### Reductio ad Absurdum

A math professor silently steps up to the blackboard and writes this formula for the students to contemplate...

a² + b² = c²

After giving them a minute or two to absorb the formula, the professor speaks.

"Who can tell me what that is?"

A smart alecky kid with red hair pipes up, "It's the precursor to Fermat's Last Theorem."

The professor rolls his eyes and waits for the one of the more neuro-typical kids to answer.

"Um, it's the Pythagorean Theorem?" ventures one of the other students.

"Very good," responds the prof. "Is it known to be true?"

More than half the class nod their heads yes.

"How do we know it's true?" prods the prof. "Is it true because Pythagoras said so?"

One of the brighter students says, "It is true that Pythagoras said so, but it's not true because Pythagoras said so. We need a genuine proof that stands on its own."

"Very good," responds the prof. "If we had such a proof, would the theorem be established as true?"

"Doesn't the 'proof' also have to be technically correct?"

The professor smiles. "Very good. A true theorem is supported by a correct proof."

Some of the students start to tune out. A few glance longingly at their iPhones and BlackBerries, hoping for an SMS on a more thrilling topic than mathematical epistemology. But the intrepid prof is not daunted. He forges on.

"Suppose we had such a correct proof. Would you then accept the theorem as true, without exception?"

"I'm not sure how we can be sure a proof is technically correct. Wouldn't we have to consult an expert mathematician to examine the proof and declare it correct? Or is that what we're supposed to learn how to do in this course?

The professor nods. "So let's say that you learn how to critically examine a proof, or even construct one on your own. Are we then home free?"

The kid with the red hair pipes up again, "I don't see where you're going with this."

The professor picks up a globe of the Earth and hands it to the student, along with a washable magic marker.

"I want you to draw a triangle on the surface of this globe. Put one vertex at the North Pole. Put one leg of the triangle along the Greenwich meridian, down to the equator. Put another leg of the triangle from the North Pole along the 90th meridian down to the equator. Now put the base of the triangle along the equator."

The other students turn their gaze up from their BlackBerries to watch.

"Is it a right triangle?"

"All three vertices are right angles. And all three legs are the same length."

"So what's wrong here?"

"The surface of a globe is not a Euclidean plane," says the redheaded kid. What we learned about triangles only applies to triangles on a flat surface. The surface of the globe is not a flat surface."

"Very good. What else do we find on the surface of the Earth that we don't find in Euclid's Plane Geometry?"

"Politics," says one of the students, just as the bell rings to end the class.

"Who can tell me what that is?"

A smart alecky kid with red hair pipes up, "It's the precursor to Fermat's Last Theorem."

The professor rolls his eyes and waits for the one of the more neuro-typical kids to answer.

"Um, it's the Pythagorean Theorem?" ventures one of the other students.

"Very good," responds the prof. "Is it known to be true?"

More than half the class nod their heads yes.

"How do we know it's true?" prods the prof. "Is it true because Pythagoras said so?"

One of the brighter students says, "It is true that Pythagoras said so, but it's not true because Pythagoras said so. We need a genuine proof that stands on its own."

"Very good," responds the prof. "If we had such a proof, would the theorem be established as true?"

"Doesn't the 'proof' also have to be technically correct?"

The professor smiles. "Very good. A true theorem is supported by a correct proof."

Some of the students start to tune out. A few glance longingly at their iPhones and BlackBerries, hoping for an SMS on a more thrilling topic than mathematical epistemology. But the intrepid prof is not daunted. He forges on.

"Suppose we had such a correct proof. Would you then accept the theorem as true, without exception?"

"I'm not sure how we can be sure a proof is technically correct. Wouldn't we have to consult an expert mathematician to examine the proof and declare it correct? Or is that what we're supposed to learn how to do in this course?

The professor nods. "So let's say that you learn how to critically examine a proof, or even construct one on your own. Are we then home free?"

The kid with the red hair pipes up again, "I don't see where you're going with this."

The professor picks up a globe of the Earth and hands it to the student, along with a washable magic marker.

"I want you to draw a triangle on the surface of this globe. Put one vertex at the North Pole. Put one leg of the triangle along the Greenwich meridian, down to the equator. Put another leg of the triangle from the North Pole along the 90th meridian down to the equator. Now put the base of the triangle along the equator."

The other students turn their gaze up from their BlackBerries to watch.

"Is it a right triangle?"

"All three vertices are right angles. And all three legs are the same length."

"So what's wrong here?"

"The surface of a globe is not a Euclidean plane," says the redheaded kid. What we learned about triangles only applies to triangles on a flat surface. The surface of the globe is not a flat surface."

"Very good. What else do we find on the surface of the Earth that we don't find in Euclid's Plane Geometry?"

"Politics," says one of the students, just as the bell rings to end the class.

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