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No by and large mathematics is not having a replication crisis. As the blog post in your link states: > Question: Was the proof in [Analytic] found to be corre

No by and large mathematics is not having a replication crisis. As the blog post in your link states: > Question: Was the proof in [Analytic] found to be correct? > Answer: Yes, up to some usual slight imprecisions. This has been the case for almost all math formalization efforts. Even when (very rarely) proofs were revealed to be incorrect, the result was salvageable.

thrown_22 5y ago
That's because most math proofs are treading on well understood grounds and only extending them slightly. E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes. When you enter truly new grounds mathematicians don't even agree if the distinctions being made have a meaning, let alone if they are true.
- dwohnitmok 5y ago
  > When you enter truly new grounds mathematicians don't even agree if the distinctions being made have a meaning, let alone if they are true. What examples are you thinking of? I can think of only one case where this has been the case (Mochizuki and the ABC Conjecture), but that turned out to have so much fanfare precisely because mathematicians did not agree on what distinctions were being made and this was the only time in living memory that this had occurred (the general consensus is that Mochizuki's proof is simply too obfuscated to make heads or tails of). As such the ABC Conjecture is not considered solved. However, that is almost always not the case, even on the cutting edge of mathematics and even when making other breakthrough discoveries (e.g. Fermat's Last Theorem). > E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes. And to be clear the major reason why fields like psychology are termed to have a "replication crisis" is because well-known results are being overturned, not just cutting-edge ones. EDIT: I see now that OP also refers to the ABC Conjecture.
  - thrown_22 5y ago
    >What examples are you thinking of? Back in grad school I was very much into types which let me do things sideways to how most mathematicians did them. The worst one was the derivative of a derivative - NOT the second order derivative. Using R for the real numbers, the type signature of first (and any) order derivative is (R -> R) -> (R -> R). The type signature for what I was talking about was ((R -> R) -> (R -> R)) -> ((R -> R) -> (R -> R)). It didn't have very many interesting properties I could find but trying to explain it to anyone else in the department was like pulling teeth. They'd start thinking about second order derivatives every time and there is no common notation for talking about third order functions like there is for first order and (some) second order functions (derivative, integral, transforms, etc). Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing, not because there is some innate quality in mathematics that pushes mathematicians towards truth.
    - grungegun 5y ago
      >It didn't have very many interesting properties I could find Usually notation is defined for higher order functionals when the need arises - when a new result is found to have interesting properties. Topological proofs on function spaces can use third order functions. > Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing I don't think any person alive can understand all the major branches of mathematics well. By that measure mathematics is very broad.
    - Jensson 5y ago
      > The worst one was the derivative of a derivative What you are talking about already exists, in many different versions at that. I think you really underestimate how varied objects mathematicians works with, they work with so many things that it is hard to come up with original ideas, this idea was already explored close to 200 years ago. https://en.wikipedia.org/wiki/Fractional_calculus#Fractional_derivatives https://en.wikipedia.org/wiki/Fractional_calculus#Fractional...
      - thrown_22 5y ago
zozbot234 5y ago
Note that fixing the "slight imprecisions" in a typical ad-hoc proof is still very valuable. Quite often it leads to a simpler, easier to understand, tighter, etc. argument as the proof can be freely "refactored" without changing its validity. This is the underlying reason why formalizing a proof that hadn't been done previously is considered actual, publishable work.
andrewjl 5y ago
Not rarely. This talk shows several examples: https://www.youtube.com/watch?v=E9RiR9AcXeE https://www.youtube.com/watch?v=E9RiR9AcXeE One of the more important results discussed was not salvageable.
- dwohnitmok 5y ago
  > Not rarely. This talk shows several examples I think you're underestimating the amount of theorems now that have been verified without issue. Those several examples are an absolute drop in the bucket. > One of the more important results discussed was not salvageable. I don't see any important result in that talk that was not salvageable. I see a lemma that had to be changed in support of a larger theorem, but again nothing that "broke downstream papers" so to speak. All the results ended up being fine for their purposes.

thrown_22 5y ago

That's because most math proofs are treading on well understood grounds and only extending them slightly. E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes. When you enter truly new grounds mathematicians don't even agree if the distinctions being made have a meaning, let alone if they are true.

dwohnitmok 5y ago
> When you enter truly new grounds mathematicians don't even agree if the distinctions being made have a meaning, let alone if they are true. What examples are you thinking of? I can think of only one case where this has been the case (Mochizuki and the ABC Conjecture), but that turned out to have so much fanfare precisely because mathematicians did not agree on what distinctions were being made and this was the only time in living memory that this had occurred (the general consensus is that Mochizuki's proof is simply too obfuscated to make heads or tails of). As such the ABC Conjecture is not considered solved. However, that is almost always not the case, even on the cutting edge of mathematics and even when making other breakthrough discoveries (e.g. Fermat's Last Theorem). > E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes. And to be clear the major reason why fields like psychology are termed to have a "replication crisis" is because well-known results are being overturned, not just cutting-edge ones. EDIT: I see now that OP also refers to the ABC Conjecture.
- thrown_22 5y ago
  >What examples are you thinking of? Back in grad school I was very much into types which let me do things sideways to how most mathematicians did them. The worst one was the derivative of a derivative - NOT the second order derivative. Using R for the real numbers, the type signature of first (and any) order derivative is (R -> R) -> (R -> R). The type signature for what I was talking about was ((R -> R) -> (R -> R)) -> ((R -> R) -> (R -> R)). It didn't have very many interesting properties I could find but trying to explain it to anyone else in the department was like pulling teeth. They'd start thinking about second order derivatives every time and there is no common notation for talking about third order functions like there is for first order and (some) second order functions (derivative, integral, transforms, etc). Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing, not because there is some innate quality in mathematics that pushes mathematicians towards truth.
  - grungegun 5y ago
    >It didn't have very many interesting properties I could find Usually notation is defined for higher order functionals when the need arises - when a new result is found to have interesting properties. Topological proofs on function spaces can use third order functions. > Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing I don't think any person alive can understand all the major branches of mathematics well. By that measure mathematics is very broad.
  - Jensson 5y ago
    > The worst one was the derivative of a derivative What you are talking about already exists, in many different versions at that. I think you really underestimate how varied objects mathematicians works with, they work with so many things that it is hard to come up with original ideas, this idea was already explored close to 200 years ago. https://en.wikipedia.org/wiki/Fractional_calculus#Fractional_derivatives https://en.wikipedia.org/wiki/Fractional_calculus#Fractional...
    - thrown_22 5y ago

zozbot234 5y ago

Note that fixing the "slight imprecisions" in a typical ad-hoc proof is still very valuable. Quite often it leads to a simpler, easier to understand, tighter, etc. argument as the proof can be freely "refactored" without changing its validity. This is the underlying reason why formalizing a proof that hadn't been done previously is considered actual, publishable work.

andrewjl 5y ago

Not rarely. This talk shows several examples: https://www.youtube.com/watch?v=E9RiR9AcXeE https://www.youtube.com/watch?v=E9RiR9AcXeE One of the more important results discussed was not salvageable.

dwohnitmok 5y ago
> Not rarely. This talk shows several examples I think you're underestimating the amount of theorems now that have been verified without issue. Those several examples are an absolute drop in the bucket. > One of the more important results discussed was not salvageable. I don't see any important result in that talk that was not salvageable. I see a lemma that had to be changed in support of a larger theorem, but again nothing that "broke downstream papers" so to speak. All the results ended up being fine for their purposes.