Is Math a Young Man's Game? 276
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."
The problem is with modern mathematics... (Score:2, Interesting)
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.
Andrew Wile (Score:5, Interesting)
Re:thelimitis30++ (Score:5, Interesting)
Demanding: Writing the GPL, starting FSF, the Hurd, travelling the world over, believing in yourself despite others jeering you - RMS age 50.
Innovating: Buying an OS from someone, putting it onto someone else's h/w, building up a monopoly, driving out others (using suspect means), releasing newer and newer OSes that do essentially the same things, generate obscene profits, etc. etc. - William Gates, Age 45 (?)
Life begins after 30, methinks.
Re:New field vs. old fields (Score:5, Interesting)
Now, as the article says, you are a graduate student -- and probably not a new graduate student -- before you're even looking at other people's cutting-edge work, let alone doing your own.
Re:The problem is with modern mathematics... (Score:3, Interesting)
In the meantime, WTF is a Lie group? WTF is an algebraic varity? Non-holomorphic sounds very impressive, but WTF is it?
You might be right; I've observed that certain Asian groups do seem to have a handle on maths that many Western brains don't, and I doubt it's entirely due to genetics. But if you actually want to change things, as opposed to sounding clever, people have to understand what you're on about. I don't, and I'm three-quarters of the way through an engineering degree. Thank you.
Life expectancy (Score:5, Interesting)
people usually didnt live
beyond 40?
competing with discoveries from the past (Score:5, Interesting)
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 [ams.org] 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).
Re:It is obvious why this is the case.. (Score:5, Interesting)
Thinking, exploration, calculation, research, experimentation--all of these take a great deal of time. Relationships with friends, your SO, and eventually kids require a great deal of this time to keep healthy and strong.
If you want smart kids/pets, that takes time as well.
No, I am not saying that one can't be productive or creative once older; however, it just becomes more difficult. Those that do it successfully usually do it though their profession. That is... you can do it though your job if they give you the freedom to do so.
I don't think all of this is so bad... most of us would rather have healthy relationships than awards/accomplishments as we get older.
Davak
Re: Whose game? And who said it was a game? (Score:3, Interesting)
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?
Re: Whose game? And who said it was a game? (Score:5, Interesting)
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.
Re:New field vs. old fields (Score:3, Interesting)
The counter-argument to that is that it is "insights" that count in making breakthrough discoveries. Since that often involves looking at things from a different direction, knowing too much about the conventional thinking within a chosen field may be a bad thing. Speaking from personal experience, as I have grown older it has become more difficult for me to recognize when my own assumptions are restricting the ways I think about a particular problem.
Finally, any field in which research requires large amounts of money is going to be problematic for young people. Raising such money requires a reputation of sorts and a network of contacts and experience, all of which take years to acquire. And people who control large sums of money do tend to be inclined to conservative approaches -- evolution, not revolution.
Andrew Wiles at age 41 (Score:3, Interesting)
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...
In the spirit of mathematics: (Score:4, Interesting)
Paul Erdös. Read about him in this [amazon.com] 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.
Interesting article on Fermat's Last Theorem (Score:2, Interesting)
A highlight:
Re:"Math" Arrrrrgggghhhh!!!!! (Score:3, Interesting)
Re:The problem is with modern mathematics... (Score:1, Interesting)
1. There has been a compromise and less of geometry is being taught, as it is considered "hard". This must be reversed.
2. I think it is feasible to segregate students based on espoused areas of interests. A couple of decades ago this would have been accused of elitism. However, with the growth of awareness among students and parents, not being especially interested in math is not tantamount to mediocrity.
Re:New field vs. old fields (Score:3, Interesting)
In the middle ages people weren't very interestes in mathematics
Then we finally get descartes, Euler, Fermat end those dudes, who finally got the math ball rolling. But it didn't get REALLY interesting until in the twentieth century.
In that light, mathematics, at least modern mathematics could be considered young in the beginning of this century.
And that's the same math that's getting old now.
Check out "A mathematician's apology" by G. Hardy (Score:2, Interesting)
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 [amazon.com] 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].
Science, Math, and Age (Score:5, Interesting)
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
Expounding against the tide (Score:4, Interesting)
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.
A poor education system does not help (Score:3, Interesting)
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?
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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.
You're looking at it the wrong way (Score:3, Interesting)
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.
same old shit. (Score:1, Interesting)
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.
Re:Science, Math, and Age (Score:2, Interesting)
A young man's game? (Score:3, Interesting)