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After getting Bachelor's degree from Computer Science I changed my field to Applied Mathematics. The previous degree was mostly programming-oriented, so I know quite a lot about software engineering, databases, Linux etc., but not so much about maths.

To be honest, we had just basic single-variable calculus, basics of linear algebra and a little of graph theory.

After finishing my B.S. I got some awards for my bachelor thesis and because of that I was offered a job in our research centre. I changed my field to mathematics to be more useful there (most of my colleagues are concentrating on computational mathematics).

The problem is, that while I'm studying for final exams of this term, I recognized, that I don't understand things thoroughly and some of them at all. The biggest problem for me is Functional Analysis, where I'm simply stuck on very basic concepts and I have to look up something for almost every lemma or proof I want to understand.

I takes tremendous amount of time and even then I feel like I know nothing about it, because in every subject there are exercises I don't understand and proofs I'm not able to invent on my own.

The truth is, I was able to pass somehow all the tests so far and to finish all the projects with full score.

But still, I feel I'm not very confident in this and so I thought about the possibility I'm simply not talented enough.

So, is there any way to find out if my problem is caused by the lack of talent or just by gaps in my knowledge I'll be able to fill one day?

I'm mean something like what amount of knowledge one should be able to grasp in one half of a year, one year etc.

I've read several questions about studying maths here on SE. For example:

Grasping mathematics

How to effectively study math?

Steps to Re-Learn Mathematic the right way

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Eenoku
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    I think many of us went through the same feelings and experience you have. I graduated with a bachelor's in math with almost straight A's and then within a few weeks of arriving at graduate school I realized that I had hardly understood anything. I think you have to study some things several times thoroughly before they really sink in. By the time I graduated from graduate school I finally did understand everything from undergrad pretty well, and I was on to worrying about the next level. I think no matter how far you go, there's always intimidating material above you. My advice, persist. – Gregory Grant Dec 28 '15 at 20:01
  • I have gone through a similar situation to yours: I started my software engineering course almost 3 years ago, knowing that I actually wanted to study mathematics much more than that, but my decision ended up being financially-guided (mathematicians almost starve in my country). So I started learning mathematics on my own, I grabbed a few books on introductory number theory, abstract algebra, graph theory, a few calculus ones, and now I'm trying to beat baby Rudin, it took a lot of effort to do this while learning C, Java and all of those things you learn in the first years of an engineering.. – YoTengoUnLCD Dec 28 '15 at 20:38
  • course, but the effort was worth it. I reaffirmed my love for mathematics, and I signed up on a 6year course. After a bit of paperwork, most professors allowed me to take the final exams to their subjects (the ones I had studied on my own before) so I could skip 1.5 years of classes. I know it will probably take me a lot of time and even more effort to ever get my degree, but I know I love mathematics, and it's worth it. I suggest you stop doubting yourself and go for it. – YoTengoUnLCD Dec 28 '15 at 20:41
  • Studying math as an undergraduate is different than studying it as post graduate. The later is much harder, more abstract and requires lots of personal effort and desire. The other thing is that Math careers may be disappointing financially in many parts of the world, are you ready for that? – NoChance Dec 28 '15 at 20:46
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    I did a double take when I saw that you were studying functional analysis straight after finishing single-variable calculus and linear algebra. You're probably missing about two years' worth of math classes. Perhaps not because you need all the material (although you do need some), but especially to build mathematical maturity. In fact, based on that amount of math, it's quite difficult to evaluate your potential ability in the subject, although being a good programmer bodes well... – David Jan 03 '16 at 19:10
  • ...At this point, I wouldn't recommend Spivak, as you need some topology immediately. I would start reading a basic analysis book like Apostol's. Rudin's might be a bit on the abstract side to fill the gap between where you are and where you need to be. And, if you feel the need for it, something short and rigorous on linear algebra (though analysis is your first priority). That could be Gelfand's Lectures on Linear Algebra. – David Jan 03 '16 at 19:17

3 Answers3

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Stop doubting yourself right now: you're talented enough. The only question is will you work hard enough? Listen to Terence Tao, the idea of lacking "talent" should be done away with: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

You've got to find the problem areas where you lack understanding and spend some time carefully learning everything. Analysis and analysis proof-writing is difficult and requires time. Sit down with a copy of Spivak's Calculus or Rudin or something and go over every part with a fine-toothed comb until you understand everything. There's no other way to do it. If you have the willpower to make yourself concentrate, and can come away with some understanding, you shouldn't have any problems.

Also, talk to people about math. Go to your math department (or a message board or something) and have a discussion about what you want to know. That's the easiest way to learn, from other mathematicians.

But, most importantly, STOP DOUBTING YOURSELF!

Jeff Strom
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    Yes although it's all fine hearing these things from the big famous names, but it'd be nicer to have the people encouraging exemplifying hard work to not very often be incredible prodigies who probably cruised through what most of us would find quite difficult... – ocg Dec 28 '15 at 20:20
  • I disagree. And doesn't the existence of non-geniuses in Math prove that you don't have to be a genius to succeed? Every professor in your local state college math department is a testament to the fact that you can be an ordinary person and succeed. And of course I'm going to quote someone like Tao: wouldn't it be incredibly underwhelming for me to give you advice from someone you've never heard of? – Jeff Strom Dec 28 '15 at 20:33
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    I agree, if anything, you should doubt your willpower, that's the only thing that could really stop you from learning. And if that happens, it's because you didn't really want to learn mathematics. – YoTengoUnLCD Dec 28 '15 at 20:33
  • If you want an example of an ordinary person who later went on to become a great mathematician, I present to you Stefan Banach: https://en.wikipedia.org/wiki/Stefan_Banach An orphan who didn't technically pass his high school graduation exam until he was 22, he worked hard regardless, and was discovered by one of the professors at the university he was at. From then on, he was catapulted into success by virtue of his hard work. – Jeff Strom Dec 28 '15 at 20:39
  • @JH Strom Never disputed the existence of successful non-geniuses nor was I looking for an example. I was saying that I personally would find it more reassuring to hear those same words from a successful, preferably well-known mathematician who wasn't a massive prodigy - such as Banach (and I am adding well-known simply because the words of someone you know are always more powerful than those of someone you've never heard of). – ocg Dec 28 '15 at 23:01
  • "If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries. " C.F. Gauss – CuriousIndeed May 05 '17 at 10:41
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Ok so 1) Dont let anybody else (exams, professors, university committiees) decide whether you are talented enough.

2)Forget the question of whether you are talented, and ask yourself if it interests you.

3) And most importantly are you willing to put in the time and effort to overcome the obstacles.

Rene Schipperus
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    In my opinion: Some intelligent people don't seam to know how life is like to the less gifted. Talent is everything in every field I can imagine. IQ is a must for success in this world. If one has high enough IQ and good talent, then the concept of 'interest' can come to play, otherwise, being interested without the rest of requirements will be a no-go. – NoChance Dec 28 '15 at 20:43
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    @NoChance IQ a must ? haha. You have surprises in store for you. – Rene Schipperus Dec 28 '15 at 20:48
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    @Nochance my brother's IQ is 20 points higher than mine. I have a PhD in math and he never graduated from high school. – Matt Samuel Dec 28 '15 at 22:56
  • IQ as measured by some tests is not the point. I meant intelligence when I wrote the letters IQ. It is quite possible that some intelligent people hate to be controlled by an educational system. Some intelligent people work on things that make them happy and can cover their living expenses. I am yet to hear of an intelligent person who is homeless even in poor countries. What is surprising is that you have such a high degree in math and don't admit that math study requires intelligence, this gets us back to my first phrase in the first comment :) – NoChance Dec 28 '15 at 23:13
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    @NoChance By the way dumb people also hate being controlled by the educational system, its pretty much universal. – Rene Schipperus Dec 28 '15 at 23:20
  • @ReneSchipperus, agreed, I know a lot about dumb people, I belong to the set. – NoChance Dec 29 '15 at 10:16
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Before beginning my mathematical studies at the university I had the same question and asked my former maths teacher if he recommended me to study maths. His answer was simply: "If you can imagine yourself working for about 80 hours per week on mathematical problems, then do it, otherwise let it be!" I thought he wanted to tell me jokes, since he knew that I was somewhat lazy...So I decided to ask a professor at the university where I was going to study maths, what I could expect. He did not know about my laziness, but told me almost the same (about 70 hours per week). Then I knew what I had to expect... Now, as a mathematician, I know what the two people wanted to say: First of all, studying maths consists of hard work, talent can be helpful but if you really enjoy solving tricky problems and puzzling, developing strategies and so on, then I guess you will also have enough talent...

Another fact which occured to me: The more you already know (in a certain field) the more you become aware of the vast amount of things you do not know yet! It is no reason for fear if you come across some open questions which you have not yet an answer for, but can be merely a sign for developing consciousness for the subject. When I was in this situation I found it very helpful to take a "bird's eye view" of the stuff, to find the "links" between the theorems and lemmas, to find out what proposition is needed for the proof of theorem xyz and so on, not to loose myself in the details. It is helpful to have a "red line" along which the different topics can be arranged in an appropriate way in order to get a closer understanding of the results and their relationship one to another.

mkk030572
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