While extremely talented, I am not surprised to find this coming from a teen. Major mathematical discoveries often have come from those in their mid 20’s with the greater discoveries being skewed towards the younger 20s and teens. I think this because pure mathematics is just so creative.
It doesn't help what we've designed a rather silly academic system where principal investigators are forced to spend a good deal of their time thinking where they'll be applying for their next grant. We also optimise the system for short term thinking rather than long games. There are some exceptions in research institutes but I think young people are the ones who have the clearest minds because of it.
I wonder how many years on average people can dedicate themselves to think deep about research problems, before they must stop thinking and start managing, or quit the race.
You finish your degree, and start your PhD. The first year, you are busy learning techniques and getting caught up with the relevant literature.
You are far too concentrated on learning new things to get any thinking done.
In your second year of your PhD, you are getting better. You can do most things without thinking about them. This frees up your brain to think about other things.
However, your grasp of the wider literature is still lacking, so you use that brainspace to optimise your current experiments (as you should).
In your third year of your PhD, you are starting to write things up: either your thesis, or your first (big) paper. You read a lot more, you know a lot more.
The deep thinking can commence.
Your first postdoc is probably your most productive time: you know what you are doing; you know the state of the literature and which parts are reliable and which aren't; you have a clear idea of what problems need solving.
You are starting to write your first grant applications, but you only need one for yourself and not several to cover the needs of a full lab. You don't have any kids at home.
This is a good time to solve some big problems. It lasts about 2-3 years.
At the start of your second postdoc, you panic. The big problem was harder than you thought and you don't have enough high-impact papers to be competitive in job applications for a principal investigator (PI) role.
You start churning out low-value fillers and collaborating with everyone and their hamster to get your name on as many papers as possible.
The rest of the time is taken up by applying for grants and PI positions. You don't even make it to the interview stage. You start pondering about life outside of academia.
You could replace “research” with nearly any term not undergirded by a direct profit motive (eg civics, politics, education, community health, urban planning). Or maybe it’s just (and you may be saying this) that one can only dedicate themselves up until a clear profit opportunity appears.
Whenever I hear this claim about younger mathematicians I wonder if it still holds true (or really did historically). For example, Andrew Wiles proved Fermat’s Last Theorem in his 40s and there are numerous examples of productive older mathematicians. But also I think the claim skews towards big flashy problems rather than the work of building mathematical frameworks, finding structural insights and finding connections between disparate areas (which requires broad experience rather than just young intensity).
That seems like a pretty weak argument. I don’t think he was just filling in easy details for the next decade plus. Also, even 33 is quite a bit older than the commonly claimed early 20s needed for great mathematical work.
> there are numerous examples of productive older mathematicians
Curious about the extreme cases. Did any centenarians ever managed to come with an outstanding original math result? If it didn't happen before, I hope to see it happening in the next decades, given current demographic trends.
I was told that a book published in honor of Oscar Zariski's 80th birthday included a paper by Oscar Zariski, either proving or at least making progress on a longstanding conjecture by Oscar Zariski.
I was in the relevant department at the time (Harvard math), but I wasn't much of an algebraic geometer, so I took that at face value without probing for details.
I think this hasn't been true in a long, long time. The most recent example of major contributions coming from someone in their 20s would be Evariste Galois around the time of the French Revolution.
Peter Scholze received his Fields Medal at 30 for work he carried out through his 20’s. However he is not the youngest recipient: J.P. Serre received his Fields medal at 27.
Before we get too excited about the Fields Medal as an indicator for age of great mathematical achievements, let's remember that it's only awarded to people under 40 for work done earlier, potentially many years earlier.
My understanding is that that’s more about encouraging younger mathematicians rather than an expectation that older ones won’t produce anything worthy.
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I've yet to run out of free image gen credits with Gemini, so I use it for any low-effort image gen like when my kids want to play with it or for testing prompts before committing my o4 tokens for better quality results.
For anyone else confused by the version number bump from 15 to 26, apparently Apple announced they are moving to version numbers that reflect the year _after_ the release date similar to car model years. This change feels odd to me, but I can't put my finger on why.
Based on early comments, which say it makes sense, I’m an outlier.
However, this shifts from something that had semantic value (not saying Apple used Semver, but there were linear major and minor versions) to marketing driven version numbers.
It makes sense to have everyone in 2026 running version 26. It’s also nice to announce 27 ahead of 2027 and people feel like they are on the bleeding edge getting it early
It makes sense to have all the product families on the same versioning scheme because they keep blurring the lines between iOS and macOS but that’s the part I don’t like as much
I don’t like my laptop/desktop OS to look and feel like a tablet interface but this is the way they have been going for a while
> It makes sense to have everyone in 2026 running version 26.
For that, they'd need to release iOS 26 for all devices, including discontinued ones, and force people to upgrade (and prevent them from upgrading to iOS 27 in September 2026!)
Since all that won't happen, no, it does not make sense.
They have control over one thing: The year in which they make a new iOS version available. That's the year they should have named it after.
Change sometimes feels odd at first, but given time it will feel normal.
The new numbering makes sense to me, considering Apple have been on an annual release schedule for their OSes for some time. The question is, will they do this for hardware too? Will the iPhone 17 actually be the "iPhone 26"?
For those on the "it makes sense" camp, remember Apple sticks a year number on their hardware to distinguish models with the same otherwise the same name, e.g. "MacBook Pro (16-inch, 2021) " [0]
And indeed, that specific Macbook was released in 2021, not 2020.
So they're breaking their own convention purely for the marketing benefits.
I don’t think that’s totally fair. The OS version number gets mentioned a lot more often than the year of a specific laptop. Furthermore it’s only made available to the general public close to the end of the year. The majority if its use is seen in a year matching the version.
I’m not saying it’s a perfect system but I can see why they prefer that then having people use iOS 25 for the majority of 2026.
Now that they are referred to by the SoC like M1 M3 etc, but they definitely were known by year models. I have a 2011 MBP, a 2017 MBP, and a 2019 MBP. I couldn’t tell you what cpu it has because Intel’s naming convention is something mere mortals do not know or care about. I know the 2011 had the last Nvidia GPU, the 2017 had the shit keyboard, and the 2019 is the last intel cpu.
To say that they did not use the years sounds like some one commenting on something they are not as familiar as they’d like the rest of the internet to think they are.
Also, people refer to the OS by the cat or California location. I couldn’t tell you what year snow leopard or mavericks came out though
> they prefer that then having people use iOS 25 for the majority of 2026.
I'm not blind to the advantage of their new naming scheme, and honestly they could name it iOS 2077 it would be their prerogative. It just sounds off to me to equate "they're just cheating a bit" to "it makes sense".
I keep pretty up to date with this stuff but I recently read a line on GitHub that said “this software requires macOS 15 or greater”. And I had no idea if that was the current version, a really old version, or the next beta.
When I see a version number, what I really care about is how new/old it is, not how many releases it’s been since the first version. Which is why one of my most common Google searches is “when did <software> <version number> release”
If they switched to the current year at time of release, there would be 6+ months each year where the current version represents the previous calendar year, making it seem outdated.
It’s not marketing fluff, it is maximizing the amount of time out of each year that the year matches the current version number, given that macOS releases are in the second half of the year and not on 1 January.
Doing it with version = release year would mean more days each year where the year != version.
Why are strategic decisions like this always interpreted in the most superficial ways? Apple does scumbag shit frequently (monopolistic anticompetitive practices, walled garden price gouging, illegal wage fixing) but this isn’t that. It’s just math.
> If they switched to the current year at time of release, there would be 6+ months each year where the current version represents the previous calendar year, making it seem outdated.
One could argue that this is simply a marketing concern.
I've never heard of car models named after the year after release. Weird. But here in Europe cars aren't changed every year anyway. I don't think they're ever even numbered.
Except for some brands like Peugeot and formerly Renault but those had nothing to do with the year.
But even even naming them after the year of release makes more sense than the year after. I think for cars it's sometimes done because people don't want to buy cars in the last 3 months because it affects the resale value.
Oh yeah we don't have dodge here in Europe. Maybe it's just a US thing?
Most US models are not sold in Europe because they're unpractically big or don't meet our standards. Even the US brands that operate here like Ford make models specially for the foreign market. Like the Ford Escort (now Focus), Fiesta or Mondeo.
Yes. The paper found 2.4 Hz frequencies to be the secret sauce which is 144 BPM and in pystrance/goa range. I would guess that is common freq range for lfos and modulations as well.
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