AChaoticEnd? 249
Statistics, in the form of hypothesis testing developed in the early twentieth century, gave a useful
way of dealing with the unpredictable; even apparently random processes are regular. We may not
know when any particular carbon 14 nucleus will decay to nitrogen 14 , but we know the statistics
of the process as applied to a large number of nuclei, and so can use it within limits for reliable
dating. Chaos theory can be seen as the mirror image of statistics, asserting that even some com-
pletely deterministic processes cannot be used for accurate prediction. How can these viewpoints
be reconciled? It depends on one’s point of view. No matter how much information one has about
the weather on July 1, one’s forecasts for July 15 will be limited by the variability which arises from
sensitive dependence—this is chaos theory’s input. On the other hand, observation of the weather
over a long period makes possible some reasonable predictions about the mean July temperature
and rainfall—this comes from statistics. Asked about the effect of the French Revolution in the
1950s, the Chinese Prime Minister Zhou Enlai replied ‘It is too early to say’; and this is certainly
a reasonable response to the problem of assessing the contribution of the very young ‘paradigm
shift’, if there is one, associated with chaos theory.
Exercise 3.(a) Consider the function f(z)=z^2 on the unit circle C, or|z|= 1 in the complex plane.
Find all periodic points of period n, that is, points z such that fn(z)=z. How many points of period 4
have prime period 4, that is, fn(z) =zfor 0 <n< 4?
Show that given z, andε> 0 , we can find z′and an integer n such that: (1) d(z,z′)<ε;
(2) d(fn(z),fn(z′)) >^12 , where d denotes distance on the circle. (This is sensitive dependence, for a very
simple map).
(b) Define g(z)=z^2 +c where z is complex. Show that g has (in general) two fixed points z 0 ,z 1 ; and
that the condition that one of them should satisfy|g′(z)|= 1 is that c lies on a curve (a cardioid) in the
c-plane. (This curve bounds the largest area in pictures of the Mandelbrot set.)
7. From topology to categories
We cannot think any object except by means of the categories; we cannot know any object except by means of
intuitions corresponding to these concepts. (Kant 1993, p. 117)
[M]athematics is about to go through a second revolution at this moment. This is the one which is in a way completing
the work of the first revolution,^9 namely which is releasing mathematics from the far too narrow limits imposed by
‘set’; it is the theory ofcategories and functors...(Dieudonné, 1961 lecture, cited in Corry 2004, p. 383)
‘Algebraic topology’, which we have seen in process of defining itself in the 1920s, had progressed
by the 1950s to a massively successful and integrated subject. Indeed as the century progressed it
was constantly growing and subdividing (like most other fields in mathematics) into further areas
of specialization. Any kind of survey which took in Smale’s horseshoe map (Fig. 5), Vaughan
Jones’s knot invariants, and the wide variety of contributions made by Michael Atiyah at the
interface of topology, differential equations and even physics would become a whole chapter in
itself. We should, though, find space to consider topology’s illegitimate child category theory which
succeeded in doing what, as we have seen, the Bourbakists failed to do: to provide a clear idea of
what was meant by ‘structure’.
- Axiomatization and structure, briefly.