That is, the concept cannot be coherently expressed using the new terminology. For example, there's no way to express coherent concepts in things like Ptolomy's epicycles or Aristole's impetus. I'm not seeing how the second sentence is an example of the criterion in your first sentence.
Using modern physics, there is no way to express the concept that Ptolomy intended when he said epicycles. More casually, modern physicists would say "Epicycles don't exist" But contrast, the concept of set is still used in Cantor's sense, even though his formulation contained a paradox. So I think the move from geocentric theory to heliocentric theory is a paradigm shift, but adjusting the definition of set is not.
I'm also not seeing what the attributes "empirical" and "non-social" have to do (causally) with the ability to form coherent concepts.
I'm using the word science as synonymous with "empirical studies" (as opposed to making stuff up without looking). That's not intended to be controversial in this community. What is controversial is the assertion that studying the history of science shows examples of paradigm shifts.
One possible explanation of this phenomena is that science is socially mediated (i.e. affected by social factors when the effect is not justified by empirical facts).
I'm asserting that mathematics is not based on empirical facts. Therefore, one would expect that it could avoid being socially mediated by avoiding interacting with reality (that is, I think a sufficiently intelligent Cartesian skeptic could generate all of mathematics). IF I am correct that they are caused by the socially mediated aspects of the scientific discipline and IF mathematics can avoid being socially mediated by virtue of its non-empirical nature, then I would expect that no paradigm shifts would occur.
This whole reference to paradigm shifts is an attempt to show a justification for my belief that mathematics is non-empirical, contrary to the original quote. If you don't believe in paradigm shifts (as Kuhn meant them, not as used by management gurus), then this is not a particularly persuasive argument.
WP lists "non-euclidean geometry" as a paradigm shift, BTW.
If Wikipedia says that, I don't think it is using the word the way Kuhn did.
WP lists "non-euclidean geometry" as a paradigm shift, BTW.
If Wikipedia says that, I don't think it is using the word the way Kuhn did.
For Kuhn, the word was, if anything, a sociological term -- not something referring to the structure of reality itself. (Kuhn was not himself a postmodernist; he still believed in physical reality, as distinct from human constructs.) So it seems to me that it would be entirely consistent with his usage to talk about paradigm shifts in mathematics, since the same kind of sociological phenomena occur in the la...
Here's the new thread for posting quotes, with the usual rules: