Series: Penn State Logic Seminar Date: Tuesday, June 14, 2005 Time: 2:30 - 3:45 PM Place: 123 Pond Laboratory Speaker: Andrew Arana, Philosophy, Kansas State University Title: Purity of Methods Abstract: The prime number theorem (PNT) says that the number of primes less than n is approximated by n / ln n. This astonishing result was proved by Jacques Hadamard in 1896 using complex analysis. Nevertheless, mathematicians continued to seek new proofs of the PNT, particularly looking for one that avoided use of complex analysis. Such a proof (called "elementary") was found in 1949, by Atle Selberg and Paul Erdös. In 1950 Selberg won the Fields Medal, mathematics' highest prize, in part for his elementary proof of the PNT. Now, mathematicians widely agree that there is nothing "problematic" about complex analysis, at least not any more so than other areas of mainstream mathematics. But then why did mathematicians look for a proof of the PNT that avoided complex analysis, and why did they lavish the praise that they did when such proofs were found? In my talk I will try to shed some light on these questions. The desire to find proofs that avoid appeal to "foreign" notions goes back to the dawn of mathematics in ancient Greece, and has never let up. Today we call proofs like these "pure". It turns out that there are sound reasons for valuing pure proofs, though there are also sound reasons for valuing impure proofs like Hadamard's of the PNT. What is called for is a finer-grained analysis of what kinds of value a proof can have than we have been accustomed to, although one can find inklings of these finer analyses in the writings of mathematicians. By making these analyses much more explicit and precise, I hope to help mathematicians become more conscious of how their methodological choices impact their practice. My finer analysis of the value of purity will make use of work from mathematical logic. In particular, I will discuss the question of whether pure proofs are more complex than impure proofs, as is often claimed. Work of Avigad, Hajek, Pudlak, and Ignjatovic sheds light on this question, when considered within the framework of reverse mathematics. Time permitting, I will also discuss cases in which purity appears to be impossible to achieve, and discuss their implications.