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I'm guessing you have not read Proofs and Refutations by Imre Lakatos. It is worthwhile.

Also if you read on the history of mathematics, the various back and forth foundational issues raised by infinitesmals, Fourier series, proof by contradiction and the like are very interesting. Sure, it is possible for a modern mathematician to look at all that and say, They just didn't know how to do it, HERE you go. But over decades and centuries people were somehow successfully continuing to do math, even though they knew that something serious was wrong in their understanding.




At least those historical mathematicians acknowledged their gaps of knowledge. Cancer research is a far cry different.

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You clearly are unfamiliar with the history.

If you can find a copy, George Berkeley's The Analyst was a cogent criticism of the foundations of calculus in his day. (He didn't have answers, but he identified real problems.) A variety of books were written by mathematicians in response. Uniformly these were much lower quality, and consisted of defenses of the foundations of mathematics, rather than acknowledgements that there were real issues. His criticisms were not taken seriously until that mathematical framework completely fell apart when Fourier series constructed "obviously impossible" things.

Today we are used to a very general notion of a "function". But historically a square wave simply wasn't a function. (Because if it was, how did you know how it interacted with infinitesmals?) And getting one out of the sum of a bunch of well behaved sins, with some simple integration, was a huge shock.

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Yikes! Thanks for setting me right.

I should have said some historical mathematicians then.

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