When I was in graduate school there was considerable interest in a then-new book, Metaphors We Live By, Lakoff and Johnson (1980). More recently I encountered Where Mathematics Comes From, Lakoff and Nunez (2001). Like Lakoff and Nunez, I believe that metaphor is central to both language and thought. I believe that mentalese is close to surface language and surface language is highly metaphorical. I have also long believed that the mathematics itself is a manifestation of an innate human cognitive architecture.
But how is it possible that thought is highly metaphorical and, at the same time, based on a cognitive architecture reflected in the precise logical structure of mathematics? The answer is, I believe, that the structure of metaphor is essentially the same as the general structure of mathematical thought. The central idea in both, I would argue, is roles and assertions.
Roles and Assertions
The concept of frame or script, in the sense of Fillmore, Minsky or Schank, seems central to the lexicon of natural language. Frames and scripts have roles. Words such as “buy” and “sell” invoke semantic roles such as the buyer, the seller, the thing being sold and the money transferred. The semantic roles associated with English verbs have been catalogued in Martha Palmer’s verbnet project and semantic role labeling has become one of the central tasks of computational linguistics.
In addition to roles, frames and scripts invoke assertions. For example, we have
before x sold y to z, x owned y
after x sold y to z, z owned y
But what is the relationship between the roles and assertions of lexical frames and metaphor? As an example, the lead story on news.google.com as of this writing starts with
The Israeli military continued its substantial military attacks around Rafah in southern Gaza on Sunday.
Consider the word “continue”. This is a very abstract word. It seems to have a role for an agent (Israel) and the activity that is continued (the attacks). But the assertions associated with this word make minimal assumptions about the agent and the activity. Some assertions might be
If during s, x continued to do y then before s, x was doing y
If during s, x continued to do y then during s, x was doing y
If we consider any two particular continuation events the relationship between them seems metaphorical. For example, consider “he continued to travel west” and “the pump continued to fail”. What these two continuations have in common is the above general assertions about contiuation. But we can think of this as forming a metaphor between traveling and the “activity” of failing. The real phenomenon, however, is that in both cases the assertions of the continuation frame apply. The assertions are polymorphic — they can be applied to instances of different types such as traveling and failing.
Object Oriented Programming
Roles and assertions also arise in object oriented programming. In a graphical user interface one writes code for displaying objects without making any commitment to what those objects are. A displayable object has to support certain methods (roles) that satisfy certain conditions (assertions) assumed by the GUI program. The same code can display avatars and pi-charts by treating both an appropriate abstract level. The GUI code (metaphorically?) treats an avatar as an image.
Mathematics
Roles and assertions arise in almost all mathematical concepts. A tree is a directed graph with nodes and edges (the roles) satisfying the tree conditions. The nodes can be anything — the concept of “tree” is polymorphic. The concept of tree is very general and can be used in many different situations. All we have to do is to find some way to view a given object as a tree. For example, a chess game can be (metaphorically?) viewed as a tree. My recently posted type-theoretic foundation of mathematics is organized around the notion of structure types (frames) each of which involves roles and assertions.
Summary
In this post I wanted to simply introduce the idea that metaphor and mathematics are related through the common structure of roles and assertions. There are many issues not discussed here such as the distinction between logical (hard) and statistical (soft) assertions and the role of embodiment (grounding). But I will leave these for later posts.
Machine Thoughts 