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.
For a real good example see https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
Honest question: what would've been the consequences of this?
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.
A reasonably recent example of this is the 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_simpl... for more.
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!
Clearly you know more about the ancient history and origins. I'd be willing to bet that you know that the ancient Greeks knew 2500 years ago what prime numbers were, had proved that there were an infinite number of them, had algorithms like the Euclidean algorithm for finding the greatest common denominator, had proven unique factorization AND had demonstrated that sqrt(2) was not a fraction. We don't actually know how much farther a lot of the knowledge goes.
On the other side he had obviously encountered cryptography, and knew that a whole lot of the necessary number theory dates back to Gauss, 200 years ago. https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae is the origin of concepts like modular arithmetic, quadratic residues, and so on. But he was not familiar with the ancient history predating that, or else he could not have thought that the study of primes only goes back 200 years!
He could have avoided the problem on his side by Googling for what he was going to say before saying things with glaring and obvious errors. Very few of us are so careful.
You could have avoided the problem on your side by giving him the benefit of the doubt and assuming that he's probably not a complete idiot, then trying to figure out what he might have meant. You might or might not have figured out "cryptography", but you could have at least made your post in the form of a much more pleasant question. However that is fairly rare to find, and doubly so online.
As for me, I'm just lucky enough to know both halves of the history, so could easily sort it out.
I had thought that it was common knowledge that (small) prime numbers had a lot of practical uses (mental arithmetic in general, arithmetic with fractions, including with vulgar fractions, gear train design, that kind of thing) but apparently I was wrong. It turns out that lots of people don't know about this. So my inference, that only a complete idiot would not know this, he did not know this, and therefore he was a complete idiot, was ill-founded.
And so I came out looking like some kind of ignorant, arrogant know-it-all. I really appreciate the feedback, Ben. Natanael_L, I'm sorry I was such a dick to you.
So you just failed to register the cryptography reference and then backsolve to what he really meant. If that's the worst thing that you did last month, then you're a better person than I...
https://arxiv.org/abs/1605.08831
from May 28, cites this paper, "Wide Residual Networks"
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.
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!
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).
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).