Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang
To give a sense of the scale of this claim: If correct, Zhang's work is the most significant progress towards the Generalized Riemann Hypothesis in a century. Moreover, I think this result would not only be a more significant advance than Zhang's previous breakthrough, but also constitute a larger leap for number theory than Wiles' 1994 proof of Fermat's Last Theorem (which was, in my opinion, the greatest single achievement by an individual mathematician in the 20th century).
Some discussion / explanation of Siegel zeros and Zhang's claim can be found here:
https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
https://mathoverflow.net/questions/433949/consequences-resul...
An account of Zhang's remarkable story (and his previous breakthrough) can be found here. Famously, prior to his breakthrough, he worked at Subway and lived in his car:
https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
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[ 5.9 ms ] story [ 295 ms ] threadDiscussion https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
https://mathoverflow.net/questions/433949/consequences-resul...
New Yorker article https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
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."
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.
I agree.
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.
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
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.'
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.
[A journal reviewer of his famous paper says]: "you should be careful. This guy posted a paper once, and it was wrong. He never published it, but he didn’t take it down, either.’ ” The reader meant a paper that Zhang posted on the Web site arxiv.org, where mathematicians often post results before submitting them to a journal, in order to have them seen quickly. Zhang posted a paper in 2007 that fell short of a proof. It concerned another famous problem, the Landau-Siegel zeros conjecture, and he left it up because he hopes to correct it.
Looks like he might have lived up to that!
All of these guys are probably a hundred times smarter than me or most of the other code monkeys working for the FANGMAN, but they're all squabbling over little 5-figure scraps of grant money.
No murderers, great success!
( Interestingly, every summary of the case in media and Wiki stops listing the evidence against him at his secretly taped confession to a girlfriend - confession that included some things absolutely not confirmed. The most convincing evidence to my eyes is the victim’s DNA in the blood found under the carpet and elsewhere there it has survived cleaning efforts. This is not mentioned anywhere except in the court recordings: https://news.wttw.com/sites/default/files/article/file-attac... . Kinda sad what is convincing these days ).
UIUC is not having a great track record wrt grad student murderers o_O
I'm glad Zhang was able to find success despite his initial setbacks and from what it seems like in his recent interviews also let go of his bitterness/resentment (holding something like that in your heart can only ever hold you back). And though the power dynamics here were clearly unequal, I don't think it's fair to blame Moh entirely for what happened at Purdue.
I think it's important to remember Moh is also human with all the complexity that comes along with that. In reading his published statement, even though there is no direct apology to Zhang, I sense that he does genuinely regret how things turned out.
Perhaps one day, Zhang and Moh will be able to meet again and resolve/rekindle their relationship.
> Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop. A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days.
https://en.wikipedia.org/wiki/Yitang_Zhang
But why is streaming rental for 24 hours? Why can't we do rentals for two weeks for videos? Is there a good reason to make it so difficult to stream? I don't want to watch it a million times. I just want to make it through once, but it takes me several sittings typically to finish a movie.
https://archive.org/details/countingfrominfinityyitangzhanga...
Money / capitalism.
Colors of Math breezes across six contemporary mathematicians.
Same thing goes for mathematics that is too "deep". Most people could not care less about prime numbers. Yes, they drive important cryptographic procedures. But we care about cryptography. We care that the message gets to its destination "safely". If its done using prime numbers, imaginary numbers, geometric numbers or fantasy numbers, it does not matter.
I think the real issue is the producers and directors and business folks in the film industry are not in that camp. They’re more likely to make a biography of Robin Williams than Paul Dirac, and they decide what gets made.
Could we do better as a society? Maybe. To do that though you’d have to bring these stories into film classrooms and get students inspired. But first you have to make the stories inspiring and a lot of them aren’t thaaat much except for the advancements they provided to the field and that’s harder to convey. Maybe Leibniz and Newton would be a good rivalry on the screen.
[1] https://www.themoviedb.org/keyword/8689-albert-einstein/movi...
https://www.dogomovies.com/einstein-and-eddington/movie-revi...
My point is even einstein doesn’t get the same treatment as, say, a politician or actor gets as far as coverage. There’s more out there about Elvis than Newton.
I mean would a biopic about Einstein touch on his many relationships (including his cousin) and the drama around that?
There are tons of excellent books on mathematicians, engineers, and scientists. Just virtually no visual media.
Many graduates from post Cultural Revolution China left the country in 90's and found the language barrier (and accompanying discrimination) too high to overcome. The man delivering your fried rice might have a PhD.
But if Newton had decided to open an inn instead of work on the Principia the world would have been far worse off for it. Possibly centuries behind where we are now.
I will apologise for calling the restaurant mediocre. I have no idea about the quality of the food, but if you think that's what this hinges on then you've missed the point. That as good a restaurant as it may be, it's not worth losing a mathematician of his calibre over.
He came back to Romania and to Bucharest, which was really good for me and for my then colleagues because he was an excellent professor.
FTA:
In Kentucky, he became involved with a group interested in Chinese democracy. Its slogan was “Freedom, Democracy, Rule of Law, and Pluralism.” A member of the group, a chemist in a lab, opened a Subway franchise as a means of raising money. “Since Tom was a genius at numbers,” another member of the group told me, “he was invited to help him.” Zhang kept the books.
Quite a different feel to that characterization.
Can't be harder than learning the meaning behind these characters: https://www.pandatree.com/book/DiaryofWorm.jpg
It's like watching a Marvel movie and not only knowing the plot of the current movie but also the deep history of each character and their relationships with other characters.
I assume the paper didn't come out of nowhere and it's based on "the shoulders of giants".
Another example is the use of commonly known constants, such as pi, i or e. No paper will define those constants and explain indepth what they mean, so you need prior knowledge or to find external resources. I think for such complex papers you will find many such cases where extensive prior knowledge is needed.
I sometimes rewrite them using A, B, C then try to understand them. This procedure should be automatic but unfortunately TeX is ancient.
A real-life Good Will Hunting, his backstory is incredible.
source: UK is my alma mater.
On the other hand, I doubt this proof will help you to build a faster gizmo or something in the real world.. especially since it's proving something we really think is true (a consequence of the Generalized Riemann Hypothesis), and for real-world applications you can just assume the thing that we think is true is actually true (even if we haven't been able to prove it for a century). (E.g., you don't need to prove that factoring is hard to use cryptography for practical purposes..)
Like, try
Congratulations, you just found a prime that is big enough for every cryptographic protocol that uses prime numbers (not counting unusual and non-deployed post-quantum proposals).Some number theory research may impact the security of cryptosystems, but not all results do.
Some crypto (namely, RSA) depends on on composite numbers being hard to factor, which is a different problem.
[1]: https://en.wikipedia.org/wiki/Prime_number_theorem
Factoring that large integer directly yields the user's private key.
That depend on large semiprimes being hard to factor.
But here is the worst (or "most mysterious," depending on your mood..) thing about Siegel zeros. Our best result about Siegel zeros (excluding for present discussion Zhang's work), namely Siegel's theorem, is ineffective. That is, it says "there exists some constant C > 0 such that..." but it can tell you nothing about that constant beyond that it is positive and finite (we say that the constant is "not effectively computable from the proof").*
So then, if you try to use Siegel's theorem to prove things about primes, this ineffectivity trickles down (think "fruit of the poisoned tree"). For example, standard texts on analytic number theory include a proof of the following theorem: any sufficiently large odd integer is the sum of three primes. However, the proof in most standard texts fundamentally cannot tell you what the threshold for "sufficiently large" is, because the proof uses Siegel's theorem! In this particular case, it turns out that one can avoid Siegel's theorem, and in fact the statement "Any odd integer larger than five is the sum of three primes" is now known https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture. But it is certainly not always possible to avoid Siegel's theorem, and Zhang's result would make so many theorems which right now involve ineffectively computable constants effective.
*Why is the constant not effectively computable? Because the proof proceeds basically like this. First: assume the Generalized Riemann Hypothesis. Then the result is trivial, Siegel zeros are exceptions to GRH and don't occur if GRH is true. Next, assume GRH is false. Take a "minimal" counterexample to GRH, and use it to "repel" or "exclude" other possible counterexamples.
Please, keep going. This is good reading.
https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
https://old.reddit.com/r/math/comments/ymlacu/professor_yita...
The class number formula, mentioned in the second comment, is one of the craziest "bridge results" in all of math (meaning a result that connects two seemingly disparate areas). The class number formula connects the values of Dirichlet L-functions at s = 1 (Dirichlet L-functions are complex functions related to the distribution of primes in arithmetic progressions), to class numbers of number fields. (Remember that the value of Dirichlet L-functions at 1 is exactly what the question of Siegel zeros concerns.)
To give a crash course on what some of those words mean:
1. A number field is what you get when you take the rational numbers, and you throw in the roots of some polynomials to get a bigger object where you can still do all of the usual arithmetic operations, in the same way that we throw in the roots of x^2 + 1 (namely, i, -i) into the real numbers to get the complex numbers.
2. The ring of integers is the right notion of the "integers" in that number field. (That is, rational numbers : integers = number field : ring of integers in that number field.)
3. The class number of a number field tells you how close you are to having unique factorization into primes holding in the ring of integers of that number field*. If the class number is 1, then you have unique factorization; if the class number is 1000, then you are very far from it.
What this connections means is that you can prove things about regular old primes in arithmetic progressions (in the integers) by proving things about these exotic / abstract primes (in rings of integers of number fields), and vice-versa.
Anyway, as a result of the class number formula, there are a lot of results about class numbers that are ineffective because of Siegel's theorem too, e.g., https://en.wikipedia.org/wiki/Brauer%E2%80%93Siegel_theorem. Zhang's result (if correct) would make all of those effective, too.
*While in the integers, it is true that every number factors uniquely into a product of primes, this is unfortunately not true in more general contexts. In fact, algebraic number theory basically began with a mistaken proof of Fermat's Last Theorem, which was mistaken precisely because it assumed that unique factorization always holds in this more general context, which is not true. (If unique factorization did always hold, then that proof of FLT would have been correct.)
Thanks, that totally failed to give any sense of scale.
I suppose superconductors. Semiconductors are well in the room temperature regions :)
If you think that prime numbers are interesting, then I can tell you that GRH is the single most central conjecture in the study of prime numbers. Personally, I think prime numbers are some of the most fundamental and intrinsically interesting objects in pure math, but of course, this is subjective!
what would be an example of "deep philosophical" implication that has no bearing on the "nature of reality"
Was the rest of the context not more enlightening?
E.g. Would constitute a larger leap than proving Fermat's Last theorem which was "the greatest single achievement by an individual mathematician in the 20th century"
I still need to read the article. The letter from Moh was interesting.
https://www.youtube.com/watch?v=RjzC1Dgh17A
If someone tells you they were a “clerk” for a Supreme Court judge don’t be surprised when later in life they have an impressive legal career
Laws of nature (and descriptions thereof) aren't patentable.
https://www.bitlaw.com/source/mpep/2106_04_b.html#:~:text=Th....
I never expected to see his name in a context like this again. I'm glad he's still being himself and working hard on what he loves.
Including Xi Jinping.
Clearly a flawed person. Not sure why he blew off his family so hardcore for so many years. Too bad to hear he was arrogant as a kid too, although a lot of smart kids are. Some of them turn out to be Peter Thiel, luckily this guy just wanted to work on math.
Anyway, I wish he had been better to his parents. On the other hand, he needed that big breakthrough to save his life as a mathematician: until that point, he was just an adjunct lecturer with no stability at all. Life is weird and complicated and we don't have full control of the choices we make, some choices can seem really hard to some people. I'm not excusing his behavior toward his family but I would be interested to know why he made that choice.
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."
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.
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 are 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?
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.
https://eater.net/quaternions
And speaking of Maxwell's Equation"s" :
http://www.av8n.com/physics/maxwell-ga.htm#sec-preview
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. There's no hyperspheres with a rotation matrix.
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.
I definitely like geometric algebra, it's a very nice continuation of topics in linear algebra and makes it clear why things like normals behave differently from standard vectors. But I don't use it every day. I use standard linear algebra every day.
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.
[0] https://en.wikipedia.org/wiki/Equivalence_principle
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.
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.
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:
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 proove his big proofs faster had his life gone otherwise.
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.
Age is (almost) not a limitation with the right tools, less in the near future.
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.
This sounds like a hard task as I couldn’t find anything online that does it :(
https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
For certain complex number inputs s, this function ZETA(s) returns zero. Riemann's hypothesis states it returns zero when the real part of the input Re(s) = 1/2, and the imaginary part Im(s) some non-zero value (the first zero occurs at Im(s) = +/- 14.135.) As far as we've checked with computers, all zeroes have Re(s) = 1/2. We are interested in these "zeros" because we can use them to construct a harmonic function (think overlapping waves) which tells us how the prime numbers are distributed.
A Siegel zero is a potential counterexample where a zero could theoretically occur for complex number with Re(s) close to 1 (i.e. not 1/2.) This is based on the study of Dirichlet-L functions which are a generalized version (i.e. superset) of the Riemann Zeta function.
If Zhang's result is correct, it simplifies the problem space for finding Riemann zeros, and thus for understanding the distribution of primes.