One of the best tools you can learn in logic is the power of contradictions. You may think contradictions are illogical, but that is exactly the point. Yes, you can use this to prove something false. This means that its opposite is true. This is a true dichotomy, meaning that if one is true, then the other is false. The ability to entertain and show the flaws in your opponents arguments is the best tool to prove your own. The ancient Greeks invented this logical tool to help them solve philosophical matters. This is known by many names, Reductio Ad Absurdum and Proof by Contradiction. I will share with you two of my favorite stories.
Back in the day in Athens, a bunch of dank philosophy bros got together to philosophize. They had a school run by this guy named Plato called the Academy. Plato was thinking real hard about what a man is. He was like, “man is a featherless biped”. This just means something that is featherless and walks on two legs. The students thought he was so clever. Now get this, a homeless man walked in off the street with a freshly plucked cock(the bird). The homeless dude was like, “Here is Plato’s man”. Was plato going to really say this featherless chicken was a man? Obviously not. This represents a contradiction.
There was a cult of mathematicians in ancient Greece, the Pythagoreans. Their leader, Pythagoras, was an excellent mathematician and the popular Pythagorean Theorem you learn in school is named after him. The pythagorean theorem. A squared plus B squared equals C squared. So back in the day they thought of numbers like this. You got 1, 2, 13, 7; These are called the whole numbers. Then you can divide them like ½ or ⅔; These are called fractions or ratios. So the Pythagoreans were interested in this idea called triples. The basic example: 3 squared plus 4 squared equals 5 squared. So you can have these triangles with perfect whole numbers. But then there were other ones which didn’t appear to have a solution. The most basic was when both the smaller sides, also called legs were equal to each other. In this case what was the larger side, also called the hypotenuse equal to?
There was a man who proved that the hypotenuse of a right triangle with two legs equal to each other did not have a numerical solution. There was no number that you could write that would solve for the length of that side. This presented a massive flaw in their understanding of mathematics. How can we create a physical shape which has no numerical solution. They decided to throw him overboard one day while at sea. This was his punishment for ruining mathematics in their mind. Today, we recognize his contributions for the creation of irrational numbers, meaning they can’t be written as a ratio. For visual sake, I have created a really cool demo that will walk you through the logic of the discovery. This will give you a feel for what their thinking was.