I feel quite confident saying that mathematics will never undergo paradigm shifts, to use the terminology of Kuhn.
It believe it already has. Consider the Weierstrass revolution. Before Weierstrass, it was commonly accepted that while continuous functions may lack a derivative at a set of discrete points, it still had to have a derivative somewhere. Then Weierstrass developed a counterexample, which I think satisfies the Kuhnian “anomaly that cannot be explained within the current paradigm.”
Another quick example: during the pre-War period, most differential geometry was concerned with embedded submanifolds in Euclidean space. However, this formulation made it difficult to describe or classify surfaces—I seem to believe but don’t have time to verify that even deciding whether two sets of algebraic equations determine isomorphic varieties is NP-hard. Hence, in the post-War period, intrinsic properties and descriptions.
EDIT: I was wrong, or at least imprecise. Isomorphism of varieties can be decided with Grobner bases, the reduction of which is still doubly-exponential in time, as far as I can tell. Complexity classes aren’t in my domain; I shouldn’t have said anything about them without looking it up. :(
Reading the wiki page, it looks like Weierstrass corrected an error in the definition or understanding of limits. But mathematicians did not abandon the concept of limit the way physicists abandoned the concept of epicycle, so I’m not sure that qualifies as a paradigm shift. But I’m not mathematician, so my understanding may be seriously incomplete.
I can’t even address your other example due to my failure of mathematical understanding.
Reading the wiki page, it looks like Weierstrass corrected an error in the definition or understanding of limits.
Hindsight bias. The old limit definition was not widely considered either incorrect or incomplete.
But mathematicians did not abandon the concept of limit the way physicists abandoned the concept of epicycle, so I’m not sure that qualifies as a paradigm shift.
They abandoned reasoning about limits informally, which was de rigeur beforehand. For examples of this, see Weierstrass’ counterexample to the Dirichlet principle. Prior to Weierstrass, some people believed that the Dirichlet principle was true because approximate solutions exist in all natural examples, and therefore the limit of approximate solutions will be a true solution.
Hindsight bias. The old limit definition was not widely considered either incorrect or incomplete.
Not true. The “old limit definition” was non-existent beyond the intuitive notion of limit, and people were fully aware that this was not a satisfactory situation.
We need to clarify what time period we’re talking about. I’m not aware of anyone in the generation of Newton/Leibniz and the second generation (e.g., Daniel Bernoulli and Euler) who felt that way, but it’s not as if I’ve read everything these people ever wrote.
The earliest criticism I’m aware of is Berkeley in 1734, but he wasn’t a mathematician. As for mathematicians, the earliest I’m aware of is Lagrange in 1797.
That’s pretty clear, thanks. Obviously, experts aren’t likely to think there is a basic error before it has been identified, but I’m not in position to have a reliable opinion on whether I’m suffering from hindsight bias.
Still, what fundamental object did mathematics abandon after Weierstrass’ counter-example? How is this different from the changes to the definition of set provoked by Russell’s paradox?
I don’t recall where it is said that such an object is necessary for a Kuhnian revolution to have occurred. There was a crisis, in the Kuhnian sense, when the old understanding of limit (perhaps labeling it as limit1 will be clearer) could not explain the existence of e.g., continuous functions without derivatives anywhere, or counterexamples to the Dirichlet principle. Then Weierstrass developed limit2 with deltas and epsilons. Limit1 was then abandoned in favor of limit2.
It believe it already has. Consider the Weierstrass revolution. Before Weierstrass, it was commonly accepted that while continuous functions may lack a derivative at a set of discrete points, it still had to have a derivative somewhere. Then Weierstrass developed a counterexample, which I think satisfies the Kuhnian “anomaly that cannot be explained within the current paradigm.”
Another quick example: during the pre-War period, most differential geometry was concerned with embedded submanifolds in Euclidean space. However, this formulation made it difficult to describe or classify surfaces—I seem to believe but don’t have time to verify that even deciding whether two sets of algebraic equations determine isomorphic varieties is NP-hard. Hence, in the post-War period, intrinsic properties and descriptions.
EDIT: I was wrong, or at least imprecise. Isomorphism of varieties can be decided with Grobner bases, the reduction of which is still doubly-exponential in time, as far as I can tell. Complexity classes aren’t in my domain; I shouldn’t have said anything about them without looking it up. :(
Reading the wiki page, it looks like Weierstrass corrected an error in the definition or understanding of limits. But mathematicians did not abandon the concept of limit the way physicists abandoned the concept of epicycle, so I’m not sure that qualifies as a paradigm shift. But I’m not mathematician, so my understanding may be seriously incomplete.
I can’t even address your other example due to my failure of mathematical understanding.
Hindsight bias. The old limit definition was not widely considered either incorrect or incomplete.
They abandoned reasoning about limits informally, which was de rigeur beforehand. For examples of this, see Weierstrass’ counterexample to the Dirichlet principle. Prior to Weierstrass, some people believed that the Dirichlet principle was true because approximate solutions exist in all natural examples, and therefore the limit of approximate solutions will be a true solution.
Not true. The “old limit definition” was non-existent beyond the intuitive notion of limit, and people were fully aware that this was not a satisfactory situation.
We need to clarify what time period we’re talking about. I’m not aware of anyone in the generation of Newton/Leibniz and the second generation (e.g., Daniel Bernoulli and Euler) who felt that way, but it’s not as if I’ve read everything these people ever wrote.
The earliest criticism I’m aware of is Berkeley in 1734, but he wasn’t a mathematician. As for mathematicians, the earliest I’m aware of is Lagrange in 1797.
I’m also curious about this history.
That’s pretty clear, thanks. Obviously, experts aren’t likely to think there is a basic error before it has been identified, but I’m not in position to have a reliable opinion on whether I’m suffering from hindsight bias.
Still, what fundamental object did mathematics abandon after Weierstrass’ counter-example? How is this different from the changes to the definition of set provoked by Russell’s paradox?
I don’t recall where it is said that such an object is necessary for a Kuhnian revolution to have occurred. There was a crisis, in the Kuhnian sense, when the old understanding of limit (perhaps labeling it as limit1 will be clearer) could not explain the existence of e.g., continuous functions without derivatives anywhere, or counterexamples to the Dirichlet principle. Then Weierstrass developed limit2 with deltas and epsilons. Limit1 was then abandoned in favor of limit2.