Ever since I determined mathematics - mainly set theory and number theory - was my main passion, and I began learning mathematics formally outside of the curriculum posed within secondary schools I have proved every single theorem I have learned, sometimes with the aid of sources, and thus kept all theorems I have learned up to Advanced Calculus in my head. However, this has took up a lot of the time I could have devoted to discovering new mathematics and progressing my studies at a further rate, but, I also can't truly advance to new mathematics without being able to formally prove all of its preliminaries as I have a physical feeling of being trapped once I do not fully understand everything about a particular theorem. So, my question is whether it is truly worth it to prove all the things you learn, although I fear I may have to stop proving things once the mathematics reaches undergraduate level. And, also, I can't physically and mentally progress into new mathematics without proving everything before it, should I report this to a doctor as I fear it will impact on me later in life.

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    $\begingroup$ It's useful to learn the proofs of theorems you learn since it helps you understand why they are true. Whether it's worth it for a particular theorem depends on what you need the theorem for and how long and complicated the proof is. $\endgroup$ – Qudit May 10 '15 at 20:39
  • $\begingroup$ The theorem can be anything and I never care about the complication of the proof, even if it ought to lead me to the necessity of learning another theorem which then requires proof. For instance, I was full bent on proving the fundamental theorem of calculus even though I had a concrete intuition and comprehension of it, thus leading me to discover Riemann integrals. $\endgroup$ – Reinhild Van Rosenú May 10 '15 at 20:42
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    $\begingroup$ Well, you might find that some theorems are somewhat more complicated than you're used to. For example, the proof of the classification of finite simple groups is thousands of pages long. I would not advise reading through that one. $\endgroup$ – Qudit May 10 '15 at 20:44
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    $\begingroup$ It depends on what your goals are. I do not think it is a good idea to try to learn the proof of everything. $\endgroup$ – Qudit May 10 '15 at 20:55
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    $\begingroup$ "We derived a lot of corollaries of Euclid's fifth postulate, by using them one can prove corollaries of Euclid's fifth postulate while proving Euclid's fifth postulate", i.e. when one know the base of the proof, it's enough to not to make "proofs" like $A\to B\to C\to A$. But intuition on how the base or the "basic idea" is used in particular proof is almost always enough to reconstruct that proof, imho. $\endgroup$ – Alexey Burdin May 10 '15 at 20:55

Ultimately, you should hopefully be able to prove the vast majority of the mathematics you know.

You do not necessarily need to prove things as you learn them, nor is it necessary or of benefit to do so. Oftentimes you will find that the proof for what you are learning is of a greater level and complexity than you can handle.

But as you learn new mathematics, the goal of a good mathematics curriculum is that you begin to understand the nuances that underpin basic math.

You begin to notice things, like the derivative of the volume of a sphere w.r.t. its radius is the surface area. Curious, no?

As you take further and further courses, into analysis, for example. In single-variable calculus, you might be taught about Taylor expansions, with some of the finer points about error estimates and derivations brushed under the carpet. Once you get to analysis you should be able to derive Taylor series.

The benefit of learning the proofs behind the maths you are learning is that you gain a deeper understanding of the "why." And in any field, whether it is math or physics or the humanities, advancing that question is the ultimate goal of further instruction.

Furthermore, I know many mathematicians who do not memorize facts or identities, but derive them on the spot. A friend of mine (a respected mathematician, whose name I will leave out for the sake of privacy) once said very aptly: "If you show a person a formula, they'll forget it within the hour. If you show them how you got the formula they'll remember it for their life." (rough paraphrase; his was more poetic)

As Qudit noted above, however, this can only last so long. If you continue into a lifetime of mathematical research (good for you!), there will always be more questions, and furthermore, you'll never have all of the answers.

I would suggest that instead of attempting to prove everything as you are told it, try and see how new material relates to what you already know. As you learn more, you should recognize more patterns and ultimately, more proofs.

And sometimes, such as in addition and multiplication, it is not worth reading Russell and Whitehead's 300 page opus to figure out that it is true.

  • $\begingroup$ I agree with you, but it's only been a couple of months since I've actually changed my learning method (similar to the one you are suggesting), and I must admit it's working better. $\endgroup$ – Reinhild Van Rosenú Feb 8 '16 at 23:54

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