What does it mean to define something?

It seems to me that there are two ways to define a term. One is to explain its meaning in terms of other terms. The intent is to convey an idea. This is the common interpretation of the term to define. For example, the American Heritage Dictionary gives these definitions (among others) for definition.

A statement of the meaning of a word, phrase, or term, as in a dictionary entry.

The act or process of stating a precise meaning or significance; formulation of a meaning.

This sense of definition assumes that we are operating within a real of beings who already have ideas and that we are attempting to convey a new (or more precise version of an existing) idea in terms of other ideas (or other experiences) that are available for use. There is no fixed set of primitives in this sense of definition.

The second is to assign some properties to a symbol within a formal symbol manipulation system. In this sense no idea is presumed. The meaning of a term is no more or less than its symbolic definition. For example here are Peano's Axioms as expressed in Wolfram's MathWorld. [Italics added.]

1. Zero is a number.


2. If a is a number, the successor of a is a number.


3. Zero is not the successor of a number.


4. Two numbers of which the successors are equal are themselves equal.


5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

Sufficiently formalized these axioms define the terms zero, number, successor, and equal. (These definitions presume that we have predicate calculus—so that we can use variables in quantified expressions and so that we know what every means—and a workable definition for set.)

They don't do it by attempting to cause ideas to arise in the mind of the reader. They do it by formulating a set of formal rules that will relate these terms to each other in ways that we find compatible with our intuitive sense of these terms.

But our intuitions aren't part of these definitions. The definitions are intended to stand on their own, i.e., to be such that a computer or other mechanical reasoning device could work with them without our assistance.

Definitions of this second sort formalize levels of abstraction. For any level of abstraction, if sufficiently well understood, it should be possible to formulate rules that relate the abstractions (types and operations) defined at that level of abstraction. Of course, most levels of abstraction are too complex for such formalizations. But the point is that even at best all one can do with a level of abstraction is to define its terms in terms of themselves and each other. One shouldn't expect new terms that come into being when representing concepts relevant to a level of abstraction to be defined in any other terms.

Since a level of abstraction is (by definition) implemented in terms of lower level operations, it will always be possible to show how the relationships among the terms are realized. But those implementations are not definitions. And as is well understood, implementations are not the point. Multiple implementations of a level of abstraction are possible as long as the required inter-relationships among the terms are preserved.

This issue comes to mind because Aaron had asked me to define qualia. I essentially gave a short and ad hoc version of the answer elaborated above, i.e., that either (a) to define qualia as an idea that someone could understand would have to rely on the fact that the person already experienced them or (b) to define qualia as a term that makes sense on the subjective experience level of abstraction, i.e., in terms of other terms at the subject experience level of abstraction, probably wouldn't (seem to) satisfy the request. (Besides, I doubt that I could do it anyway.) So either way, it would be pretty difficult to define qualia in a way that would seem responsive to the question.