"I would ask you a similar question; How many of the scientists and mathematicians, that you know, are human?"
All of them. All of us humans belong to one or more subcultures, John, whose boundaries seem opaque to outsiders.
"Further, how many of them are intellectual products of a system of education stretching back centuries, in which each generation has gradually built on the foundations laid down by previous generations."
None of them. You assume that historicism drives science -- by what evidence?
"Further, has it ever happened that within this system, it was discovered that assumptions, concepts, models and other such mental tools were found to be incomplete, misunderstood, lacking crucial insights, etc. which then required re-working and rethinking? By this, not only the physical sciences, but everything from history to geology."
Scientists and mathematicians thrive on the possibility of discovery and re-discovery. Probably a bit like prospectors who do backbreaking labor for long periods of time for small rewards.
"In what reading I have done, rarely are new ideas accepted when first proposed and frequently the rejection is valid, but there are certainly enough examples of where the outsiders were eventually accepted as having been right, that the status quo cannot always be assumed correct."
That's what makes being right so special. Once in a while, someone does discover the mother lode. That doesn't devalue the worth of the achievements and experiences of the rest of the prospectors -- on the contrary, it motivates more hope, more labor, more expectations for success. The journey is everything.
"As you state it; 'Science is a rationalist enterprise. There aren't different practices and beliefs -- it is universally practiced the same way.'
"The consequence is that one frame emerges and while that frame logically considers itself to be universally objective, the fact remains 'that no system of axioms is strong enough to prove itself.'"
That, John, you should realize is an objective statement.
"Right now, some of those on the cutting edge of theory, strings and supersymmetry come to mind, who think some of the more formal proofs of science should be relaxed."
On the contrary, string theorists are among the more rigorous theorem-provers of mathematical theorists. I think you mean that some think that string theory deserves more status in the physics community despite its lack (so far) of novel physical predictions. I expect that Ed Witten would be among those, and Joe Polchinski -- and I think they have a good case. String theory is the only mathematically complete theory that unifies all the forces of nature which are already experimentally supported.
"That, as others have strongly objected, is not a good move."
Here again, John, a reference to the literature would be helpful so that we have the advantage of knowing who said what, and what they are objecting to.
"There are a number of ideas out there that many people have devoted decades to, which could well prove to be intellectual dead ends."
Absolutely. Everyone in the scientific community knowingly takes that risk. So? The journey is everything.
"That is the nature of science. And civilization."
More risk, more reward. :-)
