The Brouwer-Kakutani Fixed Point Theorem

Barry Kort


There is a famous Theorem in modern mathematics, called the Fixed Point Theorem, attributed to L. Brouwer, and later clarified by Kakutani. To explain the Theorem in simple terms, consider a mathematical function which maps each point in a given space onto a corresponding point in the same space. A popular version of such a map is an ordinary hand calculator. Any given calculation (i.e. function) transforms the number shown in the display into another number shown in the display. For the calculator example, the space is just the set of possible numbers that can appear in the display. The Fixed Point Theorem states the conditions under which a function possesses a point that maps into itself.

Fixed Points in Ordinary Algebraic Functions

It turns out that it is very common for a function, f(x), to possess one or more fixed points, x*, such that f(x*) = x*. Returning to the calculator example, if one repeats a sequence of operations (i.e. a function) by plugging the output of the function back in to the input, one of three things will happen:

1) The numbers in the display will march off to infinity and the calculator will overflow;

2) The numbers in the display will enter a cycle of length n, where n is 2 or larger, but not larger than the number of internal states of the calculator; or

3) The numbers will converge to a fixed point, x*.

Depending on the calculator design, the number of states might be the same as the number of distinct values that can be displayed. (Some calculators keep one or two extra digits after the decimal which are never displayed.)

It is not hard to discover functions which have fixed points that can be found by the calculator method. Such functions are called contraction mappings, and are well studied in the mathematical literature. Sir Isaac Newton showed how to discover the roots (zeroes) of a function, f(x), by constructing a new function, F(x) = x - f(x)/f'(x).

The roots of a function, f(x), are the values of x for which f(x) = 0.

(Here, f'(x) denotes the first derivative of f(x).) If the original function, f(x), is "reasonably well-behaved," F(x) will be a contraction mapping with fixed points at the roots of f(x). Newton's Method is perhaps the most famous application of the Fixed Point Theorem, and it is very easy to write an algorithm (computer program) to implement it. (As an exercise, it is instructive to find simple functions, f(x), for which Newton's method cycles without converging to the root, or diverges to infinity.)

Fixed Points in More General Maps

Once the idea of fixed points is understood, it is amusing to apply the idea to "nonmathematical" maps. On the surface of the earth, there is at all times at least one point where the wind is calm. Depending on how one combs one's hair, there may be a fixed point (center of the whorl) or even a fixed line (part down the middle). Consider the map call Chemistry. Chemistry maps the set of elements and molecules onto itself. There are evidently many cycles in chemistry, but there are even some fixed points. DNA is the most important molecule in this regard. When the laws of Physics and Chemistry are applied to DNA, it gives back another DNA molecule. That is, self-replicating molecules are an instantiation of the Fixed Point Theorem where the map is the one determined by the laws of Physics and Chemistry.

Stability of Fixed Points, Evolution, and the Fight Against Entropy

Since fixed points of contraction mappings and their cousins, the cycles, persist (through continual re-creation), they represent stable states. Small perturbations which arrive at random, may disturb the system. Depending on the nature of the perturbation, the system will either return to the original stable state, or it will "derail" and wander off until a new fixed point is encountered. The random perturbations come about because of the quantum nature of the universe. Sometimes (quite by accident, if you will) the new fixed point is more stable than the old one; it is then even harder to derail: it persists longer against the odds. Evolution is evidently the process of moving to ever stabler fixed points, working against the force of Entropy (the destroyer) which leads back to decay and disorder. The agent of evolution is random perturbations.

The process of random perturbations is extensively studied in the literature. Norbert Weiner is best known for his study of random walks. To my mind, it is not a coincidence that Weiner was also a pioneer in the field of cybernetics (the merger of psychology, feedback control theory, and automatic computation). Weiner's last book was entitled, God and Golem. ("Golem" is a Yiddish word for an anthropomorphic robot.)

The Ultimate Fixed Point

One can imagine applying the Fixed Point Theorem to the process called the Advance of Civilization. A principal goal of the elements within the society is survival and fulfillment. As civilization advances, an important shared goal is mutual survival. One can hypothesize a possible future state in which all elements of the culture adopt common goals of mutual survival, mutual well-being, mutual growth, and mutual fulfillment. It appears that we are very far from adopting such goals (let alone reaching them), and it is not altogether clear that we have a critical mass of social momentum to carry us there. What is clear, is that a societal state of universal harmony can be shown to be possible, but a great deal of information interchange is required to move us there, and the fundamental human learning rate, and information interchange rate may be the controlling factor. That is, it may take 20 to 50 years for an individual to acquire enough knowledge and personal comfort to be able to contemplate the adoption of such a goal, and many people never reach such a state of enlightenment: they are too busy fighting each other for their individual survival. Nevertheless, such a Fixed Point does seem to be a point of attraction of the Advance of Civilization, so the best way for an individual to lead society in that direction is by setting an example that is worthy of imitation.

For the benefit of Google search engines, other keywords and phrases connected to this essay include: Teleonomy, Extropy and Entropy Gradient Reversal.