Page 272 - J. C. Turner "History and Science of Knots"

On Theories of Knots 263

mathematical theories have been developed to help the braider with his craft?
Especially as the process of braiding is a regular, repetitive sort of task, with
string or strings being interwoven into patterns which always have symmetries
which are capable of geometric or other sorts of mathematical description.
In this Chapter we shall describe various attempts made in this century
by knot craftsmen (artisans rather than mathematicians) to list and classify
knots by means of diagrams and reference to their uses. Then we shall outline
the work done in the last fifteen years by a New Zealand mathematician, Georg
Schaake, to provide a strong mathematical theory of braiding processes. We
shall call his work `Braiding Theory', to distinguish it from `Topological Knot
Theory'. It must not be confused with Emil Artin's Braid Theory, which was
introduced in 1925 and is part of topological knot theory.
We wish to show that the statement, often heard on the lips of mathe-
maticians, that `Knot Theory is a branch of Topology', is not only misguided
but also false. There are other theories of knots, which are not topological.
Moreover, topological knot theory deals with but a small subset of the mys-
teries of knot lore: to imply that it covers all of Knot Science is rather like
saying that Medical Science is the study of diseases of the foot.

2. What is a Science?

The question: What is a Science? is a large one. Indeed the philosophy of
science is a subject in its own right. However, simple and general notions
of what constitutes a science, leaving aside any mathematical requirements,
can be culled from any dictionary. For our starting point we shall take two
definitions from a recent Collins Concise Dictionary.
The first definition, which is too weak to be of much value (it makes
just about any subject into a science) is: A Science is any body of knowledge
organised in a systematic manner.
The second states that: Science is the systematic study of the nature and
behaviour of the material and physical universe (or any particular branch of
the knowledge gained) based on observation, experiment, and measurement.
Before a scientific theory of anything, still less a mathematical science, can
be developed, one has to settle upon a class (or universe) of objects to study.
Then one has to attempt to define objects in that universe with something akin
to mathematical precision. Thus the first step towards a Science is essentially
descriptive; it involves observing, recording and naming the objects in a chosen
universe.
Within and following the initial step, observations are made which de-
termine relationships between the objects. These enable the universe to be
partitioned in useful or interesting ways. In other words, one specifies equiva-
lence relations, which then place the objects into different classes according to