FOM: [HM] "Theory of Species" by Bergeron, Labelle and Leroux
Robert Tragesser
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RE: [HM] "Theory of Species" by Bergeron, Labelle and Leroux
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Introduction to "Theory of Species" by Bergeron, Labelle and Leroux
Advances in mathematics occur in one of two ways.
The first occurs by the solution of some outstanding problem, such as the
Bieberbach conjecture or Fermat's conjecture. Such solutions are justly
acclaimed by the mathematical community. The solution of every famous
mathematical problem is the result of joint effort of a great many
mathematicians. It always comes as an unexpected application of theories
that were previously developed without a specific purpose, theories whose
effectiveness was at first thought to be highly questionable.
Mathematicians realized long ago that it is hopeless to get the lay public
to understand the miracle of unexpected effectiveness of theory. The
public, misled by two hundred years of Romantic fantasies, clamors for
some "genius" whose brain power cracks open the secrets of nature. It is
therefore a common public relations gimmick to give the entire credit for
the solution of famous problems to the one mathematician who is responsible
for the last step.
It would probably be counterproductive to let it be known that behind every
"genius" there lurks a beehive of research mathematicians who gradually
built up to the "final" step in seemingly pointless research papers. And it
would be fatal to let it be known that the showcase problems of mathematics
are of little or no interest for the progress of mathematics. We all know
that they are dead ends, curiosities, good only as confirmation of the
effectiveness of theory. What mathematicians privately celebrate when one
of their showcase problems is solved is Polya's adage: "no problem is ever
solved directly".
There is a second way by which mathematics advances, one that
mathematicians
are also reluctant to publicize. It happens whenever some commonsense
notion that had heretofore been taken for granted is discovered to be
wanting, to need clarification or definition. Such foundational advances
produce substantial dividends, but not right away. The usual accusation
that
is leveled against mathematicians who dare propose overhauls of the obvious
is that of being "too abstract". As if one piece of mathematics could be
"more abstract" than another, except in the eyes of the beholder (it is
time
to raise a cry of alarm against the misuse of the word "abstract", which
has
become as meaningless as the word "Platonism").
An amusing case history of an advance of the second kind is uniform
convergence, which first made headway in the latter quarter of the
nineteenth century. The late Herbert Busemann told me that while he was a
student, his analysis teachers admitted their inability to visualize
uniform
convergence, and viewed it as the outermost limit of abstraction. It took
a
few more generations to get uniform convergence taught in undergraduate
classes.
The hostility against groups, when groups were first "abstracted" from the
earlier "group of permutations" is another case in point. Hadamard admitted
to being unable to visualize groups except as groups of permutations. In
the thirties, when groups made their first inroad into physics via quantum
mechanics, a staunch sect of reactionary physicists, led by the late
Professor Slater of MIT, repeatedly cried "Victory!" after convincing
themselves of having finally rid physics of the "Gruppenpest". Later, they
tried to have this episode erased from the history of physics.
In our time, we have witnessed at least two displays of hostility against
new mathematical ideas. The first was directed against lattice theory,
and its virulence all but succeeded in wiping lattice theory off the
mathematical map. The second, still going on, is directed against the
theory of categories. Grothendieck did much to show the simplifying power
of categories in mathematics. Categories have broadened our view all the
way
to the solution of the Weil conjectures. Today, after the advent of braided
categories and quantum groups, categories are beginning to look downright
concrete, and the last remaining anticategorical reactionaries are
beginning
to look downright pathetic.
There is a common pattern to advances in mathematics of the second kind.
They inevitably begin when someone points out that items that were
formerly thought to be "the same" are not really "the same", while the
opposition claims that "it does not matter", or "these are piddling
distinctions". Take the notion of species that is the subject of this
book. The distinction between "labeled graphs" and "unlabeled graphs" has
long been familiar. Everyone agrees on the definition of an unlabeled
graph, but until a while ago the notion of labeled graph was taken as
obvious
and not in need of clarification. If you objected that a graph whose
vertices are labeled by cyclic permutations -- nowadays called a "fat
graph"
-- is not the same thing as a graph whose vertices are labeled by integers,
you were given a strange look and you would not be invited to the next
combinatorics meeting.
The correct definition of a labeled graph turned out to be more
sophisticated
than the definition of an unlabeled graph. A labeled graph -- or any
"labeled"
combinatorial construct -- is a functor from the grupoid of finite sets and
bijections to itself. This definition of a labeled object is not
"abstract":
on the contrary, it expresses in precise terms the commonsense idea of
"being
able to label the vertices of a graph either by integers or by colors, it
does not matter", and it is the only way of making this commonsense idea
precise. The notion of grupoid, which is one of the key ideas of
contemporary
mathematics, makes it possible to withhold the assignment of a specific set
of labels to the vertices of a graph without making the graph unlabeled.
Joyal's definition of "labeled object" as a species discloses a vast
horizon
of new combinatorial constructions, which cannot be seen if one holds on
to the reactionary view that "labeled objects" need no definition. The
simplest, and the most remarkable application of the definition of species
is the rigorous combinatorial rendering of functional composition, which
was formerly dealt with by handwaving -- always a bad sign. But it is just
the beginning.
Species are related to generating functions in the much same way as random
variables are related to probability distributions. Those probabilists of
the thirties who held on to distributions, while rejecting random variables
as"superfluous", such as the late Aurel Wintner of Johns Hopkins, were
eventually wiped out, and their results are not even acknowledged today.
Those who reject the notion of species will suffer the fate of Professor
Wintner.
I dare make a prediction on the future acceptance of this book. At first,
the old fogies will pretend the book does not exist. This pretense will
last
until sufficiently many younger combinatorialists publish papers in which
interesting problems are solved using the theory of species. Eventually, a
major problem will be solved in the language of species, and from that time
on everyone will have to take notice. The rewriting, copying and imitating
will start, and mathematicians who capitulate to the new theory will begin
to tell us what species really are. Considering the speed at which
mathematics progresses in our day, that time is more likely to come sooner
than later.
Gian-Carlo Rota
Professor of Applied Mathematics and Philosophy
MIT room 2-351
77 Massachusetts Avenue
Cambridge, MA 02139-4307, USA
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