I first learnt about Plato’s allegory of the cave when I was in senior high school. A mathematics and English nerd – a strange combination – I played cello and wrote short stories in my spare time. I knew a bit about philosophy and was taking a survey class in the humanities, but Plato’s theory of ideal forms arrived as a revelation: this notion that we could experience a shadow-play of a reality that was nonetheless eternal and immutable.
Somewhere out there was a perfect circle; all the other circles we could see were pale copies of this single Circle, dust and ashes compared with its ethereal unity. Chasing after this ideal as a young man, I studied mathematics. I could prove the number of primes to be infinite, and the square root of two to be irrational (a real number that cannot be made by dividing two whole numbers).
These statements, I was told, were true at the beginning of time and would be true at its end, long after the last mathematician vanished from the cosmos. Yet, as I churned out proofs for my doctoral coursework, the human element of mathematics began to discomfit me. My proofs seemed more like arguments than irrefutable calculations. Each rested on self-evident axioms that, while apparently true, seemed to be based on little more than consensus among mathematicians.
These problems with mathematics turned out to be well known. The mathematician and philosopher Bertrand Russell spent much of his career trying to shore up this house built on sand. His attempt was published, with his collaborator Alfred North Whitehead, in the loftily titled Principia Mathematica (1910-13) – a dense three-volume tome, in which Russell introduces the extended proof of 1 + 1 = 2 with the witticism that ‘The above proposition is occasionally useful.’ Published at the authors’ considerable expense, their work set off a chain reaction that, by the 1930s, showed mathematics to be teetering on a precipice of inconsistency and incompleteness. READ MORE...
These problems with mathematics turned out to be well known. The mathematician and philosopher Bertrand Russell spent much of his career trying to shore up this house built on sand. His attempt was published, with his collaborator Alfred North Whitehead, in the loftily titled Principia Mathematica (1910-13) – a dense three-volume tome, in which Russell introduces the extended proof of 1 + 1 = 2 with the witticism that ‘The above proposition is occasionally useful.’ Published at the authors’ considerable expense, their work set off a chain reaction that, by the 1930s, showed mathematics to be teetering on a precipice of inconsistency and incompleteness. READ MORE...
