[HN Gopher] Teenager solves stubborn riddle about prime number l...
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Teenager solves stubborn riddle about prime number look-alikes
Author : theafh
Score : 216 points
Date : 2022-10-13 15:07 UTC (7 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| dgs_sgd wrote:
| Congratulations to him! That's an amazing accomplishment. There's
| a lesson in this line
|
| > For more than a year and a half, Larsen couldn't stop thinking
| about a certain math problem.
|
| It's rare to be able to focus on something for that long without
| giving up.
| renewiltord wrote:
| Summary:
|
| 1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p)
| for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8
| = 512-8 = 504 = 168 x 3.
|
| 2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is
| prime? No. Sometimes n is non-prime (like n=561, divisible by 3).
| We call these n, Carmichael numbers.
|
| 3. Okay, so these numbers exist. How common are they? For primes
| we know they're common. Bertrand postulated (Chebyshev proved)
| that for any n>1, there is a prime p between n and 2n. That's
| cool!
|
| 4. Is it true that there is such a bound for these pretend-
| primes? Well, we have an interesting fact that there are x^(1/3)
| of them below any x, once we pass a certain point (i.e. there
| exists an X such that there are x^(1/3) of them below any x > X)
| so that makes us think it could be true! Worth seeing!
|
| 5. But what about this common-ness measure like the B-C result
| for primes? Well, it turns out that it exists. It ain't as pretty
| as just between straight integer multiples, but the fact that it
| exists in some shape at all is cool! That's what this kid proved.
| Absolutely rollicking fact. https://arxiv.org/abs/2111.06963
| jll29 wrote:
| By the way, if you don't like reading bulky proprietary PDFs,
| there is a trick: substitute the x in arxiv.org by the digit 5,
| and you will see the paper rendered in HTML5, e.g.:
|
| https://ar5iv.labs.arxiv.org/html/1910.06709
|
| (great work by FAU Erlangen's Michael Kohlhase and team).
| renewiltord wrote:
| Good tip. I should have just linked the arxiv.org page for
| the article and not the PDF directly
| https://arxiv.org/abs/2111.06963
| vorticalbox wrote:
| Thanks for this. I find articles like this super hard to read
| due to the mixing of the topic and all the "back ground".
|
| It does sometimes feel like it's only there to bulk out the
| article.
| jbverschoor wrote:
| Lazy writers and/or who get paid per word.
|
| The writer can fix the article, or let every reader fix it in
| his mind.
| Ar-Curunir wrote:
| Maybe just consider that the audience of the article is not
| you?
| SilasX wrote:
| The article is not technically minded people who would
| like a lucid explanation of the math behind the result?
| Because they're pretty clearly trying to hit that target,
| and frequent interruption of that kind of exposition
| works against it...
| renewiltord wrote:
| One can hope that future AI autosummarizers can be aware of
| our personal level of knowledge!
| phkahler wrote:
| But from the article:
|
| >> In fact, Larsen's argument didn't just allow him to show
| that a Carmichael number must always appear between X and 2X.
|
| And yet the Wikipedia page says 2821 and 6601 are the 5th and
| 6th Carmichael numbers, which means there are not between 3000
| and 6000 (X and 2X). So is his result actually that one must
| always exist between X and 2.5X or some other small multiple?
| If so, what multiple did he prove?
| bibanez wrote:
| I only read the abstract and the result is in the same spirit
| but doesn't say exactky between X and 2X. It's between some
| more complicated expressions (using logs like usual in number
| theory)
| zeroonetwothree wrote:
| The bounds are tighter than n and 2n. So he actually proved
| a stronger result.
| zeroonetwothree wrote:
| It's true for x sufficiently large. So there will be some
| small counterexamples.
| renewiltord wrote:
| Ah I didn't mean to mislead that it was x and 2x. Corrected
| to be slightly clearer.
|
| The bound is easiest seen on the arxiv link above, in the
| abstract. The HN forum software doesn't do math very well.
| dekhn wrote:
| I've known quite a few families where both parents are scientists
| or software engineers or quantitative engineers.
|
| Very frequently, their children will have a scary intuitive
| understanding of concepts that took my many years to understand
| (I'm a slow learner; didn't really understand hash tables until
| my 30s) and then apply their abilities to be in the higher
| echelons at science in a very young age. I see a similar thing in
| the children of world-class athletes.
| misterprime wrote:
| Agreed, but this brings up the topic of nature vs. nurture, and
| perhaps even the previously (~90 years ago now?) popular
| concept of eugenics.
|
| I think society settled on a response of "yeah, but we
| shouldn't actively do anything about it as a group." It seems
| to be best left to individuals to seek out parental partners
| and enjoy the outcomes on a personal level.
| nominusllc wrote:
| I wish this could be discussed more openly. I feel the same
| way.
| feoren wrote:
| > I think society settled on a response of "yeah, but we
| shouldn't actively do anything about it as a group."
|
| I hope there comes a day when humanity is responsible enough
| to handle precise gene-tailoring of humans without it
| devolving into a cliche Sci-Fi dystopia. Until then, I'm glad
| that humanity has settled on "Eugenics could theoretically
| work but we know we're not responsible enough to do it right,
| so let's not". It feels like a very enlightened stance:
| better than pretending selective breeding would literally not
| work, and certainly better than doing it badly. Why could we
| come to such a sensible consensus on this topic but fail to
| do so on so many others?
|
| All that being said, I still want to see what would happen if
| Michael Phelps and Katie Ledecky had a child together. You
| know. For science.
| TimTheTinker wrote:
| > I hope there comes a day when humanity is responsible
| enough to handle precise gene-tailoring of humans without
| it devolving into a cliche Sci-Fi dystopia.
|
| Power itself is a bit of a paradox - having some is
| essential to a free society, but too much ends up
| corrupting its holders and results in human suffering. The
| power you describe falls in the latter category, IMHO.
| dekhn wrote:
| This isn't eugenics. People are allowed to choose their mates
| for their phenotypes, and we don't sterilize (except in very
| rare, extreme circumstances, with legal approvals) people who
| don't have specific phenotypes.
|
| It's a gedanken experiment, but I would expect that people
| who measure low on IQ tests but are raised immersed in an
| technical or artistic environment have as much potential to
| become masters as people with higher IQ, or very nearly so.
| Further, I think that by increasing the quality of the
| education system in a country, you can massively increase the
| intellectual output. There are probably enormous amounts of
| untapped potential never matched to an equivalent supply of
| knowledge and skill.
| tr33house wrote:
| very impressive to have such fundamental contributions at such an
| early age. To even know it's applicability to modern day
| cryptography is also really impressive. All the best to Daniel
| Larsen!
| goy wrote:
| Nobody is talking about the proof, but about the fact that the
| person who produced it was younger than them and they are trying
| to explain the achievement by innate abilities or parent
| influence. The fact is just that most of people don't want to do
| something like that. They just want to be someone who do things
| like that.
| 120bits wrote:
| Congratulations! This is a great achievement at his age. Maybe a
| Fields medal next?
| 4gotunameagain wrote:
| Getting older, sometimes it can be so tough to accept the fact
| that people a fraction of your age achieve things you never will.
|
| Given the extreme connectivity of the present, we are also
| exposed to brilliant minds with incredible capabilities, making
| us (me at least) feel even more incapable..
|
| I guess it is a lesson for humility.
|
| Good job Daniel, you show us !
| lo_zamoyski wrote:
| Why is it tough?
|
| This may be worth reflecting on [0].
|
| [0] https://www.newadvent.org/summa/3036.htm
| dylan604 wrote:
| Yeah, I have no problem admitting that there are people much
| smarter, more talented, better looking, etc than I am. The
| day that I assume I'm the top of whatever category is a sad
| sad day, as I must be the only one left.
| markus_zhang wrote:
| I think what really bugs me is that for some hobby I chose,
| there is a always someone who already achieved everything I
| could dream to achieve in my life.
|
| Then I figured some hobbies are better suited for my use
| case. For example science is better suited for me. I
| somehow was never bothered by the fact that some geniuses
| in 19th century already knew enough math/physics that I
| could digest in my life. Science is infinite so however
| much knowledge someone gained, it's zero comparing to the
| total amount. I guess I need that kind of comfort to commit
| to something. Fossil collection is also a good one because
| every piece I collected is MINE and unique so I don't need
| to compare with someone else.
| bluGill wrote:
| Either that or you have narrowed the field so much that you
| are the only one even doing what you do. (you might
| sometimes have a junior to help you, who then becomes the
| second best in the world)
| munk-a wrote:
| Personally, I can claim, without a doubt, to be the world
| champion at writing this comment.
| [deleted]
| 363849473754 wrote:
| Not to downplay any of Daniel's accomplishment but sometimes it
| isn't a "fair" comparison when others started younger with more
| resources. His father is a distinguished professor of
| mathematics and his mother is a professor of mathematics. When
| you have that sort of resources available at a young age and
| advanced training you'll probably accomplish more sooner than
| someone of similar IQ without those resources who started later
| paulpauper wrote:
| There are thousands of mathematicians in the US. I am sure
| many have kids. How many of those kids do even a fraction of
| what Daniel did even when having every possible advantage?
| Today, young people have assess to more resources than ever,
| yet talent is one of those things that resists this trend of
| egalitarianism seem elsewhere. More resources means that the
| super-talented will pull way ahead of the untalented or only
| moderately talented.
| 363849473754 wrote:
| To be honest I care more about the high iq kids interested
| in math who come from poor or working class families that
| either won't be identified or will be identified but not
| much can be done for them given the lack of resources. Plus
| this doesn't negate that someone of a similar IQ may
| accomplish less / seem less impressive at an early age due
| to such disadvantages.
| quickthrower2 wrote:
| Resources also include parental encouragement, not being
| bullied, not having to do stuff to get by that isn't
| delving into deep work, not trying to fight boring school
| lessons and exams in subjects not of interest, no pressure
| to shape your studies to get a job. These are not
| universal.
| jamiek88 wrote:
| I wonder how many more Srinivasa Ramanujan's died in poverty in
| a rural village?
|
| Maybe there's a path now for these kids.
| dQw4w9WgXcQ wrote:
| The set of things you will never achieve is infinite compared
| to the set of things you will, best to stay focused on the
| latter and enjoy your life. Unless of course you prefer
| drinking from an infinite well-spring of misery.
| munk-a wrote:
| I celebrate people who achieve memorable milestones like this
| or build something that revolutionizes the world - but there
| are a plethora of different ways to live life. For me, as long
| as you're happy and you're putting more happiness out into the
| world than you're consuming then you're achieving a pretty damn
| good life. We're not all responsible for the entire world - as
| long as you're leaving your little corner of it better than you
| found it then good on you.
| paulpauper wrote:
| It's not enough to make the world better or be happier, a lot
| of people also want acclaim and recognition. Why do so many
| people apply to Ivy League schools when 50-100 ranking
| schools can also provide a good education? Status is
| necessarily scarce.
| munk-a wrote:
| This may just be the odd musing of a mid-thirty year old
| but acclaim and recognition feel pretty worthless to me.
| Chasing individuality through others' approval will always
| be more difficult than just accepting yourself as yourself
| and being happy with that.
| paulpauper wrote:
| it's not like everyone can win the IQ lottery. Superior ability
| is unequally distributed.
| strikelaserclaw wrote:
| people should measure themselves by how best they utilized
| their gifts and circumstances to live a life that they are
| happy with. No point in comparing yourself to some kid who
| probably inherited an amazing mathematical brain through no
| conscious effort, moral superiority etc.. of his own.
| zeroonetwothree wrote:
| Ability to maximize your own talent is a talent in itself.
| svat wrote:
| Tom Lehrer once said: "It is a sobering thought that when
| Mozart was my age, he had been dead for two years."
| dekhn wrote:
| More Lehrerisms here:
| https://en.wikipedia.org/wiki/That_Was_the_Year_That_Was
| citizenpaul wrote:
| jklinger410 wrote:
| You are 100% correct.
|
| https://en.wikipedia.org/wiki/Michael_J._Larsen
| kevinventullo wrote:
| His mother is also a mathematician, and his uncle is a Fields
| Medalist.
|
| That said, I disagree with OP that this suggests his parents
| secretly did this work. Rather, it tells me that being
| surrounded by great mathematicians at a young age is going to
| foster success in those with raw talent much more effectively
| than, I dunno, attending some "gifted" program at a typical
| public school.
| klyrs wrote:
| That's one of two parents. Ayelet Lindenstrauss, interviewed
| in the article, is also a mathematician, as was her father
| Joram, and her brother Elon got a Fields medal.
|
| To accept OC's comment as 100% correct is to accuse two
| mathematicians of academic fraud. And if we're going that
| route, are we going to insinuate Joram Lindenstrauss
| responsible for both Ayelet's and Elon's academic works?
| forgotpwd16 wrote:
| >Let me guess, this teenager has a parent that is an
| established mathematician.
|
| But wouldn't those teenagers have higher chance and assistance
| in studying higher math compared to peers? (Though being sole
| paper author is a bit strange.)
| [deleted]
| bawolff wrote:
| People learn from mentors. Its not that surprising and doesn't
| neccesarily imply the parent did it. The parents probably did
| help by teaching them math.
| mlyle wrote:
| That's the thing.
|
| This is a spectacularly rare accomplishment for a kid that
| grew up in an academic family.
|
| But it's nearly nonexistent for kids who didn't have these
| advantages.
|
| So chalk most of it up to this kid, and some of it up to his
| rarified environment.
| strikelaserclaw wrote:
| both his parents are mathematicians and his uncle is a fields
| medal winner and his grandfather is a mathematician. I'm sure
| this kid is very intelligent, and i could even believe he
| solved most of the problem himself but in the end of the
| article stating ""He did all this without an undergraduate
| education," Grantham said." made me roll my eyes.
| nicoburns wrote:
| > He did all this without an undergraduate education
|
| I take your point, but I think it serves us to be reminded
| that formal education isn't the only place that one can learn
| things.
| Loughla wrote:
| Yes, that is the point. But it's super cringe-inducing
| because this teenager benefits, what we can assume is
| substantially, from literal generations of formal education
| in his family.
| svnt wrote:
| Is this some strange attempt to align this article with the
| "college education isn't necessary" mantra?
|
| Like college education isn't necessary as long as you have
| college professors for parents?
|
| The absurdist continuation is something like: "I'd like to
| think if I was motivated enough I could retroactively
| convert my parents from city bus drivers to tenured
| professors in a lucrative field, then I wouldn't need a
| college education"?
| nicoburns wrote:
| It's intended to push back more against "we won't hire
| anyone without a degree" type sentiments than "you
| shouldn't bother going to college" type sentiments.
| kevinventullo wrote:
| I mean it sounds like he has already received the equivalent
| of an undergraduate education.
| strikelaserclaw wrote:
| thats exactly what i mean, they made it sound like this kid
| was some average guy who at the age of 15 went to the
| public library, read books and solved some hard math
| problem. I bet this kid had an advanced math education and
| math immersion since he was a toddler.
| paulpauper wrote:
| Think of all the millions of dollars spent on immersion
| and tutoring by rich parents. How many of their kids
| produce anything of noteworthiness at any age, let alone
| so young as he did? This is 99% the product of IQ/talent.
| It's a huuuge leap to go from merely having an advanced
| math education to actually solving or proving important
| stuff. This is mathematician-caliber work, not just
| someone who took advanced courses at a university or had
| parent's help.
| strikelaserclaw wrote:
| this kid is probably talented as hell but it doesn't mean
| he didn't have a great mathematical upbringing as well.
| kevinventullo wrote:
| Eh, the kind of immersion and tutoring that rich parents
| can buy doesn't remotely compare to having two
| professional mathematicians as parents.
|
| The tutors for rich kids are likely to be local grad
| students who meet with the kids at most a few hours a
| week; you can't exactly hire a fields medalist for
| tutoring. Perhaps more importantly, those rich kids are
| not getting _singular_ training in math, they're getting
| tutored in a gazillion things so they can be "well-
| rounded". Also, those kids are not likely to develop the
| intrinsic motivation to do this stuff because their
| parents are still the ones instilling values in them.
| Those values are going to be "go to law school" or "start
| a business" or "pursue the arts" or some other avatar of
| "make as much money /social capital as possible". Those
| values are likely not going to be "study math and prove
| theorems because it's interesting".
| _zoltan_ wrote:
| I bet you could hire a Fields medalist with enough money.
| Might be a lot, but for sure you can. Everybody has a
| price.
| lern_too_spel wrote:
| And if you did, you would surely get results. That isn't
| really happening at any sort of scale today. Compare to
| Alexander the Great, who was tutored by Aristotle
| himself.
| zozbot234 wrote:
| > Think of all the millions of dollars spent on immersion
| and tutoring by rich parents.
|
| The kind of tutoring these provide is nowhere near as
| immersive as the elite one-on-one education that was
| historically common in upper-class households.
| Vt71fcAqt7 wrote:
| The article mentions him discussing infinity with his
| mother at three. It does not try to hide his background.
|
| Also, being "privileged" in this way isn't a bad thing
| (not that you have said otherwsie).
| paulpauper wrote:
| we're talking even greater than Terrance Tao level of IQ.
| No amount of parent's help can instill that kind of
| talent.
| Vt71fcAqt7 wrote:
| >greater than Terrance Tao level of IQ
|
| Can you describe how you arrived at this conclusion?
|
| >No amount of parent's help can instill that kind of
| talent.
|
| Totally agree here, but the _use_ of tallent _can_ be
| shaped by parenting. I think it 's fair to say that his
| upbringing allowed him to develop an interest in
| mathematics at an earlier age. Not to mention being able
| to discuss it all the time (making some assumptions here,
| I can only guess).
| MikePlacid wrote:
| Dunno. My granddad was a mathematician and my mother was
| a mathematician (and my father was a mathematician and a
| computer scientist, but they divorced). Still the only
| "math immersion" I got was Perelman books, scattered
| around our apartment, like Physics for Entertainment and
| Algebra for Fun:
|
| https://www.amazon.com/s?i=stripbooks&rh=p_27%3AYakov+Per
| elm...
|
| They are both challenging and entertaining and very
| simple - on the surface. Some were written in 1913 or so.
| Each American family can get such an "immersion" for
| their kids - total is less than $100 I guess. But you
| also should turn off TV and computers, so probably would
| not work.
|
| Also my daughter took major in mathematics - and I like
| my parents had no time to immerse her in anything -
| adults work, children play with fun but impractical
| problems.
|
| I would rather guess that it is some genetic defect in
| the brain causing a person to prefer playing with
| abstract problems to booze, smoke and sexual
| gratification. But I doubt that having such a guess is
| allowed.
| [deleted]
| bumby wrote:
| > _I would rather guess that it is some genetic defect in
| the brain causing a person to prefer playing with
| abstract problems to booze_
|
| They may not be as mutually exclusive as you think. I
| remember reading some research years ago that
| investigated the positive correlation between alcohol
| abuse and IQ. Their theory was that high IQ personalities
| seek novelty and that can manifest in chasing novel
| mental states.
| BaseballPhysics wrote:
| > I would rather guess that it is some genetic defect in
| the brain causing a person to prefer playing with
| abstract problems to booze, smoke and sexual
| gratification. But I doubt that having such a guess is
| allowed.
|
| Understandable given it's a comically elitist point of
| view.
|
| Fun fact: Richard Feynman experimented with both LSD and
| Ketamine, among other things. Shame, imagine how much he
| could have achieved if he had this "genetic defect" you
| posit...
| MikePlacid wrote:
| Elitist?? I doubt that any American family has a lower
| standard of living, than a Soviet math post-graduate
| student, single mother of two. We have no permanent beds
| only folding ones, I made my studies on a drawing board
| put over a sewing machine (do you know what sewing
| machine is for? It's to repair your old clothing) our
| apartment was shared by two families, it has no hot water
| and water itself was de facto rationed.
|
| In elementary school - while my mother was on the field
| trips gathering data for her PhD thesis on methan gas
| distribution in coal mines of Donbass region I was living
| with my mathematician granddad. It was a small house
| shared by three families, no running water and amenities
| inside. My grandparents grew their own vegetables since
| you can't buy vegetables on the market (since there were
| no market), and even bread was rationed - it was
| Khrushchev time. Elitist family, indeed.
|
| Btw my ancestors were slaves, freed about the same time
| American slaves were freed. And then communists killed a
| brother of granddad, and a father of my stepfather, and
| then national-socialists killed the only brother of my
| mother, my uncle. My stepfather was never allowed to
| college, as a son of the "enemy of the people".
|
| If you want to see "elitist" - look at any American
| house.
| BaseballPhysics wrote:
| There may be a language issue here, but to be clear:
| Elitist != rich. I've run across many poorer folks who
| harbour elitist attitudes (like, say, ascribing mental or
| moral failings to folks who choose to indulge in sex,
| drugs, or alcohol).
| MikePlacid wrote:
| There easily can be a language issue here, but I believe
| the only negative word I used was "defect" and I've
| applied it to myself, not to my classmates who chose to
| indulge in these wonderful things you've listed. Sex is
| indeed wonderful. Alcohol also gives you some pleasant
| feelings. I do not get nicotine at all but people seem to
| enjoy it, who am I to disagree.
|
| Note that indulging in all these pleasant activities does
| not prevent you from being successful. A lot of my
| heavily drinking classmates became rather accomplished.
| We have a head of a regional KGB office (our region is
| about 2 mln people, a dozen counties or so), a head of
| regional sanitary office (a medic who checks business
| medical safety compliance), doctors, engineers...
|
| And if I was born say a hundred years earlier there would
| be no such a crazy demand for computer specialists, and I
| would have probably landed a job of math teacher at high
| school, or if I was lucky - at community college like my
| granddad.
|
| Math teacher at high school - only some brain defect can
| lead you to give up pleasures of booze to get such a job,
| no?
| selimthegrim wrote:
| Come to New Orleans and after I am done giving you tour
| of some neighborhoods let's see what you say.
| MikePlacid wrote:
| As one Indian professor I don't know said to a Russian-
| Mexican professor I do know while the latter was giving
| the former a tour on some unhappy parts of Mexico City:
| "You want to surprise me with poverty? Look, everyone
| here has shoes".
|
| Our company moved to San Francisco in late 90s to save
| money before IPO. The location was right near Tenderloin,
| the murder capital of the US. I walked through it from
| time to time. Nothing special, a city like a city. I
| remember thinking: how nice it is to walk through
| American streets - so less aggression in the air...
|
| There was a huge homeless camp though near Civic Center.
| There was a homeless man sleeping near a back door to our
| office, he once saved a girl patronizing a local bar from
| rape. In Soviet Union these homeless would have been
| rounded up and moved to some rural location not near than
| 100 km to any big city. Some could have been criminally
| tried even, like "elitist" Soviet poet Brodsky who was
| tried and convicted for a lack of permanent job.
| "Elitist" who can be tried and convicted any time
| authorities do not like something he says? (I kinda
| worried that Americans do not feel that personal freedoms
| had much value, and are willing to trade them for some
| fashionable illusion).
|
| But I can agree that there can be found some Americans
| who are materially worse than our family was. Still,
| calling us elitists - is a sign of laughable ignorance of
| the world at large.
| [deleted]
| bumby wrote:
| You might not know the wide variability in American
| standards of living. I had a SO who did not experience
| much luxury growing up. They had to run an extension cord
| from a charitable neighbor to have electricity to do
| their homework. It wasn't uncommon for them to sleep
| bundled up in winter clothes together in the living room
| because they had no heat.
|
| I understand television may give a false impression of
| the American lifestyle, but there's a wide range of
| experience in a country approaching 400MM people.
| c1ccccc1 wrote:
| Is that really true about Feynman? I thought he wrote in
| "Surely You're Joking" that he didn't take psychoactive
| drugs because he loved thinking and he "didn't want to
| mess up the machine".
| BaseballPhysics wrote:
| My understanding is he was reluctant in his earlier days
| but did indeed experiment later in life.
|
| From: https://gizmodo.com/10-scientific-and-
| technological-visionar...
|
| > Nevertheless, Feynman's curiosity got the best of him
| when he became acquainted with none other than John C.
| Lilly and his sensory deprivation tanks. Feynman
| experimented briefly with LSD, ketamine, and marijuana,
| which he used to bring on isolation-induced
| hallucinations more quickly than he could when sober.
|
| As an aside, that page has a list of other notable
| scientists who also experimented with psychoactive drugs.
| dekhn wrote:
| The article you linked left out that Mullis not only took
| acid, he wrote a Nature paper on universal time reversal
| based on ideas he had on a trip in Golden Gate Park.
| jamiek88 wrote:
| Yep and Einstein got laaaaaaaid.
| Ekaros wrote:
| I doubt he didn't spend his own effort. But I also agree
| that he was probably heavily pushed into math, by some if
| not all relatives.
| kiba wrote:
| I wish I was that enterprising at age 15.
| BaseballPhysics wrote:
| Even just a strong encouragement to pursue those
| interests, and a willingness to push him beyond what a
| typical parent might expect, would go a long way.
|
| You see the same thing in, for example, sports dynasties
| (e.g. Earl Woods coaching Tiger from a very young age).
| sriram_malhar wrote:
| Mother (Ayelet), her brother (Elon .. Fields medalist), and
| their parents (Joram and Naomi) had their work published in
| the Mathematical Reviews! That family is something else!
| ok_dad wrote:
| I knew a guy in high school that carried around a sub-compact
| notebook and one day in science class we were learning about how
| to factor quadratic equations (a review of old math we should
| know) and this guy was not paying attention at all, just typing
| away. The teacher asked him what he was doing that was so
| important that he couldn't listen, and to please come up and
| solve the problem.
|
| This kid walked straight up to the board and explained how you
| can design a computer program to factor any polynomial equation
| string input to it, and in fact had implemented a polynomial
| equation factoring program while the teacher explained how to
| factor simple quadratic equations.
|
| Since then, I don't feel bad if someone achieves more than me,
| because clearly there are some people out there that are _born_
| to solve certain classes of problems (maybe their brain structure
| is better for those, or something, who knows).
| Waterluvian wrote:
| Someone once told me, "comparison is the thief of joy."
|
| I'm confident this is a famous quote. But once I heard it, it
| kept resonating.
|
| Then years later I watched Bluey with my kids and mom says,
| "just run your own race" and it all clicked. I'm so thankful
| that it clicked because I feel liberated from this self-imposed
| sense that I need to absolutely maximize my time here, which is
| an impossible task.
| sho_hn wrote:
| Learning a new and at the time somewhat obscure (in my
| setting) foreign language when I was around 30 taught me this
| lesson at the time. I did well in the small class I attended
| with others, but not as well as some of the far younger
| students with both more free time and perhaps better age-
| given baseline performance at language acquisition (though
| this is not entirely clear-cut scientifically).
|
| I realized it didn't matter. What mattered is that I
| progressed toward my goal at whatever pace _I_ could muster,
| not making it a race. Interestingly, this mindset has made me
| willing to take on much larger and more complicated
| challenges in general, one day at a time, without feeling
| overwhelmed by the mountain to climb ahead.
| Waterluvian wrote:
| Yes! I love that. Thank you for sharing.
|
| This is the mindset that got me to do so many projects
| where a younger me thought, "someone smarter has already
| done a better job at that..."
|
| I now write software for me. Sometimes I publish it.
| chatterhead wrote:
| I don't know about mathematics, but in orgies this is
| absolutely true.
| [deleted]
| Waterluvian wrote:
| Semantics.
| JadeNB wrote:
| > Since then, I don't feel bad if someone achieves more than
| me, because clearly there are some people out there that are
| born to solve certain classes of problems (maybe their brain
| structure is better for those, or something, who knows).
|
| As a math teacher, I very much like every part of your comment
| except for this. There are way too many people who decide never
| to try mathematics because they weren't born to it. I don't
| believe (and it seems rather insusceptible to proof) that math
| is something that you are born with a talent for, or not;
| instead, it's all about how much passion you have for it.
| Genius is any field is not born of effortless gift; it is born
| of tireless effort. The real difference is between the people
| who are willing to put in that tireless effort, and those who
| do not wish to do so (which is a reasonable choice for
| something you're not passionate about!).
| phaedrus wrote:
| What's funny is I was that guy (not literally the same guy,
| but did similar things) as GP describes who could type in a
| program implementing the math lesson in the time it took the
| teacher to describe it.
|
| And. I. Struggled. So. Much. with math classes.
|
| Could never get homework done because I couldn't focus due to
| undiagnosed (and thus unmedicated) ADHD. On tests it'd take
| me the whole hour to do three problems out of twenty because
| I had to re-derive every equation and lemma from first
| principles before I convinced myself I was doing it right.
| And then (probably again due to the undiagnosed ADHD) I'd
| transpose symbols while working the problem or make other
| working-memory errors.
|
| The best math class experience I had (later in life) was with
| a professor at a community college who was a military veteran
| (mentioned because it influenced his attitude toward
| teaching). He said, "I'm going to spend hardly any time
| explaining things; we're just going to work the problems."
| And that's what we did; we just worked problems.
|
| You'd think I'd be bored with that, but it was the opposite.
| As I think about it now to try to explain why, I realize that
| other math classes I'd been in the expectation was lecture
| was for explaining the concepts, homework was for self-
| directed working the problems, and tests were for
| demonstrating that. But if you're bright and have ADHD you're
| quickly bored with the explanation and completely UNable to
| self-direct working the problems. So the class time was
| wasted and homework sessions were hell. So this professor
| using class time to work the problems was just what I needed.
| gcanyon wrote:
| I was on my high school's math team. For three years I was
| in a special class where all we did was look at math
| problems selected from competitions and solve them. On the
| one hand, it was great and fun. On the other hand, trig
| problems were in short supply, and I came out with a less
| than excellent grasp of trig, especially identities.
| klyrs wrote:
| I struggled a lot in math too. Got a D in calculus, nearly
| dropped out of high school. I'd absorbed my dad's excuse,
| "I'm bad at math." What a crock of shit. Turns out, I was
| bad at doing work in absence of motivation. But my parents
| didn't believe in ADHD, so I couldn't have that.
|
| Went back to community college after some time as a web
| dev, and had a teacher with a booming voice and a gentle
| attitude, who explained that we were going to be doing a
| _lot_ of homework. Like your instructor, he 'd spend class
| time working problems, and then I'd go home and do an hour
| or two of homework every night while it was still fresh.
| That kicked off a trajectory that resulted in a PhD and a
| very fulfilling job as a mathematician.
| akira2501 wrote:
| In school I essentially hated all math beyond Algebra,
| because it started to become something less tangible and more
| rote. Which is a sensible way to teach it as a pure subject,
| but it completely undercuts it's utility and connection to
| real world problems and the intrigues that go along with
| them.
|
| Once I started to really want to understand and write audio
| processing code, I suddenly had a framework to not only care
| about Trigonometry, but to actually apply it and to have a
| tool that would validate my understanding of it. That short
| feedback loop got me through the subject incredibly quickly.
|
| I was completely blown away when I started reading some
| Linear Algebra material because I realized that I had
| essentially been doing it for years without even strictly
| realizing it. The subject came to me very quickly because I
| already had a solid "mechanical" understanding of how to
| implement systems utilizing a particular subset of it.
|
| I think passion for the deeper maths is something most people
| just have to discover through application, and there just
| happen to be a rare few who discover a pure love of it very
| early in life.
| ducktective wrote:
| >there are some people out there that are born to solve certain
| classes of problems
|
| exact same experience... there is literally a genetic factor
| but people are not willing to admit it
| wingmanjd wrote:
| I wasn't as cool as this kid, but I wrote a binomial expansion
| program in TI BASIC back in high school that I was pretty proud
| of. Teacher said it was neat, but then banned calculators on
| our tests/ quizzes after I demoed it.
| noir_lord wrote:
| I had a maths teacher who accused me of cheating because we
| had coursework to solve a particular problem, I solved the
| problem for that case and then the general case using math
| she hadn't taught me even had a pascal program that you could
| enter parameters and it'd give you the answer.
|
| She literally couldn't get her head around a student going to
| the library, taking out a book on maths and teaching
| themselves because they where curious.
|
| The UK has a very strange educational system, curiosity isn't
| encouraged, you learn like a good little peg in a round hole.
|
| Left a very bad taste in my mouth but at least she did
| eventually apologise.
| nickstinemates wrote:
| "Never let school get in the way of your education" is
| roughly the philosophy my parents raised me on. Precisely
| to encourage what you had done vs. what school expected.
| saurik wrote:
| I remember my teacher banned the use of software on
| calculators (not the calculators themselves) but left an
| exception for me because she knew I was writing all the
| software I was using and didn't want to discourage me using
| that to learn about both algorithms and the underlying math.
| ok_dad wrote:
| It was a while ago, and my memory sucks for those kind of
| things, so maybe it was you! Anyways, programming anything in
| the space of 5 minutes is hard enough for me, so I'm still
| amazed at that binomial expansion (which I had to look up,
| myself).
| wingmanjd wrote:
| We're you in eastern PA in high school? Would be wild if we
| were in the same class!
| sudosysgen wrote:
| I wrote a similar program in those years about when we were
| first learning of quadratics. It's not really that I'm very
| smart, my parents were just math teachers and they'd force me
| to go through more advanced math books where I learned stuff,
| and I'd get bored in math class.
|
| My algorithm was slow garbage though, so I'm sure your friend's
| was much better.
| Vt71fcAqt7 wrote:
| And, to add to this, we need to find and nurture these minds so
| they don't get bored and can grow at their natural speed.
| RC_ITR wrote:
| >Throughout high school, Kaczynski was ahead of his
| classmates academically. Placed in a more advanced
| mathematics class, he soon mastered the material. He skipped
| the eleventh grade, and by attending summer school he
| graduated at age 15. Kaczynski was one of his school's five
| National Merit finalists and was encouraged to apply to
| Harvard.[17] While still at age 15, he was accepted to
| Harvard and entered the university on a scholarship in 1958
| at age 16.[19] A classmate later said Kaczynski was
| emotionally unprepared: "They packed him up and sent him to
| Harvard before he was ready ... He didn't even have a
| driver's license."[9]
|
| But we also need to make sure that we respect the humans who
| possess those brains too...
| r00fus wrote:
| For those who, like me, were wondering - this excerpt is
| from the Wikipedia page [1] about the Ted Kaczynski, aka
| the Unabomber.
|
| [1] https://en.wikipedia.org/wiki/Ted_Kaczynski
| PakG1 wrote:
| As discussed in Good Will Hunting:
| https://www.youtube.com/watch?v=-XCsE5NffMA
| [deleted]
| AuryGlenz wrote:
| There should be some sort of university made for people
| like that - where they're learning high level stuff but
| with people their age.
| tambourine_man wrote:
| I think you'd still have a problem mixing 7, 12 and 15
| year old kids. Perhaps not as much as 15 with 20, but
| still.
|
| This growing up thing is hard.
| Avicebron wrote:
| Bards college at Simon's Rock did something like this. My
| friend went when he was 15ish. I used to visit during the
| summers and while the education was relatively
| accelerated, there are a host of other issues that don't
| often get brought up with this model.
| jetbooster wrote:
| Added bonus is they come out of it being able to cast 4
| first level and 2 second level spells each day
| Zamicol wrote:
| I don't hear this viewpoint acknowledged enough.
|
| Society wasting and discarding undiscovered talent is one of
| my deepest fears.
| doodledo32 wrote:
| Eisenstein wrote:
| "I am, somehow, less interested in the weight and
| convolutions of Einstein's brain than in the near certainty
| that people of equal talent have lived and died in cotton
| fields and sweatshops." - Stephen Jay Gould
| swayvil wrote:
| It's easy to be a genius. All you have to do is give 100% of
| your attention to your subject, indefinitely.
| novosel wrote:
| That is the definition of a genius-savant.
|
| True genius is always happening at the crossroads of his
| interests. Not linear, not monosyllabic, but divergent-
| recomposing-thing.
| thaumasiotes wrote:
| > This kid walked straight up to the board and explained how
| you can design a computer program to factor any polynomial
| equation string input to it, and in fact had implemented a
| polynomial equation factoring program while the teacher
| explained how to factor simple quadratic equations.
|
| I find this difficult to square with the well-known theorem
| that there is no closed-form solution to polynomials of degree
| five or more. (Where a "solution" and a "factoring" are, for
| polynomials, the same thing.)
| TimTheTinker wrote:
| Must have been for degree 4 or below.
| jablongo wrote:
| > A computer program to factor any polynomial equation string
| input to it: Do you mean a program to solve for f(x) = 0 using
| numerical approximation? Factorization has a specific meaning
| and is not necessarily possible for quintic and higher-degree
| polynomials / there is no closed form solution like the
| quadratic formula for n>=5.
| THENATHE wrote:
| If they're talking about the quadratic equation, this is
| likely 8th grade to 10th grade in the US education system.
| Factoring would be y^2 = 3(ab)^2 + c^2, get y all alone so
| it's y = _____
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