The Other Mathematics: Language, Logic, and Probability in the Footsteps of Boole and Venn

Leo Depuydt (Leo_Depuydt@brown.edu)

Description as of January 25, 2009

The unity of the forms of thought in all the applications of reason, however remotely separated, will one day be matter of notoriety and common wonder; and Boole’s name will be remembered in connection with one of the most important steps towards the attainment of that knowledge.

Augustus De Morgan

This independent project is human-oriented and not machine-oriented. It is also a test pilot for a possible class. It takes as its point of departure the undeniable fact that the brain produces language and thought and that we at this time have little or no clue as to how this happens. Finding out will be one of the great assignments of the twenty-first century. But that should not discourage anyone from exploring possibilities. And it goes without saying that the matter is of no small interest.

To do this, the hard evidence to which we are limited at this time is what comes out of mouths of speakers and the pens of writers. Brain scans are not detailed enough to discern the structure of thought and language and the turning on and off of microvoltages that make it happen. However, because people understand one another when they speak, there is every reason to believe that the basic structure of the brain waves that produce language and thought is reflected in the sound waves of speech, which can also be conveyed in other mediums like writing.

The basic starting hypothesis of the present project is that most everything that relates to how the brain produces language and rational thought is digital, that is, On (1) or Off (0). Emotions are something else. The flow of “juices” such as adrenaline, dopamine, and serotonin is presumably continuous and not digital. All this cannot be proven positively at this time because knowledge of the brain is still very incomplete. In any event, the mathematics needed to handle digital structure is that of Boolean algebra. It is in a sense the “other” mathematics.

At the outset of his Elements of Algebra, Euler describes the normal and well-known mathematics as the study of anything that is “capable of increase or diminution.” In the “other” mathematics, nothing ever gets bigger or smaller. For example, adding the set of teachers to the set of teachers still only gives you the set of teachers (t + t = t, not 2t). Anyone working on the hypothesis that the brain produces language and thought digitally therefore has but little choice to study the “other” mathematics and seek ways to apply it to language. Only one thing seems sure: There’s got to be a way! And in fact, it can be demonstrated that already a sizeable number of phenomena lend themselves so easily to such an analysis that there is hope for more.

Much of the present effort will be devoted to the study of two foundational works of the digital age, namely George Boole’s Investigation of the Laws of Thought (1854) and John Venn’s Symbolic Logic (1894). In 1937, Claude E. Shannon converted Boole’s ideas for electronic circuits and the digital age began. Putting these two works at the center should give focus to the project. By means of additional discussion of case studies, it would be further investigated how the other mathematics can be applied to the structure of language and thought.

The project otherwise has a certain provisional quality. It constitutes a beginning with hopes of more to come. The proposer has written on the subject and has incipient ideas (see, e.g., Name at www.gorgiaspress.com). But it lies in the nature of the subject matter that it is exploratory. It is not clear where it will lead. It is being developed as we go along. It is of course clear where efforts like this should ultimately lead, namely to turning the camera 180 degrees on to ourselves and see us function materially 100 percent as it happens with the help of genetics and Boolean algebra.

The focus will be on the structure of language and human thought and a sufficient level of Boolean algebra. But in the long run, there is more than one field to which projects like these can be of interest, just about all of which fall outside the competence of the proposer: linguistics, biochemistry, brain science, electrical engineering, and mathematics.