FOM: Harvey Friedman PR in campus newspaper
Stephen G Simpson
simpson at math.psu.edu
Sun Apr 23 21:11:47 EDT 2000
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.
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