When reading proofs, I often get confused and need to devise my own examples to understand what's going on. Is this practice ok or should I train myself to think in abstract terms?

As an example, here's something that I'd need a sketch on paper to understand.

enter image description here

  • 18
    $\begingroup$ No, it is NOT bad $\endgroup$ – kjetil b halvorsen Jun 20 at 4:58
  • 7
    $\begingroup$ I think that the bold statement by @kjetil does not give enough emphasis. It is absolutely a great idea to perform numerical tests. Both when learning and when seeking new truths. Also in contest-math. Almost everywhere. $\endgroup$ – Jyrki Lahtonen Jun 20 at 7:43
  • $\begingroup$ I believe it is more than necessary. I have been studying maths for a while and it is the only, I believe, to fully understand what is being presented. Yes, abstraction is compact and elegant but it must work when going down to the "ground level" i.e. with numerical examples with sketch. $\endgroup$ – Rene Girard Jun 23 at 19:07

That's perfectly normal. I do the same thing, and I've heard it strictly encouraged to solidify your understanding with examples. Like, you can bet the author looked at tons of examples before they even came up with the correct statement of the theorem they're proving. :)

| cite | improve this answer | |
  • 1
    $\begingroup$ Yeah that's the thing. Proofs without examples always seem to say something along the line of "this is self-evident". $\endgroup$ – Josh Ng Jun 19 at 19:21
  • 2
    $\begingroup$ @JoshNg "this is self-evident" or "this is obvious." Yeah, that's a disparaging thing many mathematicians say. They certainly shouldn't, but, maybe it's understandable? Thinking back to some math you learned years and years ago, like how to add fractions or something, that's obvious to you now, right? All math is self-evident, but only after you understand it $\endgroup$ – Mike Pierce Jun 20 at 21:43

I would say that, not only is it normal, it is recommended and also something that some people have to be taught to do. John Conway said that he worked out numerous examples with physical tokens when he was inventing his game of Life, and I got the sense he thought that kind of concreteness was important in other more conventional mathematical contexts too.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.