The Unreasonable Effectiveness of Mathematics in the Natural
"What is Mathematics?
. . . I would say that mathematics is the
science of skillful operations with concepts and rules invented
just for this purpose. The principal emphasis is on the
invention of concepts."1a
. . . [T]he mathematician could
formulate only a handful of interesting theorems without defining
concepts beyond those contained in the axioms and
that the concepts outside those contained in the axioms are
defined with a view of permitting ingenious logical operations
which appeal to our aesthetic sense both as operations and
also in their results of great generality and simplicity."1b
is Physics? The physicist is interested in discovering the
laws of inanimate nature. . .
It is, as Schrödinger has
remarked, a miracle that in spite of the baffling complexity of
the world, certain regularities in the events could be
discovered. . . The laws of nature are concerned with such
regularities. . . This property of the regularity is a
recognized invariance property . . ."1c
empirical law has the disquieting quality that one does not know
. . .
we do not know why our theories work so well. Hence
their accuracy may not prove their truth and consistency."1e
"The miracle of the appropriateness of
the language of mathematics for the formulation of the
laws of physics is a wonderful gift which we neither
understand nor deserve."1f
Italics in the original.
Eugene P. Wigner.
Effectiveness of Mathematics in the Natural Sciences.
Richard Courant Lecture in Mathematical Sciences delivered at New
York University, 11-May-1959. Communications on Pure and Applied
Mathematics, Vol. XIII, 001-14 (1960). New York, NY: John Wiley &
Sons, Inc., 1960.