Improving Math Rigour - Tips & Advice

[cdux]

I was told I lack mathematical rigour. But how do I go on improving on it? Is it only a matter of being very careful? Do I have to always support everything with a clear Euclidean succession of logical steps? Is it only a matter of 'believing' in the validity of the supporting tools? Then it's an oxymoron that while some people consider rigorous to firmly step on past tools, they mainly do it via respect to the mathematicians that invented them, rather than on a clear understanding of them.

Concerning my personal case, I think I don't lack knowledge so much on the process but rather on discipline. e.g. I was taught from a very young age the elegance of Geometrical axioms leading to a whole science but when it gets to other concepts, my mind usually flies to places that should really have a more solid basis behind them before going there.

 

[hsetennis]

Depends who's telling you this. It means different things if it comes from your chemistry professor or your math professor.
I'll assume you mean your math professor. First, understand the fundamentals of proofs: implications, contradictions, necessity/sufficiency, etc. Watch carefully how the results are proven in class and emulate the style in your exercises. What helped me was to do is to work ahead in the textbook and [thoughtfully] copy down the proofs that will be presented the next day. Once you feel like you grasp the material, go back to the theorem, cover up the proof, and try to prove it on your own.

 

[Theorem.]

As stated by hsetennis it absolutely depends where this is coming from. If it is from a math professor (as I will from here on in assume it is), then it means you have to work on justifying your arguments mathematically i.e; proofs. There are many ways you can learn about this. One is by reading through the introduction sections of elementary rigorous mathematics textbooks (like analysis, set theory, algebra and so on), and in specific going through the proofs offered by the textbooks and trying to figure out what each step means and why is it necessary. These books usually offer practice questions also where you can practice your own proofs. Another option is to read a book specifically on proofs such as "How to prove it: A structural approach"- which has been mentioned on here several times. Your professor probably means that there are "holes" in your arguments. That is, you aren't including all the necessary steps and jumping from one step to the next without proper justification.

 

[epenguin]

I was told I lack mathematical rigour.

Next time you are told ask for a rigorous proof of the assertion.