= 27d567bc2d48d9f40e9ccc20ea370b15)
paper-machine comments on Rationality Quotes March 2012 - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (525)
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.