Do professional mathematicians remember all the proofs they come across?

[Group_Complex]

Hello, I am a college freshman currently taking Real Analysis. Calculus was fairly mechanical, and dare I say it trivial, the concepts were easy to grasp and it required little memorisation. As I have began to study more abstract areas of mathematics, I have found my speed and confidence have decreased, and whilst I still understand the material, I will often doubt my memory of a definition or a proof, and have to resort to looking back at the textbook. This frustrates me incredibly as I feel it is indicative of a lack of ability and talent within the sphere of abstract mathematics. At the beginning of the real analysis course I set myself the goal of re deriving many of the proofs of the theorems presented before reading them in the text. Needless to say I have found this process to be drawn out, and in the end I have had to just read and understand the proofs from the text. Yet even then I find myself forgeting key details and having to re read the proofs and or definitions.

My question to professional mathematicians (or anyone that has taken a significant amount of pure mathematics) is this: Do you find it easy to remember every proof you read, and do you find yourself having to re-read books that you had previously mastered so as to remember key details? Is the process of continued revision indicative of the profession of pure mathematics, or is a mathematician supposed to remember key details without revision? Is this a problem that I can fix or is it a sign that I have reached my mathematical limit?

 

[micromass]

I certainly don't remember all the proofs I ever read. And sometimes I do need to go back to older books to refresh my memory. For example, when I took a Galois theory course, I did extremely well. But ask me anything about it right now and I couldn't tell you. It's immensely frustrating.

Even when studying, I couldn't remember all the proofs. I had to do proofs a few times before I really remembered it.

However, some proofs you should be able to remember. These are usually proofs in which a certain technique is used. Once you master the technique, you should be able to get all these proofs. For example, epsilon-delta proofs just require familiarity with it. Once you get it, you should be able to get all these proofs fairly well.

 

[MathematicalPhysicist]

Practice,practice practice.

As athletes keep practicing until they master their sports so it is with maths and sciences.

You can't expect people to remember every detail, but you can expect people to remember the techniques, tricks and methods that have been used in the proofs, as long as they keep using them in their research or studying.

I still sometime remember High school geometry theorems, though almost 10 years have passed since high school (and people still call me senile).

 

[Group_Complex]

MathematicalPhysicist, the more I practice the worse I feel. I feel as though I should not have to put in so much effort to succeed (besides attending lectures and doing the assigned problems) yet that does not seem to be enough.
How can it be that mathematicians like John Milnor, Vladimir Arnold, John Nash could master mathematics by the age of 19 so as to work on original work? I feel way behind even taking real analysis in my first year. It is quite a shock to come up to mathematics that requires prolonged thought to master, I spend most of my time feeling terrible and frustrated, but I can't give in, I feel a burning desire to understand.

How should I spend my time mastering the material? I do not seem to have the time to focus on doing every problem in the book, and sometimes even when I do try I spend minutes doing nothing, staring blankly at the paper trying to resolve an approach of attack. When I try to prove theorems without the aid of the book, I am always flawed in my reasoning, and find that while the analysis proofs are understandable, they are un-intuitive, and I very much doubt I could have come up with some of the proofs included given a prolonged period of time.

 

[MathematicalPhysicist]

When you read a proof of some theorem you need to be able to understand completely every step in the proof, and understand the reasoning behind every step.

For example a lot of proofs in mathematics is to work from the end backwards, i.e you know what you want to prove then start from it and then work backwards and piece together the puzzle. In a proof by contradiction you assume the consequent is false and then prove something that contradicts the premise of your theorem.

Mathematics is basically applied logic most of the time. :-)

 

[Group_Complex]

When you read a proof of some theorem you need to be able to understand completely every step in the proof, and understand the reasoning behind every step.

For example a lot of proofs in mathematics is to work from the end backwards, i.e you know what you want to prove then start from it and then work backwards and piece together the puzzle. In a proof by contradiction you assume the consequent is false and then prove something that contradicts the premise of your theorem.

Mathematics is basically applied logic most of the time. :-)

I understand that. Once I read the proof I understand, but the trouble is I doubt my own ability to derive proofs (which is something I think I should be doing).
Also I must ask, how common is it for good students to make stupid mistakes? I am always making mistakes, I try to learn from them, but I continue to make more (unrelated) ones in future. Sometimes I get definitions completely wrong and proceed to prove something, only to realize my initial assumptions were incorrect.

 

[fourier jr]

Practice,practice practice.

As athletes keep practicing until they master their sports so it is with maths and sciences.

You can't expect people to remember every detail, but you can expect people to remember the techniques, tricks and methods that have been used in the proofs, as long as they keep using them in their research or studying.

I still sometime remember High school geometry theorems, though almost 10 years have passed since high school (and people still call me senile).

I wouldn't say the goal is just memorization, it's understanding the proofs well enough to be able to reconstruct them. It sort of depends on the proof as well. Off the top of my head there are proofs in Suppes' Set Theory that are all pretty similar, so as you go farther along in the text he leaves out bigger gaps in the proofs since by that point the reader has gotten the idea. Hungerford & Kaplansky do similar things, where for some things they simply write "proof: as usual" or "proof: Zorn's Lemma" & in those cases it makes sense.

 

[nickadams]

Group Complex, your posts describe how i feel 100% as well.

It will be interesting to read the responses ITT

 

[nonequilibrium]

I think that the "professionals" distinguish themselves not by not making mistakes, but by daring to make them, and as many as are needed.

 

[Group_Complex]

How important is taking time to derive a proof of a theorem yourself?

 

[Sankaku]

I feel as though I should not have to put in so much effort to succeed...

That is your problem right there. Having cruised through High-School and Calculus, many people feel that all of math will be just as easy. It isn't. You will have to work harder than you ever imagined if you want to succeed. Do not compare yourself to fairy-stories of mythical mathematicians who "mastered mathematics by age 19." Those stories are usually exaggerated and distorted.

Everyone suffers. Everyone doubts their abilities. Get used to it!

Take-home message? Do the amount of work you have to do to learn the material properly.