The Nature of Definition

in Mathematics and Science

In natural language, a definition is intended to give sense of meaning and context to words. This is generally done by providing synonyms and examples of usage. Our paradigm for language definitions is a dictionary. Webster’s says a definition is “explanation of the meaning or meanings of a word”; The American Heritage Dictionary calls it “a statement of precise meaning”. As a second item, American Heritage adds that to define is to “describe the basic qualities of”. This latter notion is consistent with definition in sciences. Science and mathematics define objects and properties by providing criteria for identification for assessing whether an example meets, or fails to meet, the criteria to be that object or have that property.

In taxonomy, the categorization of living things is made according to presence or absence of specified basic qualities. Mammals, for example, are warm-blooded vertebrate animals that nourish their young with milk. All these characteristics are necessarily present in any mammal. When we note that mammals include such animals as cats, dogs, humans, mice, whales, we are giving examples of mammals, not defining them. All mammals have some hair; this provides an equivalently strong (necessary) identification criterion, only mammals have hair. (It is a characteristic less pertinent to the name, though.) Hairy animals have all the defining characteristics of mammals. When we note that most mammals bear their young alive, we are providing a stronger criterion that can identify some mammals that will then have all the defining characteristics of mammals

In mathematics, an even number is an integer that has no remainder upon division by the number 2. This provides testable characteristics; every number meets or fails these defining criteria. Examples of even numbers include 2, 4, 6, 0, -2, and 324. When we note that even numbers always have last digit (in the ones column) 0, 2, 4, 6, or 8, we are noting an equivalent condition that can identify even numbers. When we note that for counting numbers, a set of an even number of objects can always be arranged in pairs, we are providing a stronger criterion that can identify some of the even numbers.

The “concrete operational” response to “what is a mammal?” or “what is an even number?” will be a list of examples. Asking for clarification, “how do you know?” will generally result in a strong identifying criterion (sufficient condition). The “formal operational” stage will allow for elicitation of defining characteristics or their equivalent. The table below shows our two examples through this progression.

	Mammal	Even Number
What is a…	cat, dog, human	2,4,6, …, 10, 12
How do you know?	has live babies, not eggs like birds or reptiles or fish	that many objects can be evenly paired
How is it defined?	females have mammary glands that provide nourishment for young	division by 2 results in remainder of 0
How else can you identify it?	all mammals, and only mammals have hair … somewhere	last digit must be 0,2,4,6, or 8

In mathematics, we would call these characterizations examples, sufficient criterion, defining criterion, and identifying criterion, respectively. In mathematics texts, generally a definition is given, then some examples, then some propositions or theorems that provide alternative criteria.

Here is a typical algebra example.

Definition: An additive identity is a number that, when added to another number yields that number. For any number x, x + (additive identity) = x.

Example: Zero is an additive identity; 0+x=x.

Proposition: Zero is the additive identity;

0 = 0 + (additive identity) = (additive identity).

property of identity from example