80-110: The Nature of Mathematical Reasoning

Spring 1998

Class 5: 1/27/98

 

Definitions I

Defined and Undefined Terms

The homework assignment was to go to the library or your own bookshelves, and find two definitions, one from a non-mathematical theory, e.g., political science, sociology, drama, etc., and one from a mathematical theory, e.g., calculus, topology, etc. For each definition, write it out and identify the undefined terms just like you did with Euclid's definitions.

Several example students brought in were illustrative of exactly the ideas I wanted to get across. Two were Management Science and Gross Domestic Product (GDP). GDP is defined as the total value of all goods and services produced domestically in a given year.

What are the undefined terms, or primitives, here?

What counts as a service? What counts as "produced domestically"? Is a car whose parts are made abroad but assembled here "produced domestically"? Is a car that is assembled domestically, and whose parts are all produced here, except for the air filter, "produced domestically". The definition of GDP is only as useful as its primitives, which in this case are quite vague. If France applies a different standard for "produced domestically" than we do, than comparing our GDP to theirs is a problem.

In mathematics, this sort of defintional chicanery is a disaster and cannot be tolerated. Imagine the chaos that would ensue if all mathematicians used slightly different definitions for the same word. Suppose that Wyles, the Princeton mathematician who proved Fermat’s last conjecuture, used a theorem about non-abelian groups, but he meant something slightly different by non-abelian group than did the author of the theorem he cited. Is his proof still valid? Who knows?

Even and Odd

To get more of a feel for how definitions function, lets be more concrete. Consider two concepts you know well, odd and even.

Consider a proof of a familiar sort:

Claim: If p2 is even, then p2 is divisible by 4.

Proof

1) Assume p2 is even.

2) By lemma 1, p is even, and thus

3) there is a whole number r s.t. p = 2r

4) So p2 = 4r2

5) If r is a whole number, then r2 is a whole number, so there is a whole number q

s.t., p2 = 4q, so p2 is divisibile by 4. Q.E.D.

Notice how we used the definitions of "even" and "divisible by." We could not have done this proof if we didn't know that the definition of "even" is: a number P is even if there is another whole number R such that P = 2R. And we could not have done it if we didn't know that a number M is divisible by 4 if there is another number N such that M = 4N.

A definition is a precise reduction of a property or concept to logic and primitive properties or concepts.

For example, here are six examples of definition.

1) A number P is even if there is a whole number Q s.t. P = 2Q

2) A number P is even if it is not odd

3) A number P is even if there is no whole number Q s.t. P = 2Q + 1.

4) A number P is odd if there is a whole number Q s.t. P = 2Q + 1

5) A number P is odd if it is not odd

6) A number P is odd if there is no whole number Q s.t. P = 2Q.

Defintions are equivalent if the exact same set of objects satisfy them. For example, definitions 1,2, and 3 are equivalent definitions of even, and definitions 4,5, and 6 are equivalent definitions of odd.

A system of definitions should be grounded in primitive terms. So, for example, using defintions 2 and 5 to define even and odd would not work, because the system is cyclic and has no grounding.

In mathematics, definitions are conventions. We could just as easily have said that a number P is gork if there is a whole number Q s.t. P = 2Q + 1.

Mathematical definitions cannot be vague, i.e., it must always be clear when a definition applies and when it does not. For example, suppose I define the relation of "greater than" as: P is greater than Q if the perfectible numerical essence of P is closer to God's grace than is the perfectible numerical essence of Q.

Are Euclid's definitions adequate by these criteria?

 


Back to Lecture Notes Index