Infinity is in the Eye of the Beholder

Abstract

There are clear senses in which natural languages are finite. It is generally assumed that human brains and hence linguistic processing ability is finite. If we assume that a sentence used by a human must fit in memory, this implies that there is some upper bound on the number of sentences that a human being can understand or produce. The number would be extremely large, but would be finite nonetheless. It is possible that a person might produce or understand a sentence by processing it in pieces without ever having the entire sentence in memory, but that would require either extensive use of external storage, such as pencil and paper, or else limit the complexity of natural languages to that of finite-state languages. Moreover, other processing constraints place very small upper bounds on the sizes of sentences that humans can process. For example, humans seem incapable of processing clauses nested to arbitrary depths. Without the aid of pencil and paper, humans lose track of certain types of nested structures once the depth of nesting exceeds about five [Miller (1956), Bach et al. (1986)]. Perhaps the most compelling of all senses in which language is finite is the simple fact that, at any point in time, an individual, or the entire community of human beings, will have experienced only a finite number of sentences.

This research was supported in part by NSF grant DCR-8604031. I thank Alexis Manaster-Ramer and Gila Safran-Naveh for helpful discussions on this material. Part of this work was done while visiting the Computer Science Department at the University of Cincinnati.