Wolfram Computation Meets Knowledge

Wolfram Summer School


Thomas Ferguson

Summer School

Class of 2012


I am presently a doctoral candidate in philosophy at the City University of New York Graduate Center and received a BA in philosophy from the University of Illinois at Chicago. The topic of my dissertation is a formal study of two varieties of logics—containment and connexive logics—that have not traditionally received a great deal of attention. My main research interest is presently the formal study of non-classical logics and their applications to philosophical problems and the foundations of mathematics. I am also interested in the connection between computation and non-classical logics and the question of what role empiricism can play with respect to mathematics.

Project: Salience and Syntactical Complexity of Intermediate Logics

Suppose that one enumerates the theorems of equational propositional logic by increasing syntactical complexity and calls a theorem that is not a consequence of the set of theorems preceding it “informative.” On page 817 of A New Kind of Science, Wolfram makes the observation that these “informative” theorems are those that have historically warranted, e.g., receipt of a distinguished title such as “Law of Excluded Middle.” In other words, the set of syntactically simplest theorems sufficient to establish completeness of the system tend to likewise be historically salient.

The observation, however, is limited to the Boolean propositional logic, i.e., that whose semantics are supplied by Boolean operations. In the past century, a host of systems of logic in competition with the Boolean interpretation has been introduced, the most prominent member of which—intuitionistic logic—corresponds to Brouwer’s constructivist philosophy of mathematics. Such logics tend to be fragments of the Boolean logic; hence some theorems provable modulo the Boolean logic will not be provable modulo one of its fragments. As an immediate consequence, the list of theorems “informative” with respect to such a logic will necessarily differ from that of the Boolean case.

My project is to investigate whether the coincidence of “informativeness” and historical salience holds in general for such fragments of the Boolean logic. I plan to employ the implementation of the Waldmeister automated theorem prover in Mathematica to identify such “informative” theorems with respect to fragments of the Boolean logic and determine whether this phenomenon is a facet of something more general in the space of propositional logics or rather a feature peculiar to the Boolean case.

Favorite Four-Color, Four State Turing Machine

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