Computer Science and Mathematics
Computer science is generally thought of as the study of the theoretical aspects of the use or design of computers, and computer engineering as the study of hardware and the construction of real computing devices. This distinction is not rigorous, nor is it understood the same way by everyone. However, most people would agree that computer science is closer to mathematics and computer engineering closer to electrical engineering.
Looked at a certain way, computer science is a specialized topic in applied mathematics. Many of the topics of applied math, such as numerical analysis, are investigated almost exclusively in conjunction with computers. Other topics, such as algorithmic algebra (also called computer algebra), are devoted to mathematical analyses of computer-science problems. Still other topics, like graph theory, are studied by mathematicians as well as by computer scientists; thus it is really not sensible to say that such topics belong solely to one discipline or the other.
It is common for college courses in computer science (and other fields) to require that the student take a prerequisite course in calculus, even though computer science itself does not use calculus to any significant extent. This emphasis in high school and beginning college on traditional mathematics (geometry and calculus) is often baffling and irritating to students (perhaps parents and teachers also), who feel that their time is being wasted because hardly anyone in the real world uses all those theorems and derivations anyway. However, calculus is used in the real world, constantly: algebra, geometry, and calculus are as basic to the physical sciences (which employ many millions of people) as vocabulary is to writing, being not so much ends in themselves as the indispensable tools with which work is carried on. Second, even students who do not go on to use algebra, geometry, or calculus in their daily work often receive from these subjects their first exposure to rigorous abstract reasoning, and the intellectual skills learned in this way are liable to prove handy in virtually any other career, including computer science. Today there is much debate about how (and how much) calculus should be taught to high school and college students, but the value of the calculus in training students in abstract reasoning, and in many real-world careers, is beyond dispute.
It is often a surprise to people to learn that computer science is in a certain sense older than computers. The nineteenth-century mathematician George Boole described a logical notation for "truth" using abstract symbols. His elementary logic, which is really a form of non-numerical mathematics, is now known as Boolean algebra, and is a basis for the design of the logic circuits that are at the heart of today's computing machines. Likewise, the work of other early pioneers like Charles Babbage, Ada the Countess of Lovelace, and others helped clarify the notion of computing, which was later developed by John von Neumann and other mathematicians working in the first half of the twentieth century--largely on paper.
Mathematics has provided a basis for rigorous analyses of the techniques that are the mainstay of computer science. the ideas and abstractions that are commonly applied by mathematicians in determining--in a formal, demonstrably correct way--the truth of clearly-stated claims are also useful in computer science. Mathematics thus provides computer scientists with the tools needed for the rigorous study and presentation of their subject.
Computer science has also enriched mathematics. To the extent that it provides a new tool with which mathematical research may be conducted (the computer), it is a challenge for people interested in mathematical analyses "just for the fun of it." But that is not all; the formal results of computability theory have a great relevance to the foundations of mathematics as well. One simply cannot ultimately separate mathematics from computer science. Mathematics makes progress in computer design possible; improved computers enable mathematicians to carry out more research than their forebears. Using computers, a theorem called the Four Color Theorem, well-known for centuries, was first proved as recently as 1976. The theorem simply states that any map drawn on a plane may be colored with just four colors in such a way that no two adjacent regions have the same color. In spite of its simple statement, a proof of the Theorem had eluded mathematicians for generations, but with the help of a computer, researchers were able to reduce the problem to a large number of special cases that the computer could check for obedience to the Theorem.
This is the complete article, containing 741 words
(approx. 2 pages at 300 words per page).
