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.

 

[Matrix0]

I understand the importance of mathematical rigour in research and scientific studies. It is essential for ensuring the accuracy and validity of our findings. Improving mathematical rigour requires a combination of carefulness, discipline, and a clear understanding of the supporting tools.

Firstly, being careful and meticulous in your approach to mathematical problems is crucial. This means paying attention to details, double-checking your calculations, and being mindful of any assumptions or simplifications made. It also involves being open to feedback and continuously reviewing your work for any errors.

Secondly, discipline is essential in developing mathematical rigour. This involves following a systematic approach to problem-solving, breaking down complex concepts into smaller, manageable steps, and consistently practicing and reviewing your work. This discipline also extends to maintaining a strong foundation in basic mathematical concepts and regularly revisiting them to ensure a solid understanding.

Moreover, mathematical rigour also requires a clear understanding of the supporting tools and concepts used in your work. This means not just accepting them at face value but delving deeper to understand the underlying principles and assumptions. This understanding will help you to confidently build upon these tools and use them effectively in your work.

It is also important to note that mathematical rigour is not just about following a set of rules or steps. It involves critical thinking, problem-solving, and being able to justify your reasoning and conclusions. This is where the concept of 'believing' in the validity of supporting tools comes in. It is essential to have confidence in the tools and concepts you are using, but this confidence should be based on a thorough understanding and critical evaluation, rather than blind faith.

In your case, it seems that your lack of discipline may be hindering your ability to improve your mathematical rigour. I would suggest setting aside dedicated time for practicing and reviewing basic mathematical concepts, as well as seeking out additional resources or guidance to strengthen your understanding.

In conclusion, improving mathematical rigour requires a combination of carefulness, discipline, and a clear understanding of supporting tools and concepts. It is an ongoing process that requires dedication and continuous improvement. By following these tips and advice, I am confident that you will be able to improve your mathematical rigour and become a more confident and proficient scientist.