It is rare for someone to pose a mathematical question that elicits a variety of answers, but Philip J. Davis has done just that, commenting wryly, “There are probably more answers to this question that there are people who have though deeply about it.”
His question is this: “Why are the theorems of mathematics true?”
The following list is a synopsis of some of the common answers that Davis states may be given to this question.1
- Mathematics is true because it is God-given.
- Mathematics is true because humans have carefully constructed it; its fabric is knit from its axioms as a sweater is knit from a length of yarn.
- Mathematics is true because it is nothing but logic and what is logical must be true.
- Mathematics is true because it is tautological.
- It is true because it is proved.
- It is true in the way that the rules and the subsequent moves of a game are true.
- Mathematics is true because it is useful (or because it is beautiful, or because it is coherent).
- Mathematics is true because it has been elicited in a way that reflects accurately the phenomena of the real world.
- Mathematics is true because by agreement. It is true because we want it to be true, and whenever an offending instance is found, the mathematical community rises up, extirpates that instance and rearranges its thinking.
- Mathematics is true because it is an accurate expression of a primal, intuitive knowledge.
- Mathematics is not true at all in the rock-bottom sense. It is true only in the probabilistic sense.
- Truth is an idle notion, to mathematics as to all else. Walk away from it with Pilate.
1“When a Mathematician Says No,” Mathematics Magazine, April 1986, p. 70.
From Mathematics in a Postmodern Age. (2001). Grand Rapids, MI: Eerdmans, pg. 15-16.