Death by Proof By Contradiction

In my first post, I talked a bit about Pythagoras and his school.  One important feature of Pythagoras's belief system was that he believed that the natural world was composed entirely of positive whole numbers and ratios of such numbers (called "rational numbers" because they can be expressed as ratios).  This belief was as fundamental to the Pythagoreans as the notion of a benevolent creator is to modern Christianity.

Because of their belief that numbers defined the natural world, Pythagoras and his students put a lot of time and energy into studying the properties of numbers and operations on numbers.  (They also attributed various properties, such as masculine and feminine, to different numbers, but that's not important to this story).  Their work was based in part on real-world observations ("square" numbers derive their name from the Pythagorean observation that certain numbers of stones can be used to make perfect squares in the sand) and they wanted to use their work to understand the workings of the real world.

One day, a student by the name of Hippasus was attempting to discover the rational number that would measure the diagonal of a square whose sides were equal to 1 unit ("unit" can be any unit of measurement you like, the argument will still work the same).  The Pythagoreans knew that a square's diagonal cut the square into two identical right triangles, with both legs equal to 1 in this case and the diagonal being the unknown.

Hippasus assumed, as any good student of Pythagoras would, that the measurement of the diagonal would be a rational number of units.  We'll call it p/q units, where p and q are both positive real numbers that share no common factors with each other (this is allowed because if they shared common factors, we could cancel them and get a rational number u/v, which equals p/q but doesn't have common factors floating around to complicate things, and we would use that one).

By the Pythagorean Theorem, Hippasus was then able to say that 1²+1²=p²/q².

This simplifies (through some basic arithmetic) to 1+1= p²/q², or 2=p²/q².

Multiplying both sides by q² gives 2q²=p².

Recall that p and q are both whole numbers. This means that p² and q² are also whole numbers, since any product of whole numbers is a whole number. Then 2q² is an even whole number, since twice any whole number is even. That tells us that p² is even.

A product of two odd numbers is odd, so p itself can't be odd. It must be even.

Since p is even, let's write it as 2k, where k is also a whole number.

Then p²=(2k)² (by squaring both sides of the equation p=2k). Distributing the exponent on the right side tells us that p²=4k².

But we said earlier that  2q²=p², so we know have 2q²=4k².

Divide both sides by two, and we get  q²=2k².  Now by the same reasoning we used before to show that p was even, we can see that q is even.  But we had said (without breaking any rules of math) that p and q have no common factors, and now we have shown that they are both divisible by two.  The only assumption we made without justification was that our hypotenuse consisted of a rational number of units (or in modern parlance, that the square root of two is rational), and so this assumption must be wrong.

By essentially this same method (I updated notation and language to make it more accessible to modern readers) Hippasus had proven that one of the central tenets of Pythagoras's belief system was false by demonstrating that the diagonal of a square was incommensurable with its sides (sharing no common measure; this is how they would have phrased our statement that the diagonal is irrational), thus showing that there is some part of the natural world which cannot be described as a ratio of whole numbers under any measuring system.

Remember that this had not been Hippasus's intention.  He was simply trying to find the correct ratio, and ended up proving that no such ratio exists (it is not uncommon in mathematics for investigations in one direction lead to astonishing unexpected discoveries).  Unfortunately, that did not soften the response from the other Pythagoreans.  Like many people throughout history, they did not take kindly to having their belief system uprooted with a few strokes of a pen.  Accounts vary about how they treated him.  The kindest claim that they kicked him out, erected a tombstone for him, and never spoke to him again, while other sources claim they took him out to sea and threw him overboard.  In any case, he was punished for his discovery.