Saturday, March 2, 2013

The linear continuum

Discrete mathematics is founded on the integers, directed graphs and other discrete structures. On the other hand, continuous mathematics is based upon continuous structures the simplest of which is the linear continuum. The unit interval [0,1] is a linear continuum whose every element can be represented by an infinite sequence of binary digits the positions of which are natural numbers.

The infinite subsets of the natural numbers numbers cannot be enumerated because attempting to determine the equality of infinite data structures leads to the halting problem. Without an enumeration these infinite subsets cannot be described by discrete mathematics and herein lies the fundamental distinction between discrete and continuous mathematics.

The probabilities of statements form a linear continuum [0,1]. Negation is defined by $1-x$, conjunction is defined by $xy$, and disjunction is defined by by $x+y-xy$. The values of zero and one are the absolute certainties used in deductive logic but that aren't used in probabilistic logic.

In order to convert [0,1] to another interval [a,b] with finite length we can use a linear polynomial $|b-a|x+a$. With infinite intervals we can use the reciprocal function and the logarithm function to achieve the same effect. A discrete partition of the linear continuum is equivalent to a discrete probability distribution.