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\noindent \Er Tribute
\\ AMS San Diego, Jan 1997
\\ Joel Spencer
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Paul \Er was a searcher, a searcher for mathematical truth.
Paul's place in the mathematical pantheon will be a matter of strong
debate for in that rarefied atmosphere he had a unique style. The
late Ernst Straus said it best, in a commemoration of Erd\H{o}s's
$70$-th birthday.
\begin{quote} In our century, in which mathematics is so strongly
dominated by ``theory constructors" he has remained the prince of
problem solvers and the absolute monarch of problem posers. One
of my friends - a great mathematician in his own right - complained
to me that ``\Er only gives us corollaries of the great metatheorems
which remain unformulated in the back of his mind." I think there
is much truth to that observation but I don't agree that it would have been
either feasible or desirable for \Er to stop producing corollaries and
concentrate on the formulation of his metatheorems. In many ways
Paul \Er is the Euler of our times. Just as the ``special" problems
that Euler solved pointed the way to analytic and algebraic number
theory, topology, combinatorics, function spaces, etc.; so the methods and
results of Erd\H{o}s's work already let us see the outline of great new
disciplines, such as combinatorial and probabilistic number theory,
combinatorial geometry, probabilistic and transfinite combinatorics
and graph theory, as well as many more yet to arise from his ideas.
\end{quote}
Straus, who worked as an assistant to Albert Einstein, noted that
Einstein chose physics over mathematics because he feared that one
would waste one's powers in persuing the many beautiful andattractive
questions of mathematics without finding the central questions.
Straus goes on,
\begin{quote} \Er has consistently and successfully violated every
one of Einstein's prescriptions. He has succumbed to the seduction
of every beautiful problem he has encountered - and a great many
have succumbed to him. This just proves to me that in the search
for truth there is room for Don Juans like \Er and Sir Galahad's
like Einstein.
\end{quote}
I believe, and I'm certainly most prejudiced on this score, that
Paul's legacy will be strongest in Discrete Math. Paul's interest
in this area dates back to a marvelous paper with George Szekeres
in 1935 but it was after World War II that it really flourished.
The rise of the Discrete over the past half century has, I feel,
two main causes. The first was The Computer, how wonderful that
this physical object has led to such intriguing mathematical
questions. The second, with due respect to the many others,
was the constant attention of Paul \Er with his famous admonition
``Prove and Conjecture!" Ramsey Theory, Extremal Graph Theory,
Random Graphs, how many turrets in our mathematical castle were
built one brick at a time with Paul's theorems and, equally
important, his frequent and always penetrating conjectures.
My own research specialty, The Probabilistic Method, could surely
be called The \Er Method. It was begun in 1947 with a $3$ page
paper in the Bulletin of the American Math Society. Paul proved
the existence of a graph having certain Ramsey property without
actually constructing it. In modern language he showed that an
appropriately defined random graph would have the property with
positive probability and hence there must exist a graph with the
property. For the next twenty years Paul was a ``voice in the
wilderness", his colleagues admired his amazing results but adaption
of the methodology was slow. But Paul persevered - he was always
driven by his personal sense of mathematical aesthetics in which
he had supreme confidence - and today the method is widely used
in both Discrete Math and in Theoretical Computer Science.
There is no dispute over Paul's contribution to the spirit of mathematics.
Paul \Er was the most inspirational man I have every met. I began working
with Paul in the late $1960$-s, a tumultuous time when ``do your own
thing" was the admonition that resonated so powerfully. But while
others spoke of it, this was Paul's modus operandi. He had no job;
he worked constantly. He had no home; the world was his home.
Possessions were a nuisance, money a bore. He lived on a web of
trust, travelling ceaselessly from Center to Center, spreading his
mathematical pollen.
What drew so many of us into his circle. What explains the joy we
have in speaking of this gentle man. Why do we love to tell \Er
stories. I've thought a great deal about this and I think it comes
down to a matter of belief, or faith. We here know the beauties of
mathematics and we hold a belief in its transcendent quality.
God created the integers, the rest is the work of Man. Mathematical
truth is immutable, it lies outside physical reality. When we show,
for example, that two $n$-th powers never add to an $n$-th power for
$n\geq 3$ we have discovered a Truth. This is our belief, this is
our core motivating force. Yet our attempts to describe this belief
to our nonmathematical friends is akin to describing the Almighty to
an atheist. Paul embodied this belief in mathematical truth. His
enormous talents and energies were given entirely to the Temple of
Mathematics. He harbored no doubts about the importance, the
absoluteness, of his quest. To see his faith was to be given faith.
I sometimes think that the world of religion would have understood
better than our rationalist world of mathematics this special man.
I do hope that one cornerstone of Paul's belief will
long survive. I refer to The Book. The Book consists of all the
theorems of mathematics. For each theorem there is in The Book
just one proof. It is the most aesthetic proof, the most insightful
proof, what Paul called The Book Proof. And when one of Paul's myriad
conjectures was resolved in an ``ugly" way Paul would be very happy
in congratulating the prover but would add, ``Now, let's look for
The Book Proof." This platonic ideal spoke strongly to those of us
in his circle. The mathematics was there, we had only to discover it.
The intensity and the selflessness of the search for truth were described
by the write Jorge Luis Borges in his story The Library of Babel. The
narrator is a worker in this library which contains on its infinite
shelves all wisdom. He wanders its infinite corridors in search of
what Paul \Er might have called The Book. He cries out,
\begin{quote} To me, it does not seem unlikely that on some shelf of
the universe there lies a total book. I pray the unknown gods that
some man - even if only one man, and though it have been thousands of
years ago! - may have examined and read it. If honor and wisdom and
happiness are not for me, let them be for others. May heaven exist
though may place be in hell. Let me be outraged and annihilated but
may Thy enormous Library be justified, for one instant, in one being.
\end{quote}
In the summer of 1985 I drove Paul to what some in the audience here
tonight fondly remember as Yellow Pig Camp - a mathematics camp for
talented high school students at Hampshire College. It was a
beautiful day - the students loved Uncle Paul and Paul enjoyed nothing
more than the company of eager young minds. In my introduction to his
lecture I discussed The Book but I made the mistake of discribing it
as being ``held by God". Paul began his lecture with a gentle correction
that I shall never forget. ``You don't have to believe in God," he
said, ``but you should believe in The Book."
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