It has been said that mathematics is true because it is God-given. Mathematics is true because man has constructed it. Mathematics is true because it is nothing but logic, and what is logical must be true. Mathematics is true because it is tautological. Is is true because it is proved. It is true because it is constructed; its fabric is knit from its axioms as a sweater is knit from a length of yarn. It is true in the way that the rules and subsequent moves of a game are true. It is true because it is beautiful, because it is coherent. Mathematics is true because it is useful. Mathematics is true because it has been elicited in such a way that it reflects accurately the phenomena of the real world.
Mathematics is true 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, like all knowledge, it is based upon tacit understanding. Mathematics is true because there are numerous independent but supportive avenues to its kind of knowledge which are constantly being reconciled.
It has also been said that mathematics isn't true at all in a rock-bottom sense, it is true only in a probabilistic sense. Mathematics is true only in the sense that it is refutable and corrigible; its truths are eternally provisional. Mathematical truth is not a condition, it is a process. Truth is an idle notion, to mathematics as to all else. Walk away from it with Pilate.
Rattling off this list in a rat-a-tat-tat fashion has very likely induced some vertigo in the minds of readers and a feeling that chaos muse prevail in this most fundamental question of this most fundamental field. But the chaos is something that only philosophers of mathematics contend with. The majority of mathematicians hardly worry about it at all and often regard philosophical speculation with disdain or amusement...
— из статьи: Philip J. Davis, When Mathematics Says No, Mathematics Magazine 59/1986, p.70.
Сама статья, впрочем, довольно слабая и малоинтересная, просто это перечисление понравилось.
P.S. Не хватает варианта "Математика верна, потому что всесильна" ;)