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  • William L Culbertson

Contradicting Euclid?

Answering a question on August, 2019:

Why do high levels of maths contradict lower level maths? Why do ground rules of math get broken when you get to higher levels of maths?

Unless you had a bad teacher, what was true remains true. There are times when things might seem to change, but if you understand the bigger picture of what is going on, you will see more deeply into mathematics.

As you advance to higher levels of mathematics, you sometimes revisit a topic you covered before. This time, you’ll look at the scruffy little details that you glossed over the first time.Those general principles, introduced first, are not wrong. The comprehensive picture, however, is more complete.

One thing mathematicians often do at advanced levels is to change one of the basic rules of mathematics to see what happens. Change the rules of a game, and see what happens! Sometimes, the new rule results in a contradiction, a fundamental conflict. That shows the new rule is a mistake, a bad idea. It breaks the game. Occasionally, however, changing a basic rule creates a whole new game–a whole new area of mathematics. This is what makes mathematics really exciting!

A classic example of this in geometry is Euclid’s Fifth Postulate. This is the “rule” that parallel lines never meet. Taught in high school geometry, this rule seems to mimic our understanding of parallel lines in the real world. However, this rule bothered mathematicians. It seemed more complicated than Euclid’s first four basic rules about geometry. Therefore, they tried to prove it using only the first four rules. If they could do that, they wouldn’t need Euclid’s overcomplicated Fifth Postulate.

But they couldn’t.

So, they tried a different method, a backwards method, to prove Euclid’s Fifth Postulate—proof by by contradiction. They assumed a “wrong” Fifth Postulate, say: parallel lines will always meet. They changed a rule in the game. If they could show this changed rule led to a contradiction, if it broke the game, that would prove the original rule must have been the correct one after all. If making it false is wrong, then it must have been right to start with! If this sounds a little awkward, mathematicians don't like it either. However, it would get the job done.

Several mathematicians tried this. They got strange results—results that seemed . . . weird. At that point, they said, “See! A contradiction.” They had changed the game, but they did not understand what they saw.

Turns out that “weird” is not necessarily a contradiction. Changing Euclid’s Fifth Postulate leads to a whole series of different geometrical results—different, but not wrong. These “weird” results are perfectly provable using the altered fifth postulate. The game with the new rule is perfectly playable.

Changing Euclid’s Fifth Postulate makes what are now called non-Euclidian geometries. They are extremely useful including being necessary for Einstein’s Relativity Theory, the theory that supports our basic understanding of the universe.

For mathematics in general, just like in geometry, there are different structures that arise depending on what basic assumptions you start with. It’s like making up a game. When you change the rules, you get a different game—but it’s still a game. This is an over simplified analogy of course, but it points to the reason that mathematics is so powerful and so endlessly fascinating.

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