On 3/26/2013 1:18 PM, david petry wrote: > On Tuesday, March 26, 2013 4:05:38 AM UTC-7, Dan wrote: > >> Gödel's theorem is here to stay . > > As I have argued previously, if we treat mathematics as a > science and accept falsifiability as the cornerstone of > mathematical reasoning, then Godel's theorem is utterly > utterly trivial, while at the same time, his proof of the > theorem is not a valid proof. > > https://groups.google.com/group/sci.math/msg/25be708362cb7e? >
"When mathematicians talk, they want people to believe that they are telling the truth. Again, that cannot be denied. When they claim to have proven a theorem, they are implicitly claiming that the theorem is actually true. That is, they are claiming that proofs are compelling arguments."
Let me take "actually true" as corresponding with "metaphysically true".
In order for your statements here to be substantive, every mathematician would have to be a mathematical realist. That is not at all clear. How a mathematician views mathematical objects is a rather personal matter since those objects are abstract. Moreover, that viewpoint must change as a given object is considered. By the time some proof is being given, protocol dictates that the mathematician is playing a simple symbol manipulation game. But, in prior deliberation one should not be surprised if the object is a platonic form.
All mathematicians would want their proofs to be compelling arguments. But, since one can consider premises that are contrary to one another, the limit of a claim based on the quality of the argument only applies to the conclusion in relation to the premises.
There is a part of the foundations of mathematics that is grounded in realism. It is found in Frege and Russell. It is found in Lesniewski, until he concluded that the only sensible interpretation was nominalism. And, Lesniewski is the perfect example that your statements are too general. The search to understand mathematics need not be the same as the effort to provide mathematics with a metaphysical ground. No ground, no claims of truth in any realist interpretation of the word.