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Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry
I.E. Addendum: What are Definitions?
I am always surprised by how difficult it seems to be for students to begin
to use definitions in the way a mathematician does--as perfectly solid
and fully adequate foundations for reasoning. They commonly assume more than
what the definitions state or feel
the need to prove more. For example, a parallelogram
is by definition a
quadrilateral with opposite sides parallel.
But asked what a parallelogram is, many students give more information, e.g.,
it's opposite sides are congruent. And asked to prove that a figure is a parallelogram,
often prove more than
that its opposite sides are parallel.
Natural concepts are
hazy, flexible and redundant. There are good reasons for this, for in many
natural situations these properties are desirable. When people communicate,
to the frames of those with whom they're communicating. When faced with novel
problems, people seek solutions by modifying existing cognitive patterns. And
as any engineer knows, redundancy can protect a system from faults and errors.
In mathematics, we we set up artificial conceptual systems that are clear,
rigid, and sparse. Such systems give us the power to find truths that natural
systems cannot see. There is a cost. To use the artificial systems, we must
be far more observant of detail, far more more sensitive to fine distinctions
than most people are naturally inclined. Sometimes, we need to suppress the
thoughts that spring from older, natural habits.
Yet, when we learn to use mathematical
concepts, we do not cease to reason in the natural way. Our natural reasoning
guidance and intuition as we craft demonstrations using the formal
concepts. Both kinds of thinking are important, but they are separate.
The formal system is the final arbiter of truth, and
Readings on the idea of a definition: Here are
three readings that I think provide useful insights into these issues, particularly
in relation to mathematics education.
- Poincaré. This reading
consists of excerpts form a classic essay on definitions in education that
is included in Poincaré's famous book of essays: Science and Method.
Poincaré considers the educational problems that arise from the peculiarities
of mathematical definitions.
and Vinner. Here
is a famous exposition by mathematics educators David Tall and Schlomo Vinner
on their distinction between "concept image" and "concept
ideas seem in accord with the Murphy reading (below). Concept
image is what people tend to have in mind when they think of something--a
Concept definition is the exact formal rule for the use of a term: "a
quadrilateral is a union of four segments, each of which meets two others
at its endpoints
and only at its endpoints." It may be very different from the concept
image, yet it is what counts in
reasoning. A comparison to the law might be useful. When you think of "trespassing"
you probably have specific mental images. The law defines trespassing in
a very precise way that may not accord with your mental images. When you
go to court, it's the law that counts, not your images. (Here are additional
comments by Tall.)
- Murphy. Excerpts form Gregory
L. Murphy, The Big Book of Concepts. Cambridge: MIT Press 2002.
This is a textbook on cognitive science. This chapter demonstrates that the "natural" way
in which concepts work is quite different from the way mathematical concepts
function. A mathematical definition for a concept gives necessary and sufficient
conditions for a thing to be an instance of that concept. ("Parallelogram: a
quadrilateral in which the both pairs of opposite sides are parallel." The
conditions are necessary: if something is not a quadrilateral
or if it is, but it fails to have opposite sides parallel, then it's not
a parallelogram. The conditions are sufficient if something
has the properties in the definition, then it's a parallelogram. Nothing
more is needed.) But natural concepts (like "dog" or "game")
cannot be specified by necessary and sufficient conditions. No matter what
properties you might list in order to give necessary and sufficient conditions
for being a dog, you will likely find some dogs that don't have all the properties
on your list, or some things that do, but are not dogs.
- Wu. This is a contemporary essay by
a mathematician on the use of definitions in mathematics and math education.