Mathematics in Computer Science

Abstract

This text explains how to use mathematical models and methods to analyze prob- lems that arise in computer science. Proofs play a central role in this work because the authors share a belief with most mathematicians that proofs are essential for genuine understanding. Proofs also play a growing role in computer science; they are used to certify that software and hardware will always behave correctly, some- thing that no amount of testing can do. Simply put, a proof is a method of establishing truth. Like beauty, truth some- times depends on the eye of the beholder, and it should not be surprising that what constitutes a proof differs among fields. For example, in the judicial system, legal truth is decided by a jury based on the allowable evidence presented at trial. In the business world, authoritative truth is specified by a trusted person or organization, or maybe just your boss. In fields such as physics or biology, scientific truth1 is confirmed by experiment. In statistics, probable truth is established by statistical analysis of sample data. Philosophical proof involves careful exposition and persuasion typically based on a series of small, plausible arguments. The best example begins with Cogito ergo sum, a Latin sentence that translates as I think, therefore I am. This phrase comes from the beginning of a 17th century essay by the mathematician/philosopher, Ren e Descartes, and it is one of the most famous quotes in the world: do a web search for it, and you will be flooded with hits.