Encyclopedia of Computer Science and Technology

mathematics software 295Mathematically, a computer can be seen as a way to rapidlyand automatically execute procedures that have beenproven to lead to reliable solutions to a problem (see algorithm).Once computers came on the scene, mathematicalprinciples for verifying or proving algorithms wouldacquire new practical importance.By the early 20th century, however, mathematicianswere beginning to examine the problem of determiningwhat propositions were provable, and in 1931 Kurt Godelpublished a proof that any mathematical system necessarilyallowed for the formation of propositions that could notbe proven using the axioms of that system. An analogousquestion was determining what problems were computable.Working independently, two researchers (see Church,Alonzo and Turing, Alan) formulated models that couldbe used to test for computability. Turing’s model, in particular,provided a theoretical construct (the Turing Machine)that could, using combinations of a few simple operations,calculate anything that was computable.By the 1940s, electromechanical (relays) or electronic(tube) switching elements made it possible to build practicalhigh-speed computers. Computer circuit designerscould draw upon the advances in symbolic logic in the 19thcentury (see Boolean operators). Boolean logic, with itstrue/false values, would prove ideal for operating computersconstructed from on/off switched elements.The mathematical tools of the previous 150 yearscould now be used to design systems that could not onlycalculate but also manipulate symbols and achieve resultsin higher mathematics (see the next entry, mathematicssoftware).Mathematics and Modern ComputersA variety of mathematical disciplines bear upon the designand use of modern computers. Simple or complex algebrausing variables in formulas is at the heart of many programsranging from financial software to flight simulators.Indeed, one of the most enduring scientific and engineeringlanguages takes its name from the process of translatingformulas into computer instructions (see fortran).Geometry, particularly the analytical geometry basedupon the coordinate system devised by Rene Descartes(1596–1650) is fundamental to computer graphics displays,where the screen is divided into X (vertical) and Y(horizontal) axes. Modern graphics systems have added 3Ddepiction and sophisticated algorithms to allow the rapiddisplay of complex objects. Beyond graphics, the Cartesianinsight that converted geometry into algebra makes avariety of geometrical problems accessible to computation,including the finding of optimum paths for circuit design.Design of computer and network architectures also involvesthe related field of topology. The fascinating field of fractalgeometry has found use in computer graphics and data storagetechniques (see fractals in computing).Aspects of number theory, often considered the mostabstract branch of mathematics, have found surprising relevancein computer applications. These include randomization(random number generation) and the factoring of largenumbers, which is crucial for cryptography.Mathematics also bears on computer networking withregard to communications theory (see bandwidth andShannon, Claude) and techniques for error correction.The Computer’s Contribution toMathematicsMathematics as a discipline is thus essential to its youngersibling, computer science. In turn, however, computer scienceand technology have enriched the pursuit of mathematicaltruth in surprising ways. As early as 1956, aprogram called Logic Theorist, written by Herbert Simon(1916–2001) and Allen Newell (1927–1992) demonstratedhow a program (that is, a collection of algorithms) couldprove mathematical propositions given axioms and rules.While these early programs worked on a somewhat hitor-missbasis, later theorem-solving programs producedsolutions different from the standard ones known to mathematicians,and sometimes more elegant. Thus the computer,which began as an aid to calculation, became an aidto symbol manipulation and to some extent an independentcreative source.Further Reading“Computers & Math News.” ScienceDaily. Available online.URL: http://www.sciencedaily.com/news/computers_math/.Accessed August 14, 2007.Henderson, Harry. Modern Mathematics: Powerful Patterns inNature and Society. New York: Chelsea House, 2007.Maxfield, Clive, and Alan Brown. The Definitive Guide to HowComputers Do Math. New York: Wiley-Interscience, 2005.McCullough, Robert. Mathematics for Computer Technology. 3rded. Engelwood, Colo.: Morton Publishing, 2006.Took, D. James, and Norma Henderson. Using Information Technologyin Mathematics Education. New York: Haworth Press,2001.Vince, John. Mathematics for Computer Graphics. 2nd ed. NewYork: Springer, 2005.mathematics softwareAs explained in the preceding article, computer sciencelooked to mathematics to create and verify its algorithms.In turn, computer software has greatly aided many levelsof mathematical work, ranging from simple calculations tomanipulation of symbols and abstract forms.At the simplest level, computers overlap the functionsof simple electronic calculators. Indeed, operating systemssuch as Microsoft Windows and UNIX systems includecalculator utilities that can be used to solve problemsrequiring a basic four function or more elaborate scientificcalculator.The true power of the computer became more evident toordinary users when spreadsheet software was introducedcommercially in 1979 with VisiCalc (see spreadsheet).Spreadsheets make it easy to maintain and update summariesand other reports generated by formulas. Later versionsof spreadsheet programs such as Lotus 1-2-3 and MicrosoftExcel have the ability to create a wide variety of plots andcharts to show relationships between variables in visualterms.