Why should we prove obvious things?

Obviously, there are obvious things in mathematics. Why we should prove them?

  • Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$?
  • Prove that $f(x)=x$ is continuous on $\mathbb{R}$?
  • $\dotsc$

Just to list few examples.

Answer

Because sometimes, things that should be “obvious” turn out to be completely false. Here are some examples:

  1. Switching doors in the Monty Hall problem “obviously” should not affect the outcome.
  2. Since Gabriel’s horn has finite volume, then it “obviously” has finite surface area.
  3. “Obviously” we cannot decompose sphere into a finite number of disjoint subsets and reconstruct them into two copies of the original sphere.
  4. Since the Weierstrass function is everywhere continuous, then “obviously” it must have at least a few differentiable points.

Of course, mathematics has shown that switching doors is to the player’s advantage, that Gabriel’s horn actually has infinite surface area, that you can indeed get two copies of the original sphere (see Banach-Tarski paradox), and that the Weierstrass function is everywhere continuous but nowhere differentiable. The point being, there are many things out there which are “obvious” but actually turn out to be entirely counterintuitive and opposite what we would otherwise expect. This is the point of rigor: to double check and make sure our intuition is indeed correct, because it isn’t always.

Attribution
Source : Link , Question Author : x.y.z… , Answer Author : Kaj Hansen

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