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

Where I try to explain this stuff and hopefully don't completely screw it up.

In the preface to their book, Lakoff & Nunez have this to say about how our ideas are shaped.

One of the great findings of cognitive science is that our ideas are shaped by our bodily experiences – not in any simpleminded one-to-one way but indirectly, through the grounding of our entire conceptual system in everyday life. The cognitive perspective forces us to ask, is the system of mathematical ideas also grounded indirectly in bodily experiences? And if so, exactly how?¹

Their explanation of how we conceptualize and handle mathematical sets is placed within this idea with the necessary grounding of the concepts rooted in conceptual metaphors. You’re probably thinking of metaphor as you learned it in high school English. This is similar, but different. These metaphors are part of your unconscious cognitive structure – you are unaware of them, but they structure how you think and relate to the world.

So, what are the conceptual metaphors and how are they grounded? This is, I hope, a simple explanation. Let’s start with image schemas. Think of a schema as a kind of script or guide for understanding how the world works, how you interact with the world. An image schema is used to describe and understand spatial relationships. They are apparently universal conceptual primitives.² The particular image schema of importance to the understanding of sets is that of the container schema.

The Container schema has three parts: an Interior, a Boundary, and an Exterior. This structure forms a gestalt, in the sense that the parts make no sense without the whole…This structure is topological in the sense that the boundary can be made larger, smaller, or distorted and still remain the boundary of a Container schema.³

Lakoff & Nunez further state that image schema have a special cognitive function in that they are both perceptual and conceptual, thus providing a bridge between our language and reasoning and our vision and visual/spatial perceptions. You can see milk as in the glass (with glass as the container) or bees in the garden (where the garden is a non-physical container). Lakoff & Nunez further note that this type of image schema has a built in spatial logic because of its image-schematic structure.⁴

The following is provided as an example, with an illustration that includes Venn diagrams (remember those?), of the built in spatial logic of the container schema.

1. Given two Container schemas A and B and an object X, if A is in B and X is in A, then X is in B.
2. Given two Container schemas A and B and an object Y, if A is in B and Y is outside of B, then Y is outside of A.

When you look at the Venn diagram that illustrates these two statements, the spatial logic should be obvious and the schemas, with their built-in logics, can function as the concept and these schemas can also be used directly when reasoning about the concept. In theory, you can replace the various mathematical symbols that indicate membership in a set with this container image schema.⁵

Lakoff & Nunez continue with a description of the roles of various neural circuitry in generating these image schema before returning to the idea of containment.

The concept of containment is central to much of mathematics. Closed sets of points are conceptualized as containers, as are bounded intervals, geometric figures, and so on. The concept of orientation is equally central…The concepts of containment and orientation are not special to mathematics but are used in thought and language generally. Like any other concepts, these arise only via neural mechanisms in the right kind of neural circuitry. It is of special interest that the neural circuitry we have evolved for other purposes is an inherent part of mathematics, which suggests that embodied mathematics does not exist independently of other embodied concepts used in everyday life. Instead, mathematics makes use of our adaptive capacities – our ability to adapt other cognitive mechanisms for mathematical purposes.⁶

Now let’s take a look at the metaphorical constructs Lakoff & Nunez use when talking about mathematics in general and sets in particular. These constructs come in two flavors: grounding metaphors and linking metaphors. Although both are defined below, only the grounding metaphor will be used here when describing the cognitive/conceptual ideas underlying sets.

Grounding metaphors yield basic, directly grounded ideas. Examples: addition as adding objects to a collection, subtraction as taking objects away from a collection, sets as containers, members of a set as objects in a container. These usually require little instruction.

Linking metaphors yield sophisticated ideas, sometimes called abstract ideas. Examples: numbers as points on a line, geometrical figures as algebraic equations, operations on classes as algebraic operations. These require a significant amount of explanation.⁷

These metaphors preserve inferences and ground the understanding of mathematics in prior understandings of every day physical activities. A primary method for metaphor to preserve these inferences is through the image-schema structure. The example given is that of forming a collection or pile of objects and then conceptualizing that collection as a container – “that is a bounded region of space with an interior, an exterior, and a boundary – either physical or imagined.” Lakoff & Nunez suggest that when numbers are conceptualized as collections, the logic of collections is projected onto the numbers.⁸

This should lead to the metaphorical concept of a set as first a collection and then projecting the container metaphor onto the numerical collection. But a further metaphor is needed. One that allows us to look at how sets are defined as classes within the container metaphor and one that allows us to understand the set axioms within this framework.

Let’s back-up a few steps and look at the idea of conflation. Conflation is the simultaneous activation of two brain regions – two kinds of experiences – which when activated at the same time produce a single complex experience (i.e., affection as warmth). Conceptual metaphor creates a cross-domain mapping that explains this experience and preserves the structure of image schema. In terms of sets, what is and how does this cross-domain mapping occur? Begin with the spatial inference – container – and map this to an abstract idea – category. The mapping occurs in this manner. (Note that mappings are unidirectional. You go from the concrete to the abstract.)⁹


Categories are Containers

Source Domain: Containers	-->	Target Domain: Categories
Bounded regions in space	-->	Categories
Objects inside the bounded regions	-->	Category members
One bounded region inside another	-->	A subcategory of a larger category

You can further delineate and map the container concept to the abstract concept of class. The mapping, shown below, brings us closer to an understanding of the metaphor/image-schema and underlying cognitive process that provides the understanding and manipulation of the abstract notion of sets. Starting with the container schema and its representation as a bounded region of space, move to the intuition that something is either inside or outside of this region. You now have a metaphorical concept of class where something inside the container belongs to the class while something outside the container does not. Through a metaphorical mapping, a grounding metaphor of Classes as Containers arises.¹⁰


Classes are Containers

Source Domain: Container Schemas	-->	Target Domain: Classes
Interiors of Container schemas	-->	Classes
Objects as interiors	-->	Class members
Being an object in an interior	-->	The membership relation
An interior of one Container schema within a larger one	-->	A subclass in a larger class
The overlap of the interiors of two Container schema	-->	The intersection of two classes
The totality of the interiors of two Container schema	-->	The union of two classes
The exterior of a Container schema	-->	The compliment of a class

Lakoff & Nunez observe that the Zermelo-Fraenkel axioms with the Axiom of Choice do not characterize sets as containers. Sets and membership within sets are undefined primitives. However, they argue that most of us conceptualize sets in terms of a container schema and that this conceptualization is consistent with the axioms. In addition, the Venn diagrams used when first learning set theory explicitly make use of the container schema. Lakoff & Nunez also point out that the use of the container schema removes the paradox addressed by the Axiom of Choice. If a set is conceived of as a container, a set can not be a member of itself since it is nonsensical to conceptualize a container as being inside itself.¹¹


All references are to: Lakoff, G. & Nunez, R.E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books: New York: NY
¹p. xiv
²p. 30
³pp. 30-31
⁴p. 31
⁵p. 31-33
⁶p. 33
⁷p. 53
⁸p. 54
⁹p. 43
¹⁰p.122-123
¹¹p. 145

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 19^th century and early 20^th 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 20^th 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. (http://en.wikipedia.org/wiki/Image)

Have we got outlines

I thought that presenting three outlines might give you an idea of where I've been going over the past couple of years. I could make myself more crazy than I am trying to get these outlines to format properly, but I'm not going to.

The first is the outline for my original presentation. The presentation very closely followed this outline.

Development of Mathematical Thinking and Its Effect on Teaching Math to Adolescents
April 20, 2004

I. Introduction
II. Definitions
III. Evolution & Biology
IV. Biological Hardwiring
V. Is there a Math Gene?
VI. From Arithmetic to Mathematics
VII. How we create and do mathematics
VIII. How the embodied mind brings mathematics into being
IX. Mathematics and the embodied mind
X. Conclusions
XI. Teaching Mathematics
XII. If Devlin and Lakoff/Nunez are right . . .
XIII. The Mathematician’s Brain (this was a joke)

The second is the original outline for a paper I wrote during the spring 2005 semester (completed just a week less than a year after the first presentation). This was a starting point for the paper, but the end result was somewhat different - a lot more stuff from anthropologists and archaeologists as well as a brief analysis of mathematics as language. However, you can still see where I was going a year later.

I. Introduction
a. Appearance of Abstract Symbols
i. Where
ii. How common are they?
b. Do they indicate cognitive processes?
c. Do evolutionary psychologists address this question?
II. Presentation of research on Paleolithic cave art
a. What appears in the art
i. Figural/animal representations
ii. Types of abstract symbols
b. Frequency of representations
III. Presentation of research on Paleolithic artifacts
a. Counting bones
b. Sculpture and marked stones
IV. Evolutionary Psychology
a. Presentation of research
V. Conclusion
a. Do Paleolithic symbols indicate the start of mathematical cognition?
b. Does the literature support this conclusion?

And finally the original outline for my project proposal written in December 2005. As I started writing the paper, this outline began to resemble more of a sketch than an outline. As the paper has gone through various revisions and additions, the whole process has become more detailed and included more information from people who actually know what they're talking about (unlike me).

I. Introduction
a. Mathematical cognition – theories
i. Majority look at number sense and numerosity
ii. Lakoff and Nunez (2000) – theory of embodied cognition applied to mathematics
1. metaphor
2. schema
3. blending of constructs
b. specific construct
i. containment
ii. containment as class
c. underlying cognitive construct for set theory
d. previous studies focus
i. most are linquistic
1. studies show existence
2. studies are based on interpretation of English language texts
ii. study of containment concept in young children (Casasola & Cohen, 2002)
1. evidence for in children at ages 10 months and 18 months
2. study shows development from ages 9 – 11 months to 17 – 19 months
iii. study by Fischbein & Beltsan (1998)
1. flaws
a. using collection instead of containment – different metaphor
b. testing set knowledge instead of cognitive process
2. study results
a. in students
b. in teachers
c. shows correlation between student knowledge and teacher knowledge
e. present study
i. investigate presence of containment and containment as class metaphors
1. general student population
2. math majors before and after mathematics class that includes set theory
II. Methods
a. Participants
i. Approximately 40 students
1. total determined by number of students enrolled in math-225 (includes introduction to set theory)
2. have equal number of students not in math program
ii. half of subjects are math majors enrolled in math-225
iii. half of subjects non-math majors – control group
1. must not have taken math-225 or equivalent
b. Materials
i. Test designed to show evidence of the containment and containment as class metaphor
1. test to be created
2. reliability study – test-retest on general student population
3. validity study – method to be determined
ii. watch or stopwatch
c. procedure
i. two subject groups tested independently at start of semester and again at end of semester with same procedure
ii. provide informed consent
1. if professors have agreed, inform subjects of extra credit for participating
2. explain that test will be given a second time, may opt out at any time
iii. describe test without indicating what looking for – do not want to prime subjects
iv. give test to each group
v. note time to complete (start to completion by last subject) for each group
vi. collect tests
vii. thank subjects
viii. repeat steps ii through vi for retest at end of semester
ix. analyze data and write-up findings for presentation
1. apply findings to math education
x. meet with each group
1. explain cognitive process looking for
2. present brief description of results
3. thank subjects for their participation
III. Expected findings
a. Analysis
i. Results of test and retest to be analyzed between groups and within groups for evidence of containment and containment as class
ii. Use a factorial design (group x test) – look at results within and between students and control group
b. Hypotheses
i. Test(math-225) < Retest(math-225)
ii. Test(math-225) = Test(control)
iii. Retest(math-225) > Retest(control)
IV. References

So there you have it, the growth of an obsession. *grin*

An Introduction or maybe an explanation

In my last semester of the math degree I took a child and adolescent psychology course to complete my general education requirements. The first half of the semester consisted of lectures from the professor. The second half of the semester was a series of presentations by the students. We were supposed to work in groups of 3 or 4, research a topic (approved by the professor), and do a 1 hour presentation. I really really really did not want to work with a group of students on this. I already had a class where every week I had to meet with a different group of students to prepare and do a presentation on the week's topic and a math class where I was working with another student on a presentation. Yet another student group presentation was straining my patience and my schedule - work so gets in the way sometimes. *grin* And, to be perfectly honest, the topics the kids in the psych class wanted to talk about bored me to tears. It's not that eating disorders, body image, and sports aren't important, but ...

So I talked the professor into letting me do the research and the presentation on my own and got him to approve "mathematical thinking" as the topic. I had a blast and no trouble whatever talking for an hour on the topic. Unfortunately for the world at large, a new "passion" was awoken. The semester ended, I graduated, and I kept on researching. However, a small problem arose - I was reading stuff I didn't always understand. I decided to fix that by going back to school for a degree in psychology. And whenever possible I used the research I was doing on my own in assigned papers. The honors research project grew out of that.

The project started with a book I read for the original presentation. The book is Where Mathematics Comes From by Lakoff and Nunez (2000).

As time went on, I ended up with a few problems with this book. I felt like the authors had jumped past a few basics by going to more advanced mathematics. Why weren't they talking about the maths, like Euclidean geometry, that are the oldest and most basic? I was starting to feel as if the act of communicating math was being confused with the act of doing math. And finally, after a lot of reading, I was feeling as if one of the authors was trying to be all things to all areas of psychology. On top of all of this, the authors said that there were studies proving their claims, but I wasn't finding the studies. At least not specifically in relation to mathematics.

So when it came time to present a proposal for a study, I decided to choose one of the ideas from this book and see if I could "prove" it. Because I love Cantor's transfinite sets, I chose a piece of their ideas on how we conceptualize sets.

And that's my story. Next time I'll explain the ideas behind sets a la Lakoff and Nunez.