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.
posted by Moulton | 6:29 AM
1 Comments:
Sam Chan pasted this blog post into ChatGPT, requesting a poem:
«A math professor stood with a grin,
Ready to teach the class, let's begin!
He wrote a formula on the blackboard high,
A puzzle for the students, he let out a sigh.
"a squared plus b squared equals c squared,"
He challenged their minds, left them undeterred.
The smart kid with red hair, oh so bright,
Claimed, "It's Fermat's Last Theorem, right?"
The professor rolled his eyes, not impressed,
He wanted a different answer, no jest.
The other students pondered, one took a chance,
"It's the Pythagorean Theorem, at first glance!"
"Very good," said the prof, filled with glee,
"But tell me, my students, is it true, you see?
Do we trust it blindly, for Pythagoras' sake?
Or do we seek a proof, one we can make?"
A bright student raised their hand in the air,
"It's true Pythagoras said it, that's fair.
But a genuine proof we need to find,
To stand on its own and leave no doubt behind."
"Very good," the professor praised with delight,
"A true theorem needs a proof, shining bright.
But even with proof, do we truly know?
Must the proof itself pass the expert's show?"
Some students drifted, lost in their thoughts,
Daydreaming of topics that excited them lots.
But the professor continued, undeterred by the rest,
Unwavering in his quest, putting them to the test.
"Imagine we possess that perfect proof,
Reviewed and approved, nothing aloof.
Would you then accept the theorem, so grand?
Without a single exception, across the land?"
"But how can we be sure a proof's technically right?
Shouldn't we consult experts, shining their light?
Or should we learn to examine and construct,
To unlock the secrets, our minds conduct?"
The professor nodded, acknowledging the thought,
Critical examination, a skill to be sought.
"But even with your knowledge expanded,
Are we truly at the end, where understanding is handed?"
The red-haired kid questioned, unsure of the aim,
Perplexed by the professor's puzzling game.
The professor reached for a globe, round and wide,
A hands-on demonstration, taking them for a ride.
"Draw a triangle upon this Earth's sphere,
From North Pole to equator, without fear.
Follow the meridians, make it precise,
And lay the base along the equator's slice."
The students watched, their attention drawn,
Curious about the lesson soon to be drawn.
"Is it a right triangle?" the professor inquired,
With all vertices right, their interest was fired.
The red-haired kid spoke up, a voice so clear,
"The Earth's surface isn't flat, that's what I fear.
Our triangle's truth, it cannot hold,
On a curved world, different stories are told."
"Very good," the professor exclaimed with a smile,
"You've discovered a truth, going the extra mile.
On Earth's surface, much more we find,
Not bound by Euclid, it's a different kind."
"Politics!" one student cheekily chimed,
As the bell rang, ending the class in its prime.
The professor chuckled, knowing they understood,
In life, not all truths fit neatly as they should.
So they packed their bags, with knowledge in their minds,
Ready to explore, where truths intertwine.
For the world is diverse, with lessons yet learned,
And the journey of knowledge, forever concerned.»
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