Yiting Zhang as discussed in the comments of the MO thread, is one of the most important contemporary examples. He worked as a temporary lecturer and being aged 58 produced one of the finest works in number theory. He solved the long-standing problem infinitely re-occuring consecutive prime numbers (a weaker formulation of twin prime conjecture) [0]. This will be regarded as one of the most important breakthroughs of 21st century in mathematics. New Yorker had an in depth article about him[1]
There has never been more of a Real One than Yitang Zhang. He proved a theorem that had stumped the most eminent professors. While keeping books at a Subway.
To translate his result for people not familiar, unique factorization means that a number uniquely decomposes into primes. You almost certainly learned this happened in school for integers (Z), but it does not apply to all cases.
Quadratic Numbers in Algebraic Number Theory terms are Q[sqrt(-d)], that is, a+b\sqrt(-d) where a and b are rational numbers. d=5 is the first number we can pick where unique factorization does not hold.
In fact, the Stark-Heegner theorem tells us something even more powerful: if d is squarefree, the only imaginary quadratic fields containing unique factorization are when d=1, 2, 3, 7, 11, 19, 43, 67, and 163. Any other choice (or any choice containing a square of any prime, e.g. 4=2^2), and unique factorization will fail.
I've left out what a prime, or indeed an irreducible, mean in this case, but what's astounding at least to me is that there are only 9 such numbers where it works, and this is provable. Heegner did that aged 50+.
Attributing "young man's game" to G. H. Hardy, he'd had a heart attack the year before he published A Mathematician's Apology, and by all accounts, had lost much of his drive and energy.
This really seems like a detail that should be emphasized. One famous man uttered a very quotable line that was more reflective of his mood at the time than of some universal truth. I don't get how the culture of science sometimes has this tendency to fetishize things like youth or pedigree. I guess it's the classic fallacy of confusing averages with maximums or of thinking that summary statistics preclude the possibility of individuals with unusual characteristics.
Makes all of us (since we will all be 30+ one day if not alright) happy to know that Einstein wasted a lot of time trying extremely hard to disprove quantum mechanics because "God doesn't play dice with the universe."
I think you'll find many mathematicians agree with it.
For example, The Abel Prize [1] does interviews with their award winners every year and quite often they are asked about this question, and most of them agree that as a mathematician your peak "mathematical powers" so to speak and energy are at their highest in your 20s and 30s, while afterwards you have to rely more on experience to make up for it as you grow older.
Curious how much of this is truth. I am sure in your 20s or 30s your brain functions better per say, but how much decline is there really and how fast? And surely experience does make up for it in most fields.
I read Hardy's book when I was in my late 20s, as an aspiring scientist. I do remember coming across this quote and thinking it did not apply to software. Last forward 20ish years, and I think it is absolutely true for me. I used to be a code monk/guru in my younger days .. having responsibilities has truly blunted this. The opportunity cost of any project feels too much. Beyond that, I feel I am less foolish and this is problematic. In my younger days, I was code first, and think later. This resulted in things getting built and iterated on. Now, I think a bit too much and this slows down (or completely eliminates) the building part. I also remember a quote from John Carmack about how with unlimited pizza and soda, you can build anything. This felt so true in my teens and 20s .. just not attainable for me now. I think having a somewhat high salary has something to do with it too (it makes one scared) to go on adventures. I think none of my points generalize beyond myself but something feels broken inside me ... maybe just burnout from tough jobs, responsibilities, etc. If anyone has insight on how to overcome, I'm all ears.
There's a quote, usually attributed to Einstein: "the only difference between genius and stupidity is that stupidity has no bounds." As you said, you've put limits on yourself to keep yourself from being stupid.
Remove your limits and you'll be as productive (if not more productive) than any 20-something, though you'll probably do stupid things from time-to-time. Don't be afraid of being stupid, after all, it has no bounds... :p
In all seriousness though, just spend an afternoon coding something you wouldn't normally code without thinking about it first. As you get older, it's harder to just cowboy code something without stopping to think about it. With some practice, you can bang out some pretty high quality code very quickly. And then you'll eventually learn when to cowboy it and when to think.
>This isn't exactly what you were asking for, but Littlewood himself, after overcoming depression at age 72, did good mathematics throughout his 80's--it's hardly a young man's game.
Of course that's not really true. The origin of the story is that so many papers were published by Hardy and Littlewood, that an Indian mathematician thought that Littlewood was a pseudonym that Hardy used for his inferior work.
Littlewood was reported to be quite amused when he heard this.
And of course the story might be apocryphal. Especially since it's also attributed to Edmund Landau and Norbert Wiener.
the majority of the damages of a stroke are generally induced in the following weeks of the event, e.g. via extremely high oxidative stress and apoptotic signaling and impaired bioenergetics.
Those issues are trivial to fix pharmacologically speaking and indeed there are countless studies showing a very potent protection against damage including neurons death, unfortunately doctors have not the required erudition nor do they care to save those lives and therefore people are left helpless and suffering.
I have a small one every few months.
If I’m quick with blood thinners it Takes a couple days to a couple months to get back to normal. Typical average is about a week after an event.
I'm surprised Erdős was so far down the list. He very famously didn't die until he left[0].
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[0]: Erdős had a notoriously quirky way of expressing himself in ordinary conversation. To "die" in Erdős-speak is to quit doing mathematics, while to "leave" is to actually pass away.
I believe an unmentioned counterside to the above question is: how old can you afford to start getting into mathematics in order to still be able to make great discoveries?
I did not see this mentioned so far, so I'll suggest that perhaps older mathematicians are more inclined to offer statements of solvable problems to their students and colleagues rather than jumping in and trying to knock it out all by themselves. (And after all, simply setting up a problem in a form that points to a plausible solution is at least half the battle.)
Surely by the time they reach 50, mathematicians will have the experience to know that mathematics is a collaborative field, and that working alone to get ahead is the less efficient tactic for proving theorems. Some of the most productive mathematicians seem to work with armies of graduate students, and even though an older mathematician may be at the height of their abilities, they will likely be even better at stating unsolved problems that just need more eyeballs.
What do you all think about the role of genius in making discoveries in mathematics today? Are you screwed if you're not a genius? Or is working hard, good collaboration, and wide-reading all that is needed?
Anyone who makes a remarkable discovery in mathematics is going to be considered a genius nearly by definition, so by this, "genius" is all that is involved.
But oppositely, the definition of genius overall is fairly fuzzy.
And especially, people have made important contributions to mathematics without showing "early promise".
I think a genius is a lucky person whose hard work is not very blatant and has got a good hype-man.
If you're unlucky you work hard on a dead end, if people can see you work hard all the time your results are not effortless enough to be the work of a genius, and if you don't have a hype-man you'll at most be seen as very good.
You 100% need to have a significantly above average IQ, just to get to the level where you can start working on any given known mathematical problem.
Once you are at that level, then one could conceivably think that luck+hardwork might be enough to get you results, but when you are considerably smarter, you can work much faster than others, and maybe even make connections that others will never be able to make.
is this a trick question. of course u have to be a genius to make interesting, original research. it was that way 200 years, why would it be different now.
We've built upon the collective knowledgebase, true. But it is as hard for a person today to do arithmetic and logical reasoning as it was in the 1800s.
More kids learn those than before. But that is better explained by everyone spending thousands of hours learning math today, kids spending that much time on math 200 years ago likely learned algebra as well.
My experience in academia has taught me the following: Any slowdown in productivity among mathematicians as they age is more a result of increased administrative duties, childcare, or loss of interest than it is cognitive decline. The slowdown is real, but the commonly suggested reason (biological aging) is not quite right.
You can look up cognitive decline studies and see that it really doesn't hit in full force until the 60s and 70s, with lots of heterogeneity. I've seen plenty of sharp 70-year-old academics doing great work (though I can't remember someone at 80 who I thought was still going full steam).
I think it is way too common to look at studies of the median person and then generalize to older.
Major discoveries are made by top, top people - it seems not unlikely that past a certain age most people cannot remain top. The same is true with chess, which does not have the same administrative duties problem.
I think there are a lot of relevant dis-analogies between math and chess. A chess match at the professional level is a grueling, multi-hour contest, while math research is a lot more chill. Also, the time limit matters a lot in chess.
Anyway, you can look at the pages of Annals of Math and Inventiones and see that there is a good mix of ages.
But I think the OP has a point that falling out of the top has nothing to do with cognitive ability. For instance, many researchers become known for certain contributions and then eventually expected to remain in that field. If the landscape changes (say neural networks become the hot topic) then your research might fall in prestige but the level/complexity/quality does not. There are things beyond their control to prevent quick pivots let alone large pivots. A biologist cannot start doing NLP stuff. Then there is the problem of your graduate students who are doing other things and so the momentum to switch is very real.
Hm. If someone in their 20s is able to make a major discovery for a problem they only heard of when they were 18, that is maximum 12 years of that line of research, which could certainly be replicated by someone older.
you also see people in their 20s make big contributions to many disparate fields, like Tao.
I think you are misunderstanding me. I'm saying that having time in the field is the exact problem. If you have established a pipeline of graduate students and you have 5 of them at various stages of completion. It's difficult to convince new students to join you in a new endeavor because no one in your group does that. Keep in mind, your role as a professor is to take on students. You would also face pressures from your current students to stay in the forefront of your field as that is what they are banking on getting advised from you. If you're 20 years old, you don't care about any of that, you just switch fields.
However, I strongly doubt the various parts and cells erodes any differently. Brilliant minds, bog standard glial cells. Their parts come with the same warranty we receive.
My experience is that ageing is a very similar process.
Acuity and ability in old age follows a very similar trajectory. Use it or lose it.
Whether you're an outlier or the mean, use it or lose it applies just as much.
The difference is, with top minds I find they receive so much external attention that it keeps them focused on their work in their 50s/60s/70s.
All the external plaudits and celebration simply fuels use it or lose it. The brilliant thing is, we can all leverage use it or use it, we just need to maintain our abilities to leverage it.
I am not saying it erodes differently, I am saying measuring median people on median tasks they can perform when they start the study is going to be different from measuring top people on the pinnacle of intellectual achievement, and for those right tail activities you might see degradation earlier that is not caught on median, relatively simple cognitive tasks. Nothing about their brain chemistry being fundamentally different.
There's also an argument to be made about bias - the younger you are the less you 'know,' which can lead to more experimentation and creative solutions to problems.
There's also the 'hunger' argument that dovetails with yours - are younger people just more incented to 'prove' themselves? (since, as you imply, most older people realize a supportive family makes them happier than a well-regarded publication does)
An interesting question to address is 'Is the status quo just fine? Will any solutions to this just make more academics die as virgins or can we actually improve the output of older academics?'
In my experience as a 60 year old, I can recognise cognitive decline in myself compared with younger me. Examples include finding it hard to remember language constructs and APIs, and taking longer to navigate window systems and hierarchical menus. I suspect there are maths equivalents that would impact problem solving performance.
This is in addition to changes in lifestyle and life perspectives that have changed the way I prioritise my tasks and interests.
I'd add disease to that list. Once I got past 30 and put in some weight, just sitting to study became harder, as back pain/discomfort comes real quickly. Around my 20s I just sat hours straight and read entire books on a computer screen, or spent a entire afternoon doing math exercises. Now with a single hour sitting my back already starts hurting, and my focus is broken.
> Sorry to hear that. Have you seen a doctor? This is not something that typically happens to 30somethings.
Yeah. Got pretty standard advice from him. Exercise more, lose weight, book some pilates sessions, pause more often, stretch more often... The point is, your self-maintenance burden only grows as you age, and it starts chewing away time you could spend otherwise. When you're younger, given you have the basics of life taken care by your family, it's easier to give a ton of attention to a few things (say, videogames or even the next big scientific breakthrough) at expense of everything else.
"The point is, your self-maintenance burden only grows as you age, and it starts chewing away time you could spend otherwise"
A 44 y.o. here who spends, say, 15-20 hours weekly on self-maintenance. That includes a lot of walking (to the train station to commute and back etc.) where slightly faster transport alternatives are available.
Yes, this time slice tends to grow slightly, but the rewards are worth it. For example, I used to have back pain that limited me much like you; it is gone.
> your self-maintenance burden only grows as you age
People of all ages ought to exercise. Young people often play sports, no?
> given you have the basics of life taken care by your family, it's easier to give a ton of attention to a few things (say, videogames or even the next big scientific breakthrough)
I'm not aware of any big scientific breakthroughs achieved by teenagers living at home with their parents.
> > given you have the basics of life taken care by your family, it's easier to give a ton of attention to a few things (say, videogames or even the next big scientific breakthrough)
> I'm not aware of any big scientific breakthroughs achieved by teenagers living at home with their parents.
As a 41-year-old, I'm incentivized not to believe that math is a young person's game, and so agree with you; but one could try to disagree with you by suggesting that it's a sweet spot: the years, say, mid-20s to late-30s are the time when the accumulating benefit of knowledge and experience has not yet been outweighed by the accumulating burden of age and responsibility. (And one could, of course, argue against this argument; I think that there is no valid conclusion but "it depends.")
> And one could, of course, argue against this argument; I think that there is no valid conclusion but "it depends."
Not sure why you were arguing yourself in circles here. Yes, it depends. Everyone's life and life circumstances are different. It's a mistake to overgeneralize about what life is like at certain ages, and use that as the basis for ageism. "Older people are more distracted" just seems ridiculous to me.
Some people have no children. People who do have children have them at various ages, sometimes very young. Some people ignore their children. Children tend to leave home within 20 years or so, leaving an "empty nest".
> paying the mortgage
How is that different from paying rent? And how exactly does it "distract" you? My mortgage payment was automatically deducted from my bank account, I didn't even have to think about it. Not to mention, eventually you pay off your mortgage and don't have to pay anymore.
> taking care of elderly family
This can also happen at almost any age. Or never.
What about the distraction of raging young hormones. The distraction of personal uncertainty about life and the future. The distraction of financial insecurity.
I think a more apt description is "owning a home". Owning a home brings all kinds of complications beyond just paying a single bill. One of the major benefits of renting is that you effectively outsource all the maintenance headaches, for example.
> One of the major benefits of renting is that you effectively outsource all the maintenance headaches, for example.
One of the downsides of renting, very often, is that you actually don't outsource headaches, you just add an extra layer of complication on top of them.
You quoted someone else so I can’t speak to their claim.
But mine is that it’s easier to succeed if you’re focused and it’s easier to focus when you have a simpler life. I was just clarifying an important part of the context of the mortgage argument because you seemed to strawman it to be only about the monetary transaction. For some, renting helps simplify their life. It’s not a difficult through line to follow and I don’t think it’s extraordinarily controversial, despite your consistent mischaracterization. (I never said, for example, that “only” renters can succeed).
Do you think it’s easier to succeed running one business or a dozen? Why?
> I don’t think it’s extraordinarily controversial
Do you have any evidence that people generally believe that renting is more conducive to success than homeowning? Should we take it as advice? "If you want to maximize your chances of success, don't buy a house!"
Strangely though, most successful people do own a home. Sometimes more than one! When exactly is it ok to buy a home, and why do successful people do it, if it makes them less focused?
> Do you think it’s easier to succeed running one business or a dozen? Why?
A dozen, of course. Because if you're running a dozen businesses, you likely have a lot of financial resources. And since many businesses fail, you have a greater chance of at least one of them succeeding. (Do I even need to mention that the richest person in the world has multiple businesses and is trying to buy yet another?)
[0] https://annals.math.princeton.edu/2014/179-3/p07
[1] https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty