Benjamin Peirce (father of Charles Sanders Peirce) on induction, deduction, and the role of math in the scientific process:
Observation supplies fact. Induction ascends from fact to law. Deduction, by applying the pure logic of mathematics, reverses the process and descends from law to fact. The facts of observation are liable to the uncertainties and inaccuracies of the human senses; and the first inductions of law are rough approximations to the truth. The law is freed from the defects of observation and converted by the speculations of the geometer into exact form. But it has ceased to be pure induction, and has become ideal hypothesis. Deductions are made from it with syllogistic precision, and consequent facts are logically evolved without immediate reference to the actual events of Nature. If the results of computation coincide, not merely qualitatively but quantitatively, with observation, the law is established as a reality, and is restored to the domain of induction.
Via The Metaphysical Club, p. 155-156. The original text is from 1881.