One of the laziest last resorts of internet dullards is the argument that "you can't prove something does not exist" or even simply that "you can't prove a negative".
There are two reasons to hate this: because it's just plain wrong, and because, even if it were true, it's just plain dull. There's no real point wasting bandwidth on the second point ... so I'm going to out of spite. I'm not even going to idly refer people to Russell's teapot. How's that for decadence?
Why is this dull? Because it leads nowhere. It's the intellectual equivalent of jogging on a machine. Delving into the logic may exercise the brain muscles, but at the end of the day you're still wearing lycra and breathing other people's sweat when you could be out for a nice stroll in the countryside. I broke my metaphor. Sorry. What I'm saying is that this is an argument for the sake of an argument. It's a distraction away from whatever the original discussion was. You've stopped making progress towards anything new and resorted to trampling over old ground where it's easier to score imaginary debating points without actually thinking.
Why is it wrong?
A rational expression for Pi does not exist.
It's going to come as a tremendous disappointment to messers Lambert and Legendre to discover that their proofs of this statement are "impossible". Or it would be a disappointment if they hadn't both been dead for a couple of centuries.
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