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> The paper soon set off a cascade of what Ellenberg called “math at Internet speed.” Within 10 days, Ellenberg and Dion Gijswijt, a mathematician at Delft Uni

> The paper soon set off a cascade of what Ellenberg called “math at Internet speed.” Within 10 days, Ellenberg and Dion Gijswijt, a mathematician at Delft University of Technology in the Netherlands, had each independently posted papers showing how to modify the argument to polish off the original cap set problem in just three pages. This is a very interesting trend I've noticed. I wanted to cite a few papers on arXiv in one of my own research papers recently, but my advisor commented that none of the articles had been peer reviewed (since arXiv is a preprint server). I told him that in the last year alone, six papers on arXiv follow a "research trail" (i.e. a paper is put on arXiv in May that builds on results from a February paper that builds on results from December, etc.), and that the most recent peer-reviewed article in a published journal is so far behind the state-of-the-art that completing my paper without any mention to the arXiv works would put me significantly behind the rest of the field. Of course, these papers all relate to math and computer science — whether a new algorithm or proof works is (usually) immediately evident upon implementation, and the papers on arXiv include the complete algorithm and often link to the author's code. Peer-reviewing their work yourself often takes no longer than a half hour or so (unlike, say, a research article in materials science, where a complete replication study could take over a year).

Ankaios 10y ago
That has been common in physics for a long time, too. For hot new topics, the conversations are often happening through preprints rather than peer-reviewed articles. (The papers are usually eventually published in peer-reviewed journals, though.)
- MichaelBurge 10y ago
  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.
guelo 10y ago
I'm not familiar with academia but I was surprised that the discovery was made by three mathematicians at three different far flung universities. Is that type of collaboration also being accelerated by the internet?
- sdenton4 10y ago
  Collaborators will often meet at conferences or similar, and then go out of their way to attend the same conferences. There's a big difference between the fast in person work that lays the ground work, and the refinement that goes into finishing the paper. The internet is way more useful for the latter, imho. It's much easier to share ideas quickly on a black board...
personjerry 10y ago
> whether a new algorithm or proof works is (usually) immediately evident upon implementation I would argue that this is a dangerous claim, because an algorithm working generally is different from proof that it always works, and this can lead to serious issues. I would say that this is one of the big reasons peer review exists!
vintermann 10y ago
Yup. An example from machine learning: This paper, "Weighted Residuals for Very Deep Networks" https://arxiv.org/abs/1605.08831 https://arxiv.org/abs/1605.08831 from May 28, cites this paper, "Wide Residual Networks" https://arxiv.org/abs/1605.07146 https://arxiv.org/abs/1605.07146 from May 23. Less than a week between them! And it's with good reason too, because they want to compare with state of the art on a certain problem/dataset, and that improved five days ago. Both of these papers are about very basic advances, in a sense: If you have a residual network implementation lying around (and there are plenty of open source ones), it's trivial to implement the improvements they propose. So it's not as crazy to cite a 5 day old paper as it seems.
A1kmm 10y ago
The situation could be even better if the proposition and proof could be written in a machine verifiable form using a proof assistant like Coq. Potentially you could have a section in a maths journal which 'papers' (in the form of computer readable propositions and proofs) are immediately reviewed for correctness by a computer. Post-publication, humans may then assess the impact / significance by voting (perhaps with propositions from significant conjectures potentially configured to instantly go to the top of the significance rankings).

Ankaios 10y ago

That has been common in physics for a long time, too. For hot new topics, the conversations are often happening through preprints rather than peer-reviewed articles. (The papers are usually eventually published in peer-reviewed journals, though.)

MichaelBurge 10y ago
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.

guelo 10y ago

I'm not familiar with academia but I was surprised that the discovery was made by three mathematicians at three different far flung universities. Is that type of collaboration also being accelerated by the internet?

sdenton4 10y ago
Collaborators will often meet at conferences or similar, and then go out of their way to attend the same conferences. There's a big difference between the fast in person work that lays the ground work, and the refinement that goes into finishing the paper. The internet is way more useful for the latter, imho. It's much easier to share ideas quickly on a black board...

personjerry 10y ago

> whether a new algorithm or proof works is (usually) immediately evident upon implementation I would argue that this is a dangerous claim, because an algorithm working generally is different from proof that it always works, and this can lead to serious issues. I would say that this is one of the big reasons peer review exists!

vintermann 10y ago

Yup. An example from machine learning: This paper, "Weighted Residuals for Very Deep Networks" https://arxiv.org/abs/1605.08831 https://arxiv.org/abs/1605.08831 from May 28, cites this paper, "Wide Residual Networks" https://arxiv.org/abs/1605.07146 https://arxiv.org/abs/1605.07146 from May 23. Less than a week between them! And it's with good reason too, because they want to compare with state of the art on a certain problem/dataset, and that improved five days ago. Both of these papers are about very basic advances, in a sense: If you have a residual network implementation lying around (and there are plenty of open source ones), it's trivial to implement the improvements they propose. So it's not as crazy to cite a 5 day old paper as it seems.

A1kmm 10y ago

The situation could be even better if the proposition and proof could be written in a machine verifiable form using a proof assistant like Coq. Potentially you could have a section in a maths journal which 'papers' (in the form of computer readable propositions and proofs) are immediately reviewed for correctness by a computer. Post-publication, humans may then assess the impact / significance by voting (perhaps with propositions from significant conjectures potentially configured to instantly go to the top of the significance rankings).