"Recently, yet another category of low-hanging fruit has been identified as within reach of automated tools: problems which, due to a technical flaw in their description, are unexpectedly easy to resolve. Specifically, problem #124 https://www.erdosproblems.com/124 was a problem that was stated in three separate papers of Erdos, but in two of them he omitted a key hypothesis which made the problem a consequence of a known result (Brown's criterion). However, this fact was not noticed until Boris Alexeev applied the Aristotle tool to this problem, which autonomously located (and formalized in Lean) a solution to this weaker version of the problem within hours."
That doesn’t seem very fair. The problem was stated, and remained unsolved for all this time. You can’t take away that accomplishment just because the solution seems easy in hindsight.
It's technically true that this version of the problem was "low-hanging fruit", so that's an entirely fair assessment. Systematically spotting low-hanging fruit that others had missed is an accomplishment, but it's quite different from solving a genuinely hard problem and we shouldn't conflate the two.
My point is stronger than that. Some things only appear low hanging fruit in hindsight. My own field of physics is full of examples. Saying “oh that should’ve been easy” is wrong more often than it is right.
It’s a combination of the problem appearing to be low-hanging fruit in hindsight and the fact that almost nobody actually seemed to have checked whether it was low-hanging in the first place. We know it’s the latter because the problem was essentially uncited for the last three decades, and it didn't seem to have spread by word of mouth (spreading by word of mouth is how many interesting problems get spread in math).
This is different from the situation you are talking about, where a problem genuinely appears difficult, attracts sustained attention, and is cited repeatedly as many people attempt partial results or variations. Then eventually someone discovers a surprisingly simple solution to the original problem which basically make all the previous paper look ridiculous in hindsight.
In those cases, the problem only looks “easy” in hindsight, and the solution is rightly celebrated because there is clear evidence that many competent mathematicians tried and failed before.
Are there any evidence that this problem was ever attempted by a serious mathematician?
Sure, but unless all solvable problems can be said to "appear as low-hanging fruit in hindsight" this doesn't detract from Tao's assessment in any way. Solving a genuinely complex problem is a different matter than spotting simpler solutions that others had missed.
In this case, the solution might have been missed before simply because the difference between the "easy" and "hard" formulations of the problem wasn't quite clear, including perhaps to Erdős, prior to it being restated (manually) as a Lean goal to be solved. So this is a success story for formalization as much as AI.
One of the math academics on that thread says the following:
> My point is that basic ideas reappear at many places; humans often fail to realize that they apply in a different setting, while a machine doesn't have this problem! I remember seeing this problem before and thinking about it briefly. I admit that I haven't noticed this connection, which is only now quite obvious to me!
Doesn't this sound extremely familiar to all of us who were taught difficult/abstract topics? Looking at the problem, you don't have a slightest idea what is it about but then someone comes along and explains the innerbits and it suddenly "clicks" for you.
So, yeah, this is exactly what I think is happening here. The solution was there, and it was simple, but nobody discovered it up until now. And now that we have an explanation for it we say "oh, it was really simple".
The bit which makes it very interesting is that this hasn't been discovered before and now it has been discovered by the AI model.
Tao challenges this by hypothesizing that it actually was done before but never "released" officially, and which is why the model was able to solve the problem. However, there's no evidence (yet) for his hypothesis.
Is your argument that Terence Tao says it was a consequence from a known result and he categorizes it as low hanging fruit, but to you it feels like one of those things that's only obvious in retrospect after it's explained to you, and without "evidence" of Tao's claim, you're going to go with your vibes?
Smalltalk Environment - the dawn of IDEs - "Don't mode me in"
"Novices are not the only victims of modes. Experts often type commands used in one mode when they are in another, leading to undesired and distressing consequences. In many systems, typing the letter "D" can have meanings as diverse as "replace the selected character by D," "insert a D before the selected character," or "delete the selected character." How many times have you heard or said, "Oops, I was in the wrong mode"?"
It's worth noting that the debunking has been debunked (it's much more nuanced but essentially you can take the quotes, experience of others and the height of the bridges at face value):
"I recorded clearances for a total of 20 bridges, viaducts and overpasses: 7 on the Bronx River Parkway (completed in 1925); 6 on the initial portion of the Saw Mill River Parkway (1926) and 7 on the Hutchinson River Parkway (begun in 1924 and opened in 1927). I then took measure of the 20 original bridges and overpasses on the Southern State Parkway, from its start at the city line in Queens to the Wantagh Parkway, the first section to open (on November 7, 1927) and the portion used to reach Jones Beach. The verdict? It appears that Sid Shapiro was right."
"Overall, clearances are substantially lower on the Moses parkway, averaging just 107.6 inches (eastbound), against 121.6 inches on the Hutchinson and 123.2 inches on the Saw Mill."
If buses have always about 118" that would be effective.
I can’t read the second article, but the first does not cite the later work and makes exactly the same mistake as Winner: sitting at a desk, assuming that (per Caro) parkways were the only way to get there, and looking only at the parkway bridge heights. The point is that parkways were by definition not for commercial traffic and you could get there by routing around the parkway overpasses, and the contemporaneous bus schedules show that they did.
Since busses aren't ”commercial”, I'm not sure why we need to go around measuring bridges to decide if this was an anti-poor design decision (which in many areas and times in the US would also make it a racist one).
Video from 1979 about F International (Dame Shirley's company) - working from home, as a computer a programmer, is the greatest revolution for working women since the pill. https://www.youtube.com/watch?v=b6URa-PTqfA&t=295s
No, the 14% figure is for people that self-requested testing. You can see it as an upper boundary, but it won't be the actual figure, which is likely around 8% or so.
If I were being charitable, the article is saying "Yes, they were grossly and intentionally negligent, but it was only because they were rationally maximizing profit given the market and regulatory environment."
https://mathstodon.xyz/@tao/115639984077620023