same strategy. It's quite false to say that mathematics is now done only in a formal logic. The vast majority of mathematical proofs are presented informally (even in textbooks on logic!)
>How so? Even early results like Godel's incompleteness theorems look strikingly like other mathematical proofs.
I guess the best example is Frege, whose work was ignored by mathematicians and logicians alike because it looked completely alien. But I was thinking more of, say, proofs of completeness for systems of propositional logic, which are highly notation-dependent and consist in running through a bunch of cases in a boring mechanical way. Peirce and Gauss seem worlds apart.
> Frege (as well as the others you listed) at the very least thought the right way to study logic was through mathematics.
I'd say he thought that the right way to study mathematics was through logic. In other words, he was trying to put mathematics on proper logical foundations. (This is the project that Russell tried, and essentially failed, to complete.)
>how many are exploring something new? I don't mean "something new" as in standing completely alone and without any references to past works (I've yet to see such a paper in math or science), but simply drawing conclusions that are something other than a response to an age old argument. And as I said before, how many use formal logic as opposed to informal logic?
Most conclusions in philosophy are responses to age old arguments for the very simple and obvious reason that people have been thinking about philosophical problems for a long time. Occasionally people do come up with new problems, of course. The philosophy of language would be one example where lots of genuinely new questions have arisen and been given interesting answers (partly as a result of developments in mathematical logic).
It's quite common for philosophy papers to use formal logic, but only when it is helpful. In exactly the same way, mathematical proofs are presented formally on occasion, when it is helpful, but not as a matter of course.
I don't see any distinction between the two disciplines here. Philosophers make use of formal logic when it makes sense to do so (lots of discussion of the ontological argument makes uses of formal logic, for example [1]). Mathematicians follow exactly the same strategy. It's quite false to say that mathematics is now done only in a formal logic. The vast majority of mathematical proofs are presented informally (even in textbooks on logic!)
>How so? Even early results like Godel's incompleteness theorems look strikingly like other mathematical proofs.
I guess the best example is Frege, whose work was ignored by mathematicians and logicians alike because it looked completely alien. But I was thinking more of, say, proofs of completeness for systems of propositional logic, which are highly notation-dependent and consist in running through a bunch of cases in a boring mechanical way. Peirce and Gauss seem worlds apart.
> Frege (as well as the others you listed) at the very least thought the right way to study logic was through mathematics.
I'd say he thought that the right way to study mathematics was through logic. In other words, he was trying to put mathematics on proper logical foundations. (This is the project that Russell tried, and essentially failed, to complete.)
>how many are exploring something new? I don't mean "something new" as in standing completely alone and without any references to past works (I've yet to see such a paper in math or science), but simply drawing conclusions that are something other than a response to an age old argument. And as I said before, how many use formal logic as opposed to informal logic?
Most conclusions in philosophy are responses to age old arguments for the very simple and obvious reason that people have been thinking about philosophical problems for a long time. Occasionally people do come up with new problems, of course. The philosophy of language would be one example where lots of genuinely new questions have arisen and been given interesting answers (partly as a result of developments in mathematical logic).
It's quite common for philosophy papers to use formal logic, but only when it is helpful. In exactly the same way, mathematical proofs are presented formally on occasion, when it is helpful, but not as a matter of course.
[1] https://philpapers.org/archive/OPPOTL.pdf