# Types of Definitions

Share Cognitive Science Implications for Teaching Mathematics
•
•
•
•
•
•

Scholars frequently organize definitions into five categories: Persuasive, Precising, Theoretical, Lexical, and Stipulative. The last two are of interest to the mathematics teacher and student. A common name for a lexical definition is an extracted definition because it is extracted from common actual usage. Extracted definitions have a truth value—they can be true or false. Definitions familiar to beginning students are generally extracted definitions.
Definitions used in mathematics are very different. Definitions in mathematics are always stipulative definitions. They are stipulative in the sense that they specify usage rather than report usage. Stipulative definitions do not have a truth value. They are neither true nor false—they just are! Beginning students are generally completely unfamiliar with stipulative definitions.
A definition in mathematics does not announce what has been meant by the word in the past or what it commonly means now. Rather it announces (stipulates) what will be meant by the word (or term) in the present work.
Unlike extracted definitions, stipulative definitions cannot be reliably learned by repeated exposure to instances of the definition. They must be memorized.
In normal everyday communication the receiver can use context to gain some understanding of an unfamiliar word. This is not the case with mathematical communication because the only definitions used in mathematics are stipulative (as opposed to lexical). Moreover, the definitions in mathematics are extremely precise. Therefore definitions in mathematics must be memorized to facilitate simple communication.

## 2 thoughts on “Types of Definitions”

1. Michael Bächtold says:

Interesting. It’s the first time I hear of this. Could you give an example of an extracted definition that is true vs an extracted definition that is false?

1. Pluto is a planet. A false lexical definition.
Pluto is a dwarf planet. A true lexical definition.

A bird is a creature that can fly. A false lexical definition.
A bird is any warm-blooded vertebrate of the class Aves, having a body covered with feathers, forelimbs modified into wings, scaly legs, a beak, and no teeth, and bearing young in a hard-shelled egg. A true lexical definition.

A real number is a rational number. A false lexical definition frequently used by beginning algebra students.
A real number is either a rational number or an irrational number. A true lexical definition.
In mathematics a lexical definition is true or false depending on its comparison with the relevant stipulative definition.

Lexical, or dictionary, definitions are reports of common usage (or usages). Such definitions are said to be reportive (alternatively, reportative) definitions. They are true or false depending on whether they do or do not accurately report common usage.
I have always considered this a philosophical consideration. The false lexical definition rarely occurs because we all try to insure that common usage conforms to reality.