How to deal with self-doubt in mathematics?

[cbarker1]

TL;DR

Dealing with Self-doubt in mathematics in general situation.

Dear Everyone,

I am wondering how to deal with the self-doubt in proof-writing in general situation like on exam or homework question.

Suppose I want to prove Theorem B. I assume the hypothesis. Then I apply the right mathematics definition. I am hesitant on the next step; I have the feeling that I can use Theorem A. But I can't explain why I can. What to do?

Thanks,
Cbarker1

 

[Office_Shredder]

Mathematics is literally the study of explaining why. Step 1 is probably read some examples of similar proofs to get a feel for how this typically goes. If you had a bit more detail of the specific problem you are thinking about we could help a bit better.

 

[cbarker1]

But I want to prove on my own because it is best way to know the material. Suppose that I did that before I started the proof.

 

[Office_Shredder]

Did what before starting the proof? Mathematics is the accumulation of four thousand years of human knowledge. The idea there's nothing to learn from seeing how the rest of humanity did it is assuming a lot about your own intellectually ability :)

 

[cbarker1]

I know that math is cumulative. I am not trying to be arrogant in my previous post; I am trying to think about a thrm in a different way (like the fundamental theorem of algebra proofs). I understand that it is hard to know what to do without a concrete example of a proof that would help you understand what I am asking to the community. But I wonder if you can give me some strategies to move forward with the feeling of hesitant in the proof in general.

 

[jedishrfu]

In proof based classes, like high school geometry in the good old days, we started with some basic axioms and definitions. From there we proved some theorems.

Later we were able to use the theorems to prove other theorems. All we had to do was explain why the theorem applied.

Often our proofs would start with the givens, and then proceed to step by step statements backed up with a reason: given, axiom X or definition Y or theorem Z and also why.

You should be able to talk your way through the proof to teach someone its validity.

An aside:

One time in my twelfth grade math class, I had to miss it because of a music lesson. My classmate asked to borrow my proof for a class problem. I later heard he got called to the blackboard to write down the steps and then was asked what was the reason for the first couple of steps and he couldn't answer because I gave no reason, panic set in and he froze.

Later he yelled at me for not providing the reasons and I calmly said but Don those were given statements no further reason was needed. Oops. Needless to say he did not become a mathematician but had a great career as a newspaper journalist.

 

[cbarker1]

Ok. I see. But how should I proceed without being hesitant in my proof?

Summary:: Dealing with Self-doubt in mathematics in general situation.

Suppose I want to prove Theorem B. I assume the hypothesis. Then I apply the right mathematics definition. I am hesitant on the next step; I have the feeling that I can use Theorem A. But I am not sure if it is the correct theorem...

I don't want to look up the solution of the proof. I have a habit that I struggled in a problem. I just looked it up online and not really think about the skill that I want to learn.

 

[fresh_42]

I tried to answer your question in an insight article.
https://www.physicsforums.com/insights/how-most-proofs-are-structured-and-how-to-write-them/

Maybe it can help you, although it certainly could have been done better.

 

[wrobel]

I think that a self-doubt is a good feeling especially in science.

 

[cbarker1]

I think that a self-doubt is a good feeling especially in science.

I agree with you that some self-doubt is good thing. But if you have too much, you feel like you want to stay still and not proceed.

 

[jedishrfu]

We all have self doubt in some form or another.

Mathematicians will run their proof through others especially parts that feel sketchy. They will polish it like a fine gem and then release it to the general population in the form of a paper defending it, fixing an errors and then rebelling in its acceptance.

However, some future mathematician may discover a flaw and the process repeats.

It takes loads of practice, focus and confidence on what you are stating and why to remove most self doubt.

Think back to your algebra derivations where you applied the various associative, commutative and distributive laws to come to a solution. That is a form of proof. How did you erase your self doubt?

Practice, practice, practice.

 

[cbarker1]

ok. I see. How to gain confidence in my skills?

 

[nuuskur]

It takes time. Working through many proofs, techniques and mastering the Art(?) of Predicate Logic. Second guessing your claims is a good habit to have. Explaining why step X follows from step Y is, in a nutshell, what we do.

I have had these thoughts, too and many of my students do, as well. The eventual demon is going to be "am I even cut out for this..?" I can say all the nice things in the book, but the truth of the matter is you'll have to conquer those demons, yourself. The path of the mathematician is a steep climb.

How do you become more confident behind the wheel, say? Well, you practice driving. Skiing? Put on some skis and off you go. Math is mental gymnastics, prepare to get ripped, brother!

 

[Stephen Tashi]

ok. I see. How to gain confidence in my skills?

If you only study the technique of writing proofs by trying to prove statements you know to be true, you don't truly test your ability. You should try studying material where the assignment is: prove something or disprove it by counterexample. Invent some mathematical conjectures and investigate them. Look at material about counterexamples as well as lessons about proofs. There are various compilations of counterexamples with titles like "Counterexamples in Analysis".