4 ms·

Building on research seems like it's a form of peer review, especially in math where you'd need to understand the entire proof before making incremental improve

Building on research seems like it's a form of peer review, especially in math where you'd need to understand the entire proof before making incremental improvements to it.

abritinthebay 10y ago
While this can be true it's also common to have "assuming X is true" where X is some very complicated hypothesis. We saw this with Fermat's Last Theorem and it was with a great sigh of relief that it was finally proven in the 90s. If the inverse was true then entire fields of Mathematics would have collapsed.
- throwaway676565 10y ago
  > If the inverse was true then entire fields of Mathematics would have collapsed. Honest question: what would've been the consequences of this?
  - schoen 10y ago
    Maybe this is more true for research that assumes the truth of the Riemann Hypothesis than research that assumed the truth of Fermat's Last Theorem? Maybe there was some significant body of research before 1995 that assumed the Taniyama–Shimura–Weil conjecture, a more powerful statement that implied Fermat's Last Theorem and was ultimately proven as a way of proving it. https://en.wikipedia.org/wiki/Modularity_theorem https://en.wikipedia.org/wiki/Modularity_theorem
  - btilly 10y ago
    The consequence is that a whole body of work winds up coming with an asterisk until people figure out what they can and can't trust. Papers may be looked at for inspiration, but won't be quoted for results. Eventually some of it gets proved properly, and the rest is abandoned. After that the older papers become mere historical curiosities. A reasonably recent example of this is the https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge.... A possible place where this could happen is the classification of finite groups. It has been "proven", but the proof is long, technical, and never was adequately reviewed. Lots of papers these days start off using the classification in interesting ways. However there is an open program to produce an actual reviewed proof. If in the process of doing that, we found that the original result was long, there would be a fairly large project to figure out the consequences. See https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl... for more.
    - tamana 10y ago
      But when the results are useless anyway, it doesn't really matter if they are right or wrong...they just may be speculative of some alternate universe, or may still contain ideas that are applicable elsewhere.
      - Natanael_L 10y ago
        Prime numbers used to be useless when first researched (edit: during the previous two centuries, when their properties was studied). We don't always know in advance what will turn up useful.
        kragen 10y ago
        You say, "Prime numbers used to be useless when first researched," but when the Middle-Kingdom Egyptians were doing their initial research on prime numbers, they needed them for the algorithms they used to calculate with fractions. These were used in the Rhind Papyrus to calculate things like the volumes of granaries. You could hardly have picked a worse example.
        btilly 10y ago
        No, he picked a famous and perfect example. He just didn't specify it well enough. Over the last 2 centuries, number theorists developed the theory of large prime numbers. The numbers that they were dealing with were so large that they had no conceivable use in describing the physical universe. Famously one prominent number theorist, G. H. Hardy, wrote A Mathematician's Apology, a book describing and justifying his life. In it he famously described his field as being utterly useless with no practical applications. Then cryptography came along, and the mathematics of finding large prime numbers, and factoring hard to factor large numbers, turned out to have practical applications of great importance!
        kragen 10y ago
        Please don't blame me for refuting what he did say, instead of what he would have said if he'd known what he was talking about.
- xamuel 10y ago
  FLT is a poor example. Very little depended on it, and if it weren't for its colorful history, it wouldn't ever have gotten the exposure it did. For a real good example see https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...

Building on research seems like it's a form of peer review, especially in math where you'd need to understand the entire proof before making incremental improvements to it.

abritinthebay 10y ago

While this can be true it's also common to have "assuming X is true" where X is some very complicated hypothesis. We saw this with Fermat's Last Theorem and it was with a great sigh of relief that it was finally proven in the 90s. If the inverse was true then entire fields of Mathematics would have collapsed.

throwaway676565 10y ago
> If the inverse was true then entire fields of Mathematics would have collapsed. Honest question: what would've been the consequences of this?
- schoen 10y ago
  Maybe this is more true for research that assumes the truth of the Riemann Hypothesis than research that assumed the truth of Fermat's Last Theorem? Maybe there was some significant body of research before 1995 that assumed the Taniyama–Shimura–Weil conjecture, a more powerful statement that implied Fermat's Last Theorem and was ultimately proven as a way of proving it. https://en.wikipedia.org/wiki/Modularity_theorem https://en.wikipedia.org/wiki/Modularity_theorem
- btilly 10y ago
  The consequence is that a whole body of work winds up coming with an asterisk until people figure out what they can and can't trust. Papers may be looked at for inspiration, but won't be quoted for results. Eventually some of it gets proved properly, and the rest is abandoned. After that the older papers become mere historical curiosities. A reasonably recent example of this is the https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge.... A possible place where this could happen is the classification of finite groups. It has been "proven", but the proof is long, technical, and never was adequately reviewed. Lots of papers these days start off using the classification in interesting ways. However there is an open program to produce an actual reviewed proof. If in the process of doing that, we found that the original result was long, there would be a fairly large project to figure out the consequences. See https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl... for more.
  - tamana 10y ago
    But when the results are useless anyway, it doesn't really matter if they are right or wrong...they just may be speculative of some alternate universe, or may still contain ideas that are applicable elsewhere.
    - Natanael_L 10y ago
      Prime numbers used to be useless when first researched (edit: during the previous two centuries, when their properties was studied). We don't always know in advance what will turn up useful.
      - kragen 10y ago
        You say, "Prime numbers used to be useless when first researched," but when the Middle-Kingdom Egyptians were doing their initial research on prime numbers, they needed them for the algorithms they used to calculate with fractions. These were used in the Rhind Papyrus to calculate things like the volumes of granaries. You could hardly have picked a worse example.
        btilly 10y ago
        No, he picked a famous and perfect example. He just didn't specify it well enough. Over the last 2 centuries, number theorists developed the theory of large prime numbers. The numbers that they were dealing with were so large that they had no conceivable use in describing the physical universe. Famously one prominent number theorist, G. H. Hardy, wrote A Mathematician's Apology, a book describing and justifying his life. In it he famously described his field as being utterly useless with no practical applications. Then cryptography came along, and the mathematics of finding large prime numbers, and factoring hard to factor large numbers, turned out to have practical applications of great importance!
        kragen 10y ago
        Please don't blame me for refuting what he did say, instead of what he would have said if he'd known what he was talking about.
xamuel 10y ago
FLT is a poor example. Very little depended on it, and if it weren't for its colorful history, it wouldn't ever have gotten the exposure it did. For a real good example see https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...