Are mathematicians born not made?

[Group_Complex]

Hello.
It concerns me, that I may lack the creativity to pursue my interests in Pure Mathematics. I do not believe I am any more intelligent than average, Yet for some reasong I love the deductive method and beauties I find in Mathematics.

I was reading a short article http://journalstar.com/news/local/math-whiz-gives-lecture-at-unl/article_aacec19e-e75d-5537-9742-92efc517b3a7.html In which Michael Atiyah (Who I look up to very much as a mathematician) claims that Mathematicians are born rather than made. This dissapointed me greatly and for a few days I was considering giving up my goal of becoming a pure mathematician.

The reason I do not believe I am creative in mathematics is that I cannot prove theorems presented In textbooks, without reading the proof in the text (Real and Complex analysis). This has led to a reduced confidece in my mathematical abilities, which was already quite low due to poor performances in mathematical competitios and olympiads.

I am not striving to be a fields medalist or ground breaking mathematician (those were once my immature goals), but rather to contribute to research somehow, and present at least one creative proof in pure mathematics (It does not need to be a "proof from the book"). I really admire Raoul Bott, and would like to work in a field such as his, but I am unsure how to do this, if I lack creativity and insight at the undergraduate level.

I can understand all parts of the mathematics I study (Real Analysis, Complex Analysis, Abstract Algebra) with enough head banging, but I can rarely do the harder exercises without looking at the solutions, seeking aid, or re-reading the text.

Any advice?

 

[chiro]

Hey Group_Complex.

You might want to take a look at this thread:

https://www.physicsforums.com/showthread.php?t=530550

This is actually a common topic from time to time and I think reading this might give you some perspective from both people who are aspiring to be mathematicians and people who are mathematicians.

Hope it helps.

 

[RoshanBBQ]

The problem seems to be your self-worth, not your creativity. For example, I approach any task I do with the mindset that I can create perfection with enough time and hard work. After creating something, I will sometimes look at it throughout the day to marvel at its excellence.

 

[Pyrrhus]

It is true that some are born with a propensity to learn quickly some specific subjects. However, it doesn't mean that those without such propensity may not be able to learn them at all. It just takes more effort. I believe it is similar to sports. Some people are just better at playing some sports without previous training than others. It does not mean that without hard work, and perseverance those without such propensity cannot learn to be good at those same sports. It also means that depending on your passion those without the propensity might quit because of all the additional work.

 

[Number Nine]

In a semi-inspirational example, I neglected mathematics almost all throughout high-school, and barely passed in the few instances I actually bothered to take any (basic algebra; nothing approaching calculus). I (very) reluctantly took pre-calculus and calculus-lite (for the social sciences!) in University, since it was required for my major (psychology), performing terribly in both (actually failing calculus once). I found calculus mildly interesting, however, and somehow discovered the notion of pure mathematics while on wikipedia. I took an intro to proofs the next semester and found that I actually enjoyed it enough to follow it up with courses in linear and abstract algebra, in both of which I made A's.

I don't think I have an unusual aptitude for any of these subjects; I just think about them often enough to be perfectly comfortable with the material, even if it's quite difficult.

 

[micromass]

I don't there there are many mathematicians out there who never struggled before. I don't think your situation is very abnormal. Many people struggle with proofs in analysis and abstract algebra. When I took the course, I couldn't find the proofs in my course from scratch either. But after the course, I could do it. I learned how to do the proofs.

Seriously, if you could do all the proofs in your course without looking, then what's the point of taking the course?? It's supposed to be a learning process.

Try to learn as much as you possibly can and you'll get there.

 

[Whovian]

Agreed, it's not too unusual. Just to give you my story, I am horrible at Number Theory and anything that involves integers, even sums of sequences. My first thought when I see $\displaystyle\sum_{k=\text{some integer}}^\infty$ is "Uh-oh." Analysis hasn't been too hard, but I had some trouble with the concept of a limit when starting out.

So yes, don't get too discouraged.

 

[homeomorphic]

Atiyah isn't a psychologist, and even psychologists have limited knowledge, so I wouldn't take him too seriously about this. The way you study math makes a huge, huge difference. There are also all different levels of mathematicians. Some people just stumble into the right way of doing things without thinking about it too much, so they don't realize what they are doing that other people are not doing, but COULD do if only they knew what to do.

 

[thrill3rnit3]

for argument's sake, let's say that they are born. What are you going to do about it now?

 

[diligence]

Textbooks are collections of the most important theorems since the birth of modern mathematics. It's taken ALL OF HUMANITY HUNDREDS OF YEARS to compile these results, yet you expect to be able to prove them all on your own in the amount of time you spend on your studies (i.e. much less than a single human life-span). That's quite ridiculous.

 

[diligence]

Atiyah isn't a psychologist, and even psychologists have limited knowledge, so I wouldn't take him too seriously about this.

This is the pure truth right here. Just because some person is an expert on a topic (any topic) doesn't make he or she an expert on what it takes to become an expert (i think that makes sense). True, he has an IDEA of ONE way to become an expert (his way), but there are many paths to any single destination.

For some reason ego seems to accompany the perception of mathematics for many students. Indeed, I still struggle with myself over this nonsense too often. It's not about how smart you are, it's about what you're interested in and your level of interest! Furthermore, any expert who uses his knowledge and/or genius as a display of superiority is simply insecure... and that really has nothing to do with math at all.

 

[lisab]

Textbooks are collections of the most important theorems since the birth of modern mathematics. It's taken ALL OF HUMANITY HUNDREDS OF YEARS to compile these results, yet you expect to be able to prove them all on your own in the amount of time you spend on your studies (i.e. much less than a single human life-span). That's quite ridiculous.

Excellent point.

I remember watching my teachers do what looked mathematical contortions to solve algebra problems. It made me frustrated! How could I have ever figured that out on my own?! It all looked like tricks!

But now I realize, the purpose of learning those "tricks" is to take them in and make them part of your mathematical toolbox.

 

[homeomorphic]

Textbooks are collections of the most important theorems since the birth of modern mathematics. It's taken ALL OF HUMANITY HUNDREDS OF YEARS to compile these results, yet you expect to be able to prove them all on your own in the amount of time you spend on your studies (i.e. much less than a single human life-span). That's quite ridiculous.

I don't think that's exactly what he's saying.

A good math student ought to be able to prove the easier theorems by themselves once they have reached the appropriate level of mathematical maturity. The hard theorems, no, unless you get tons of hints. But it takes time.

Landau used to read papers by trying to prove all the results for himself. But he was Landau.

I do that sometimes, too, but it's not that I try to do everything that way. I just like to try to ATTEMPT a proof, just to get more engaged with the material before I read it. Just start thinking about how to go about it. Not go all the way through with the whole proof. Also, you can do that with parts of a proof. Or just use the proof for hints when you get stuck. Then, you have a better chance of seeing the idea of the proof.

 

[Group_Complex]

I don't think that's exactly what he's saying.

A good math student ought to be able to prove the easier theorems by themselves once they have reached the appropriate level of mathematical maturity. The hard theorems, no, unless you get tons of hints. But it takes time.

Landau used to read papers by trying to prove all the results for himself. But he was Landau.

I do that sometimes, too, but it's not that I try to do everything that way. I just like to try to ATTEMPT a proof, just to get more engaged with the material before I read it. Just start thinking about how to go about it. Not go all the way through with the whole proof. Also, you can do that with parts of a proof. Or just use the proof for hints when you get stuck. Then, you have a better chance of seeing the idea of the proof.

Hmm, the thing is I often forget the intricacies of a longer proof after a while, and when I attempt to prove the result again, I get stuck and become quite frustrated with myself.

 

[Group_Complex]

for argument's sake, let's say that they are born. What are you going to do about it now?

If it were true that mathematicians were born, not made, I suppose I wouldn't bother going to graduate school in pure mathematics. I mean, what is the point, if you are always going to be getting stuck and needing to go to others for help and advice?

 

[homeomorphic]

Hmm, the thing is I often forget the intricacies of a longer proof after a while, and when I attempt to prove the result again, I get stuck and become quite frustrated with myself.

If you want to remember something, all it takes is some review. But you should question whether it's actually worth remembering or not.

If I really want to remember an argument, there are two things I do. One is to summarize it. What were the key ideas? What's the kernel of the proof? Secondly, I will just rehearse the argument in my mind, until I remember it.

Actually, there are more ways of remembering, too. You remember things better if they are connected to other things. So, that dictates a lot of how you should learn.

 

[mathwonk]

no one, no matter what favorable star they are born under, will become a mathematician without extensive work toward that end. As to whether those of us born with ordinary abilities can train hard enough to become a mathematician? the answer is yes.

 

[R.P.F.]

If it were true that mathematicians were born, not made, I suppose I wouldn't bother going to graduate school in pure mathematics. I mean, what is the point, if you are always going to be getting stuck and needing to go to others for help and advice?

So you pretty much just said that 98% of the pure math PhDs should never have gotten their PhDs. One of my math professor once said that 'grad school is feeling stupid 90% of the time and spending the rest 10% wondering why the heck you are so stupid.'. If you are not a genius and decide to go into math, then chances you are going to get stuck pretty often. If you can't take that, then maybe as you said, you should not go to grad school. Maybe math is not for you.

 

[RoshanBBQ]

If you want to remember something, all it takes is some review. But you should question whether it's actually worth remembering or not.

If I really want to remember an argument, there are two things I do. One is to summarize it. What were the key ideas? What's the kernel of the proof? Secondly, I will just rehearse the argument in my mind, until I remember it.

Actually, there are more ways of remembering, too. You remember things better if they are connected to other things. So, that dictates a lot of how you should learn.

You forgot the most sure approach to remember something, understand it.

 

[Group_Complex]

You forgot the most sure approach to remember something, understand it.

I do understand though, at least upon reading the proof. I will read a proof and think "That is simple, why didn't I think of that?".

 

[homeomorphic]

You forgot the most sure approach to remember something, understand it.

I thought it was too obvious to mention. That goes without saying for me.

And understanding doesn't just mean you can regurgitate it.

 

[Group_Complex]

I thought it was too obvious to mention. That goes without saying for me.

And understanding doesn't just mean you can regurgitate it.

Those are my thoughts exactly. I can follow a proof, but I cannot write down some of the longer ones from scratch without relying upon something other than memory.

 

[homeomorphic]

Those are my thoughts exactly. I can follow a proof, but I cannot write down some of the longer ones from scratch without relying upon something other than memory.

Sometimes, it can take time to understand something really well. You might come back years later and it will just be obvious once you have more experience in the subject. Doesn't always happen if you don't work towards it.

 

[Group_Complex]

Sometimes, it can take time to understand something really well. You might come back years later and it will just be obvious once you have more experience in the subject. Doesn't always happen if you don't work towards it.

Yes, but I want to do more than just understand other mathematician's work. I want to create something original myself. I don't think I have ever done a piece of creative mathematics in my life. I have simply modified other people's definitions and proofs to get marks on exams, but this obviously is not real mathematics.
I read in a "beautiful mind" that John Nash avoided reading textbooks, as he thought it would stifle his creativity, and preferred to rederive mathematics himself (such as Fermat's little theorem) from quite a young age. Of course I don't think I am anywhere near Nash's level, but I think his method was right. The thing is, I just can't do it. I will sit down, attempt to beat the textbook to a proof, get stuck, give up fairly quickly, and just read the textbook's proof, and the cycle begins again. : (

 

[homeomorphic]

You have to get some input from textbooks or talking to people, sometimes, but it's good not to rely on them too much.

Actually, sometimes, when I come with a solution, it works, but it's not the most instructive way to do it. So, sometimes, even if you CAN do it by yourself, you can still gain something from seeing how someone else did it better than you did.

Getting stuck is part of the game. If you want to solve a hard problem, don't expect to be able to do it one sitting. Think about doing it in a week, instead of an hour. You have to mull over it for a while, try lots of different things that don't work, let your brain work on it while you do other things, and maybe inspiration will strike or after taking a break, you'll do better with a fresh start once you have the problem firmly planted in your mind.

 

[homeomorphic]

A very successful mathematician once said 99% of what he tries doesn't work.

 

[Group_Complex]

A very successful mathematician once said 99% of what he tries doesn't work.

I have read that too (was it Isadore Singer?). I think that was false modesty.

 

[homeomorphic]

I have read that too (was it Isadore Singer?). I think that was false modesty.

I forget who it was. What reason would you have to believe it was false modesty? No reason. You're just being pessimistic, now.

 

[Group_Complex]

I forget who it was. What reason would you have to believe it was false modesty? No reason. You're just being pessimistic, now.

99% is an incredibly large proportion. Eve then, I think exploring most of those dead ends were actually more worthwhile than anything I myself could achieve.

Also, these are dead ends in breaking edge research, not Real or Complex analysis. I make most of my errors on things I thought I actually understood.

 

[mathwonk]

that time in my career when i was feeling most stupid, it was because i was hanging out with absolutely brilliant people who made me better. it occasionally also dawned on me, if i was a stupid as i felt, why would let me hang around?

 

[homeomorphic]

Also, these are dead ends in breaking edge research, not Real or Complex analysis. I make most of my errors on things I thought I actually understood.

Yeah, but when he first studied those, he was probably hitting dead ends.

 

[Group_Complex]

Yeah, but when he first studied those, he was probably hitting dead ends.

I doubt it. I mean, I still make errors with epsilon delta proofs back from calculus 1. It is like some things just won't stick in my mind.

 

[Group_Complex]

What is more, I am 19 years old. Most mathematicians seem to finish their undergraduate by the time they are 20 years old, I feel way behind despite taking complex analysis...

 

[homeomorphic]

What is more, I am 19 years old. Most mathematicians seem to finish their undergraduate by the time they are 20 years old, I feel way behind despite taking complex analysis...

Hogwash. I am 30 and struggling to finish my PhD. I'll be 31 when I'm done. We had an 18 year old in our program (who either flunked out or chose to leave, I don't know which), but no one who graduated under age 20, by the way. The standard age to be done with your PhD is more like 27.

 

[Group_Complex]

Hogwash. I am 30 and struggling to finish my PhD. I'll be 31 when I'm done. We had an 18 year old in our program (who either flunked out or chose to leave, I don't know which), but no one who graduated under age 20, by the way. The standard age to be done with your PhD is more like 27.

An 18 year old doing a PhD? wow, I am envious of people who get this far ahead. Are you at one of the top 10 US graduate schools in Math?