Consider this mathematician, with her standard-issue infinitely sharp knife and a perfect ball.
She frantically slices and distributes the ball into an infinite number of boxes.
She then recombines the parts into five precise sections.
Gently moving and rotating these sections around, seemingly impossibly, she recombines them to form two identical, flawless, and complete copies of the original ball.
This is a result known in mathematics as the Banach-Tarski paradox.
The paradox here is not in the logic or the proof— which are, like the balls, flawless— but instead in the tension between mathematics and our own experience of reality.
And in this tension lives some beautiful and fundamental truths about what mathematics actually is.
We'll come back to that in a moment, but first, we need to examine the foundation of every mathematical system: axioms.
Every mathematical system is built and advanced by using logic to reach new conclusions.
But logic can't be applied to nothing; we have to start with some basic statements, called axioms, that we declare to be true, and make deductions from there.
