Proofs, Exercises & Mathematica - Training the same skills?

[Kolmin]

First post on this forum, that IMO is amazing!

I was reading the introduction of the book “A gentle introduction to the art of Mathematics” and I was wondering about what the authors wrote on whom the book is for. In particular he stated that the book is in particular for people who can easily solve exercises (i.e. computation) and want to make the next step, which is doing real maths in terms of proving stuff.

Now – and that’s not a rhetorical question, cause I really don’t know – if somebody is pretty good at math in terms of computations and usually he can solve problems easily, but at the same time he can get stuck by tough integrals and other tough computations, is it a problem or not?
Are making computations (like calculating integrals) and writing proofs related to the same skills?


Considering that there is a software like “Wolfram Mathematica” around that can do the computation for him, if doing tons of basic computational exercises is important in order to make somebody starts to think like a mathematician (without doing him/her a mathematician), if somebody has already that kind of mentality, but at the same time has some minor problems in terms of computations, can he skip the problem altogether and focus on proofs (and functional forms) instead of problematic exercises?

In other words: Is a necessary condition for writing proof doing every possible calculation without problems?

Looking forward to your opinions!

PS: I was not sure on which part of the forum this thread should be part of…I hope I didn’t make mistakes.

 

[Number Nine]

In a sense, your ability to deal with things like complex integrals is a measure of your comfort level with the material, but I certainly wouldn't say that it's a prerequisite for dealing with theoretical mathematics. I have a pretty good intuition for abstract algebra, but my ability to do basic calculus borders on incompetence.

 

[Kolmin]

That's interesting and you are not the first one who tells me something like this, but I tend to think it's something related to people that work on algebra and logic (others who told me the same were working on those fields as well).

So, what about real analysis?
I mean, is a prerequisite to prove stuff in that field being able to kill integrals or things like these?

 

[Nano-Passion]

First, when I say thinking mathematically-- I mean thinking in terms of patterns and relationships, among other things.

With that said, I think doing some computation helps you build up your conceptual framework of thinking mathematically (this is geared more towards Calculus II). You learn some techniques and concepts while you are doing computation-- take for example the washer method and substitution. One gives you the basic concept that its more than just variables, that these variables actually describe certain things like volume as they are manipulated. While the latter makes you face the importance of simplification. Even though, it is a whole different beast from proofs-- it has a lot to offer for thinking mathematically.

 

[Matrix0]

As a scientist and mathematician, I can say that being able to solve exercises and write proofs are definitely related skills. However, they are not necessarily dependent on each other. It is possible to be good at one and struggle with the other.

In terms of using software like Mathematica to do computations, it can be a helpful tool but it should not be relied on completely. Doing basic computational exercises is important in developing a strong understanding of mathematical concepts and problem-solving skills. Skipping these exercises and focusing only on proofs may lead to gaps in understanding and potentially hinder progress in more advanced mathematics.

That being said, it is not necessary to be able to do every possible calculation without problems in order to write proofs. Being able to recognize patterns and apply mathematical concepts is more important in writing proofs. However, it is important to have a solid foundation in computational skills to be able to effectively apply them in more complex problems.

In summary, while being good at computations and writing proofs are related skills, they are not necessarily dependent on each other. It is important to have a balance between practicing both and not relying solely on software for computations. Developing a strong foundation in both areas is crucial for success in mathematics.