I have an abiding (if amateur) interest in mathematics, one reason being that (in all modesty) I am good at it.
While my interest a decade back was in problem-solving, I find I am unable to concentrate hard enough to dispatch the problems that would have hardly teased me back then. It’s a worrying trend, and one reason why I want to do the Ironmind.
But I digress. My interest now is primarily in what I would call the ‘philosophy of mathematics’ and this post is about a very interesting proof to an interesting theorem. I first encountered this theorem in high school mathematics, so it’s hardly the dizzying heights of mathematics.
The problem is simple enough to state: Prove that the product of ‘r’ consecutive numbers is divisible by r! (r factorial).
For example, if we pick 5 numbers 83, 84, 85, 86 and 87, we need to prove that their product (83x84x85x86x87) is divisible by 5! (1x2x3x4x5).
The textbook proof to this theorem is using a double-barreled induction, once over the starting number in the sequence and once over the number of numbers in the sequence. It’s a bit tedious, but otherwise not particularly difficult.
I only state this because I want to communicate that yes, this theorem is correct, that it can be proved without any controversy.
Now onto the controversial part.
When I was in class 12, my mathematics teacher gave us a very cute proof of this theorem.
He said, consider the value:
If we can prove that this is an integer, it means the numerator is divisible by the denominator.
Now for a moment, let’s consider a different problem. Say we have 87 people in a room, and we need to pick 5 of them for a lunch treat. How many ways do we have to do this?
We can pick the first person in 87 ways, the second in 86, the third in 85, the fourth in 84 and the fifth in 83 ways. But if we do that, we’ll have repetitions because the order in which we pick people is not important. How many repetitions will we have? This is the number of ways in which we can arrange 5 people, which is 5 times 4 times 3 times 2 times 1.
So the effective number, which is known in high-school mathematics as 87 C 5, or in notation form,
It’s the same number!
And because it represents the number of ways in which we do something, it has to be an integer. You can’t have 8 and a half ways of choosing people, right?
So it’s a neat little proof.
For a very long time since I heard this proof, there was something gnawing at the back of my head about it. I felt there was something a bit spooky about this proof, but I couldn’t put my finger on it.
And then I realized something weird.
This is a different proof because it is an experimental proof of a theorem.
Can you believe that? An experimental proof of a mathematical theorem.
The proof doesn’t base itself on the underlying axioms of mathematics.
It bases itself on the underlying realities of our universe.
The final statement in the proof – the climax – is not that this theorem is the last link in a chain of deductions reasoned from a set of axioms using a set of logical steps.
The climax in the proof is: this number represents a countable quantity in the real world, so it must be a countable number.
Which is very, very weird.
What do you think?