abstractmath.org
GLOSSARY
Posted 13 January 2009
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
generalization
To generalize a mathematical
concept C is to find a concept C' with
the property that instances of C are also instances of C'.
Examples
a)
,
for arbitrary positive integer n, is
a generalization of
.
One replaces the ordered pairs in
by ordered n-tuples,
and much of the arithmetic and spatial
structure (except for the representation of
using complex numbers) and even some of our intuitions carry over to the more general
case.
b)
The concept of abstract vector space is a generalization of . To get it you forget that the elements
of
are n-tuples
of real numbers and your produce a different, more abstract definition obtained
by finding properties of
such as the existence of addition,
multiplication, scalar multiplication, and so on, and using these properties as
axioms for the new concept of vector space.
Expansive generalization
Example (a) is an example of an expansive generalization, obtained by changing a datum in the definition of a concept (the dimension 2 in this case) into a parameter.
Reconstructive generalization
Example (b) is
obtained taking the concept and reconstructing
it, producing a
very different-looking definition that includes all the original examples and
others as well. This is reconstructive generalization.
Both expansive
and reconstructive generalizations, if done correctly, take a concept and introduce
another concept that includes all the examples of the original concepts and (in
general) others as well. These are legitimate
generalizations.
Generalization from examples
It can happen that all the examples you know of a specific concept have a certain property P not required by the definition. If you conclude from this that all the examples of the concept have property P you are engaging in generalization from examples.
Generalization from examples is an illegitimate mathematical method.
It does not always give true statements about math objects.
Example
The statement “An infinite sequence gets close to its limit but never reaches it” is a MYTH. Many newcomers to abstract math believe it because in the typical examples you see of limits of sequences the elements of the sequence in fact are never equal to the limit. For example
but 1/n is never equal
to 0.
On the other hand,
but every fourth term is zero. The sequence starts out like this:
The names “expansive” and “reconstructive” are due to Harel and Tall.
Remark
The usual meaning of the word “generalization” in colloquial American English is generalization from examples, often used in a derogatory way. When mathematicians refer to a generalization it usually means an expansive or reconstructive generalization and has no derogatory intent. See cognitive dissonance..
give
Give is used in several ways in math English. often with the same sense it would be used in any academic text (“we give a proof...
”, “we give a construction ... ”).
General academic meaning
Example
“We give a proof…”
This is seen in general academic prose.
Produce a description of an example
To “give a object” means to describe it sufficiently that it is
uniquely determined. A phrase of the form “give an X such that P”
means describe a object of type X that
satisfies predicate P.
Example
“Problem: Give a function of x that is positive at x = 0.”
Correct answers to this problem could be “the
cosine function” or “the function ” .
Example
You could provide an answer for the problem in the preceding example by
saying:
“the function given by f(x) = 2x
+ 1.”