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Polynomials and Derivatives
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P p-sylow with $ P ⊂ Z(G) $

I solve mathematical problems every day, some by hand and some with the computer. I wonder: What will I lose if I start doing the mathematical problems only by computer? I’ve read this text and the author says that as techonology progress happens, we should change our focus to things that are more important. (You can see his suggestions on what is more important, in the suggested text.)

I’m in search of something a little more deep than the conventional cataclysmic hypothesis: *“What will you do when a EMP hit us?!”* and also considering an environment without exams which also exclude the *“Computers are not allowed in exams”* answer.

This is of crucial importance to me because as I’m building some mathematical knowledge, I want to have a safe house in the future. I’m also open to references on this, I’ve asked something similar before and got a few indications.

- Learning mathematics as if an absolute beginner?
- Studying mathematics efficiently
- Is it possible to practice mental math too often?
- How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?
- Bad at computations… but not math?
- How do I sell out with abstract algebra?

- Vivid examples of vector spaces?
- Why do we care about dual spaces?
- Examples of “transfer via bijection”
- What is the importance of eigenvalues/eigenvectors?
- Surveys of Current (last 50 years) Mathematics at Graduate / Research level?
- Reference request: calculus of variations
- What is an Isomorphism: Linear algebra
- Formulae of the Year $2016$
- Can the Bourbaki series be used profitably by undergraduates?
- How important is programming for mathematicians?

This highly depends on the level at which you want to solve problems. You will be able to solve linear algebra and calculus problems with a computer, but you will not be able to prove theorems and various results. However, there are very good reasons for using a computer in mathematics. In particular, you can make very technical computations and save a lot of time, but again, these computations **always** rest on a good conceptual understanding.

There’s nothing wrong with performing calculations with computers.

The thing is that performing calculations is ineffective as pedagogy.

Another important thing is to remember that computers should be regarded as tools, and that they don’t substitute at all for thinking ability.

I don’t fully understand the question, but let me say a few things roughly related to the question and hope this is useful.

In my opinion, one of the things that draws me to mathematics is understanding the hidden relationships between structures. For me, the most interesting part of a proof is that it can show you why something is true. If the proof doesn’t help you understand why it is true, then either you don’t understand the proof or the proof is unsatisfying — either of these mean that more work is needed. I remember attending a lecture by Gromov in which he said, no one knows why Gauss’s reciprocity law is true. A very precocious undergraduate student raised his hand and proceeded to give a proof. Gromov smiled and said, yes, I know many proofs, but I still don’t know why it is true.

A calculation can also be similar. There are some calculations that tell you absolutely nothing. There are other calculations that are enlightening. Sometimes the skills you build in doing calculations that a computer could have done enable you to do calculations the computer could not have done… or maybe allow you to see a pattern that you would not have seen otherwise. Other times, you are just wasting your time doing straightforward computations and it would have been better if you had delegated the work to the computer. For instance, in my work, I usually have the computer find eigenvectors of matrices, do all plots, and integrate anything I need to integrate. The semesters I teach a class that does one of these by hand, my skills in that area dramatically increase.

Finally, there is a lot of merit to being technically proficient. All the greats of the past were monsters at computation (Newton, Gauss, Riemann, …) Allegedly, Serge Lang would give his Calculus students a quiz in basic high school algebra at the beginning of the semester. Those who could solve the problems immediately, he predicted would do well; those who had to think, he predicted would not. (Here is the only reference I could find for this story, http://www.joelonsoftware.com/articles/GuerrillaInterviewing3.html). This matches my experience, to a certain degree, though I have no explanation as to why this might be true.

My conclusion then is that (1) it definitely makes sense to delegate some work to a computer, at least once one has mastered the concept, (2) mathematics, as I understand it, cannot be done by computer entirely, and (3) developing some proficiency at computation is essential so that you are able to understand the steps involved.

Yes you can solve most of the problems by computer but you will lose your critical thinking ability, you are going to be an operator not a creator eventually!

Perhaps you’d be interested in the TED Talk by Wolfram.

http://www.youtube.com/watch?v=60OVlfAUPJg

There are more materials on the Wolfram web site.

HTH, as this is an area I have interest in and have often wondered how to change the way we explore, learn and interact with Mathematics.

Well I guess it depends what kind of math you do. If, for example, you mean simple arithmetic, what you are losing is speed. If you are talking about deriving or integrating, then the problem gets a little worse because after a while you might not remember how to do it. I think that the harder something gets the more you need to practice it to REALLY get the hang of it.

This is all just my opinion, but remember the saying that says practice makes perfect? I know for a fact that it is also true in math, and if you don’t practice you can’t be perfect. However I think that its still a good idea occasionally when its really just a tedious thing you have to do in order to get a greater intuition on a more abstract concept. Hope this helps you.

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