Is Struggling with Mathematical Proofs Normal for a Math Major?

[Bleys]

Hey,

I'm sure this must have been asked before, but I couldn't really find anything specific using the search tool;
I'm a second year maths major and I love maths and would really like to pursue a career in mathematics. My problem is, often I can understand a proof (whether easily or not depends on the proof and tools used); what I find frustrating is that I very often could probably never contruct such proofs! And I heard somewhere or read on these forums that I should be trying and most importantly be able to prove theorems, corollaries, lemmas and such already. Now ofcourse I try to prove things I come across, but I very often find I have trouble with it. Some books like set theory or discrete maths ones have preliminary chapters explaining proof strategies and such; and while they certainly provide some practice, it still remains mind-boggling when I try to go about a proof myself in other theorems which require more abtract thinking (so I'm not talking about trivial proofs of the kind where you easily use the definitions used and logically deduce the conclusion; like even no+even no=even no; or proving identities or properties like vector product).

Should I be worried that I find I have trouble with proofs at my level and should just give up on this too ambitious dream? Or should I become comfortable enough once I've explored more maths?

 

[Tac-Tics]

There is no general strategy for finding proofs efficiently. If there was, mathematicians wouldn't have a job to do.

Proofs of nontrivial theorems are like discoveries in science. They either come unexpectedly out of no where or they come only after generations of frustrated geniuses struggle with them. You shouldn't feel deficient or anything if you can't come up with a proof of a nontrivial theorem, because that's the hardest part of the job description.

A proof answers a question. But mathematics is interesting in that you can make up the questions. It's often just as respectable to come up with a good problem as it is to come up with a good proof. The trick is to come up with systems which strike a balance. You come up with a problem which *seems* simple, but is full of emergent complexity. My favorite example of this is cellular automata (http://mathworld.wolfram.com/CellularAutomaton.html).

 

[Hurkyl]

Suppose I asked you to prove that
$$(x + y)^2 - 4x y = (x - y)^2$$

Would you be able to prove this identity? Almost surely. The reason why is because you understand quite well the domain of this problem, and you have a lot of experience using the tools that are good for this problem.

But if you weren't well-versed in high-school algebra and symbolic arithmetic, wouldn't you have some difficulty with this proof?



I've heard the process of devising new theorems described as follows:
* First, figure out what should be true
* Only then do you figure out how to ensure that it really is true

For example, you might be doing a calculation, and wind up with an expression like
$$\lim_{t \rightarrow 5} \int_0^t x^2 \sqrt{1 + \sin x} \, dx$$

and you mull it over a bit, and you decide that it really should be true that you can simplify this expression to
$$\int_0^5 x^2 \sqrt{1 + \sin x} \, dx.$$

(Do you agree?) So, you probably abstract it a bit to make the problem 'simpler', and make the following conjecture:
$$\lim_{t \rightarrow b} \int_a^t f(x) \, dx = \int_a^b f(x) \, dx$$

and then you set about to try and prove this theorem... or at least a special case that's relevant to the problem at hand. Can you do so? I claim that if you really know how to use the tools from your calculus classes that this is actually a straightforward exercise, at least in a special case that includes the case we want. Actually, you might have already seen the theorem in your calc class!

 

[Bleys]

Ofcourse I understand there's no set guideline for going about a certain proof, but that is one of the things that fascinates AND scares me the most. I find it amazing how some proofs are constructed and even thought of in general; the simplicity of some is remarkable in the poweful statements they prove. Yet the frightnening part is that I feel I lack the innovation and originality to do so myself.

The examples you provided are, like you said, straightforward if I know how to use the tools given properly (at least I hope I used them correctly). Fair enough, you can't, for example, ask me to prove anything in topology because I'm not familiar with any of its basic theory and definitions. But suppose I do have the tools to construct a proof for something, yet it requires me to use it in a "non-conventional" way, then I'm usually as lost as a law major in physics.

I guess an example of what I'm finding hard is Euclid's Theorem on the infinity if primes. Simple enough to understand, but could I have ever come up with it? Hardly... Like I said, I'm afraid maybe I'm unable to think outside the box when it's required. Or maybe a better example is proving the irrationality of $$\sqrt{3}$$ using the well-ordering property. Or proving the well-ordering property itself.

I suppose maybe I have some misconceptions on the field of work. Lectureres understand what they're talking about. But assume for example they need to prepare a lesson on something. Do you think they could prove the material they are suppose to give? Almost certainly! But then, if I aspire to become a mathematician, shouldn't I also by now be able to do so as well? I often find myself thinking after a lecture "could I have been able to prove that if it was given as an assignment?". More often than not the answer is no.

I guess I'm just worried because I know proofs are a fundamental part of being a mathematician.

 

[mathwonk]

start asking the question: why? when you see any assertion. read a book on logic, the one i read as a high school student was by ALLENDOERFER AND OAKLEY, called principles of mathematics. start thinking, and stop being satisfied with just making simple minded calculations in your math courses. ask why the formula works. think it through.

 

[CRGreathouse]

I just wanted to say that I really appreciate this thread. I'm a(n off-again, on-again) graduate student, but I often feel I'm not that good at writing proofs. I'm not creative in my choice of the type of proof, I favor RAA too much, and I omit steps. It's good to read advice on proofs!

 

[Ben Niehoff]

The first step in figuring out a proof (for something you don't already know how to prove) is just to play around with the math. Use the definitions given, write some particular examples, and see what happens when you put things together according the rules.

This can give you insight into how to use those definitions to construct a proof (and if the thing you're trying to prove is not true, maybe you'll run across a counterexample this way).

 

[monotune]

I think, one has to develop a critic mind first, and then learn how to criticize in a precise style/way. I remember getting really pissed off many times, bec I thought that those guys who had written those textbooks want me to "believe" them. Actually, you could say, that I just didnt understand or misunderstood something (yet, sometimes I was right), anyways this created a sort of "ambient". It was the ambient of being present, participation and interaction. (Hell,:D I was passionate!) This is when you start to form questions on your own. Then you try to answer them, and later the statement might turn out to be a theorem, a lemma or just "fun". You probably know most of the mathematical arsenal/ tools, tools in detail and many theorems and proofs as examples. Thats useful. But motivation is the essential part, whether you get eager to know that the given statement is true or not (or something else).

 

[Tac-Tics]

I thought that those guys who had written those textbooks want me to "believe" them.

In my high school experience and up through multivariate calculus in college the books required you to believe them. It was only a convenience if you ever noticed certain things were true.

 

[monotune]

I was referring to my university yrs. (Primary and high school are something very different, and Id rather not mix the maths studied there with the higher maths studies.) Sure you have to give some sort of credit to teachers, books in order to be able to develop, and I was talking about participation and interaction with them and their ideas, not rejection. I usually got pissed off during the first semester of analysis. And in order to do my exams I had to -at least temporarily- accept some (for me 'disturbing') things. Then in the second semester we studied set theory, and then those 'disturbing' things got explained. Eg the continuum hypothesis turned out to be 'relief'. Later I became more patient.

 

[Tac-Tics]

I was referring to my university yrs. (Primary and high school are something very different, and Id rather not mix the maths studied there with the higher maths studies.) Sure you have to give some sort of credit to teachers, books in order to be able to develop, and I was talking about participation and interaction with them and their ideas, not rejection. I usually got pissed off during the first semester of analysis. And in order to do my exams I had to -at least temporarily- accept some (for me 'disturbing') things. Then in the second semester we studied set theory, and then those 'disturbing' things got explained. Eg the continuum hypothesis turned out to be 'relief'. Later I became more patient.

What kinds of ideas did you have to accept without proof?

And what was the continuum hypothesis doing in an analysis class? =-o

I'm impatient like that too. Quick and dirty mathematics is unappealing, even though it produces correct answers a lot of the time. That's why I stopped doing calculus altogether after my freshman year of college up until I graduated =-P

For the record, I majored in computer science, and if you confuse the function "f" with the function f evaluated at a point x, "f(x)", in your code, you get an error!