Actually I think Math is more or less a young people's game is because whence someone be super successful and famous it's kinda difficult psychological to retain the previous mental state and push out similar results.
https://mathoverflow.net/questions/25630/major-mathematical-...
https://www.quantamagazine.org/20170328-statistician-proves-...
https://www.quantamagazine.org/marjorie-rices-secret-pentago...
Her discoveries first mentioned in a 1988 magazine when she was 65.
I agree.
Disclosure: a Brit who does not see Americans oppressed by compatriot affectations.
That house price appreciation must just be that good.
(Also, it was reported he "worked at a Subway" but IIRC he was actually the accountant for a friend's Subway franchise.)
Source: https://yewtu.be/watch?v=88Q2v6FTSBI 49:03 - 49:56
The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute anything novel, statistically speaking. And as population seems to be close to peaking, we will also have less chances of exploring the extremes of mathematical dexterity.
Perhaps we could then rely on computer assisted theorem provers. Or life extension, as long as intellectually productive years are also increased. Or we will need to focus and specialize kids earlier on.
>The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient
Hmm. I had never thought of it like this. Is it possible for human knowledge to become so advanced in a single subject that it takes a persons entire life to learn just one subject? That's already true of something like the human body, which is why doctors specialize in organs or regions of the body. The more human knowledge expands, the more individuals specialize into increasingly smaller niches.
Feels like attention would become the limiting factor.
I've read it, lost it, and have been searching for it for a few years now.
There's limit to human knowledge, but we are getting more and more efficient at communication, both with other humans and with machines.
Even without something like Neuralink we are creating abstractions and interfaces allowing us to quickly connect our work with others. E.g. especially with ML you can lazy load all necessary context.
The cognitive burden reduces via the new foundations not requiring deep understanding of the old ones to be able to make new steps forward.
But there is no universal law that says that the foundations of a field must be simple enough for one human to understand them in 70 years. It may well be that the simplest possible statement of a field of knowledge is still too complex for a single human to understand it in a normal life-span (not to mention that our capacity for storing information is limited - you can't actually continually learn new things for 70 years without forgetting much of what you learned initially).
Even today, you could spend literally your entire life trying to learn everything we know about the human body and you would almost certainly die before having learned everything. Now, fortunately, there is plenty of real work, both as a doctor and as a researcher, that can be done by focusing on just one aspect of the body (say, the circulatory system) and having only relatively shallow knowledge about the rest. Still, there is the possibility already that a mind that could build on all of the deep knowledge we have could come up with new ideas in medicine and biotech that we are unable to because of this silo-ing.
No, that's exactly the point. You can thereby drop the cognitive load of C++ and build greenfield on the Rust foundations.
People talk about this a lot. While I think it could happen for certain subdisciplines (it already takes essentially an entirely PhD's worth of time to learn all the necessary background to be an algebraic geometer, so most algebraic geometry PhD students publish nothing besides their thesis during their PhD studies), it can never happen to mathematics as a whole. If one part of math gets too deep, you can always go somewhere else, where the water is still "shallow."
Math as a whole may last longer, but this list reminds us how far we’ve come in a mere few millennia: https://usercontent.irccloud-cdn.com/file/SaI50Q1d/166786520...
On the timescale of civilization, it seems less and less likely that lone mathematicians can revolutionize the field.
We’re fortunate to have been born so early, relatively speaking.
Mathematics is not just a small integer multiple larger than this.
It's not that math is vastly larger (or more sophisticated) as a field. Both fields infinitely large in many senses. Rather, the number of respectable starting points where you can do interesting things is much larger, orders larger in math.
Looking from the outside, physics suffers a lot from fashion/hot trend tendencies, where you need to be doing the "hot" thing to make the jumps necessary to the coveted Tenure Track — and otherwise, you get kicked out.
Which inspires the question: how much can cutting edge math be parallelized?
Maybe some CAS-assisted work gets us into feedback loops allowing us to go indefinitely, as in a technological singularity.
But the "shallow" part is also quite wide.
You can teach people what you've learned forever, for instance.
I don't know what field you are in so it may or may not be helpful to you.
Yes, but the shallow areas aren't very interesting, which is why people work in the deep areas.
The simplest example that comes to mind is that you can learn group theory without really needing to know anything about Galois theory. I also imagine there's a lot of good math that has shed vestigial physics...
Maybe we will keep forever discovering new shallow areas but I suspect this is not the case. In any case this is a phenomenon that I think will play out in the next few hundred years, not sufficiently impactful in the next few decades but more and more noticeable.
So even if world population declines for a while, I suspect it'd be possible to enjoy more mathematical talent in the coming decades / centuries.
Another angle here, I remember seeing a study where some children were just... taught algebra. Like given high school algebra classes in 3rd grade, and kids were able to absorb all that abstract reasoning "just fine" (according to the study).
Of course there's only so much abstraction that can be done, but I think we shouldn't assume we are at the end of history on much of anything (except for parsing algorithms)
There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them.
In fact, Maxwell's Equations were originally derived with quaternions.
And speaking of Maxwell's Equation"s" :
With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning.
Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns.
Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions.
The abstractions need to be understood, but potentially only to a level of n-1. Isn't this one of the core principles of proofs?
I.e. Today's world wouldn't support an individual who could have such an impact on most math/scientific disciplines?
General relativity was much more difficult. It took Einstein about a decade to develop it and he had to learn differential geometry in order to do so. This work was undoubtedly more profound and required much more advanced mathematics merged with deep insights such as the equivalence principle [0]. This was what made Einstein so successful: the combination of mathematics with physical insights that no one else had put together at the time.
Newton's Gravity was known to be incomplete for many years, just by looking at Mercury. It took several carefully derived experiments studying light for 100s of years that led us to Maxwell's equations in the 1860s that became the basis for Einstein's special relativity in the 1910s. You don't need a billion dollar particle accelerator to come up Maxwell's Equations (2 of them were done by Guass in the 1700s!). 20 or 30 years (and talent!) studying calculus or physics could get you something.
Today we know Einstien's theory of gravity is incomplete but the only places it is incomplete is inside of a black hole (good luck running a test in a blackhole) or at very tiny scales where gravity's affect is minimal. Today, I don't know how you would even come up with a competing theory without having millions to spend on a particle accelerator. While einstein famously just did a thought experiment on what should happen if the speed of light is constant for all observers, most "discoveries" are done now by smashing particles in billion dollar tubes.
Theoretically speaking, it's not that hard to run a test inside a black hole (getting to one is the hardest part that we know of). It's communicating the results to anyone else that's "slightly" harder.
Jokes aside, in physics we have both a problem with what can be tested, but also one on the theory/mathematics side. Even for a relatively technical problem, we don't have any good way of solving the actual equations of any slightly complex system in either GR, QM, even Newtonian mechanics. We are actually always relying on numerous approximations and simplifying assumptions, and some of these could themselves lead us astray in some cases.
There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them.
In fact, Maxwell's Equations were originally derived with quaternions.
So like, the food is poisonous on strobes 3 and 8 of a 14 strobe cycle, and understanding that is the key to staying alive in their environment.
IMO, an under appreciated dynamic across the board of human endeavours.
With increasing complexity comes the need for more time to understand and master anything.
What does that mean?
Additionally, the belief that intelligence alone will improve the world is misguided imo. You need empathetic, intelligent people.
* (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th century) political philosophy.
* Everybody posting on this board is using a machine that only existed in the mind games (and works) of math philosophers just over a century ago.
* The philosophical foundations for contemporary science are very young, practically within living memory. Popper died less than 30 years ago!
Christianity, like just about every religion, has made invaluable contributions to philosophy. But it's ahistorical, to an embarrassing degree, to attribute such broad concepts and bodies of inquiry to it.
(Just a single, rather obvious example. Easy enough to find others.)
Was this philosophy finding truth that guided policy, or was it making up rhetorical arguments in a power struggle on behalf of the rising middle class and its policy?
In absence of criteria of non empirical truth that are independent of the very non empirical truths they are criteria for I tend to believe the latter.
In addition to this inherent weakness philosophical arguments are not for the masses anyways. Not even philosophers usually agree what they mean or entail. How could those idea(l)s have guided the revolution as more than just superficial self justification?
Remember that the enemies of the French Revolution had their philosophers too, they just lost.
Historical events usually have very mundane reasons, class struggles, geopolitics etc. Ideas may play a part too. But I believe not in the sense that they are the horses in front of the cart.
To me, the idea that ideas don't play a major role in world history and politics seems odd. There's nothing more fundamentally human than trying to understand the world around you and wanting to shape it in a certain way. Of course material conditions are important; the French Revolution wouldn't have happened had the state not gone bankrupt. But it also wouldn't have happened in that way without Enlightenment ideas. Very basic principles of the revolution, some of which were kept by Napoleon and in subsequent generations and still live on today were formulated by those enlightenment thinkers. Just consider the declaration of human rights.
I’m saying that the enlightenment ideas happened because the uprising middle class needed justifications in their power struggle. Not that the ideas formed a middle class that then acted upon them.
I agree with you that humans need ideas to tell a story of themselves. And that story better paint them as good! But that story is usually not the driving force.
So I fail to see progress as I don’t see a criterion to measure progress (in general. Aristoteles‘ logic is a tool. Here I see how to compare it to FOL and judge which is better or worse for specific use cases).
As for human rights: do I agree with them? Yes! Can I justify them? No! Do I know where to exactly draw the lines when two rights collide? No! Is it important to justify them in a philosophical way? I doubt.
The claim that propositional calculus is in the domain of philosophy and mathematics is not controversial in either domain. Nor is it unusual: the history of philosophy is the history of discharging subjects once they develop a field of their own.
I am not sure what that claim means precisely. Popper saved science from… a certain group of philosophers?
> when we “do” science, the epistemic model we use is less than a century old
The epistemic model is whatever the scientist in question uses. One can be a Popperian, another can claim they are guided by God, and someone else would just say “shut up and calculate.”
Popper didn’t like Bohr and his Copenhagen interpretation from the philosophical standpoint yet he didn’t deny that Bohr was a good physicist.
> The claim that propositional calculus is in the domain of philosophy and mathematics is not controversial in either domain. Nor is it unusual: the history of philosophy is the history of discharging subjects once they develop a field of their own.
The context of this discussion is that philosophy progresses over time. You cannot say that it progresses just because we put a label “philosophy” on every new thing that isn’t formalized yet. In that trivial sense, of course, philosophy had great progress; especially in its subfield of natural philosophy also known nowadays as science. But did Frege really stand on the shoulders of Hegel, Spinoza and Kant and improved on their ideas?
Ok, sure.
> allowing science to progress without imploding under the demands of positivism
How is giving a description of something necessary for its continued progress? Were people not having sex before evolutionary psychologists and biologists elucidated how and why?
https://aeon.co/essays/what-albert-einstein-owes-to-david-hu...
> * (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th century) political philosophy.
A large number of countries are ruled by dictators and their close supporters. Sometimes they position themselves within the boundaries of a more or less modern ideology but ultimately care only about being the one in command and use that ideology as lipstick. This is a straight continuation of prehistory, not something stemming from 17th century.
...and I disagree with the parent's parent's point on progress in philosophy: all three examples are not progress of philosophy but at best influence or application of it, and in the last case did not involve trained/professional philosophers (e.g. Turing was a mathematician/chemist-turned-cryptoanalyst/computer scientist).
A second example is that of recent advances in virtualization, where the last decade of advances in things like cgroups, namespaces and containers were all done by people who I assert had no training in IBM MVS/zOS and therefore weren't building on what went before (imho, to their detriment).
A lot of mathematics is cumulative, though.
And even if you can go further by going thinner (specialising more), maybe breakthroughs require lateral thinking and connections that are predicated on not being too specialised, but having breadth also. If that is the case, sooner or later we might be in trouble.
Otherwise, I’d say our two thoughts are connected. With increasing difficulty in understanding new progress, there could be an inertial tendency to over emphasise the importance of old knowledge because it’s comforting/easier/pragmatic for teachers and parents.
When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable.
For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas.
At university I used to have a teacher that started with: history of mathematics, why they were invented, made a point about its primarily practical origins.
To explain things, he could most of the time come with real-life instances of application and there were much more often intuitive or geometric interpretations of the techniques used much more often even before starting the explanation itself to have an intuitive idea and visualization of what you were achieving.
After that, I noticed that to learn math, the first thing is to develop an intuitive, non-mathy idea of what you are doing and later formalize it.
At school and high school they just taught it as almost-memorize tables, apply formulas.
Talking about Spain, btw.
Or do you mean mulitplying out the terms:
(a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2
?We should be explaining the story of maths and how it benefitted society. We should be asking kids about their interests and then showing them how mathematical tools can be used in those areas. We need to show kids how maths is tied to real life rather than just presenting them with a boring formulas to memorise.
I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society".
There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)
To put the blame on didactically seems to miss the more important factor that humans just aren’t that intelligent, save the one in a million genius who might have the intellectual capacity to learn something so difficult at such a young age.
(genuine question, as I don't know anything about it)
Over time I think we need to either create a GAI or enhance ourselves(via genetic engineering, brain-computer-interface, eternal youth or other ways) to overcome this fundamental human limitations.
I don’t foresee AI overtaking human cognition for at least a decade or two.
That's not really the case here. Zhang spent 10 years out of school working in fields in the cultural revolution, and didn't start college until he was 23. After his PhD, he couldn't get an academic job for 8 years and ended up delivering food and working as an accountant. He only got a part time lectureship after that. He wasn't made a professor until his big proof at the age of 58.
It seems incredibly likely he'd have been able to his big proofs faster had his life gone otherwise.
https://www.goodreads.com/quotes/99345-i-am-somehow-less-int...
In a field where finding every genius really matters because of the difficulty of expanding the frontiers it's heartbreaking to realise most just never get the opportunity of good mathematical education in the first place.
However, isn't it possible that giving great teaching resources to people who aren't already at the top level in terms of raw talent, allows them to succeed at a very high level? If so, it's not a zero sum game - the 'averagely clever people who have received great tuition' and who go on to succeed might be a sign of the education system working, not its failure. We might focus more on making sure a wider group of 'averagely clever people' get such good tuition, rather than on the finding and promotion of undiscovered geniuses.
Two examples that I often think about in this context: firstly the Budapest school of mathematics in the early 20th century. A hugely disproportionate number of the world-class mathematicians from that era were directly taught by Fejer and/or part of his seminar and mentee group. (He stands out as a supervisor to a far greater extent then any mathematicians stands out from the pack for his own achievements.) Is it possible that while one or two of this group might have been 'born geniuses', it also includes several who might otherwise have been second-rate, but who became world-class because of the influence both of Fejer and of their high-performing colleagues?
Secondly, George Harrison of the Beatles. In the second half of the Beatles' recording career, he wrote some of their best-loved songs, several of which are regarded as absolute classics. Did the Beatles happen to contain three inherently gifted songwriters, by sheer coincidence? Or is it more likely that working for a decade alongside two extremely talented and successful songwriters nurtured and elevated Harrison far beyond what might otherwise have been expected?
There's a catch in it as well, which is that you cannot really identify a genius up-front. A genius is, by definition, someone who thinks differently. A lot of people think differently. But the thing with the genius is that he thinks differently and correctly. The latter part won't be obvious until after the fact.
When Alex Ferguson started out as a Manchester United manager he probably already was a genius, but it wasn't obvious until many years later.
Same with Sergey Brin and Larry Page. They probably had a genius vision for the company right from the start. But it's only in retrospect that we recognize that it was, in fact, a genius vision.
Finding a genius is, I think, almost by definition, impossible.
He was just a good man-manager in what is a sea of managerial mediocrity - football management is largely restricted to ex-pro-footballers who can't do anything else. Rinus Michels was a genius, Arrigo Sacchi was a genius - they actually advanced the sport.
It's an interesting counterfactual: without Page, when would Page Rank have come around? The idea that the stationary distribution of a Markov Chain (under certain conditions) is given by the eigenvector to the (largest) eigenvalue 1 is certainly decades old, if not a century.
That seems inconsistent with reports that they tried to sell early to Yahoo! for $2m.
The failure is that mediocre students which come from an environment which helps them (good school, private tutors, stable home life) can be successful, while "naturally gifted" yet disadvantaged students don't.
The basic fact that parental income is still the best predictor of educational achievement should tell you everything you need to know.
I think you have rephrased what I said to again push the idea that it's a zero sum game. If mediocre students can be successful, given the right educational inputs (for some definition of 'successful' and 'mediocre'), isn't this a huge opportunity? Perhaps we should improve society as well as social justice by trying to promote mediocre students from poorer backgrounds to this same level of success, rather than by working on the problem of undiscovered genius.
> The basic fact that parental income is still the best predictor of educational achievement should tell you everything you need to know.
My whole point is that this does not tell you everything you need to know. If low-parental-income and high-parental-income students had exactly the same potential to succeed, the fact that parental income is a good predictor of achievement might be telling you either (or any combination) of two things: that wealthier students are achieving beyond what should be expected (and so we are ending up with incompetent people in very senior positions because they did not get them on merit), or that poorer students are achieving below where they should be (and so we are wasting the potential of talented people). These are really different problems! And probably have quite different solutions.
Two other hypotheses are that the money itself is not important, but rather the familial transmission of certain cultural traits (which also often lead to high incomes), and/or that, similarly, the causal factor is genetic transmission of some cognitive or personality traits. Note that a factor being genetic does not mean it is immutable - for instance, there could be a genetic factor that leads to students doing poorly in school because they are intolerant of sitting still for hours in desks, which would become irrelevant if one stopped requiring them to sit still for hours.
A mistaken idea of the causal explanation for the correlation with income would lead to interventions that don't actually work.
Is that controlled for genetic influence though? Suppose smart people 1) have high income, 2) have smart kids.
Disentangling the various causal factors is substantially harder than your quote above suggests.
Most students with creative insights that could help advance the field tend to wind up one of two ways: (We need to create a viable, replicable third way - today it seems to require exceptional personalities for both student and teacher...) 1) They slog through the existing math education system, but it crushes their creativity and insight, leaving them mostly useless husks (this pit catches most mathemeticians, IMO), or, 2) They become disgusted by the incomprehensibility of higher mathematics and abandon the field entirely (this is a huge chasm of a pit, that prevents most all engineers and scientists from ever becoming more than moderately competent in mathematics and methods.)
The data could be analysed further to see if there were certain mathematical learning styles (I suspect there are). One cohort of students might be better off on the visual/geometry heavy track, one might be better on the abstract track etc.
It should be initiated at the national government level and mandated to use in schools but the actual building of the application should be contracted out to a company from Silicon Valley/Roundabout/$COUNTRY_TECH_HUB.
Until this happens or something similar we’re just going to continue having this grossly unequal and unfair state of affairs where access to good mathematics education is complete geographical potluck for anyone whose parents can’t afford tuition. The amount of wasted potential is staggering.
I also like to think that intelligence is everywhere, but the persistent pursuit of truth is difficult. Zhang embodies a man determined to work on his particular pursuit.
I hope I can be as persistent over time.
Why do you think that? It takes sometimes years (!) until you get used tona certain mathematical theory. And how long it takes isa very personal thing.
Remember von Neumann, who that that we don't learn nu mathematics, but only get used to it.
I think the only thing one can say was incredibly likeli is that Zhang, while not formally employed as a mathematician, kept thinking about mathematics (a rather common for people with mathematical training, though many just stick to do occasional problem solving).
During that time he definitely spend less time on maths than he could or would in a better situation.
Especially cultural revolution oppression was monumental waste of human lives. See https://en.wikipedia.org/wiki/Cultural_Revolution#Death_toll alone
Or AI, particularly if we figure out AGI.
On the other hand, it's extremely plausible that math as an abstract idea could build arbitrarily complex structures that would be of interest if we were able to learn to model them - so, almost necessarily, within the limitations of current human cognition and/or lifespan, there will be some structures that it is impossible for a human to learn.
Of course, in some arbitrary future we may find that those limitation of human cognitition/life-span can themselves be overcome - if that's the case, then the argument changes significantly.
Question is, are we dedicating enough effort to this endeavors? Is rewriting known math under a different guise sufficient to survive the publish or perish attrition? And will these efforts keep returning results?
I suspect the answer is no to the first two questions, and I don't know the answer to the third.
Only in very few exceptional times in history it was been used for something else, being the largest exceptions the industrial and scientific revolution, and these usually did not require breakthroughs: just apply existing knowledge.
It’s never been a young person game.
Analogously, I've heard transhumanists make the argument that as fluid intelligence declines as crystallized intelligence grows, we have yet to see a human mind in its full power.
So we could solve this a few ways:
- Continue to extend the length of human life. Living to 150 will be the norm in a few generations.
- Create AGI that are smarter than us and make the advances for us
- Genetically engineer humans to be twice as smart, and learn as much as an average 30-year-old in 15 years
Nonsense.
First, the frontier of mathematics is far closer than you think. Number theory is one of the oldest branches of mathematics, going back thousands of years. There are many more branches that have originated in the latter half of the 20th century. And new ones are coming into existence all the time.
Second, we don't need to focus and specialize — we need to do the opposite. Mathematics is about seeing connections and patterns. We need to teach philosophy and critical thinking, and we need to give people exposure to the vast universe of unexplored — and fun — world of mathematics so that they can go towards the frontiers and push them, rather than spend half a lifetime going in the well-trodden direction.
Finally, undergraduates are still producing new results, every year, in numerous REU programs around the US alone. What gives?
The problem isn't that math is so well-studied that you need so much education to do it. The problem is that we aren't teaching people to do math, we teach them about math that they may or may not use elsewhere.
We don't encourage (or give space) for them to play, experiment, explore, wonder, ask questions, venture into the unknown, and be surprised by what they see (outside the aforementioned REUs).
So many people simply don't get to even start doing mathematics until their third year of graduate school, simply because that's when the structure of our education allows them to.
None of that is necessary. We do it that way because professors are underfunded and mentorship is not rewarded (publish or perish), among other things.
The situation is due to structural problems in academia, not mathematics, humanity, or the advances we made.
Signed, —your neighborhood mathematics PhD
Speaking as a former undergraduate math student producing “new” results in REU programs, all but a few of these results are completely negligible. There’s still a genius here and there though.
I would say that it supports the idea that only young people are able to change the fundations of maths, while older people can still use the existing techniques to push the bord, specially on those super technical fields where experience and knowledge are an advantage.
Note that, universities could accept people who did not attend school if they passed their university entry exams because so many people were unable to attend schools because they were all closed and teachers purged during the Cultural Revolution.
I would say he "matured" later mainly because he did not have the right opportunities because he could not go to high school and after his university graduation, had no good opportunities because many good professors were purged during the Cultural Revolution so he fled to the US for a better life.
Source: https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
And I quote from the above source which is from a 2015 New Yorker interview with Zhang:
'I asked Zhang, “Are you very smart?” and he said, “Maybe, a little.” He was born in Shanghai in 1955. His mother was a secretary in a government office, and his father was a college professor...As a small boy, he began “trying to know everything in mathematics,” he said. “I became very thirsty for math.”...The [Cultural] revolution had closed the schools. He spent most of his time reading math books that he ordered from a bookstore for less than a dollar.'
As well:
'...when he was fifteen he was sent with his mother to the countryside...where they grew vegetables. His father was sent to a farm in another part of the country. If Zhang was seen reading books on the farm, he was told to stop...After a few years, he returned to Beijing, where he got a job in a factory making locks. He began studying to take the entrance exam for Peking University, China’s most respected school: “I spent several months to learn all the high-school physics and chemistry, and several to learn history. It was a little hurried.” He was admitted when he was twenty-three.'
It is remarkable how often we see examples of government programs or laws that end-up being exactly the opposite of the deceitful names given to them. I won't get into details here in the US, we do have an embarrassing number of examples of this.
It's the old "I'm from the government and I am here to help" joke.
What else is the government for? I understand it's a popular Reagan talking point but I don't really grok it.
If you assume the govt doesn't work for us, particularly if you're in politics, it becomes a self-fulfilling prophecy.
Anyone can make a government that sometimes does good. The challenge is to create a government that cannot do unfathomable evil.
This is a nuanced and thoughtful position. The statement "The 9 words you never want to hear are I'm from the government and I'm here to help"" is not nuanced and thoughtful, it doesn't suggest governments do good and bad, it positions governments (and their agents) as incapable of doing good. That someone from the government trying to help you must not be, for whatever reason.
It's like don't talk to Cops or corporate HR works for the company and not you.
If you were to take it at face value and debate it against it, you would be making the fallacy error of secundum quid. If you want to have a real discussion about how much good and or bad the government does, you need to find a real human to have that discussion with. The saying alone is a not complete self-contained philosophy.
https://en.m.wikipedia.org/wiki/Secundum_quid#:~:text=Secund...
People in government will abuse their positions as much as possible.
So limiting government as much as possible is the goal.
Good example is pandemic. Governments saved a small number of people by collapsing economies. Quit possibly killing more people then were saved. There is a very good argument that the best thing governments could have done is “nothing”.
People constantly rail about the evil wrought by big business, and they're right to do so, but what is the government (in the US anyway), but the biggest (and perhaps most corrupt) business of all? Except it's not constrained by the requirement to turn a profit, so it doesn't suffer the bad consequences of failure, because you can always print more money.
And you're absolutely right about the pandemic response.
Meiji did this in Japan, to the point where many B. temples were burnt down (thankfully not as bad as Mao), and the whole of Japan's culture considered obsolete. KMT boys were doing the same thing in China before Mao. Nehru and his Congress cretins who inherited the British state, did the same, and India continues these "destroy Bharata" policies to this day.
In fact, since a lot of Asian culture is tied to India, one might call this an extension of the millennia long war on India from the West, but now on the realm on the realm of ideas. Not surprising that the occidental academia has deep-hatred for India's old-culture and everyone who imbibes it (at the same level as medieval anti-semitism from my own experience). They also have the victim-complex of Islam (and Xtianity to a lesser extent) to their support now.
You might argue it's a difference in degree, but if the Cultural Revolution actually succeeded (or not reverted), it would be the difference between zero and non-zero cultural heritage left.
Said another way - I've known quite a few people like him to a point - with the difference that none of the others ever produced good mathematics, much less solved a major problem.
In Zhang's case, I believe his doctoral thesis actually proved the Jacobian conjecture... but his thesis was relying on an incorrect result given by his advisor's own paper (presumably at the guidance of his advisor).
Maybe he loves what he's doing and that's the root of being stubborn and "genuinely oblivious to ordinary material feedback". Although love or passion can be overrated or too general to describe his attitude toward problem-solving, I think people can't be just stubborn, there's a drive that holds them to a higher standard.
[1]https://pandaily.com/mathematician-yitang-zhang-confirms-par...
So I expect them to stick to their rule.
Intelligent people will end up learning something profound when they are young.If they find something else interesting enough at a later stage in their life, they apply some transformation learning.
Leibniz did not start his training in Math until he was ~30
> Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus.
Not to mention that up until 1910s, life expectancy was under 50.
Being able to work in academia as a tenured professor/researcher probably resulted in drastically different life expectancy, compared to being forced to do something else. I think it's safe to say that Zhang would been forced to live as a peasant, had he been born 120 years ago.
I think the actual truth is more like, "big breakthroughs mainly happen early in one's career". Most mathematicians start their careers young, therefore they publish breakthroughs while young. Zhang started quite late so his innovations are later in his life, but still early in his career.
And it makes sense, everyone has a slightly unique way of thinking, and long-standing problems will only yield to unique thinking. Eventually someone will come along that has just that right type of unique thought process that will find a hole to solve such a problem.
1. Zhang posted an attempt at solving this problem in 2007 that he later more or less admitted was flawed: https://mathoverflow.net/questions/131221/yitang-zhangs-2007.... But speaking with mathematicians who are intimately familiar with Zhang's previous work, there seems to be good reason to be optimistic nevertheless. First, the idea behind Zhang's proof is similar to the zero-repulsion ideas appearing in known results about Siegel zeros, and is thus reasonable. Second, Zhang seems to have matured late, and unlike the flawed 2007 paper, his 2013 paper on bounded gaps in primes is meticulously written. He came a long way between those two papers, and he may have come even further since then.
2. Zhang is 67 years old. If the paper is correct, then Zhang constitutes a strong counterexample to G.H. Hardy's famous claims that "mathematics is a young man's game" and nobody alive today could say, as Hardy did, that "I do not know an instance of a major mathematical advance initiated by a man past fifty."