Contradictions Learned at School by Orlando Bartro
“Miseducation at School?”
The school year is approaching its end, a good time to reflect on all the ways we’ve been miseducated.
Aristotle said in his Prior Analytics, “You can’t believe that something is true and not true at the same time.”
But I disagree with Aristotle.
All of us (or almost all of us) believe contradictory things.
Why?
We believe contradictory things because we’ve learned contradictory things at school, and we believe what we’ve learned at school on trust. We usually don’t analyze these beliefs to determine if they are contradictory or not.
For example, we all learned at school that 0 + 0 = 0.
That seems easy and obvious. Nothing added to nothing equals nothing.
We also learned that zero added to zero as many times as you wish will always still equal zero.
That is, nothing added to nothing even a zillion times will still be nothing. Simple!
However, we also learned that a line is composed of points, and that a point has zero width.
Just a moment, please!
If a point has zero width, then how can points be added together to make a line that is anything longer than zero length?
Aristotle made this objection to the explanation for the construction of a line more than two thousand years ago.
He said that you can’t make a line by adding points together.
Henri Poincaré
And thus, we arrive at a deep mystery of mathematics.
Non-mathematicians are usually unaware that mathematics doesn’t always result in undisputed answers.
The great Poincaré would agree with Aristotle. He’d say that points of zero width can’t be added to make a line.
But the great Hilbert disagrees. He says that an uncountable number of points of zero width can be added to make a line of any length you wish.
David Hilbert
This dispute between Poincaré and Hilbert leads to a discussion of the correctness or incorrectness of Cantor’s famous diagonal proof, and to the truth or untruth of the Axiom of Choice, and to the validity or invalidity of the concept of a “completed infinity.”
But this dispute is left for the readers of this brief article who are curious enough to search these issues through the internet.
The lesson of this little article is only this: Don’t be so sure of what you know because you might have been miseducated.
Happy school year!
* Orlando Bartro is the author of Toward Two Words, a comical novel about a man who finds yet another woman he never knew, available at Amazon. He is currently writing two new novels and a play.