FOM: Harvey Friedman PR in campus newspaper

[Stephen G Simpson]

Here is a public relations article about Harvey Friedman's work on
Boolean relation theory.  This was published recently in ``On
Campus'', a newspaper for faculty and staff at the Ohio State
University.  The on-line version of the article is at
http://www.osu.edu/oncampus/v29n19/thisissue_5.html#friedman

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 *Friedman crusades for new axioms in mathematics*
                            
 By Melissa Weber           
                            
 Harvey Friedman -- a self-proclaimed philosophical mathematician and
 mathematical philosopher -- is approaching a peak moment in his
 30-year career.
                            
 Friedman explores mathematical logic and the foundations of
 mathematics, delving into the very core of mathematical
 reasoning. The "foundations of mathematics"field is considered highly
 interdisciplinary, crossing the lines of mathematics, philosophy, and
 now computer science.
                            
 "It's really not properly contained in either field -- mathematics or
 philosophy,"Friedman said. "It's definitely unusual for the American
 Philosophical Association to hold a three-hour symposium based on the
 work of a mathematician, as they're planning to do this year."
                            
 Friedman is a popular speaker this year. Before that December
 symposium in New York, he will speak at the end of April in Leeds,
 England, on "The Mathematical Meaning of Mathematical Logic"and in
 June he'll join a panel discussion in Illinois on the need for new
 axioms and rules in mathematics.
                            
 All during his tenure as Distinguished University Professor of
 mathematics, philosophy, computer science and music (he toyed with
 the idea of becoming a concert pianist before his fascination with
 logic took over), he has had one long-term pet project: to
 demonstrate to the mathematics community that normal mathematics
 needs new axioms.
                            
 "This project has been a preoccupation for 30 years, and it is now
 ready for a crescendo,"Friedman said.
                            
 The talks he'll be giving, he said, cover "work which may be leading
 towards a major expansion of the accepted axioms and rules of
 mathematical reasoning -- the first since the full formulation of the
 present rules in 1925."
                            
 This banner year factors into a long list of Friedman's honors, which
 includes becoming the youngest professor in recorded history (age 18,
 Stanford University), and receiving the prestigious Alan T. Waterman
 Award given annually by the National Science Foundation to a single
 scholar in all of mathematics, science and engineering.
                            
 To understand the significance of his achievement requires a
 distinctly historical perspective. Mathematics operates under
 definite axioms and rules that provide the currently accepted
 standard for rigorous proof, which dates to the late 1800s.
                            
 "These are the axioms and rules that guide mathematicians with
 absolute confidence and certainty through a maze of complex problems
 that drive modern technology, such as the development of computer
 algorithms,"Friedman said.
                            
 The assumption has always been that rigorous mathematics is
 consistent (no contradictions), and mathematicians have always held
 this on faith.
                            
 Enter Kurt Godel. In the 1930s, Godel tackled the question: Where is
 the proof that exists to show consistency?
                            
 In the most famous paper ever written in mathematical logic, Godel
 established that there is no proof within mathematics that
 mathematics is consistent. Or, more accurately, he established that
 if there is a proof within mathematics that mathematics is
 consistent, then mathematics is in fact inconsistent.
                            
 Work of Godel (1940) and Paul J. Cohen (1962) showed that a famous
 problem in abstract set theory called the continuum hypothesis
 couldn't be proved or refuted within the usual axioms and rules for
 mathematics. This created something of a sensation -- even fear -- in
 the math community, because of the widely held belief that every
 important math statement could be proved or refuted.
                            
 This incompleteness phenomenon of Godel threatened to force a change
 in the cherished and venerable axioms and rules of mathematics. But
 because of the remoteness of abstract set theory from normal
 mathematical concerns, the sensation -- as well as the fear --
 quickly died down.
                            
 "For 70 years, mathematicians have chosen to ignore Godel's
 incompleteness phenomenon,"Friedman said.
                            
 "Mathematicians continued to defend their adherence to the usual
 rules by declaring that these Godelian ideas were basically
 irrelevant philosophical conundrums. I was convinced otherwise, and
 for me this became a single-minded intellectual crusade."
                            
 He embarked on a program of establishing the necessary use of new
 axioms from abstract set theory in normal mathematical contexts --
 the kind of contexts that cannot be ignored by normal mathematicians
 doing normal mathematics.
                            
 "I'm developing what I call Boolean relation theory, and it lives in
 the integers,"he said.
                            
 "Boolean relation theory is a very simple basic theory involving
 sets, transformations and Venn diagrams, which is readily accessible
 at the math undergraduate level,"he said.
                            
 "Furthermore, it is expected to have significant points of contact
 with virtually all areas of mathematics. Yet it is fraught with
 difficulties that can only be gotten around through the use of
 powerful new axioms for mathematics.
                            
 "Of course, it is too early to tell what the ultimate significance of
 this work will be. Stay tuned."
                            
 Weber is director of communications and outreach for the College of
 Mathematical and Physical Sciences.