Scientific American Magazine

Presenting an issue on mathematics in both its pure and applied aspects. Although the two are sometimes separated, a line cannot be drawn between them, and they traditionally fructify each other

With geometry it is one of the two pillars at the base of mathematics. The concept of number is enlarged by building up number systems and by seeking to break them down into their most primitive elements

For 2,000 years geometry meant Euclidean geometry. Then it was found not only that other geometries described physical space equally well but also that geometry was properly the study of all possible spaces

Elementary algebra deals exclusively with the general properties of numbers. Higher algebra can deal effectively with anything and on occasion is pursued without reference to anything in particular

The real world confronts the mathematician with events that are not strictly predictable. The methods he has developed to deal with such events have opened new domains of pure mathematics

When a new mathematical idea is found to be useful, a superstructure of mathematics rises from it. Later the original idea may prove to be shaky, but it must be repaired without destroying the superstructure

The outstanding examples of the power of mathematics to relate the facts of nature can be found in physics. The latest example is the use of group theory to relate the fundamental particles

Biologists use mathematics, but the complex systems they study resist mathematical description. The kind of description that might someday be helpful is suggested by the abstract analysis of self-reproduction

Certain social sciences such as economics deal with facts that are usually represented by numbers. New techniques for relating these numbers point toward the mathematical analysis of entire societies

In technology the most advanced applications of mathematics are in the design of machines that control themselves. The same methods are relevant to the control mechanisms of living organisms and societies

These machines that do arithmetic at high speed have evolved in response to the need to apply mathematics. Now they also promise to play a role in the progress of mathematics itself