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The Comedy of Mathematics
  • Maths and Sciences
Melissa Neacsu
“But I don’t want to go among mad people," Alice remarked.
"Oh, you can’t help that," said the Cat: "we’re all mad here. I’m mad. You’re mad."
"How do you know I’m mad?" said Alice.
"You must be," said the Cat, "or you wouldn’t have come here.”
― Lewis Carroll, Alice in Wonderland
Math is the funniest subject. Its austere presentation and rigid rules are the nesting ground for the most satisfying of shenanigans. Who doesn’t love to point out the flaws in a tyrannical system and “stick it to The Man?” Who doesn’t feel a wee bit gratifyingly superior when they realize that what is taken as absolute certainty relies on a seemingly arbitrary set of assumptions? And what is humor if not revealing the absurd in an accepted state of affairs? Secretly we are all rebels because we crave Truth.
The madness and absurdity of mathematics lies in the fact that it is founded on what seem to be arbitrary definitions, notations, and unproven assumptions. We must “begin at the beginning,” and the beginning of mathematics is language, experience, and axiomatic proposition. Some would argue the necessity of experience, but I think this is inescapable. Students are taught that the square root of 4 is 2. But why not -2, since (-2) also squares to 4? Why is the inverse cosine of 0 equal to 90 degrees and not -90 degrees? I know this has kept you up at night.
The truth that I love to reveal to my students is that these are both examples of, to a large extent, arbitrary assignment and notational choice in very much the same way we say “let i be the number whose square is -1.” They are exploratory “what if” propositions similar to choosing a seed in Minecraft and seeing what strange new world emerges according to the rules of the algorithm. The goal is often the creation of the most elegant world possible.
In Geometry, we allow Euclid to postulate that “the whole is greater than the part,” that the piece of pie is always less than the whole pie. In Algebra II, I explain why the set of integers {0, ±1,±2, ±3,…} has the same size as the set of natural numbers {1, 2, 3, …}. The whole is equal to the part! Again, in Euclidean geometry, we prove that the angles in a triangle always sum to two right angles, but in the College we discuss spaces in which this angle sum is always less than two right angles. The mad mathematical world in which we find ourselves is a reflection of our underlying assumptions (What is equality? What is a line?) and is not always intuitive.
This makes mathematics the funniest of subjects, the sheep in wolf’s clothing. My dear child, did you accidentally write that 9+6=12? You are correct in some mathematical world and should explain this to your teacher (you were working modulo 3, for example). We explorers of mathematical worlds may―nay, must!―take the liberty of being cheeky when the computational landscape becomes too drab and rote. Things are not as black and white as they seem, and mental rigor requires a release of tension. A straight line may be the “shortest distance” between two points, but as Victor Borge puts it, “laughter is the closest distance between two people.”