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benelliott comments on Rationality Quotes March 2012 - Less Wrong
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Would the whole Russel's paradox incident count as a mathematical paradigm shift?
Reading Wikipedia, it looks like a naive definition of a set turns out to be internally inconsistent. Does that mean the concept of set was abandoned by mathematicians the way epicyles have been abandoned by physicists? That's not my sense, so I hesitate to say redefining set in a more coherent way is a paradigm shift. But I'm no mathematician.
Its a matter of degree rather than an absolute line. However, I would say a time when even the very highest experts in a field believed something of great importance to their field with quite high confidence, and then turned out to wrong, probably counts.
I don't think "everyone in field X made an error" is that same thing as saying "Field X underwent a paradigm shift."
Why not ? That sounds like a massive shift in the core beliefs of the field in question. If that's not a paradigm shift, then what is ?
The "non-expressible in the new concept-space" thing that you think never actually happens.
Isn't that exactly what happened? The phrase "set of all sets that do not contain themselves" isn't really expressible in Zermelo-Fraenkel set theory, since that has a more limited selection of ways to construct new sets and "the set of everything that satisfies property X" is not one of them.
I don't think it's terribly useful to frame the discussion in terms of concepts that never actually happen :-)
This looks very like trying to define away something that sure felt like a paradigm shift to the people in the field. Remember that "paradigm" is a belief held by people, not a property inherent in the universe.
Perhaps this is a limitation of my understanding of Kuhn, in that I'm misusing his terminology. I am unaware of mathematics abandoning fundamental objects as inherently misguided the way physics abandoned epicycles or impetus. I expect physics will have similar abandonments in the future, but I expect mathematics never will. The difference is a property of the difference between mathematics and empirical facts. This comment makes the argument I'm trying to assert in slightly different form.