Becoming a mathematician - I am so depressed

[H2Bro]

I agree that the motivation and confidence are huge factors. They are actually well-documented in the animal kingdom as well.

Ivan D. Chase is a professor at Stony Brook university who does research on the formation of transitive hierarchies of dominance in animal groups. It's fascinating stuff, if your interested I suggest checking it out.

The short and sweet of it is, winners go on to win more because they learn aggressiveness pays off, while losers learn deference because previous attempts resulted in failure. This is actually noticeable in the academic and job world as well - get a couple bang on papers early in your career, and you get more positions, offers, experience, which begets better offers, and so on.

 

[StatOnTheSide]

I will check it out. Thanks for pointing it out H2Bro.

Even though it is true for animals, I am sure humans are not the same as animals. We are different. There may be common factors but there are a lot of factors not common to animals. That is why we call ourselves as being different from animals!

As teenagers, a lot of it is like in animals. But as you grow up, you will enter the adult world. The reason why there are exceptions in mathematics is that there are people who have pursued math without worrying too much about any of the above.

If a pin pricks you when you are looking at it, you will feel the pain.When it pricks you when you are not looking at it, it just goes un-noticed!

Long story short, you just do math if you like it and like the idea of thinking about formalizing the logic hidden in the mathematical structure. If you enjoy it, it means that you understand it and love to do more of it. It has a positive slope and will lead you to success some day. As long as you enjoy it, why do you care right?

 

[mathwonk]

this thread is based almost entirely on one of the biggest difficulties facing those who aspire to do well in mathematics. namely the confidence problem. we all know the extreme unlikelihood that we will ever do anything comparable to the work of abel, gauss, galois, riemann, dedekind, etc, etc... indeed it is discouraging even to begin to list the names that will almost certainly always be bigger than our own.

but the same problem faces people who want to become athletes, or artists, or newspapermen, or politicians, or salesmen. so if we still love our career path and want to pursue it, we have to summon up the courage to embark on a very difficult venture in which we have every chance of falling far short of our hopes and dreams. we have to maintain optimism and objectivity, to take satisfaction in small successes, and we need to learn not to let our mental health depend completely on every little sign of success or failure.

one thing that helps some of us, especially oldsters, is trying to teach and help other younger people. this helps us realize the long spectrum of learning that exists, and that we are not at the bottom of it, even if we are also not near the top.

ultimately we learn to work "for the glory of god", i.e. for enjoyment, not for personal aggrandizement. also it helps if we have a plan B, i.e. if not all our eggs are in the same basket. even david hilbert took a teaching certificate in case research did not work out for him. one good thing that can come from competition, or comparing our work with other stronger workers, is that it can teach us how to improve, and inspire us at least to do our own best.

i hope the OP has had some chance to grow and flourish, but i was frustrated that his high school was so short sighted as not to allow him to attend the university class that suited him better than high school. i would have appealed this with help from the university professor. perhaps he has since suffered some more setbacks and again faced the ongoing challenge of finding his place. i wish him well. he certainly has youth and intelligence on his side.

 

[H2Bro]

+1 for mathwonk. I think you definitely hit the nail on the head.

Physics and math might more than other subjects make us uncertain or doubt ourselves because the ability is so quantifiable, i.e. physicist X or Y mastered calculus by age 13, or independently thought up a proof or theorem at age 10. There's a lot of these anecdotes floating around, especially the one about Gauss as a child, which make us think "gee, I never did that!"

Something that helps me is to know that I simply have a different kind of learning curve than a lot of people. My intellectual side didn't fully blossom until my 20's, but now its full steam ahead and I'm starting to notice the difference between me and the younger people around me in terms of drive and motivation. I think acknowledging that different people have different styles of learning, and that sometimes its not clear cut if one is definitively 'better', helps to calm down the voice saying 'you won't make it because you're not a prodigy, or 99th IQ percentile, etc'.

Even though it is true for animals, I am sure humans are not the same as animals. We are different. There may be common factors but there are a lot of factors not common to animals. That is why we call ourselves as being different from animals!

Humans are animals, actually. The most important factor in the 'winners keep on winning' pattern of behavior among animals is the observance of winners by bystanders. Observing another win increases your perception of their capacity and potential, and reduces your self-confidence when it comes times to challenge them.

There is an interesting line of research on this, focusing on people in discussion groups which over time produce transitive hierarchies of interaction participants. If you like PM me and I can link you to some more sources that explore it a bit more fully.

 

[StatOnTheSide]

I do not challenge anybody regarding this. In this matter, I would much rather trust my experience than the research out there. Please do not be offended by that statement. I am quite sure that the researcher has done good research and has sound conclusions but that increases his conviction but not mine.

I have a friend who is a PhD from U of Chicago and is into Representation theory. He is one of the few from India to have a gold medal at the IMO. You may or may not know it but there is no training in India for IMO and he for sure did not receive any help. He had just one book that his uncle gave him who is in the US and he made it to IMO just by solving the problems in that book. On top of that, he did receive help as soon as he cleared the regional round.

I have actually observed him and his friends and the dynamics in the group. I do agree with you that as long as you are a subordinate to people like him and tell him "you are a genius and I won't equal you ever", you will not bring out the full potential in yourself. There is no doubt about that. If a researcher has come to the same conclusion, I am not at all surprised as I knew that for a long time. But amongst his friends, there is another guy who made it well into mathematics and is still doing well. He is not a PhD from U of Chicago or anything but he nevertheless got a PhD in mathematics and is a faculty member somewhere. They may not be "equal" but it is not like one has made it way better than the other. He is happy with his teaching job and gets to do mathematics which is all that matters.

As Mathwonk has pointed, there are some people who are way beyond others and I just acknowledge that. Many are actually dead. The main problem arises with the peers. There is always a huge ego clash between the so called friends who are in the same field atleast till they establish themselves. Sometimes it continues well into their old age.

When I say that we are different from animals, I do believe that it is possible for some humans to not get bogged down by the immense success of their so called friends. They keep going at their own pace and maybe in secrecy because they do not want to confront their friends. It might take time but then one day, they come up with something substantial and then their friends are indeed surprised at that time. They go "he is after all not that bad". I have known such instances.

I still think that doing what we love without getting bogged down by the success of a so called "friend" is the key to success in anything in life. In this aspect, I am pretty sure that we humans are different from animals as I haven't heard of animals succeeding once the trend has been set during childhood. I am not saying this to be optimistic and show sympathy to the OP. If it was the other way round, I would much rather tell the OP about the truth rather than lie in order to be encouraging. I do not think that approach works. I truly believe it. I have seen it and experienced it in my life based on my observation of the members of my group. I hope that OP takes the approach of just doing what he loves without worrying about IQ and all the other factors.

 

[Mdhiggenz]

Dude you are 17 relax. Oh and your iq tends to increase as you progress uni. I suggest you take the summer of maybe meet some girls and party a bit before you start uni.

 

[hurdler1]

I am also trying to be a mathematician. But I have realized something. It is a love, not a state of being. Unless you are very intelligent, which I think most mathematicians are not (necessary), doing math research as a job is more of a habit than a state of being.

i second the sentence that success is not the right mindset to have. I had an iq of 150 as a second grader. I don't know what my iq is right now. But i have failed at learning math because I was focused on trying to do things fast like many people at mit. However, I am not a fast thinking. I like to think slowly and deeply and work on a problem for many hours if I can. However, this is hard to do at MIT. I don't think you have any iq to get a phd in math. I think you have to work hard, and have a non-proud mindset constantly. There are tons of smart mathematicians out there. However, if you know what you are doing when you solve differential equations, then that is sufficient. As long as you want to study mathematics and work at it for the love of it, then it doesn't matter if you aren't smart. Many of the professors at MIT are bad at computation, but good at what they do.

ALso, I think many mathematicians have some idea of what other people are doing, but I think overly stressing about how other people are doing is also not helpful, and actually harmful.

 

[modnarandom]

I guess this relates to the balance between doing something you love and being realistic. Like everyone else already said, you shouldn't study something unless you enjoy it a lot. However, especially with math, I found that it was like a lot of people who didn't have a lot of background in math from high school were left behind in college in comparison to those in other majors. I'm not sure why this is the case, but this really is something to consider. As for being realistic, math isn't the most "practical" or applicable to other fields and this is something to consider especially if you aren't all that interested in applied math. So, this is something the OP (and anyone else considering the major) needs to think about carefully.

As Mathwonk has pointed, there are some people who are way beyond others and I just acknowledge that. Many are actually dead. The main problem arises with the peers. There is always a huge ego clash between the so called friends who are in the same field atleast till they establish themselves. Sometimes it continues well into their old age.

I don't really understand why this should be a problem. At first, it was intimidating to be surrounded by these people, but I'm constantly amazed by the people around me and it is exciting to learn from them.

 

[StatOnTheSide]

I am not commenting on any particular case. I have been told that each person is different and it may not be the same for everybody. I have certain beliefs based on my experience and it is very strong. It is nothing more than a mammalian trait. If you are a person who has a set of friends and if you are not at the top of the ladder in that group, then it affects your performance. It is like a bunch of lion cubs. The most aggressive of the bunch will eventually be the leader. You can tell by looking at the bunch in their early childhood developmental phase. Same with humans. A kid who is way better than his peers in math will remain there. The only difference is that some of them drop out of the group not liking the hierarchy that exists in that group. Why? It helps immensely if you are in a group where you get the ego boost that you are the best in that group. Or else it does not make sense to be in that group. Why be an ego booster for someone else?

The fact of the matter is that it is all about competition. The confidence that you develop is cumulative and will results in a strong conviction that you are the best. Next time there is a competition, you start out with that mindset. Others look at you believing the same thing. This fact works against them in the competition. Nature, this way, accomplishes selection of the best. When you are running a race and see that others are taking over, just that fact works against you. So even if you have the potential to run faster, you will be looking at others thinking that they will remain ahead and there is not point running the race anymore which feeds on itself and eventually you will lose.

I do not intent to contend anyone here. modnarandom might be in a place where the only thing that people care for is learning and very little to no emphasis on competition. All the places that I have studied, it was all about competition. That being the case, it works against you if you are not running as fast as others. Even though a career in math or engineering is more like a marathon, it still is a competition. I was talking more in the context of undergrad in math/engineering or like later part of high school where you write math olympiad and other contests. Those are 100m races. It is all about psychology. Typically the one who gets the initial lead wins.

I have to admit that in a marathon, the case is different. Most winners do not have a lead initially but win it eventually.

 

[42Physics]

You are obviously capable of achieving in the field of mathematics. I wouldn't let petty occurrences discourage you from an ever-rewarding field. We are very close in age and seem to share our talent. I myself wish to be a Physicist; perhaps one day we will work together.

 

[StatOnTheSide]

Thanks for your kind words 42Physics. Please do not take my theory about ego boost etc too seriously. I do believe that it is different for older people. To be a grown up has its advantages and this is probably one of them. I was talking about my experience during my undergrad/high school days and it might be different for you. Not that those people have changed but just that now there is lot more freedom to choose only the nice people, like yourself, to be a friend.

BTW I am 32 years old :)

 

[42Physics]

I appreciate your time greatly. I'm just trying to help birth the people who make the tomorrow of science and math come. I am 14 years old btw

 

[Jimmy84]

You may very well be right - but the issue is, that most high-iq people say this :) I do not know whether is should take it as the truth or just modesty. I really do hope you are right - maybe all is not lost for me.

Levis, In my own opinion one of the most important factors in excelling in a given subject is first the motivation, that you obviously have, and second, having time. You obviously have both.

The more time you invest learning, the more you will be able to learn deeply and intimately a subject.

One of the biggest advantages that young people have now is the huge amount of information on the internet.
The best mathematicians like Hilbert, Grothendieck, Gauss etc They never had access or an easy way to find 200 graduate math books online, or online lectures, wikipedia or websites that could orient you.
If you can exploit that to teach yourself additional subjects, if you have some orientation and if you administer your time. You would be able to learn many mathematical related subjects in a deep way.

 

[JimmBean]

I would trade 85 years of life with my slow brain, for just 15 years of life with the brain of this guy: http://newsfeed.time.com/2011/03/26/12-year-old-genius-expands-einsteins-theory-of-relativity/
How wonderful it must be to be so intelligent - to reach such levels of enlightenment is just fantastic, at age 12! When i was 12, i was simply playing with sticks. I want to do things like that, but this stupid vessel of a body is not capable!

Apologies for bumping a somewhat old thread, but I really couldn't let this go. That "child prodigy" you refer to in your post, Jacob Barnett, is basically a fake. He has a good memory no doubt (possibly autistic?) and so is able to memorize a great deal of intelligent sounding phrases - even though he knows almost nothing about what he is talking about. In fact his Wikipedia page has been deleted as a result. Watch this:

http://www.liveleak.com/view?i=91e_1301861454

EDIT: Here is some additional material. Watch from 8:31 onward...

This more candid video (filmed by his Mom) is disturbingly stupid, with he only reciting patently obvious facts and hinting towards his new theory which "he cannot talk about on video". He obviously understands very little about the math/physics he is reciting. I get the feeling that this kid is vaguely intelligent with a good memory and has been unfortunately taken advantage of by his parents in a pathetic attempt for their 15 minutes of fame. I actually feel sorry for the kid.

Moral of the story Levis2: don't believe everything you read in the media - most of it is sensationalized to the extreme. Do what you enjoy and don't let others (e.g. the media) tell you what you can and can't do. And get some professional help if you are feeling really depressed - that won't help anything.

 

[SophusLies]

Did the people here that are mathematicians know that they were capable of doing such things early on? Or did it come as a surprise?

I ask this because even though I do well in math classes, even at the PhD level, and I have put in a lot of hard work into math I still feel like I am incapable of coming up with new ideas for math. I'm not trying to make it sound like there's a math gene but for whatever reason I don't see "new" math. When I decided to go to graduate school I was accepted into several math and physics PhD programs. Several were very good, but my confidence was too low to even attempt a math PhD for creativity reasons.

OTOH, I think physics comes rather naturally for me in a creative way. Even when I was young (probably 10 or so) I still remember trying to come up with models for anything physical: water flow, collisions, rotations, etc. My school district was incredibly poor so I was never taught the math I needed to fully develop my ideas until later but I was able to come up with graphs and basic equations on my own to explain the things I was observing. During my senior high school year, I found out what I was doing was called Physics, lol.

I feel that mathematicians don't look at the world this way instead they are atop a mountain of concepts looking down on the relationships. I know there's many types of mathematicians but the ones that are truly magicians to me are the ones who make powerful generalizations. I honestly don't know if I could ever do that. It doesn't bug me because I know what I'm good at but it's fascinating to me how these people view the world.

 

[neyzentanburi]

you don't have an IQ problem. your IQ is very good.
however, you have a mental illness problem because your thinking pattern is completely irrational.
visit a psychiatrist as soon as possible.

 

[PlayingMonk]

Apologies for bumping a somewhat old thread, but I really couldn't let this go. That "child prodigy" you refer to in your post, Jacob Barnett, is basically a fake. He has a good memory no doubt (possibly autistic?) and so is able to memorize a great deal of intelligent sounding phrases - even though he knows almost nothing about what he is talking about. In fact his Wikipedia page has been deleted as a result. Watch this:

http://www.liveleak.com/view?i=91e_1301861454

EDIT: Here is some additional material. Watch from 8:31 onward...

It seems this Jacob Barnett kid has a twitter account as well: https://twitter.com/PwningEinstein

Yeah... it's not making him look any smarter.

 

[gsmith]

I have a measured IQ of 127 (granted, this was measured when I was 10), and have been recruited heavily to study physics at a major research university. In all seriousness, if you were able to independently prove the Taylor Series, you have talent beyond anything I have ever been able to imagine. What it comes down to is hard work. Sure, innate intelligence is needed to an extent, but it isn't everything. Look at my IQ, 127, I believe that one day I can accomplish something, and if I can, you certainly can. The only thing standing in your way is your own self-defeat.

 

[jimmyly]

Just work hard at what you love. Simple. Forget all this useless worrying about iq and all that nonsense.

 

[Joseph King]

IQ tests are fundamentally flawed. I have a friend who is autistic with an IQ of ninety-something, but can recite the first 100 digits of pi from memory. He is the smartest person I know, so, don't sweat it.

 

[SolsticeFire]

Just work hard at what you love. Simple. [Forget about everything else ]

This.

SolsticeFire

 

[mathwonk]

I got to tell you, I find the longevity of a thread on how depressing it is to try to become a mathematician, pretty depressing in itself. What say we stop navel gazing and get back to work? (doing math?)

To be depressingly explicit, I am guessing the less time you spend on this thread the more likely you are to become a mathematician.

 

[plife]

I just had to put in my 2 cents here:

One very good friend of mine has PHDs in Math, Computer Science, and Physics. He had to take his first algebra class 5 times! He even had to take it at another university and transfer it back in order to pass...He had to take one of his calculus classes 3 times before he passed. But, he learned the material and moved on. He now works as a very high level mathematician at a very important US Government facility in Virginia. His determination and his persistence paid off... The drive to achieve what you dream has got to be stronger than your willingness to throw up your hands and give up, no matter what.

Secondly, A different (not very smart) man I know wanted to be a member of MENSA (the "genius" organization). He took the IQ test several times. He bought "how to increase your IQ" books and he took practice IQ tests. He improved his score on the IQ test to the point that he was able to prove his "high IQ" and is now a member of MENSA and active in many of their organizations. I know this man personally, and know that he is NOT a genius...just determined be prove he was one.

I am not trying to take anything away from the true geniuses who deserve our respect and admiration. But, if you want something; just go for it. Don't let negative thoughts or the words of others discourage you.
Quote by jimmyly : Just work hard at what you love. Simple. [Forget about everything else ] - This says it all.

 

[DrummingAtom]

I just had to put in my 2 cents here:

One very good friend of mine has PHDs in Math, Computer Science, and Physics. He had to take his first algebra class 5 times! He even had to take it at another university and transfer it back in order to pass...He had to take one of his calculus classes 3 times before he passed. But, he learned the material and moved on. He now works as a very high level mathematician at a very important US Government facility in Virginia. His determination and his persistence paid off... The drive to achieve what you dream has got to be stronger than your willingness to throw up your hands and give up, no matter what.

You can't be serious.. 3 different PhD's?? Doesn't each one take 4-7 years?? Considering this person took a calc class 3 times that already put them behind 3 semesters. Minimum age would be 33 finishing all of this, max would be 40. Did this person do post-docs too?

 

[1MileCrash]

I don't see how IQ plays apart, as long as you do not have any severe learning disability.

This game is 99% work and dedication. Some students seem brilliant in class and grasp something immediately during the lecture. I go home and read about it until I also understand. Big whoop.

If you LIKE math, you will do well in it.

 

[plife]

You can't be serious.. 3 different PhD's?? Doesn't each one take 4-7 years??

Yes, he is actually in his 50's now and is working on a Masters in Geological Information Systems. I guess he is "addicted" to homework - lol!

 

[A Dhingra]

To the OP,
Just to say the same all have been saying.. If you really enjoy the subject engross yourself into it so much so that you see mathematics everywhere and gradually you will learn to come up with ideas of your own. Even if you don't, the journey of learning it will be so enriching that at the end you will be satisfied with your work, which is all that matters...
IQ tests are no guaranteed criterion to say if you can be a mathematician, these tests are created for general people not customized for specific individuals so they can tell where you stand in general not what are your real strengths, so ignore them..
It is always said, "Genius is 1% inspiration and the rest 99% perspiration." So anyone ready to devote a long time can become a genius in their respective field..maths is no exception.
(I am no expert, i just presented what I feel as per what I have learned from people on PF)
All the best..

 

[DeeAytch]

I am a 172. Formally tested five times and only one test was bold enough to put it at 172. The rest had me at 160+.I didn't have an education but I got a GED, scoring top percentile. I studied
for the SAT, did very well, and started my math at calculus 1. I've since aced the calculus series.

But, higher level proof writing is very difficult for me. If you can prove like you say you did, then don't worry about your fluid intelligence. You have a talent nonetheless and would make a fine mathematician, perhaps a doctorate indeed. However, becoming a professor is to aspire to be better than all the other doctorates

 

[CosmicKitten]

Not knowing calculus at age 12 doesn't make you stupid. It means you probably weren't interested in math back then.
At age 12 I hated math but I was a savant at playing Pokemon. In fact I think that helped my math abilities to suddenly blossom when I entered high school.

Confidence is more important than intelligence. Let me tell you, when I was in kindergarten I thought I was too stupid to learn how to read or do math... I was reading chapter books by the end of the year.

Lastly, it cannot be stated enough... creative thinking is of utmost importance. Math, as taught in school, is just a language. Some pick it up quicker than others, but it takes creativity to make poetry with it, and sometimes those that are slower wind up the best poets, so to speak. It's a shame that too many think they are talented just because they are fast parrots...

 

[ktheo]

lol. Us mere mortals can't possibly know what it's like to be a genius like this self-proclaimed 17 year old prodigy. OP I really hope in 5 years you find this thread and laugh; it'll mean you've morphed into a normal human being.

 

[Pine1315]

Apologies for bumping a somewhat old thread, but I really couldn't let this go. That "child prodigy" you refer to in your post, Jacob Barnett, is basically a fake. He has a good memory no doubt (possibly autistic?) and so is able to memorize a great deal of intelligent sounding phrases - even though he knows almost nothing about what he is talking about. In fact his Wikipedia page has been deleted as a result. Watch this:

http://www.liveleak.com/view?i=91e_1301861454

EDIT: Here is some additional material. Watch from 8:31 onward...

This more candid video (filmed by his Mom) is disturbingly stupid, with he only reciting patently obvious facts and hinting towards his new theory which "he cannot talk about on video". He obviously understands very little about the math/physics he is reciting. I get the feeling that this kid is vaguely intelligent with a good memory and has been unfortunately taken advantage of by his parents in a pathetic attempt for their 15 minutes of fame. I actually feel sorry for the kid.

Moral of the story Levis2: don't believe everything you read in the media - most of it is sensationalized to the extreme. Do what you enjoy and don't let others (e.g. the media) tell you what you can and can't do. And get some professional help if you are feeling really depressed - that won't help anything.

AWW - you feal sorry for the kid. First, I doubt it is just memory. He has published research in "Physical review A"; look this up:

"Origin of maximal symmetry breaking in even PT-symmetric lattices"


He also was accepted at Permiter Institute.

This was posted on his Facebook:

It is the next step for Jacob!.. Today we were notified that Jake has been accepted to the Perimeter Institute for Advanced Studies! In the words of the director there after review of his pre work courses..."We have determined that it is very obvious that Jacob will make significant advancements to science and therefore we would like to accept him to the programs here!"

He does not have a Twitter account it does not take a genius to know that one is a fake. Further, in the video he was just saying that some of the claims in the media are incorrect (that he was going to prove Einstein wrong.)

HA HA.

 

[HakimPhilo]

IQ becomes this days a source of depression... I laugh.

 

[Positron137]

Hey Levis2! I understand your passion for mathematics - I have a similar one as well! I took Calculus BC last year (in 10th grade) and now I'm taking linear algebra and multivariable calculus as a junior. Differential equations is one of my favorite subjects in calculus and I'm hoping to either major in mathematics or theoretical physics in college.

My IQ is pretty high (around 155), but that doesn't tell you very much about the potential I might have in the future as a physicist or a mathematician. Similarly, don't take IQ scores literally. They are not a very good, accurate measurement for your intelligence. I will tell you something that is a good measurement of intelligence: TIME. You've already gotten your passion for mathematics. Now all you need is time and practice to develop those skills. Geniuses may have a bit of raw talent, but most of it isn't magic. That's what I've learned. I have the exact same problem you're detailing out, and I realized that associating Mathematics PhD's with IQ's of like 160 and 170+ is not very accurate. There are hundreds of mathematicians who have made significant contributions and who probably don't have IQ's of 160+ for the most part. And Levis2, one thing is for sure: just as what micromass said, you should ENJOY doing mathematics. Don't involve yourself with you know, wanting to be a math genius or learn tensor calculus or Galois group theory at age 14 or something like that. You already have great potential for mathematics - just play around with it, immerse yourself in it, and hone your skills. You'll definitely become a great mathematician in the future! :)

 

[Windows]

Quoting a 'TRUE' big mathematician about the "Genius and Mathematics":

Does one have to be a genius to do mathematics?

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities.

The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts – is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. (This is for instance the case with Wiles‘ work on Fermat’s last theorem, or Perelman‘s work on the Poincaré conjecture.)

Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since nobody is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness. (If someone affects to do so, I advise you to be very sceptical of their claims.) The pressure to try to behave in this impossible manner can cause some to become overly obsessed with “big problems” or “big theories”, others to lose any healthy scepticism in their own work or in their tools, and yet others still to become too discouraged to continue working in mathematics. Also, attributing success to innate talent (which is beyond one’s control) rather than effort, planning, and education (which are within one’s control) can lead to some other problems as well.

Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems. Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.

Further reading:

“How to be a genius“, David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
“The mundanity of excellence“, Daniel Chambliss, Sociological Theory, Vol. 7, No. 1, (Spring, 1989), 70-86. [Thanks to John Baez for this link.]

_____________________________________________________
~ Terence Tao.

 

[Crake]

Quoting a 'TRUE' big mathematician about the "Genius and Mathematics":

Does one have to be a genius to do mathematics?

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities.

The popular image of the lone (and possibly slightly mad) genius – who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts – is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew. (This is for instance the case with Wiles‘ work on Fermat’s last theorem, or Perelman‘s work on the Poincaré conjecture.)

Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since nobody is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness. (If someone affects to do so, I advise you to be very sceptical of their claims.) The pressure to try to behave in this impossible manner can cause some to become overly obsessed with “big problems” or “big theories”, others to lose any healthy scepticism in their own work or in their tools, and yet others still to become too discouraged to continue working in mathematics. Also, attributing success to innate talent (which is beyond one’s control) rather than effort, planning, and education (which are within one’s control) can lead to some other problems as well.

Of course, even if one dismisses the notion of genius, it is still the case that at any given point in time, some mathematicians are faster, more experienced, more knowledgeable, more efficient, more careful, or more creative than others. This does not imply, though, that only the “best” mathematicians should do mathematics; this is the common error of mistaking absolute advantage for comparative advantage. The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research. As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems. Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.

Further reading:

“How to be a genius“, David Dobbs, New Scientist, 15 September 2006. [Thanks to Samir Chomsky for this link.]
“The mundanity of excellence“, Daniel Chambliss, Sociological Theory, Vol. 7, No. 1, (Spring, 1989), 70-86. [Thanks to John Baez for this link.]

_____________________________________________________
~ Terence Tao.

I'll provide a link, if you don't mind. Also, one may not need to be a genius to succeed in math (or physics) but I think one needs to be obsessive about it.

Link: https://terrytao.wordpress.com/career-advice/

You've many interesting career advice topics, written by a giant in his field.