## UNIVERSITY OF PENNSYLVANIA

## Department of Mathematics

Fall 2002 -Hans Rademacher Lectures in Mathematics## Department of Mathematics, The Ohio State University

Harvey M. Friedman

will deliver four lectures on

## Demonstrably Necessary Uses of Abstraction

Abstract:There are many familiar theorems whose proofs use methods which are in some appropriate sense substantially more "abstract" than its statement. Some particularly well known examples come from the use of complex variables in number theory. Sometimes such abstraction can be removed - for example by the "elementary proof of the prime number theorem" - and sometimes no appropriate removal is known. The interest in removing abstraction typically varies, with no agreed upon criteria for appropriateness. E.g., the removal might sacrifice naturalness or intelligibility, or the result of the removal criticized as being merely a thinly disguised form of the original.These Rademacher lectures focus on cases of demonstrable unremovability of abstraction, primarily (but not solely) in the context of discrete mathematics. These cases rely on a sharp, fully formalized, criteria for removal, where a proof of unremovability has been found. The issue of "natural" removal is finessed, as there is no removal, natural or otherwise.

More specifically, in each case we begin with a theorem whose known proofs use methods that are unexpectedly abstract relative to its statement. Next, we delineate flexible and comprehensive methods of lower abstraction. Then we present the result that the original theorem cannot be proved using only these methods of lower abstraction.

"Demonstrably Necessary Uses of Abstraction"

Tuesday....September 17, 2002....4:30pm

"Polynomials, Termination, Hilbert Bases, Degrees"

Wednesday....September 18, 2002....4:30pm

"Comparison of Blocks, Trees, Graphs, Countable Pointsets"

Thursday....September 19, 2002....4:30pm

"Borel Diagonalization, Borel Selection, Boolean Relation Theory"

Friday....September 20, 2002....4:00pm

All lectures will be held in room A-6 of the David Rittenhouse Laboratory,

corner of 33rd and Walnut Streets, Philadelphia, PA.

Tea: 4E17 David Rittenhouse Laboratory, preceding the lectures at 4:00pm. Tea on Friday will be at 3:30pmFor further information, please call the Department of Mathematics at the University of Pennsylvania - 215-898-8627.