Purposes of this Course

The discipline of computer science seems to draw mostly upon constructs and operations from discrete mathematics, such as the propositional calculus (logic, syllogisms, truth tables), set membership & relations, and combinatorics. The fundamental notion of an algorithm is a discrete sequence of discrete operations. The elementary hardware devices that implement algorithms are discrete gates, governed by Boolean algebra, and the elementary entities that they manipulate are bits, which are discrete states. Information is generated, transmitted, and stored in discrete form, and everything that happens in computing happens at discrete sequences of points in time - the edges of clock ticks.

So why study continuous mathematics?

Answer 1:

Because the natural world is continuous. It is the discreteness of digital computing that is unnatural! If we want to model a world that is governed by the laws of physics, we must come to computational terms with continuous processes.

Answer 2:

Because the distinction between discrete and continuous processes is illusory. Just as every continuous process can be approximated by discrete ones, every discrete process can be modeled as a continuous one.

Answer 3:

Because the two domains are inextricably intertwined, mathematically or physically. Semiconductor devices such as TTL logical gates really operate through continuous quantities (voltage, current, conductance); continuous theoretical constructs such as differentials and derivatives are only defined in terms of limits of discrete quantities (finite differences); etc.

Answer 4:

Because some of the most interesting and powerful computers that we know about are continuous. Non-linear dynamical systems in continuous time can be viewed as automata having great computational power; and the most powerful known ``computer," the human brain, has the following properties that distinguish it from a digital computer: it lacks numerical calculations; its communications media are stochastic; its components are unreliable and widely distributed; it has no precise connectivity blueprints; and its clocking is asynchronous and extremely slow (milliseconds). Yet its performance in real-time tasks involving perception, learning, and motor control, is unrivaled. As computer scientists we need to be able to study neural processes, and at many levels this requires continuous mathematics.

This short course is intended to be a refresher on some of the major ideas and tools used in continuous mathematics. Its practical purpose within the CST curriculum is to serve as groundwork for the following Pt. II and Diploma courses: Information Theory and Coding; Neural Computing; and Computer Vision.