Theorem states that for no integer are there positive integer values of x, y, and z such that . Before Andrew Wiles proved this for all in 1995, it had been proved for many values of n. These proofs inspired further research in the area, and formed a quasi-empirical consensus for the theorem. Even though such knowledge is more conjectural than a strictly proved theorem, it was still used in developing other mathematical ideas. The Quine–Putnam indispensability argument has been extremely influential in the philosophy of mathematics, inspiring continued debate and development of the argument in contemporary philosophy of mathematics. According to the Stanford Encyclopedia of Philosophy, many in the field consider it the best argument for mathematical realism. Prominent counterarguments come from Hartry Field, who argues that mathematics is not indispensable to science, and Penelope Maddy and Elliott Sober, who dispute whether we are committed to mathematical realism even if it is indispensable to science. Mathematics and computer science Putnam has contributed to scientific fields not directly related to his work in philosophy. As a mathematician, he contributed to the resolution of Hilbert's tenth problem in mathematics. This problem (now known as Matiyasevich's theorem or the MRDP theorem) was settled by Yuri Matiyasevich in 1970, with a proof that relied heavily on previous research by Putnam, Julia Robinson and Martin Davis. In computability theory, Putnam investigated the structure of the ramified analytical hierarchy, its connection with the constructible hierarchy and its Turing degrees. He showed that there are many levels of the constructible hierarchy that add no subsets of the integers. Later, with his student George Boolos, he showed that the first such "non-index" is the ordinal of ramified analysis (this is the smallest such that is a model of full second-order comprehension). Also, together with a separate paper with his student Richard Boyd and Gustav Hensel, he demonstrated how the Davis–Mostowski–Kleene hyperarithmetical hierarchy of arithmetical degrees can be naturally extended up to . In computer science, Putnam is known for the Davis–Putnam algorithm for the Boolean satisfiability problem (SAT), developed with Martin Davis in 1960. The algorithm finds whether there is a set of true or false values that satisfies a given Boolean expression so that the entire expression becomes true. In 1962, they further refined the algorithm with the help of George Logemann and Donald W. Loveland. It became known as the DPLL algorithm. It is efficient