Mathematicians are already and will be increasingly faced with the
need to broaden the conception they have of their discipline and
of the way in which it should be practiced. There are purely internal
reasons for this: mixing mathematical genres often becomes the key
to obtaining new results. This applies to the unexpected rapprochement
between geometry and statistics, between quantum field theory and
algebraic geometry, and between negative curvature geometry and
complexity, etc.
This motivation from within is far from being the only motivation;
we live at a time when, with the establishment of a society that
is partially delocalized and dominated by large systems, the areas
of life in society in which mathematics is involved in a non-trivial
(but usually hidden) way are increasingly numerous. This phenomenon
is recognized by the more dynamic entrepreneurs and captains of
industry.
In fact, we are witnessing a proliferation of new areas of interaction
for mathematicians. Some are scientific (the potential partners
come, as they have traditionally, from physics in many forms, or
from economics, but today also from biology, and chemistry, and
tomorrow from medicine); and some are more directly technological.
Today, mathematics is an essential part of many products from industry
and many services, and this goes from finance to the Global Positioning
System (GPS), and from the scanner to the design and safety of automobiles.
Many of these fields of adventure require genuine investment before
pertinent modeling emerges prior to implementing various mathematical
techniques, but it is often a question of defining new concepts
and new tools. The challenges involved in structuring and organizing
the gigantic mass of data from the sequencing of the genome are
a fine example of this. These new frontiers are worthy of exploration,
but it may be observed that documents about them are rare and circulate
poorly. The variety and the diversity of these situations merit
greater publicity and justify greater curiosity.
A primary issue is, of course, to prepare students. It has to be
recognized that today's students will have to use their knowledge,
enriched with experience in the field, in situations that most of
their teachers and lecturers are completely unfamiliar with. They
will work in an environment where the most varied software will
be in daily use. This situation is largely new, and mathematicians
are not well-prepared for it. They must be aware that a timid retreat
back to familiar ground, quite arbitrarily ejecting from mathematics
certain fields that are actually purely mathematical (such as automatic
control or combinatorics, to mention but two), would simply result
in "letting go" these expanding fields, and in ignoring new challenges
which will rightly attract inquisitive students.
Fundamentally, this broadening in scope does not free mathematicians
of their traditional obligations, namely explaining tirelessly how
apparently gratuitous questions ultimately give answers to other
questions that were not asked at the time when these methods appeared
(e.g. ellipses and their use in celestial mechanics, or complex
numbers and their use in the theory of functions), and repeating,
until exhaustion ensues, the genuine meaning of abstraction.
The coherence of the mathematical edifice always causes surprise,
but to think or to make people believe that it is complete would
be a dreadful mistake. That would be to deny that mathematics is
a science and a modern and living one at that, a double truth that
is often hidden.
