Purity of Methods

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.