According to Gian-Carlo Rota, mathematics has had a pernicious influence on philosophy (in The Pernicious Influence of Mathematics on Philosophy). I agree with the overall theme of the essay, yet most of the details I take issue with. First off, I think subjective phrases such “This confused state of affairs makes philosophical reasoning more difficult but far more rewarding [than mathematics],” need to be cut. First off, I think most people would say that mathematics is more difficult than philosophy. Then, how much reward you take in doing it is completely dependent upon the person. OK. On to real issues, though.
Rota claims that this pernicious influence of clearly defining terms and using mathematical logic to create rigorous arguments is a purely 20th century invention. We should return to our philosophical roots where this was not done. I disagree. Look at Spinoza. I’ve never even read a mathematical paper that rigorous before. I’m sure there are more as well. They just aren’t coming to mind. Thus, I believe that mathematical logic has been present all along. We just didn’t really have a name or symbolic form for it.
I really liked the idea of circularity in mathematics. This pernicious influence is not the fault of mathematicians. It is the fault of philosophers that misunderstand what it is that mathematicians do. They don’t randomly create definitions, and then randomly manipulate logically. Definitions are motivated by need and justified by use. At this time I would like to point out that this is the precise argument that Corfield makes in Towards a Philosophy of Real Mathematics to shift from the group definition as primary to monoid as primary since this is more natural. I wish mathematicians would let go of tradition in this sense and allow the better definitions to come forward (like categories as opposed to sets and monoids as opposed to groups).
My last comment is on the fact that the main argument seems to be that philosophers don’t need the same precision and logic as mathematicians, because they are different disciplines. I disagree in a sense. I believe that philosophers do need the precision, but there is a problem: language is inherently imprecise (see the post Uncertainty II), so there is no possible way to do what mathematicians do. This isn’t bad or anything. I just think that that should have been the way to argue it. It is impossible to properly define terms as opposed to it is unnecessary to define terms.