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To learn mathematics, I want to get familiar with formulas/theorems by taking one and just analyze it and try to manipulate it to understand it better. I wanted to ask you whether this is a good/efficient idea to become proficient in math.

My theory behind this question is that a theorem reflects several things in mathematics.

  1. It reflects the existence of a category of problems/ideas that is meant to deal with.

  2. It provides a method or part of a method to solve a problem.

  3. It reflects rigorous mathematical expression of the insight of a mathematician. This means that a theorem is the crystallized form of expressing an idea.
  4. Being able to prove or to completely understand the proof leads to enhanced knowledge of methods used in proving mathematical ideas and give a greater clarity in what exactly the use is of the theorem/formula. (I contrast this with solving equations with a formula, which you can do without completely understanding the formula)

If this is not a good/efficient idea, would you be able to specify why not? I'm only asking about this as a complementary method.

St.Clair Bij
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    "One theorem a day" seems like a very inconsistent unit of learning; not all theorems are created equal. Furthermore, a lot of learning happens when you take an idea that you tried to understand before and come at it again. – Ben Grossmann May 27 '15 at 21:17
  • It is much more important to try to solve problems. A theorem is only useful when it can solve a problem or answer a question. That said, the study of theorems can help you approach problems. What sort of math do you want to learn? It's a big world out there, and it will take more than a day to understand many theorems. – Joel May 27 '15 at 21:18
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    Maybe a better approach would be to try to solve problems on StackExchange? Many people have taught themselves a lot of math through this website. – Joel May 27 '15 at 21:19
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    I voted to close this question because I think it's too broad and primarily opinion based. Do you have a particular branch of mathematics in mind? I don't think most branches of mathematics admit a list of theorems which are universally considered to be fundamental, especially more advanced or recent branches of mathematics. I also don't know if it is meaningful to break up mathematics into isolated formulas/theorems; mathematics is a story and how you tell that story matters. – Amitesh Datta May 27 '15 at 21:25
  • Yes, I've heard time and time again "do all the problems in the book". There is a certain concreteness solving a problem gives you beyond understanding a theorem. Maybe in the modern day, Joel's suggestion is a great way to augment that. – muaddib May 27 '15 at 21:25
  • @ Joel: A theorem does describe a sort of problem. It seems possible to me not to know at all that a certain category of problems do exist. By knowing the theorem that deals with a problem, you learn about the category, furthermore you learn about the discoveries and methodologies of mathematicians. – St.Clair Bij May 27 '15 at 21:34
  • @ Amitesh Datta: Would it be better to get the concrete part of a 1 per day out of the question, and instead focus on the methodology proposed for self-study of mathematics? – St.Clair Bij May 27 '15 at 21:36
  • Hi @St.ClairBij, regarding self-study of mathematics, I think there are numerous questions and answers already on this website on the topic. You might be interested in click me for example. A complete list of questions with the tag self-learning is available at click me. Finally, there are numerous questions on this topic listed under "Related" on the right-hand sidebar. – Amitesh Datta May 27 '15 at 23:31
  • Hi thanks, I know, I have read the self-learning section, however I have not come up with an answer to this question. If you would argue that this question is too vague as it does not specify one branch in math, and this method might work different for different types of math, it is a bit vague on that. Nevertheless I do think that a large part of the question can still be meaningfully answered and argued for logically by competent mathematicians who understand math and who know how they got to understand it. I also think that this question is way more narrow than: How to study maths? – St.Clair Bij May 28 '15 at 08:05

1 Answers1

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Spending one day on each mathematical theorem is like spending one day in each settlement in the world. It is

  1. Impossible, because there are simply too many theorems.
  2. Impractical, because it makes no sense to spend one day in a 1000-people town, and one day in London. In the same way, some theorems deserve far more than a day, and others can be handled in a matter of minutes.
5xum
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    Thank you for your answer.

    With respect to 1. I know a person would not be able to deal with all theorems. However I do not see this as a reason not to do it. In chess a person cannot study all openings, nevertheless they must study openings in order to become better at chess.

    With respect to 2. You are right about this. A day is not exactly a right period for any given theorem. Maybe to focus more on the methodological aspect of it: Would a person be able to get a good overview of mathematics and a working proficiency within fields of mathematics by studying important theorems?

     – St.Clair Bij May 27 '15 at 21:43