Anthony Pasqualoni grew up in Durham, Connecticut. He studied philosophy as an undergraduate and library science and computer science as a graduate student. He now works as a freelance programmer, mainly developing databases and websites for nonprofit organizations.
Project: Natural Language and Set Theory
This project is to understand the relationship between natural language and set theory. Two questions serve as avenues of approach:
How are notions underlying set theory (e.g. collection, membership, union, intersection) expressed in everyday speech?
How is natural language used to explain set theory and vice versa?
Regarding the first question, consider common statements such as “she loves me, she loves me not.” In this case “she” can refer to a set of an infinite number of possible real or fictional women. “Loves me” comprises a set of objects that is disjoint with “loves me not,” and both sets can be seen as subsets of the set of all possible relationships.
Regarding the second question, reflections on both set theory and everyday language employ overlapping terms. For example, we speak of “natural numbers” (which Cantor opposed to “real numbers”) and “natural languages.” Both expressions suggest that a common ground underlies both set-theoretical language and everyday speech. This is also reflected in the fact that the act of counting is expressed by commonplace sets of words such as “one, two, three.”
Writings by the founders of set theory indicate a deep connection between natural language and the language of mathematics. The following is a quote from Cantor:
The conception of number which, in finito, has only the background of enumeral, splits, in a manner of speaking, when we raise ourselves to the infinite, into the two conceptions of power… and enumerable…; and when I again descend to the finite, I see just as clearly and beautifully how these two conceptions again unite to form that of the finite integer.
Conversely, studies of natural language, including those involving everyday speech, use Cantor’s set-theoretic notions. In “Three Models for the Description of Language,” Chomsky states in his introduction that “a grammar is based on a finite number of observed sentences (the linguist’s corpus) and it ‘projects’ this set to an infinite set of grammatical sentences….” In addition, Turing machines are defined by a finite set of symbols written on an infinite tape. So on the one hand natural language is used to define set theory; on the other, set theory is used to model and describe natural language both as an object of empirical observation and as the subject of computation.
As a preliminary exploration it might be useful to measure the length of English names for the natural numbers. I suspect that the graph of the number of letters in each name will reflect a tension between the mathematical principles underlying our language and the more random (and less artificial) aspects. For example, the name “one hundred twenty” is longer than “twenty” by nature of the fact that it represents a larger quantity, but the name “one” has just as many letters and phonemes as the word “two.”
The above can serve as an introduction to a more specific question: how natural language interacts with the language of set theory. One possible approach is to list all possible symbolic expressions with a limited number of variables and primitive operators (union and intersection) and generate English-language sentences that conform to each expression. This could give ideas on how to further pursue the connection between set operators and English sentences.
Hasse Diagram of Power Sets
Favorite Three-Color Cellular Automaton