Introduction: The Mathematical Image
Certainty The theorem proving the infinitude of primes seems established beyond a doubt. The natural sciences can't give us anything like this. In spite of its wonderful accomplishments, Newtonian physics has been overturned in favour of quantum mechanics and relativity. And no one today would bet too heavily on the longevity of current theories. Mathematics, by contrast, seems the one and only place where we humans can be absolutely sure we got it right.
Objectivity Whoever first thought of this theorem and its proof made a great discovery. There are other things we might be certain of, but they aren't discoveries: 'Bishops move diagonally.' This is a chess rule; it wasn't discovered; it was invented. It is certain, but its certainty stems from our resolution to play the game of chess that way. Another way of describing the situation is by saying that our theorem is an objective truth, not a convention. Yet a third way of making the same point is by saying that Martian mathematics is like ours, while their games might be quite different.