Wednesday, June 14, 2006

Mathematical cognition and sets a la Lakoff & Nunez: Part I, the mathematics of sets

It occurred to me as I got ready to write this post that starting with a relatively brief description of set theory and a bit of its historical development might be in order. Or at least provide you with a basis for looking at the set concept in Lakoff & Nunez.

Modern set theory, which is what we are concerned with here, is a creation of the late 19th century and early 20th century mathematicians. Set theory is concerned with sets of numbers. Numbers in this case are not only the common integers (positive and negative), but all number types including the reals, irrationals, algebraic, etc. However, there was a problem since the number types had not been defined in any rigorous manner. This meant that various mathematicians took it upon themselves to define the different numbers.

The first mathematician we are concerned with is Dedekind. Dedekind, in the introduction to his work Stetigkeit und irrationale Zahlen (1872), states that “…and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus…depend upon theorems which are never established in a purely arithmetic manner...” With the perception of this lack, Dedekind embarked on a research program into the arithmetization of analysis. This program derived and proved the theorems of analysis from basic postulates of the integers and then from the principles of set theory.1 Oddly enough, since nothing here seems to indicate it, Dedekind defined the irrational numbers. Part of this was his definition of something called a cut. It’s cool, but we’ll pass on the explanation. Just remember that set theory has crept into the mathematical lexicon.

The next mathematician in this short journey is Cantor. I love Cantor. I love Cantor’s infinite sets. Some of this showed up here when showing the cardinality (size) of different sets. At any rate, Cantor’s set theory appeared in print at the end of the 19th century (1895 and 1897). In this work, Cantor defines sets as, “any collection into a whole M of definite and separate objects m of our intuition or our thought.2 Keep in mind that Cantor has defined a set as a collection. This will be important later on.

Cantor’s set theory as well as his rather loose definition of a set led to a number of problems and paradoxes. Early in the 20th century, mathematicians began a formal axiomatization of set theory including the proof of the well–ordering theorem and the axiom of choice both attributable to Ernst Zermelo in the period from 1904 to 1908. The creation of the axioms of set theory eliminated the difficulties caused by Cantor’s earlier definitions. These axioms also became the basis for defining a mathematical set.3

According to Lay4 it is essentially impossible to define a set. It is, however, possible to describe the properties of sets and to indicate what things are not sets. There is a moving away from an intuitive definition towards a formalized mathematical system. This formalized system is the Zermelo-Fraenkel Axioms. First defined by Zermelo in 1908 and modified by Fraenkel in 1922, it is the basis for modern set theory. These axioms depend on two primitive ideas, both of which are undefined. They are the concept of “set” and the concept of “membership.”5 The concept of “set” has now moved from Cantor’s “collection” to an undefined primitive.

There are ten axioms that describe the properties of sets. The tenth is the axiom of choice which may or may not be necessary. It is a mathematical discussion that we won’t go into here. The ten axioms are as follows.6

1. The axiom of extension. Two sets are equal if and only if they have the same elements.

2. The axiom of the null set. There exists a set with no elements, and we denote it by Ø.

3. The axiom of pairing. Given any sets x and y, there exists a set whose elements are x and y.

4. The axiom of union. Given any set x, the union of all the elements in x is a set.

5. The axiom of the power set. Given any set x, there exists a set consisting of all the subsets of x.

6. The axiom of separation. Given any set x and any sentence p(y) that is a statement for all y member of x, then there exists a set {y member of x: p(y) is true}. (Note that this set is a subset of x so that Russell’s paradox does not apply.)

7. The axiom of replacement. Given any set x and any function f defined on x, the image f(x) is a set. (from Wikipedia: an image of a function consists of the output values of the mathematical function.7 Or if you prefer, think of the output as the results you get when you "do" the function.)

8. The axiom of infinity. There exists a set x such that Ø is a member of x, and whenever y is a member of x it follows that y union {x} is a member of x. (This guarantees the existence of an infinite set.)

9. The axiom of regularity. Given any nonempty set x, there exists y member of x such that the intersection of y and x equals Ø.

10. The axiom of choice. Given any nonempty set x whose members are pairwise disjoint nonempty sets, there exists a set y consisting of exactly one element taken from each set belonging to x. (This means, more or less, that you can pick any element out of each of the nonempty sets and combine them into a new set y.)

This pretty much covers the basic information you need to know on how mathematicians define sets. The only other thing of interest are Venn diagrams as they are a pictorial representation of sets. Read the Wikipedia entry here to learn more about these diagrams and to see some examples of Venn diagrams. Really, read this because the information will be useful when looking at the Lakoff & Nunez explanation of how we conceptualize sets.

1 Katz, V.J. (1998) A History of Mathematics. Addison-Wesley: Reading, MA. p 729-30.
2 Katz, V.J. (1998) A History of Mathematics. Addison-Wesley: Reading, MA. p 734.
3 Katz, V.J. (1998) A History of Mathematics. Addison-Wesley: Reading, MA. p 801-814.
4 Lay, S.R. (2005) Analysis: with an introduction to proof, 4th edition. Pearson Prentice Hall: Upper Saddle River, NJ. p. 90
5 Lay, S.R. (2005) Analysis: with an introduction to proof, 4th edition. Pearson Prentice Hall: Upper Saddle River, NJ. p. 91
6 Lay, S.R. (2005) Analysis: with an introduction to proof, 4th edition. Pearson Prentice Hall: Upper Saddle River, NJ. p. 92-95
7Wikipedia (2005). Image: specialized meanings. (


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