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Is Math a Young Man's Game?
Bamafan77 writes "Slate has an interesting article on the relationship between the productivity of mathematicians and age. The conventional belief is that most significant mathematical leaps are all made before the age of 30. However, the author gives pretty compelling reasons for why this once may have been true, but is definitely not the rule now. Two of his more interesting pieces of evidence include Grigori Perelman's (probable) proof of the Poincare Conjecture at 40 and Andrew Wile's proof of Fermat's Last Theorem at 41."
I think 40 is probably the peak between the tradeoff between knowledge accumulation and physical decline. But stand for a psychologist or neurologist to correct me.
A bit like athletes maybe... experience vs. physiology results in a trade off.
I completely agree that math is a young man's game.
I'm so old, I lost count. Damn wippersnappers and their meaningless symbols.
When you get married and have some kids it is real hard do get any work done..
"Okay Dear I'll mow the lawn now"
I also suspect the growing complexity of screensavers as a factor..
is there anything really brain demanding or innovating you can do after 30?
Two of his more interesting pieces of evidence include Grigori Perelman's (probable) proof of the Poincare Conjecture at 40 and Andrew Wile's proof of Fermat's Last Theorem at 41.
Yes, but at the tender age of 22, I can not only add my bar tab together, but also figure an appropriate tip.
Young people can't do hard math my ass.
A century ago, mathematics was primarily a new field. New fields are characterized by inventiveness and a lack of prerequisite knowledge -- there isn't a lot of background to learn, and if you look at problems "the right way" you can get results very quickly. Most of mathematics is no longer a new field; in most areas, one must spend years studying before one can do anything new, and even then it's likely to be the result of long hard work rather than a quick new insight.
Computer science is moving in the same direction, but is many years behind. Thirty years ago, computer science was a new field; there were few if any courses teaching necessary background material; and someone with the right insight could find very important work very easily. Now, we're starting to see movement away from that -- there is a body of important work to build upon, and anyone who hasn't studied that work will have "new insights" which simply reinvent already existing work.
Mathematics is no longer a young man's game, and this is probably the last generation when computer science has been a young man's game. Next generation, the young will find a new field to excel in -- perhaps genomics?
Tarsnap: Online backups for the truly paranoid
0 to 5: Curious phase :-) ...: Irrelevant phase (atleast that's how it's treated by others)
5 to 15: Productive phase
15 to 40: Reproductive phase (some like to begin early and post longer
40 to 60: Consumer phase
60 to
Just remind me. The old guy, lots of white hair and a big moustache, worked at Princeton. Ein something or other, what was his name ?
if (story_id mod 3) = 1: [duplicate]
After that brainwashing people aren't simply able to do anything outstanding anymore. There are some accidential great scores, but they are very rare.
I think we should change our mathematics education to tackle with this problem. And we should indeed already start in school were the first and the most foul foundations are laid. Instead of teaching children basic counting, set theory and algebra which draws in the whole rubbish of non-intentionistic mathematics, we should start with Lie groups and algebraic varities. Indeed most "Joe Adverage" problems can be reduced to Lie/algebraic geometry problems.
I can give a simple example why this is necessary:
Imagine the Kleinian bottle in R^4.
You'll say now: "That's not possible nobody can visualized 4 dimensional spaces."
But this is only because your basic mathematical education fucked up your brain.
If a decent education would start like mentioned above, we all would have no trouble at all to visualized arbitrary n-dimensional spaces.
And because of using different logical concepts wouldn't have to use the problematic axiom of choice. So, no trouble with the Banach-Tarski paradox, inmesaurable sets and non-holomorphic refractions in H^p_2.
This is even a serious political issue. Anyone into math research will agree with me that in the last 15 years we saw a rise of a generation of brilliant new chinese mathematicians. And why did we saw it ? Because China went back to its Confucian tradition in teaching which avoids the above mentioned problems in Western math education. So, if we don't act now we'll loose our technological leader within the next 30 years forever.
Owner of a Mensa membership card.
Easy.
:)
You ever try to sell a kidney on ebay? You know how they stop you real quick?
Both filters use the same algorithm...
Frank Lloyd Wright did his most celebrated work after the age of fifty.
-kgj
Also worked on the proof for Fermat's theorem for 7 years in secret(which in the mathematics community is a rather odd thing to do). He was dreaming of solving it while he was still a child. There is quite a good book on the subject for anyone with any level of knowledge called fermats last theorem. I'd give you a link but i'm tired..
Why me?
Definitely this is the women-not-invited dept., as billed, but it reminds me of a conversation I had with a 98 year old woman in 1982. I was 28, had a toddler and an infant, and was very much afraid that motherhood would be the end of any other kind of creative work for me. (The exhaustion factor alone was daunting.)
Miss Mae said to me, in a Miss-Daisy sort of Southern accent, "Honey, women are not like men -- we get better with age. After all, you can't think straight until your parts settle. I promise, when you are 45, you'll know what you want to do with yourself, and it won't have anything to do with diapers."
She was right about women, or about me, at any rate. I'm 48 and in my first year of professional school while the "baby" is at his first year of college. (What this has to do with my "parts" I am less sure.)
What I notice is that my younger colleagues are quick and bright, but that what I lack in speed I make up in context. And all of us are passionate about what we are doing, but the flavor is a little different depending on age. When we are working well together, the combination of gifts is truly wonderful. Perhaps instead of framing the "game" (of math or of anything else) as a contest, we ought to be looking at ways to make progress that makes use of both the experience of age and the quickness of youth.
OK, now what?
Let's not forget that most pure mathematicians are University faculty members, and that the longer you're on faculty, the more committees you sit on and the more non-research responsibilities you end up stuck with.
The real question is whether or not great discoveries in a field come from someone being young and having therefore enough mental clarity or from an amount of exposure to a field, resulting a certain level of understanding.
Could it be because not so long ago
people usually didnt live
beyond 40?
What about young women?
/., is primarily an activity of men.
I know, I know: math, like so many of the things discussed here on
But it seems to me that we would be much better served if we talked about how to get more women in the field, not how we could keep old men in it. I mean, aren't there enough old men around anyway?
(spoken by a future old guy - hopefully)
It's simple: Young mathemetician's aren't getting laid -- so they work like hell on on their maths. Since male sex drive peaks at 18, the less sex drive you have, the less driven you are to find another way to spend the time.
Or maybe they got married and their wife nags at them to death and ruins their concentration.
When visiting mathtutor one can see that even 200 years ago, many important discoveries were done in the later stages of the Mathematicians career. Stories like the ones about Abel or Galois distort the picture.
More and more discoveries of younger mathematicians are achieved through collaboration or by standing on the shoulders of people with more experience (who tend also to be more generous with sharing their ideas without expecting credit).
Mathematical knowledge continues to accumulate in a fast pace and only few of this knowledge has been absorbed in books. Chances grow that a young mathematician will discover something already known or to be a special case of a much more general result. Fortunately, there are better and better online databases but it also needs more and more time to dig through that material.
The most productive age for a mathematician will grow also in the future. The same will happen in physics or computer science (as a previous post has pointed out already).
I can't believe that statement! I'll have you know that at 38 I'm just as...um...uh...what was I going to say? Hey, today's Saturday! The buffet has the early bird special today for dinner at 4pm! I'd better get the oil changed in my Oldsmobile first...
The truth is I don't feel any older than I did at 25 (still like the same age women as a matter of fact), I'm in better shape than I was then, and if coding skills are any indication I'm sharper than my 20-ish coworkers. So there!
Now if you'll excuse me I have to knock back my Ensure before I chase the kids off my lawn.
It can definitely be said that some mathematicians produced work at an early age. As the article said, many died early, some continued to produce work throughout their lives. And the body of maths has increased so much that it's much more work getting an good overview of a field.
Note also that before the 19th century, scientific research didn't have the same place in society: it has grown quite a lot.
But regardless of the mathematician's age, what has to be taken into account is the relationship between groundbreaking work, and sturdy, low-profile, everyday work that is achieved by the mathematics community as a whole.
Without that, the breakthrough cannot happen: it loses its value, as it has no ground to stand on.
This is of course relevant physics and astrophysics as well: if you didn't have people studying and cataloguing stellar spectra, you couldn't develop theories about distances, and, more crucially, n-dimensional cosmological models. Now remember, stellar spectra themselves are boring as hell, so are atomic spectra (the spectra that prompted quantum mechanics, etc.)
There are a lot of romantic ideas in the non-scientific public about science: I meet them every day. Sometimes they are just funny, but other times you wonder about the image that society has of your work. Of course I am by no means degrading the value of scientific breakthroughs and original thinking: any deep thought is a process that I consider to be mysterious in essence.
yours ever, fz.
>>In other words, hurry up and die. Your life past this point is merely an exercise in selfish indulgence.
I used this weeks mod points about 12 hours ago. If I had them now, I'd send you down to -15 you prick.
So at what age are YOU planning to die? Bastard.
He wrote about humanity's cleverness having outstripped its wisdom. In the story his hero sets up a foundation to retard the progress of scientific knowledge, to give our wisdom a chance to catch up.
About the widely spread notion that math, physics etc, are fields were only the young come up with the paradigm shifting insights... I have also read the suggestion that it is new arrival in the field that really counts, and that the older person who switches fields can come up with the paradigm shifting notion too.
My knowledge of pure math is not sufficient to know this. Are these two recent, famous developments really paradigm shifting? Or are they admirable accomplishments, but more developments of existing ideas? Can anyone set me straight?
Your life past this point is merely an exercise in selfish indulgence.
...?
And yours is an exercise in
OK, now what?
Perhaps you should realize that since you've fulfilled your primary purpose as a human being (reproduction), all you're doing is taking up space and resources needed by the next generation to raise its offspring.
In other words, hurry up and die. Your life past this point is merely an exercise in selfish indulgence.
I assume this was just a joke, but...
Au contraire. Given that there are 6 billion people and growing on this planet, and given that a depressingly large fraction of them live in crushing poverty, overpopulation is a huge problem, and it's only getting worse. The solution? Fewer offspring. Nowadays, the selfish indugence is having kids. Sure, we want the species to continue, but there's no worry about that at the moment. (It's like spaying your dog or cat; there's no anger that there won't be kittens and puppies, so it's best for all concerned to spay.)
I'm not saying nobody should have kids. But if we want to have any hope of the people on this planet living in relative comfort and prosperity, we need to overcome that evolutionary programmed urge to procreate-- which is selfish on a species level, if not an individual level. Sure, evolution designed us so that our purpose is to reproduce, but unless we want the whole world to live in squalor, we now have to redefine that purpose.
So go on to professional school and develop your brain when you're older. Learn math, contribute to human knowledge even when you're past the age when "tradition" dictates you can make your best contribution. Bettering ourselves and our world should be the purpose of existence now, not just producing more and more kids to use the dwindling resources of this planet. Meanwhile, we need to figure out a way to seriously limit the number of kids produced each year while preserving as much personal freedom as we can.
-Rob
Nobody can visualize n-dimensional geometry if n is greater than four. You can imagine a 3-dimensional retine and proyect on it 4-dimensional geometry. You get a 3-dimensional projection of a 4-dimensional object, which your brain can handle. But it's not the same than projecting n-dimensional objects on (n-1)-dimensional retina cuz' your brain can't visualize it, it's just not made for that.
If you do so, probably you are neither visualizing the clasic hypercube correctly. It's not about a theorical visualization but a real one. It's easy manipulate n-dimensional spaces, but it is biologicaly immposible to visualize it if n is greater than four, as i said, your brain is not made for that...
In a nutshell the grandmother can provide additional food resources to the weaned children of her child or her childrens mates (to increase their fertility) since she no longer has to provide those resources to her direct children and can produce excess to what she consumes.
Thus there is an evolutionary advantage to women surviving following their fertile years, and this advantage likely continues in different ways now.
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It took Andrew Wiles seven years to write a rigorous proof for Fermat's Last 'Theorem'. If he had started when he was 23 instead of 34, he would have proved it while he was 30, instead of 41.
The real problem, of course, is that it wasn't until Andrew learned about the Taniyama-Shimura conjecture that he figured out the method for proving Fermat's Last Theorem. He then waited for 2 years before starting.
Who I think is a better example of mathematician burnout is Yutaka Taniyama himself. He started his career at 28 - way old for a mathematician - and killed himself at age 31. A year after his mathematical prime. Coincidence? Maybe. But you never know...
A counterexample:
Paul Erdös. Read about him in this book.
The man did math until he died of old age, at a pace of about 18 hours per day. He cared not for material things, as he lived out of a suitcase. He cared not for life's physical pleasures, as he (almost!) never even had a girlfriend, or boyfriend for that matter. He had his doctor perscribe speed to him, so he could work more hours on mathematics.
An amazing read about a guy who I am amazed by, but also whose qualities I am glad I don't have.
No, back to studying linear & nonlinear programming, stochastic processes, dynamic programming, and queueing theory for my qualifier on Monday.
Support a few technologists in Washington.
A highlight:
Small potatoes make the steak look bigger.
In a nutshell the grandmother can provide additional food resources to the weaned children of her child or her childrens mates (to increase their fertility) since she no longer has to provide those resources to her direct children and can produce excess to what she consumes.
Interesting. That supports my current favorite perception about menopause, which is that it actually seems to make a woman operate more efficiently in a lot of ways. "Gains weight easily" translates to "needs less food." "Insomnia" translates into "needs less sleep." Hot flashes, however, only have utility in the wintertime.
OK, now what?
OK, I've got karma to burn so mod me down, but...
The abbreviation "math" really grates on me (outside the US it's called "maths"). It's not mathematic, it's mathematics.
Don't get me started on sulfur either...
Bob
Listen to my latest album here
A lot of very tallented mathematicians go down a dark road in their 20s, trying to prove the impossible, giving up prime years to fail at something and a few actually do prove something important and then are spent. Godel was nuts to start with and the work he did in his 20s pushed over the top.
not a lot of people ever achieve anything after the age of 30... but then again; not a lot of people ever achieve anything before that either!
Of course, mathematics is a young man's game. But it's also old man's game. If you're willing to devote yourself to mathematics, it's yours!
In most cases when people "get old" they just tend to drop mathematics to spend some time with their kids or whatnot. It's not like they lost their ability to think.
Want proof?
I can't give you one, but here's a conjecture.
Paul Erdös!
Of course the real reason that scientists might make more discoveries at advanced age than in past times is simple. Viagra. What's more inspiring than getting some tail?
No, I didn't think so. You call it "econ" like the rest of us. There are countless additional examples as well, but going into them would be a waste of time. Believing that your abbreviation is the only correct one is both naive and arrogant. It's obvious that ours is the only correct one ;-)
"The reader who's seen other nontechnical accounts of the subject will forgive me, I hope, for perpetuating the fiction that the whole field of topology is actually confined to the study of spheres and doughnuts. There are other shapes, I promise: They're just harder to describe."
snif, snif... is there a conspiracy against the this is the klein bottle second time on slash dot that it is expressively not mentioned ??
"Almost all the rich men have become rich late in their lifes"
Well to just take Gates example from the parent post, how old was he when he made is first 10 M$ ?
For great insights into the mind of a world class mathematician, please read A mathematician's apology by G. H. Hardy. Hardy was one of the top mathematician's of his era (1877-1947). Hardy is perhaps most famous for his discovery of Ramanujan and "A mathematician's apology" has a great Foreword by C. P. Snow documenting some of the details of the Hardy-Ramanujan collaboration.
Here are some nuggets from "A mathematician's apology". (Hope the copyright police are busy elsewhere.)
"No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game." [Section 1.4, page 70]
"Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty."[Section 1.4, page 71]. Also see Men of Mathematics for more on Galois.
"I do not know an instance of a major mathematical advance initiated by a man past fifty." [Section 1.4, page 71].
And later in the book,
"There are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call the 'trivial' mathematics, for want of a better word" [Section 1.28, page 139].
Anand Rangarajan anand@cise.ufl.edu
As far as age and mathematics go, though, I'd have to agree that the effects of age are, if not disappearing, then at least being shifted back a number of years. Not long ago, I had the fascinating realization that after 3 years of college, I know more mathematics than Euclid, Diophantus, al-Kwahrizmi, Fermat, Newton, Leibniz, Euler, Hamilton, and Abel. This is not because I'm some sort of mathematics genius (I'm not even a math major), but rather because there is simply more mathematics to learn now, and I merely came later than those guys. For centuries, the situation was such that almost all of the human race's mathematics knowledge could exist in few enough books to carry in your hands- namely, Euclid's Elements and Diophantus's Arithmetica, eventually followed by a few others like Fibbonacci's Liber Abacci. In the 17th-19th centuries, mathematics used these simple foundations to create an incredible wave of new mathematics. (Just take a look at Fermat's annotated copy of the Arithemetica.) Now the number of books written on some specialized part of mathematics like Lie algebras or K-theory could fill a library.
Also, mathematics works a bit differently than the natural sciences- it's harder to create a general survey course in mathematics. Just look at the way these subjects are taught- you generally take high school science courses in physics, chemistry, and biology, but math courses in algebra, geometry, and calculus. The specialization has to start much sooner because eachthing builds off of the previous. In my high school chemistry courses, I remember covering some basic p-chem, some orgo, etc, and in my physics courses there was mechanics, E&M, optics, etc.. I of course returned to all of these in excrutiating detail in my college course, but the simple point is that you couldn't do a similar thing with math. In physical sciences, you can give a broad overview of a subject, and then later reurn in depth, because there isn't such an elaborate hierarchy connecting all of the fields. Conversely, mathematics works more like a pipeline, shuttling students from simpler subjects (basic arithmetic, simple Euclidean geometry) to harder ones (integral calculus, diff eq, set theory). The pipe opens up at the top- areas of specialization become apparent, and a frontier is reached where knowledge in one field is not necessary for knowledge in another.
In fact, there are so many fields and subdisciplines now that it has become incredibly difficult to become a polymath (in the quite literal sense of the term) in the vein of Euler or Gauss or Riemann. The idea of a single person making revolutionary discoveries in both, say, topology and number theory is steadily becoming more remote. If this were to happen, it would have to be someone who spent a long time mastering several disciplines, i.e., an old person. It's a sublime paradox- in the past, incredible leaps of insight that would connect disparate theorems and fields of math could only be made by the young mathematicians with the creativity and the daring to do so (or, if you're cynical, the neuronal plasticity), but now such individuals will still be in grad school learning the ropes.
Look at Andrew Wiles- it took him years to learn enough a
"FDA staff reviewers expressed concern about the number of patients who were left out of the study because they died."
The Book is "Fermat's Enigma" by Simon Singh. I highly recommend it. Singh has a talent for writing about deeply analytical subjects. He also wrote "The Code Book" about the history of cryptography, and he's written a Nova episode or two.
I wish he'd written more books; an Amazon search turns up little else than these two.
Andrew Wiles' proof of the famous x^n + y^n = z^n equation having no proofs wasn't really just a breakthrough at the age of 41. He'd caught interest on this equation at the tender age of 10, and had been working on the thing his entire career. This was probably the dedication required to solve such a proof. Most people would have given up in the time it took him.
Anyway, read Fermat's Enigma, It's a great book, even though it's about math, it is surprisingly interesting
Anthropic principle: We see the universe the way it is because if it were different we would not be here to see it.
Fermat had said it was simple !!! And he didn't think it needed as much space as While took, just a little bit more than the margin, that's all it needed
I think that the proposition that mathematical breakthroughs are predominantly made in youth, whether true or not, relates not to the vigour of youth, but to the settling in of dogma.
I've seen this proposition about physicists in more than one lay venue. It was made clear that most breakthroughs in physics were made by minds that had the flexibility to "think outside the box." The gist of the "youth" paradigm is that the more years dedicated to a subject, the more that the thought patterns get set in their ways, precluding the intuitive leaps that change the intellectual landscape.
That being said, Wiles didn't just make some brilliant leaps. He worked damn hard on the details. It may have been more than 10% inspiration for him to prove Taniyama-Shimura (the real achievement for which Fermat was a by-product). Still, from what I've read about his accomplishment, his work was definitely more than half perspiration.
Is Math a Young Man's Game?
Well, if virgins are men, then yes.
The coolest voice ever.
Yes, we can learn the already discovered algorithms by the time we have a Math BS, but by then we are around 22. Our current system does not allow the best to advance at their own pace.
I was reinventing Calculus by 8th grade. I was about to win second place in an international math contest. (I was beaten by a 9th grade Canadian.) I usually ignored whatever was being taught in Math class, since I could literally get an A without waking up.
I was attempting to find the area under a curve defined by a formula. It seemed appropriate to do the work in math class. One day, my eight grade math teacher asked what I was doing. I showed him my current theory. He told me that there was already a proof that it was impossible, so I moved it from active work to the "known impossible, but cannot stop trying" category that includes a simple formula for discovering factorials.
If he had mentioned the word "calculus", I would have researched what was already done and continued with new discoveries. Or he could have encouraged me to repeat the discovery. Instead, he told me it was PROVEN IMPOSSIBLE.
Personal note: This was an important event in my life, because a few years later they tried to teach Pre-Calculus. I immediately absorbed the entire book, and then taught myself Calculus. But I could have done that a few years earlier. And it was the first time that I had proof an authority figure lied to me. The realization that adults have no clue even in their specialty was a major part of my maturing. Now I question facts even when the person giving them is the "top authority".
If our education system helped students that showed an aptitude for math to advance at their own rate, they would probably be finding better algorithms for known problems, with the possibility of discovering something new, as a teenager. Tiger Woods specialized in golf starting at age 3. Most Ice skaters, gymnasts, and dancers start before they are 6. Why should mathematicians need to wait until college before specializing?
---
Off-topic details: I was reinventing Newtonian Calculus. Newton invented a system about the same time the current system was discovered by the French. Both systems were used for a time, but further advances (Differentials) were only possible using the French version, so Newtonian Calculus was dropped. So it was unlikely my redicovery would help advance today's knowledge, since it was on a dead branch.
I spend my life entertaining my brain.
I've worked in and around Academic departments and I can tell you that you can sure see it. The Assistant Professors are busting their butts, late nights and weekends on their research and that immediately changes the day they get tenure.
Some tenured Professors work hard on their research, those that really love the field. People who really love their field are what we should be encouraging in Academia, they also make the best teachers, but the current tenure system doesn't really select for this very well.
I'm just ranting. I don't really have any good ideas on what to do about it.
Maybe there should be some way that good pedagogical performance should be factored into whether tenure is granted, but in most higher education settings I've seen, being a good teacher is considered a stain on your Academic Credentials.
Notable counterexamples are Haydn of the Classical period, who started writing his best symphonies after 50. Also, there's Beethoven, who wrote the 9th Symphony when fairly old and stone deaf.
A quote attributed to Marvin Minsky: "Ever notice that mathematicians tend to be good at music, but musicians tend to be bad at math?"
(ah, which I just discovered *IS* referenced in the article... oh well!)
It's an interesting read when you're stalling on doing your calc homework. Especially if you're not doing well at it :^)
When a mathematician is in grad school or fresh out of it, she wants to publish as much as humanly possible, because having a 15 page CV helps one get tenure at a good university. So just about any thought she has that adds a tiny bit to the sum knowledge of humanity, she'll send to a journal. This is not to say she's not doing good work, just that she's publishing early and often. But that's what the tenure granting committees look for, so what else should she do?
But when she gets older, she can settle down and try to tackle harder and more time-consuming problems--that's one of the reasons for the tenure system, after all. So she may not look as productive, but she's contributing her time to mathematics in just as important a way as she did when she was younger. Also, her experience will allow her to supervise research more effectively, and she'll find that her time is well spent supervising a number of graduate students, giving them advice and help in their research.
On another note, remember that the vast majority of professional mathematicians will never solve a famous problem. And yes, every young mathematician tries to solve the Riemann hypothesis, but as he grows older he learns to spend less time on problems on which he's unlikely to make progress. There are exceptions to this, like Andrew Wiles. (And personally, if I had been on his post-tenure review committees during those 7 years, I'd have wanted to know what he was doing to justify a salary: mathematicians very rarely keep their work secret like that.) But while a mathematician in his 20s may be encouraged to try long-unsolved problems, he tends to grow out of it unless he's brilliant enough to have success with it.
Gates' Law: Every 18 months, the speed of software halves.
His beard is a sophisticated anti-ageing and life-support system which permanently keeps him at an unknown age between 30 and 44, thus keeping him alive forever and reducing his Dirty Old Man quotient. You could eject him into space and he could live from the air bubbles and dropped food trapped deep inside his beard for weeks.
Prime numbers are exactly what Alan Greenspan says they are -S. Minsky
Obviously what is needed is the ability to patent everything in mathematics. Innovation would clearly pick up if this were to happen. Just look at what it's done for the computer industry.
"But this is only because your basic mathematical education fucked up your brain."
No, actually, it is because of our world and our perceptual makeup. We live and interact in 3 normal dimensions (time is special form a perceputal point of view). When you look at something in the real world, you see three dimensions. Be it an inherant thing, a learned thing, or some combination of the two, you are equiped to deal with 3-dimensional perception.
Whenever you deal in higher space, you are limited by that in terms of visual representations. If you want to look at a 4D fractal you have to do it in 3D. You can do it is a bunch of 3D slices, a 3D image that you can dolly around the 4th axis, whatever, but you are still only going to see a 3D slice of it since there is no way to directly percieve more.
What I find interesting about this is that Mathematics and Mathematicians often has the opinion about itself that is above and beyond real world constructs such as "age" and "gender" and stereotypes of the like....which is why whenever I cry sexism in my math classes no one listens. How can math be gender biased? How can math be age biased? The structure in which we do math causes it to be so.
But here we see the perception that math is a young men's game argued and articulated, where it really could be something that is a result of the assumption that it is and our academic culture.
Math didn't really start to distinguish itself as a distict field of study until the last 50-60 years, and previously to that it was mostly seen as a tool used by physicists and engineers - so alot of the progress made in mathematics could be a result of people learning these "tools" early in their education, and then going on to research something else for awhile.
Also, the way the accrediation system is structured, in order to get a doctorate you HAVE to show some genuis at an early age.
Not from me, but from my dad who knew Wiles when they were both young mathematicians.
My dad says the implication of the article, that Wiles become very smart at 40, is bogus. Everyone, even when Wiles was 20, spoke of how he was very smart and "one to watch for." That he was doing super-important work back then.
In other words, it is not that Wiles became suddenly smart... it is that he suddenly became a celebrity at 40. His past work was impressive, too, but in reality the public has no chance of understanding his past theorems, let alone their proofs.
The other part, he says, is that a young graduate student can not take a nebulous task like "Fermat's last theorem"; you would never graduate, never publish. It's a post-tenure job, when you can deveote your energies as you please. In that context, 40 is the appropriate age to be solving something as large as this.
Isn't this idea an insult to all the doctors who have nearly doubled the human lifespan in the past century?
No way, dude. The original poster who said "A century ago, mathematics was primarily a new field" was way off base, and the follow up isn't any closer. Sophmore engineering students are pretty amazing, I know -- check out those concrete canoes! -- but their math curriculum encompasses about one percent of the math available a century ago.
The last person who might possibly have mastered the whole of mathematics as it existed in his era was Henri Poincare'. Incidentally, he did much of his most memorable work just about 100 years ago. The suggestion that today's undergrads might have a comprehension comparable to his, is just silly.
It's an unpopular opinion because we've let females gain too much independence too quickly,
No, the opinion you expressed is unpopular because the business lobbies, through their servants, the media, have MADE it an unpopular opinion. THey want women to work because it provides more consumers. THey like consumers.
but I'm sure in 20 years we'll be looking back and saying "You know, he was right!". Get those women pregnant,
No, reproduction will be a thing of the past in 20 years, or maybe a little longer. Maybe 75 years, or whenever people get educated enough to realize that we are programmed to pass on our genes, but that this is a trap of biology.
Sig:
Navy nuke sub lifestyle?
But, 10% isn't that bad.
So, I don't think it's biological. I think it's more to do with stuff like spare time, having a drive to do something, looking at new material rather than being stuck doing the same thing all the time and so on; having children to look after etc. etc.
It's a software problem, not a hardware problem. And people can rewrite their software, and it gets rewritten by people around you all the time (although their are limits!)
-WolfWithoutAClause
"Gravity is only a theory, not a fact!"I mean, Poincaré didn't have to allocate any brain processing capability to this task. If it was quietly computing in the background (subconciously), it wasn't consuming any attention or decision-making ability from the brain, unlike the demands on the processor by software such as distributed.net and SETI@Home (which are not AI programs, of course; just using them for this point on background processing).
The problem is not that people are having too many kids, it's that its the wrong people -- or at least the people in the wrong place -- are having most of the kids. Up here in Canada, we have near limitless space and natural resources, but a reproductive rate that will not fill it before the Sun dies out. So we are the world's secondary largest (Australia, another empty country, being first) importer, per capita, of humans.
What we need to do is give the world the tools to control their reproduction, and then educate them about when reproduction is a good idea. And specifically in the empty countries we need to figure out social engineering techniques to allow our countries to accept as many immigrants as possible without becoming ugly melting pots like the US or losing our national identity.
You can learn more about it from this book.
Happy people make bad consumers.
The article has a valid point. There is indeed an uphill battle against the popular notion that mathematicians are worn out by the time they are forty. But there is a reason for it to be this way: history. The fact remains that many of the legendary exploits of the people in the pantheon of mathematical heroes, particularly in the seventeenth through nineteenth centuries, were accomplished when the men (sadly, with few notable exceptions like Agnesi, Marie du Chatelet, and Sophie Germain, women remain anonymous) were in their late teens and early twenties. Gauss established most of the underpinnings of modern number theory in his mid-teens, publishing his authoritative tome on the subject, Disquitiones Arithmeticae, when he was only sixteen years old. Newton derived the vast majority of his relevant work in his early twenties and spent the rest of his life ruminating on religious matters, holding political office (including a seat in Parliament and running the Royal mint), and dabbling- and eventually poisoning himself into insanity- in alchemy. Similar stories hold for other eminent mathematicians (e.g., Pascal, Descartes, Riemann, Ramanujan, just to name a few) some of whom died while young. What is so often overlooked is that many of the prominent mathematicians in history- e.g., Newton, Gauss, Euler, and several of the Bernoullis, and Weierstrauss among them- remained mathematically formidable in their later years (Newton's invention of the calculus of variations relatively late in his academic career is one such example) and contributed many results that have revolutionized numerous fields in mathematics and physics. This oversight, in the popular consciousness at least, perhaps illustrates the fundamental flaw of relating age to mathematical brilliance: mathematical history has been transmogrified into mathematical mythology. That this should happen should come as no surprise. After all, is there not something romantic about a young mathematical hero opening up new vistas in knowledge (that there are instances of them dying in a metaphorical "blaze of glory" while still in their ascendancy compounds the heroic and tragic elements of the story) in comparison to what's viewed as to the steady, plodding course of old and stodgy men firmly rooted in the 'establishment'? The real tragedy in all of this is that the tenure processes at certain institutions of higher learning are still premised on a gross over-prizing of the brash intuition of youth and the promise of things to come over the gradual maturation and refining of talent; the environment is as unforgiving as it is off-putting and one can only wonder at how many accomplishments of the so-called "late-bloomers" have been denied to society as a result of it.
The article has a valid point. There is indeed an uphill battle against the popular notion that mathematicians are worn out by the time they are forty.
But there is a reason for it to be this way: history.
The fact remains that many of the legendary exploits of the people in the pantheon of mathematical heroes, particularly in the seventeenth through nineteenth centuries, were accomplished when the men (sadly, with few notable exceptions like Agnesi, Marie du Chatelet, and Sophie Germain, women remain anonymous) were in their late teens and early twenties. Gauss established most of the underpinnings of modern number theory in his mid-teens, publishing his authoritative tome on the subject, Disquitiones Arithmeticae, when he was only sixteen years old. Newton derived the vast majority of his relevant work in his early twenties and spent the rest of his life ruminating on religious matters, holding political office (including a seat in Parliament and running the Royal mint), and dabbling- and eventually poisoning himself into insanity- in alchemy. Similar stories hold for other eminent mathematicians (e.g., Pascal, Descartes, Riemann, Ramanujan, just to name a few) some of whom died while young.
What is so often overlooked is that many of the prominent mathematicians in history- e.g., Newton, Gauss, Euler, and several of the Bernoullis, and Weierstrauss among them- remained mathematically formidable in their later years (Newton's invention of the calculus of variations relatively late in his academic career is one such example) and contributed many results that have revolutionized numerous fields in mathematics and physics. This oversight, in the popular consciousness at least, perhaps illustrates the fundamental flaw of relating age to mathematical brilliance: mathematical history has been transmogrified into mathematical mythology.
That this should happen should come as no surprise. After all, is there not something romantic about a young mathematical hero opening up new vistas in knowledge (that there are instances of them dying in a metaphorical "blaze of glory" while still in their ascendancy compounds the heroic and tragic elements of the story) in comparison to what's viewed as to the steady, plodding course of old and stodgy men firmly rooted in the 'establishment'? The real tragedy in all of this is that the tenure processes at certain institutions of higher learning are still premised on a gross over-prizing of the brash intuition of youth and the promise of things to come over the gradual maturation and refining of talent; the environment is as unforgiving as it is off-putting and one can only wonder at how many accomplishments of the so-called "late-bloomers" have been denied to society as a result of it.
I expect your post to disappear as a troll, and since you are a Coward, you will not receive notification about this post, but I would like to know how you arrived at your conclusions.
I seem to be a typical Slashdotter. Programming is both my career and my hobby. I have no television by choice, although I like "Buffy" and "The Sopranos". I read Slashdot because it is the best site to discover news that interests me; most techie sites tend to be too focused. OTOH, women like me.
I mentioned the math contest to demonstrate that someone in the system knew I had ability. I could have mentioned my 7th grade SAT scores. I do not even know what the contest was; my mother probably has the certificate. The school pulled me from classes to take the test because they expected me to win. I had no warning: just "Here are the questions. Fill in the answerse." It could have been one of those IQ tests they kept giving me. I did not know that it was a contest or the scope of it until they told me I was beaten by a Canadian.
I did not reinvent Newtonian Calculus. That was the point. I PROBABLY would have made the leap within the next year, but the math teacher discouraged me. If he had said anything but "It has been proven impossible," I would have continued trying. (And I had already had sex by this time. As I said, women like me.)
I am modest, but I have never claimed to be humble. Modesty is about knowing one's abilities. I do.
I am trying to improve the world. That would not be possible if "everything had been discovered."
Factual errors in his posts (particularly on historical topics) come only second to a mediocre grasp of English. The SAT vocabulary combined with mediocre sentence structure and atrocious punctuation form the True American Nerd.
The only history I mentioned is my own, and I did not provide enough information to check it, so you cannot claim to have found any "factual errors."
I do not expect everybody to like my writing style. None of the words I used should be unfamiliar to programmers interested in mathematics; if there were any you did not understand, look them up. Much of my recent writing is targeted at business people, so it may have been "dumbed down" from how I wrote in college, but you understood me. Right? So it was successful.
The punctuation was a little poor. The comma should be removed after "One day". And I tend to start sentences with a conjunction. That is how I talk. I would fix that for a professional report, but it is not worth the time for posts.
I spend my life entertaining my brain.
I had always heard people refer to them as Newtonian Calulus and French Calculus, so I checked. See History of Calculus.
Pierre Fermat and Gilles Roberval were French.
Gottfried Leibniz was born a German, but his early contributions to Calculus happened while he was living in Paris. He returned to Germany in 1676, and did not publish until later. You are correct that I was thinking of his work.
I do not know why I believed it was referred to as French Calculus. Today it is called "Integral Calculus", or just "Calculus". Does someone know the correct name that distinguishes Leibniz's work from Newton's?
I spend my life entertaining my brain.
I attended a talk by Sharon Stephenson on women in physics. She said that in Japan, there is a significant rise in productive publication from middle-aged female researchers. It was due to two factors: the fact that their children grew up, and the fact that sexism kept them off of many committees and such. Both factors conspired to give them a lot of free time to research.
... that is almost totally abstract from math, but a valuable life lesson. "Authority figures" can and will lie to you, either a lie of ommission, a lie through ignorance (as your case sems to be), or a deliberate lie from another agenda you may not be privy to. With myself at a young age it was politics and "the news". What clued me was what I read and the "popular perception" that "everyone knows",as opposed to then getting the real information from some connected people who would be classed as "insiders" in government, some relatives, some just interesting adults who I think the notion of someone so young being interested in some subjects was enough to throw them off and perhaps they told me things they wouldn't have told an adult, but..I remembered, added it to the mix. Once your eyes are opened, you may see clearer. Removing the blinders is the hardest part for most people I think,or to even notice they have the blinders on.
Some people never even do that.
By Simon Singh. Would appeal to non-math majors, too. Inspired me to seriously consider Math as a major here at Berkeley. Now if only the department wasn't so nerdy...
After years of banging my head against this age limit (mostly from hearing about great physicists making their big discoveries at age 26 or so), I came up with a different interpretation. Most breakthroughs come about after many researchers have been taking a crack at it for a long time. The one who finally gets through is the one whose education is idiosyncratic enough that he is able to see the problem in a unique way that leads to a solution. So it might not have to do with a fresh brain so much as fresh point of view.
I think math is mainly a smart man's game. Age doesn't have nearly as much to do with it as intelligence, education, and drive. I'm 21 now and there's no way in hell i'm going into a math based career; I find math to be agony.
In terms of output, you'd probably see more in a young man as they have to prove themselves. Ph.ds aren't given out at the drop of a hat you know, professors make you pay your dues. When you're old and tenured there's less of a need to be in the lab at all hours.
Hell, with the commig bitech and nanotechnologies, questions like this one will become moot and part of history, when the technology of life extention becomes availible for most people, big changes will happen. We could have this technology much sooner if we stopped or slowed the usual worl-wide arms races/dissagreements. It amazes me all these super-rich people who have vast amounts of money who have yet to realize that a relativly small investment in cash flow today, will bring immense pay-back in lfie extention technology (who really cares about money and gold etc, when your getting old..), besides, it would be great for intelligent people everywher to have to stop worrying about some arbitrairy clock that dictates when you can't innovate any more in your feild or when you have to retire....
You are welcome. Even posts that you have difficulty understanding can bring joy.
This is my first flame war, and will be the last one with an anonymous opponent. It has been fun, but it would be better if there was a person on the other side. Slashdot is available internationally, so I cannot expect you to understand English. I wish my writing was more humorous; I am sorry you could not make it funny by taking my statements out of context.
I wish there was a way to start this post at Score:0 so the readers who browse at Score:1 would not see it. I do not post anonymously, but I would prefer not to attract attention to this thread.
I will not be adding to this thread unless another real person does.
I spend my life entertaining my brain.
In a nutshell the grandmother can provide additional food resources to the weaned children of her child or her childrens mates (to increase their fertility) since she no longer has to provide those resources to her direct children and can produce excess to what she consumes.
Interesting addendum: it is now firmly established that this sort of support or its modern equivalents are stronger from maternal than from paternal grandmothers.
The reason is that paternal grandmothers cannot be 100% sure that the grandchild is really theirs. In most societies genetic testing reveals that 5-10% of all people are not the biological child of the person they believe is their real father.
Tor
I've tended to think that the reason that young mathemeticians have been so successful is actually because young mathmeticians have something to prove (to themselves and to others) and because older academic mathmeticians can't spend so much time doing research as they have to take up responsibilities of running their department. most of the older more established professors i know spend a considerable amount of time doing admin. work. another possible contributing factor is that if you've had your own ideas for a long while, its hard to give them up without a loss of pride.
Logic, macros, and more
Galois died when he was 21 in a duel.
The mathematical community already recognizes you're probably "washed up" by age 40.
In fact, the Fields Medal, which is recognized as the equivalent of a "Nobel Prize in Mathematics", has the condition that "...the awards recognize both existing work and the promise of future achievement, it was agreed to restrict the medals to mathematicians not over forty...".
Supposing you were a typical Math graduate student who finally gets awarded his/her Ph.D. by age 26 or so, that only leaves about ~14 years to figure out something sufficiently mathematically mindblowing enough to earn a Fields Medal for, in between dealing with the hassles of competing for any of the scarce number of entry-level math postdoc/researcher/assistant professorships that you hope will eventually lead to a tenure-track full professorship.
No wonder some of the math professors I know just go around muttering bitterly about their work and the only real hope lies in recruiting new graduate students with the hope that a Good Will Hunting-type genius will show up.
the article has good points, but misses the mark slightly. It uses examples of people solving problems. Granted, extremely complex problems, yet the claim that most mathematical discoveries are made before age 30 i think applies more to mathematical discoveries where the discoverer opens new fields of mathematics.
This continues with the winner of the siemens-westinghouse science/math high-school awards going to a senior who developed a new theory called poset-game theory. This is a rundown of it from the regional finals. Pretty cool stuff.
~skeeter
It has always seemed to me that Ken Ribet has been bright green with envy ever since Andrew proved Fermat's Last Theorem. I can't count the number of times he said that it was 'audacious' that Andrew did this in secret. I think Ken thinks he should have solved Fermat but he thought it was impossible to solve at the time.
"sweet dreams are made of this..."
Ok, I'm a man, and I'm getting oldish (40) but it seems to me that the most brilliant of all the mathematics students (undergrad and post-) at "my" university are mostly female. And not necessarily under 40, either.
Did you read what you wrote?
...)
the meaning of modesty is not "knowing one's abilities" but is in fact "freedom from conceit or vanity". Conceit is "excessive appreciation of one's own worth or virtue" and vanity is "inflated pride in oneself or one's appearance".
Equate "abilities", "worth", "virtues", and "appearance" since we are using them to mean "qualities" of a person. "Oneself" is referring to those qualities directly.
So "modesty" is:
freedom from
excessive OR inflated
appreciation OR pride
of/in
one's (your own)
qualities (abilities, worth,
Am I stretching anything yet?
To decide if someone has modesty, you need to compare their opinion of themself with their qualities. Is their opinion excessive or inflated? How can you decide if you do not know their abilities?
A man who can lift 20 pounds is modest when he says he can lift 20 pounds. He is immodest if he says he can lift 100 pounds.
A man who can lift 100 pounds is modest when he says he can lift 100 pounds. The first man may hear the claim as bragging since it is outside his own abilities. The second man knows his abilities, is not inflating them, and may drop a 100 pound rock on the first man as proof if the first man continues to annoy him.
---
I am confident in my abilities. I have proven them repeatedly in the corporate world. I do not inflate my worth, because I often have to deliver. Therefore I am modest.
I have noticed that I work very well with other confident people. I become annoyed with people who state they can do things and cannot deliver. And people who are not happy with their own abilities often resent mine. I believe that if they can only move 20 pound rocks, they should be happy about that, and be happy the second man is around when they need a 100 pound rock moved.
Also, the first man may have abilities that the second man does not.
- I currently work with MSAccess programmers. I have little ability with that product. I rely on them, and I do not resent their abilities. I even learn from them.
- I just had new windows installed. (The physical kind, not the MS software.) The installer did a great job. I do not resent his abilities. I watched what he did, and learned that I never want to install windows.
- My car needs maintenance. While I rebuilt an engine many years ago, I barely recognize that the thing under the hood of my current car has any relation to what I know as an engine. I am very happy that the dealership has people with the ability to care for it. I do not resent them.
When it comes to my specialty, I am much better than them. The MSAccess programmers are happy I am available to handle the integration with the front-end systems. The others do not care that I know how to boot a computer. None of them resent me for my abilities.
Be happy with who you are. If you find someonne whose states their abilities exceed yours, do not try to deflate them; try to learn from them. If they are immodest, you will know not to give them responsibilities they cannot handle. If their opinion was accurate, then you will improve yourself.
I spend my life entertaining my brain.
according to the good rabbi Dr. Doron Zeilberger:
. ht ml.
http://www.math.rutgers.edu/~zeilberg/Opinion46
This (the whole set of opinions) is a wonderful collection of ideas from a mathematician who has been arguing the (convincing) idea that in the 21st century real mathematics will be an experimental science, dominated by the study of computer algebra and experimental combinatorics. Zeilberger's computer, named Shalosh B. Eckhad, is actually cited as an author on some of Zeilberger's more computationally intensive papers -- it has been speculated that he has done this because he believes that when computers are our masters, Shalosh's descendants will have mercy on him. I became enamored with Zeilberger's humor when I first read "opinion 1 -- topology, the slum of combinatorics". His April Fool's Day opinions, of which there are several, by his own admission contain some of his best half-formed ideas, including:
1. Resolve the P/NP problem by showing any proof of P != NP would be NP-hard.
2. Storm the gates of MI6 to find Turing's counterexample to the Riemann hypothesis, classified for fifty years by the British government.
To try and stay on topic -- I think that trying to decide the peak age for mathematical creativity and the threshold for the decline of mathematical power is characteristic of the fratboy nature of the community. The concept of generalization as a means to solve a problem (ie, i cant prove A, but maybe i can lift A to some equivalent B that is easy, and includes A as a special case) has snowballed out of control. With few exceptions, contemporary research is so far from the source of mathematical inspiration (ie, physics and computer science) that all direction is lost, and all that matters is that a researcher say something about what someone else has said, not about the core (ie, hardest) problems in the field. So, you have inbreeding, cliques, "hot" fields, "mainstream" mathematics, and concepts like mathematical "talent" which are really quite ridiculous and counterproductive to the science. The young can think quickly, but the old can think deeply -- both are needed to tackle the really important problems, and to push knowledge further. Collaboration, not competition.
Anyway, my original intention is to plug Zeilberger and to let all know that there is a community of mathematicians who are about solving problems, not building castles of abstraction for the sake of giving each other awards. The eminent W.T. Gowers has described it best, so point your browser to:
http://www.dpmms.cam.ac.uk/~wtg10/papers.html
and look for "The Two Cultures of Mathematics". This paper set off a firestorm in the community, but for me it was reassuring and motivating to realize that there are still mathematicians who care about the field, and not about how smart they are or how smart they appear, much less if they are too "washed up" to make an impression on the apprentices.
Just some picky comments on the slate article and the posting. First, the man's name is Grisha Perelman, not Grigori. He is Russian, not Italian. (Even the MIT Math department's Seminar Page gets this one wrong). Second the work spoken about at MIT was written up in two preprints (here and here -- I guess I should say don't even bother reading them without a graduate education in mathematics).
FYI, this work is based on a prescription for proof of the Geometrization Conjecture (which implies the Poincare Conjecture), done by Prof. Richard Hamilton, who was at one of the UC schools at the time, but is now at Columbia University. Professor Hamilton was over 40 when he published his work on the Ricci flow, which is the basis for Dr. Perelman's recent work.
Then why are famines and abject disease ridden poverty incidental to underpopulated third world dirt holes while Tokyo and NYC enjoy the highest standards of living in the world?
The establishment will say that knowing how to find 34 * 87.33 is 'practical' whereas knowing how to prove something mathematically does not have real world use. I disagree. A mathematical proof is the canonical example of an airtight argument. All the words are precisely defined and rigorous logic can be examined and practiced. In everyday speech nothing is precise and logic fails because the words ( like 'happiness' or 'fast' ) have shifting meanings depending on usage. Words like that can not be used to make a truly airtight argument because they are inherently leaky. Exposure to mathematics helps people analize what they hear/read and see flawed logic covered up by imprecise terms or tight logic that depends on a precise meaning of a word which is then applied to make a false statement using the word in a broader sence. This kind of insight is also known as reasoning skills.
Of course everyone should know their times tables and be exposed to the long division algorithm, but they should be able to come up with it themselves. Instead of being taught a sequence of steps students should be guided toward coming up with it themselves and then should be required to prove that it always gives the right answer.
Word problems should not be where students try to apply the algorithms they are supposed to memorize for a test ( which they will forget immediately afterwards ) after applying that algorithm to umpteen thousand similar ( non-word ) problems at the beginning of the worksheet they were assigned for homework, word problems should motivate the search for an algorithm and motivate lessons where those algorithms are rediscovered and reproven to work - by the students.
Teachers should ask themselves this: If I were a cave-man on an island and wanted to come up with how could I come up with it? They should take the class on that same journey of discovery. I ended up majoring in math once I saw, in college, it presented that way.
But I always flunked/did poorly in math from about 4th grade until I went to college. I hated it with a passion. Just thinking about it gave me crossed eyes and my mind wretched at the drudgery of doing 'problems' where my silly error prone pen and paper and my sometimes imperfect memory/understanding of the algorithms used to do the problems combined to make the work impossible for me to do. 'Practice' the teachers would say when I would ask 'how did you know to try that?' while they solved algebra problems on the board. And I don't think they had a better answer than that. Yet there is a precise answer since computers can do any HS algebra.
I took a class early in college that used the textbook 'Transition to Higher Mathematics'. Most of the contents of that book should be covered - sprinkled in - in K-12 math classes. Just think, if you asked high school seniors what 1/0 equalled probably half would give the right answer: undefined and half would say 'infinity'. Why not teach gradeschoolers that infinity is not a number, and show them the proofs that there are as many natural numbers as integers and as rationals, but that there is no injective(1-1) and surjective(onto) mapping from the naturals to the reals by diagonalization. These are proofs kids can understand, and would make math interesting - like a NOVA episode, instead of boring like the phonebook.
Teachers should see what's out there for computer aided math.
Eat at Joe's.
to name but a few.
[1] having kids
In three dimensions, we see a 2D cross-section, rather than the true 3D of it. And it's not a planar cross-section, but an angular one: ie we see whatever is occupying phi/theta than what's at a constant y.
Of course, the visualisation of 4D is the like of watching all bits of a movie projected into solid space or something: that is, ye see the wholeness of the figure, rather than some 3D slice.
The same is true of hyperbolic space. I have little trouble with some figures in 4D hyperbolic space, for example. The most common projections of hyperbolic space are the poincare disk and the klein disk. When i do sketches of what i see, i use either a cylinderical or orthographic projection, depending on whether i am moving or stationary.
Unlike watching a movie, it is more a case of creating the image to watch as ye do it. So while i can watch things in 4D in this manner, the plot of the story needs to be created in the mind, rather than watching someone else's impression of it.
5D and 6D are a bit harder, but can be done.
On the question of the 4D Klien bottle, I have seen pictures of it in 3D, and i can mentally crall over its surface, and i know there is a transformation of its surface into a square-shaped thing, but i am unable to make that topological transformation. But then, i am unable to make the transformation in 3D of turning a torus inside out.
OS/2 - because choice is a terrible thing to waste.