MAT 15 - Matilde - Dansk Matematisk Forening

to intellectual ideas and concepts. Even as natural languages
such as German or Chinese embody cultural treasures,
mathematics does also. But one has to learn the
“language” of mathematics to appreciate these treasures
and to learn the history of mathematics to appreciate its
growth within the history of general ideas.
Looking about us, if we are perceptive, we can see not
only the natural world of rocks, trees, and animals, but
also the world of human artifacts and of human ideas.
The ideas of Karl Marx (and of other thinkers) explain
the world along certain lines, each necessarily limited, but
these ideas have often changed the world beyond recognition.
Mathematics also, is capable of explaining the world
in remarkable but limited ways. It is capable of formatting
our human lives in useful ways, and it is capable,
again in a limited way, of allowing us to look into the
future and to make prudential judgements. Every educated
person should achieve some appreciation of the historical
role that mathematics plays in civilization in order
that the subject be given both intelligent support as
well as intelligent resistance to its products.
Q4. What have been the dominant trends?
A4. First of all, the amount of new mathematical theories
created since 1900 is enormous. During that time,
mathematics has become more abstract and deeper than
it was in the 19th Century. By this I do not mean that
simple things such as arithmetic have to be viewed by
the general public in a more abstract or deeper way than
before, but that the conceptualization of old mathematics
and the creation and applications of new mathematics
by the professionals have had that character.
Mathematical logic is now firmly on the scene as is
set theory. A new and more abstract algebra has grown
mightily and consolidated itself. Infinitary mathematics
(i.e., the calculus and its elaborations) has grown by leaps
and bounds, but has had increasingly, as opposed to
the 19th Century, to share the stage with finitary mathematics.
Geometry, which began in antiquity as visual and numerical
(lengths, areas, volumes) moved into an abstract
axiomatic/deductive mode and now embraces such studies
as algebraic, combinatorial , and probabalistic geometry.
A major trend since the late 1930’s is to view mathematics
as the study of deductive structures. Mathematics
has shared the structuralist point of view with numerous
non-mathematical disciplines ranging from linguistics to
anthropology to literary criticism.
There is no doubt that the electronic digital computer
which emerged in the middle 1940’s, and is in many ways
a mathematical instrument, has changed our day to day
life noticeably. The computer has revitalized or widened
the scope and increased the power of a number of traditi-
onal branches of mathematics. It has also created new
branches of the subject, and all in all, has had a revolutionary
effect throughout science and technology.
Q5. Tell me about the “chipification” of mathematics.
A5. A good fraction of the mathematics that affects our
day-to-day life is now performed automatically. It is built
into computer chips that are in our wristwatches, automobiles,
cash registers, ATM’s, coffee machines, medical
equipment, civil engineering equipment, word processors,
electronic games, telephone and media equipment,
military equipment, ID plastic... the list has become endless
and the word “chips” and the idea of inserting them
to do clever things has become commonplace.
The mathematics in chips is hidden from view. The
average person, though greatly affected by it, need pay
no conscious attention to it as mathematics. This would
be the job of a corps of experts who design, implement,
monitor, repair, improve such systems. The “chipification
of the world” is going forward at a dizzying rate. One
of the disturbing side effects of this trend may very well
be that in the near future Sally and Johnny will not have
to add or read in the conventional way. Will this put us
back several hundred years when literacy and numeracy
were relatively rare achievements? Not at all. Modes of
communication, interpretation and of social arrangements
have changed in the past and will change in the future. It
does not mean that they will disappear.
Q6. Where are the centers where new mathematics is
developed? Who is “hip?” A fan of good rock music
knows where to go to hear it. Where would a math fan
go these days?
A6. There are mathematical centers in all the developed
countries of the world. They are located in universities,
governmental agencies, research hospitals, scientific, economic
or social “think tanks”, industries. All of these centers
tend to be organized into groups and tend to concentrate
their research on very limited and specific problems.
There are also many mathematical researchers
who work independently and productively outside of
groups.
Depending on what sort of “mathematical music” you
would like to hear, you would visit one of these groups
or individuals. Although these groups exhibit a great deal
of esprit de corps and self esteem, and can be influential
beyond their own walls, there is, in my judgement,
no single group that is predominates. And this diversity
is a very good thing.
Much more close to home, there are now websites that
cater to every level of mathematical interest and engagement.
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