Mathematics

The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.

“It’s magic,” the chief cook concluded, in awe.
“No, not magic,” the ship’s doctor replied. “It’s much more. It’s mathematics.”

The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.