Thoughts on proofs

Question

I am a math and statistics 1st year student.

Calculus, linear algebra, etc I am very good at and do well. This semester I have a mathematical reasoning proof class which I am really struggling with. It is a completely different way of thinking in my opinion and I enjoy it, but struggle.

It has only been 2 weeks since I started this class, but even with some of the basics of Euclud's infinitely many number of primes proof and some absolute value proofs I am spending hours to fully understand.

Anyway, my question is does it get easier and should I be putting this much time into understand basic proofs when in a couple of weeks they will probably be very easy to understand as the progression of any type of learning usually ends up this way.

Answer 1

Yes, it gets much easier. It is similar to learning anything else, whenever you begin practically every significant result is daunting. It will take a bit of time to get used to reading and writing proofs. There are some threads on this site with suggestions of books you could read to increase general mathematical thinking, proof reading, and problem solving skills. I wouldn't worry about how long it takes you to understand results right now. Just think about them until you completely understand the proofs. This will help you adjust from the types of math classes you are used to into a more rigorous type of mathematics.

Answer 2

Short answer: yes, it will get easier.

Longer comment. Write out as much as you can with words instead of symbols - for example

If $a$ is an even number, there must be another number $b$ such that $a= 2b$ (that's the very definition of "even").

rather than

$a$ even $\implies \exists \ b \text{ such that } a- 2b"$.

Imagine that your proofs are notes to yourself explaining why you know each sentence you right is correct.

I tell my students that they don't need to prove things for me - I know they are true - they need to convince me that they have convinced themselves for good reasons.

Sometimes the best questions are not "prove this" but "prove or disprove this".

Answer 3

Mathematics of all types, including proofs, tends to build on itself. If you understand the basics then they will help you as you progress. It does sound like you are putting a lot of time into some that should not take quite that much time, particularly if you understand the background behind the problems.

Make sure you understand all the information you need, though, and do not try to skip or neglect the basics, even if it takes you a lot of time. You might want to see if some one-on-one time with the teacher or another student that understands the material well can help clear up some of the underpinning principles that might be confusing you. Google and this forum are good places to find answers also.

Note that proof writing is about making sure that the logic clearly flows from the axioms and premises to the conclusion without making any logical loopholes or committing any logical fallacies. It is also about making sure that everything is well defined. The emphasis in a proof writing class is typically more on these than on actually solving major mathematical problems.