Let's begin by defining “science.” According to the Collins dictionary, science is “systematized knowledge derived from observation, study, and experimentation carried on to determine the nature or principles of what is being studied.” Under this definition, linguistics, the study of language, is certainly a science, for it is 1) systematic; 2) based on study, observation, and experimentation; and 3) focused on determining the nature and principles of its object of study. We'll examine each of these characteristics in turn.
First, linguistics is systematic. Linguists examine the various elements of language in an orderly, precise, well-regulated manner. Some key elements include sounds and sound systems (phonetics and phonology), the forms and structures of words (morphology), the relationships between words in phrases and sentences (syntax), the meanings of language (semantics), the use of language in context (pragmatics), the role of language in human development and function (psycholinguistics), and the development of language over time (historical linguistics).
These systematic examinations draw their data from the observation of language, both spoken and written. Linguists study speakers directly as well as historical and contemporary texts to create hypotheses about language, to uncover evidence that supports (or refutes) these hypotheses, and to draw conclusions. Some linguists even conduct experiments when, for instance, they study how language develops in children or when they examine linguistic patterns. In these activities, linguistics is very much like other sciences.
Ultimately, linguists seek to discover the nature and principles of language. They want to find out what language is, how it works, why it has meaning, how it changes and develops, how it is used, and how it affects and is affected by the human beings who use it. New discoveries and theories are constantly shifting the picture of language as linguists practice their science and learn more and more about the ways humans communicate.