**What do mathematicians
do?**

By
Professor A J Berrick

As
a break from the tradition of this newsletter, this article is meant to provoke
discussion. Most research mathematicians are quite passionate about their
subject. Yet they are aware that their enthusiasm is not shared (to put it
mildly) by the public at large, and even, in many cases, by research
scientists. Is this just a case of “love is blind”, or is it possible that
mathematicians are aware of something about mathematics that outsiders are not?
I’d like to investigate this matter in this article. I hope that in doing so I
will stimulate others, both mathematicians and non-mathematicians, to think
about these questions, and maybe even contribute their thoughts to later issues
of the newsletter.

I
think it’s very important to start by asking the right question. Typically,
academic disciplines are defined by their subject matter. So, to ask what a
geologist does is more or less the same thing as asking what a geologist
studies. Thus, for the Oxford English Dictionary, it is “the science which has
for its object the investigation of the earth’s crust, of the strata which
enter into its composition, with their mutual relations, and of the successive
changes to which their present condition and positions are due”. Similarly, for
the OED, biochemistry is “the science dealing with the substances present in
living organisms and with their relation to each other and to the life of the
organism”. Moving away from science, we have the OED’s definition of architecture:
“the art or science of building or constructing edifices of any kind for human
use”, economics: the study of “the development and regulation of the material
resources of a community or nation”, linguistics: “the study of languages”.
These examples were chosen at random. In every case I expect that the reader's
definition would be very similar to the one given by the dictionary.

Well,
if such definitions are so easy for other disciplines, why not for mathematics?
Like most people, the OED assumes that mathematics too can be defined by its
subject matter, and tries the following: “the abstract science which
investigates deductively the conclusions implicit in the elementary conceptions
of spatial and numerical relations, and which includes as its main divisions
geometry, arithmetic, and algebra; and, in a wider sense, so as to include
those branches of physical or other research which consist in the application
of this abstract science to concrete data”. A good effort, but one gets the
strong impression that whoever wrote it was struggling! The “spatial and
numerical relations” obviously cover geometry and arithmetic, but then algebra
had to be added because it wasn’t dealt with. However, that’s nowhere near good
enough. Important “divisions” like analysis, probability, set theory and
operational research are completely ignored by this definition, so clearly it’s
very inadequate. Should we compensate by listing the titles of, say, all
mathematics modules taught at NUS, in the hope that we’ll cover the subject
that way? That attempt is doomed too, because a glance at the list soon reveals
courses on topics like filter banks, chaos and fractals, cryptography, game
theory, etc., that weren’t there ten or twenty years ago. If the subject is to
be defined by a list constructed at a certain time, then after that time no
newcomers can ever join. However, it’s clear that mathematics is continuing to
grow, its tentacles finding their way into areas of investigation previously
thought beyond its reach.

The
first attempt I heard to define mathematics by what it studies was by T G Room
(a geometer commemorated by Room squares in combinatorics). He reckoned that
mathematics is the study of relationships between concepts. Although this is
helpful to the non-mathematician, it is clearly inaccurate. There are many
concepts, like punishment and retribution, love and fidelity, whose
relationships have failed to attract mathematical interest. In order to nail
down those concepts that might yield mathematical investigation, the topologist
D H Gottlieb claimed that mathematics is the study of well-defined things. This
notion has some appeal to mathematicians, for whom the expression
“well-defined” is part of the lingo, and for whom it excludes the above
philosophical concepts. Yet I fear that an attempt to explain it to a
non-mathematician would result in a “well-defined thing” as being “one that is
amenable to mathematical inquiry”. In other words, mathematicians study what
mathematicians study.

Mathematicians
tend to be pretty stubborn (we like to say that we persevere), but there comes
a stage when one has unsuccessfully battled against a tough question for so
long that one realises that the difficulty was simply that it was the wrong
question in the first place. I believe that’s what’s happened here.

We
shouldn’t ask *what *a mathematician
studies, we should ask *how*.

Put
another way, instead of the question What do mathematicians study? We should
ask What do mathematicians do? Interestingly, when one examines the OED's
attempt at a definition, one sees that, in contrast to the definitions of the
other disciplines, there’s a how answer only partly suppressed: “investigates
deductively the conclusions implicit in the elementary conceptions ...” Since
mathematicians get their fingers into pies that often have names very different
from geometry, arithmetic and algebra, it’s more fruitful to clarify the
process of doing mathematics. Then, when a new topic is proposed for the next
revision of the mathematics curriculum, one has some hope of answering the
question: But is it mathematics? My guess is that the correct answer would be:
Yes, provided you look at it the right way. No geology lecture would be about
zebras, or blood vessels, or language grammars, or DNA. (Or even could be - by definition
it would fail to be a geology lecture.) However, I’ve known mathematics
lectures about all four topics.

Specifically,
the lecture about **zebras** was
interested in how they and other quadrupeds move. Different speeds of walking
or running result in different sequences of hooves hitting the ground. Which
sequences can occur, and what is the relation between the sequence and the
speed?

**Blood vessels**
can be studied for the way in which cells move along them; this is the dynamics
of fluid motion where the walls are not rigid. And what governs the shape of
the vessel itself? Can one predict when the forces will be so great as to lead
to rupture?

Are
there common rules of manipulation of words and phrases that apply across
different **languages**? What does the
similarity of such rules suggest about the cultural or genetic links between
the speakers of such languages?

It’s
recently been discovered that in the process of replication, the enormously
long **DNA** molecules get tied into
knots, which partly dissolve and recombine as different knots. By inspecting
the knots that appear, one can attempt deductions about the biochemical process
that is leading from one knot to the next.

There’s
a pattern to what is happening in each of the above examples. The mathematician
immediately ignores many specific features of the object in question. He or she
is unlikely to care about whose body the blood vessel inhabits, or the age of
the zebra. But pretty soon (s)he may even forget that it’s a blood vessel or
zebra that’s being studied, and may talk to a colleague about fluid in a tube
or configurations of moving rods. The process of abstraction (OED: “of
considering ... an attribute or quality independently of the substance to which
it belongs”) takes on a life of its own, so that before long two mathematicians
may be discussing the problem in such a way that a third mathematician
listening in would find it difficult to guess its physical origins. (The degree
of difficulty is probably the distinction between pure and applied mathematics.
Put like this, it’s apparent that the distinction is more arbitrary and less
clear-cut than generally recognised.)

After
reading the above, the Japan-based mathematician A Kozlowski observed: “I think
it a very important point that mathematics is probably the only subject whose
content could change entirely and yet we would still recognize it as
mathematics. We would probably recognize mathematics of beings from another
universe, though we may have problems in distinguishing their physics from
their philosophy, their history from their mythology etc.”

I
believe that the process of abstraction is a vital characteristic of
mathematical thought, probably more distinctive than the method of deduction
that the OED emphasises. Most scientists practise deduction, although not
necessarily to the extent of mathematicians. However, other disciplines are
comparatively restricted in the amount of abstraction that they allow
themselves. While physiologists might be comfortable with the general
properties of all blood vessels belonging to humans, or of all those with a
certain condition, would they still be at ease with a level of abstraction that
considered equally other fluids moving in inorganic tubes? Or would they feel
that such generalizations were no longer in the realm of physiology?

In
fact, the desire for abstraction seems to be an essential part of a
mathematician’s psyche. It’s not just a matter of abstracting from the physical
world to the mathematical; many mathematicians commence work only long after
that process has been completed. Within mathematics, researchers are all the
time striving to find just the right level of abstraction for a given setting,
seeking the perfect balance between the twin goals of utility and generality.

Another
feature of scientific method is of course induction, the attempt to generalize
conclusions from a number of particular instances. Mathematicians practise this
more often than is usually realised, however, in a special way. For a scientist
such a conclusion has the status of a probationary law. If it stands the test
of time, that is, accords with, and even predicts, subsequent observations,
then it becomes more widely accepted. This tends to be a gradual process that
can be partly or wholly reversed (medical science provides many examples of
reversibility and controversy). For a mathematician, the result of induction is
just a hunch. A strongly held hunch is honoured with the title of conjecture.
For example, there was the Fermat Conjecture:

If
*x,y,z,n* are whole numbers greater
than 1 and *x ^{n} + y^{n}
= z^{n}*, then

Evidently,
Fermat produced this statement by induction after examining many specific cases
(with no help from an electronic computer). (In the event that *n = 2*, then for any whole number *k* we can always take *x = 2k +1, y = 2k ^{2}+2k *and

We
have now distinguished three modes of thinking that highlight the difference
between mathematics and other disciplines:

Abstraction, Deduction, Induction.

They
are listed above in decreasing order of importance to mathematical research,
but I would guess in increasing order of importance for scientists generally.
One might object that if deduction and induction are seen as opposites, then
why doesn’t an opposite of abstraction appear in the list? Well, since
abstraction consists in seeing common properties and patterns among different
situations, then once one has obtained conclusions about the general, abstract
setup, the process of deduction is enough to get one back to conclusions about
the original, more concrete setting. So our list of three seems to be enough.

Just
what the process of abstraction involves is a big topic, and not central to
this article. At its heart seems to be one of the greatest joys of mathematical
research - pattern recognition. The patterns are not usually the visual ones of
everyday experience. Recall that humans also get excited by more subtle threads
of similarity, for instance in hearing in a Wagnerian opera a musical motif
that occurs elsewhere in the Ring Cycle. Mathematicians have available for
painting their patterns the whole canvas of human experience. For example, the
kinship ties of the Warlpiri people of the Australian outback exhibit the same
pattern as the symmetries of a square (known algebraically as the dihedral
group of order 8).

The
drive to find common themes from disparate areas seems to be part of the
mathematician’s temperament. At its most banal, it’s a source of painful puns
(like the one I had to resist earlier, after including the words “pie” and
“fruitful” in the same sentence). Used more creatively, it helps to explain
what has been called the “unreasonable effectiveness” of mathematics. Consider
the following title of a research paper, by R Ghrist, that reached me today.

*Configuration spaces ...*
it begins. I think: Yes, it’s about topology.

*... and braid groups ...*
Okay, about algebra too.

*… on graphs ...*
Combinatorics as well. This is getting pretty interesting. But now for the
knockout blow ...

*… in robotics.*

The
discovery of a unifying pattern can be like lightning flashing from one
discipline to another. The difference is that it can illuminate both subjects
forever. So there is a simple message for the nonmathematical researcher
reading this article. When all seems cloudy, contact a mathematician (as
broadminded as possible). Then stand by for flashes of lightning!

**Biographical**

Prof
Jon Berrick joined the Mathematics Department of NUS in 1981, after working at
Oxford and Imperial College London. His research interests include algebraic
topology, algebraic *K*-theory (where
he has authored a monograph and two graduate texts) and cohomology of groups
(in which area he is an editor of *Journal
of Group Theory*). He has also participated in joint research work with
neurophysiologists, dentists and botanists.

**Contacts:**

A
J Berrick

Tel: +65 874 2747

Fax: +65 779 5452

Email: berrick@math.nus.edu.sg

Web: http://www.math.nus.sg/~matberic/