Difference between rigorous proof and intutive understanding

Question

As the title suggests I am confused between what arguments will qualify a explanation as a proof and when does the intuition betrays us. Here is the question that made me think about this:

On a certain planet Tau Cetus more than half of its land is dry. Prove that a tunnel can be dug straight through center of the planet (assuming their technology is sufficiently advanced) starting and ending at dry lands.

Isn't it obvious? My main motive is too understand proofs and methods of arguments better.

Answer 1

On one extreme, you have "it is obvious that". On the other, you have formal logic where a statement is proved to be true if and only if you write a sequence of special symbols obeying the rules of the logic system. (Let's ignore Godelian incompleteness ideas, interesting and ultimately important though they are.)

The question is whether or not you know it's possible to expand on your explanation to reduce it to a more detailed logical idea that you can reduce to another idea... to end up with a purely logical argument at the end of the day. If you don't know how to do that without relying on statements that "are just obvious" (although it's okay to rely on a theorem somebody else proved), then you haven't proved the statement and you might be wrong!

There are huge numbers of paradoxes in mathematics which arise precisely from just asserting that something is "obvious" without understanding how to build it out of elementary ideas. A simple example is the Monty Hall problem.

---

As you know, your example has a sensible proof, given the natural assumptions one makes in interpreting the words in the problem:

If they can't, there's ocean opposite all the land, so there's at least as much ocean as land, a contradiction.

This is hugely better than "it's obvious", but easy to say, so say it instead! The reason why it's better is that it is easy to expand upon. A more clear version of the same argument would be:

Suppose there are no possible places for tunnels. Then let $A_L$ be the area of the land. We know that there is ocean opposite all of this area, so the area of the ocean is $A_O \ge A_L$. But then the total area of the sphere is $A = A_L + A_O \ge 2A_L$ contradicting the assumption that $A_L > A/2$.

An even more formal version (I don't know if you know about integration, especially on curved spaces, but it doesn't matter too much if you don't -- think of it as formalizing the idea of area) might be something like:

Define a function $f: \text{planet} \to \{0,1\}$ taking the value $0$ on the ocean and $1$ on the land. We assume that it is an integrable function, and that $A_L = \int f > \frac{1}{2}\int 1 = \frac{1}{2}A$. Define $g(\mathbf{x}) = f(\mathbf{x}')$ to be the nature of the land at the point $\mathbf{x}'$ opposite $\mathbf{x}$. We want to prove that $f(\mathbf{x}) = g(\mathbf{x}) = 1$ for some $\mathbf{x}$. We need to assume that $\int f = \int g$; i.e. it doesn't matter whether you count up the area near you or opposite you as you go around the planet.

Now $\int (f + g) = 2\int f > A = \int 1$. But $\int (f + g - 1) > 0$ is impossible unless $f + g - 1 > 0$ somewhere, which means that $f(\mathbf{x}) = g(\mathbf{x}) = 1$ somewhere.

This relies on very simple standard theorems about integration and nothing else.

---

Notice that in the formal presentation, a hidden assumption that is needed for the computation becomes clearer: we need area to mean the same thing on the 'opposite' side of the planet. This is the odd-looking $\int f = \int g$ assumption. Here's a stupid counterexample (in cross-section/one dimension fewer) that exploits this otherwise:

If you have a really big hill, then it has lots of surface area but is opposite only a small amount of ocean, which means that you can avoid being able to build a tunnel.

Of course, now someone points this out it's obvious (and mathematicians get really good at revising their intuition to cope with new discoveries like this), but formalizing it more points you much to understand what you are saying much more carefully. Really, the original problem isn't correct as stated! We need to assume that the planet's surface has a symmetry under inversion through the center.

This is the power of formalizing your arguments. You will often discover things you overlooked. And they're not always as easy to understand as these examples!

Answer 2

You are asking one of the hardest questions of mathematics. Much about mathematics is seeing the reason for something to be true and then carefully writing it out, because very often the sheer complexity of a problem makes you overlook certain things that could go wrong. There are many statements that are obvious in two dimensional euclidean geometry that just fail to hold in three or four dimensions. A proof should be like a mechanical recipe to verify that the intuition is right. However, a proof need not be a mechanical string of formal symbols: If the explanation is worded carefully with unambiguous words, it can often suffice, because it can easily be translated to formal symbols by anyone who would want that.

With that said, the point of a proof is to make other people agree with you that your reasoning is correct. How precise and detailed you have to be depends a lot on who the reader is! Researchers sometimes state that things are obvious that would take weeks for a student to prove, because they have seen the problem before and/or just feel very confident that they can write it out. Conversely, when you prove something in a course, it is important to ground every line of reasoning in something related to the course or its prerequisites.