Intuition is not proof

I’ve been reading Amir D. Aczel’s The Mystery of the Aleph, a popular math book recounting Georg Cantor‘s work on “transfinite” numbers, which are (loosely speaking) numbers which represent various sizes of infinity.

Georg Cantor’s story is a fascinating (and somewhat sad) one in its own right, but in this post I want to emphasise an important point: Intuition is not proof. Just because something seems to make sense doesn’t mean that it is true. (In a similar vein, just because something doesn’t make sense doesn’t mean it’s false! Cantor, upon discovering that there is a 1-to-1 correspondence between points on the unit line segment and all the points in an n-dimensional cube, wrote to Dedekind that “I see it, but I don’t believe it!”)

I had a math professor who used to say that “You haven’t solved the problem if you can’t write down the solution.” And indeed, there were many times when, working on a problem, I went out for a walk and came up with an idea in my head, only to find some flaw when I began writing out the details.

Here are some counterintuitive results in math that I can think of that drive home this message. Do contribute others if you can think of any!

1. Euclid’s parallel postulate (also Euclid’s fifth postulate) states that in 2-dimensional geometry, “if a line segment intersects 2 straight lines forming 2 interior angles on the same side that sum to less than 180^\circ, then the 2 lines, if extended indefinitely, meet on that side on which the angles sum to less than 2 right angles”. For many years (centuries!) it was thought that this postulate could be derived from the other 4 postulates as it seemed a lot more complicated than the rest. However, it turns out that that’s not the case: there are consistent geometries where the 4 postulates hold and the fifth doesn’t! (These are known as “non-Euclidean geometries“.)

2. Let’s say I have a circle and n points on its circumference. What is the number of pieces the circle is cut into if we join the n points pairwise, and no 3 of these lines are concurrent? (This is also known as Moser’s Circle Problem.) By trying small cases of n, the first few values we obtain are 1, 2, 4, 8, 16, \dots. We may be tempted to then infer that the answer to the problem is 2^{n-1}. False! The answer is actually \displaystyle\binom{n}{4} + \binom{n}{2} + 1.

3. The Monty Hall Problem shows how it is very difficult to get probability right. It is notable that several highly educated people argued against the correct solution even though the solution is easily obtained by drawing out the probability tree for the game.

4. Cantor proved that there are “as many” rational numbers as there are integers, in that we can define a 1-to-1 correspondence between rational numbers and integers. This is counterintuitive since the integers are a proper subset of the rational numbers!

5. In 1924, Stefan Banach and Alfred Tarski proved the Banach-Tarski paradox which states that given a solid sphere in \mathbb{R}^3, one can “decompose” the sphere into a finite number of disjoint subsets which can then be put back together in a different way to give 2 identical copies of the original sphere! (The proof crucially depends on the Axiom of Choice, which seems quite self-evident.)

6.The Curse of Dimensionality“. In some sense, when the dimension n is large, nearly all of n-dimensional space is “far away” from the centre. More specifically, “space” can be thought of as a hypercube while “the centre” can be thought of as the hypersphere inscribed inside the hypercube. The larger the volume of the hypersphere is compared to that of the hypercube, the “closer” points are to the centre. It can be shown that as n \rightarrow \infty,

\displaystyle\frac{\text{volume of hypersphere}}{\text{volume of hypercube}} \rightarrow 0.

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