An Intuitive Understanding Vs. Analytic Proofs

[vanmaiden]

Hey Physicsforums,

This is something I run into quite a bit in my study of mathematics. Proofs are very important - that much is true. However, there are many instances when I don't need a proof to understand a concept; I just understand it.

For example, I don't need the epsilon-delta form of a limit to understand the concept of a limit, nor do I need a proof to explain to me why,

as n approaches infinity, $\sqrt[n]{n}$ approaches 1

Would you advise anybody interested in mathematics to still review such proofs even though that person may already have a strong grasp on the concept without them? My only quandary with reviewing such proofs is that they tend to be ones that are the most difficult to grasp.

 

[arildno]

Well, if the proof is "difficult" to grasp, you haven't understood it, nor the problem. You just have a feeling that you understand it.

 

[SteveL27]

Hey Physicsforums,

This is something I run into quite a bit in my study of mathematics. Proofs are very important - that much is true. However, there are many instances when I don't need a proof to understand a concept; I just understand it.

For example, I don't need the epsilon-delta form of a limit to understand the concept of a limit, nor do I need a proof to explain to me why,

as n approaches infinity, $\sqrt[n]{n}$ approaches 1

Would you advise anybody interested in mathematics to still review such proofs even though that person may already have a strong grasp on the concept without them? My only quandary with reviewing such proofs is that they tend to be ones that are the most difficult to grasp.

You're not wrong. You're just 200 years too late.

The modern insistence on logical rigor down to the very last detail really got going only 100 or maybe 150 years ago. Newton did not know how to properly define a limit; though his papers clearly show that he understood the problem and struggled with it throughout his career.

But these days if you can't do the proof, then you didn't understand the theorem. With the advent of computational methods and experimental mathematics, perhaps in another 100 years, nobody will care about proofs any more. "Proof-based math" will be an obscure sub-specialty in mathematical logic, while "real" math will be done by using computers to make plausible guesses, then proceeding to obtain practical results. Intellectual trends are subject to the whims of human nature, and anything's possible.

But today, you need to know your proofs!