Is it worth proving every theorem you learn?

Question

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.

Answer 1

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.