Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Cemela Summer School
Mathematics as language
Fact or Metaphor?
Diversions
John T. Baldwin
September 13, 2008
Goals
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Formal languages arose to remedy the lack of precision in
natural language.
1
Motivate with classroom examples the reasons for
developing a formal language for mathematics.
2
Interweave the definition of a first order language
adequate for mathematics
3
The interplay between natural language, ‘regimented
language’, and formal language
Language
Cues
Diversions
Outline
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
1 Language Cues
Language
Cues
Diversions
2 Diversions
parsing a non-native tongue
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
A rhombus is four sided figure with all sides of the same length
Which of 1-4 is not true in every rhombus?
1
The two diagonals have the same length.
2
Each diagonal bisects the two angles of the rhombus
3
The two diagonals are perpendicular.
4
The opposite angles have the same measure
5
All of 1-4 are true in every rhombus.
parsing a non-native tongue
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
A rhombus is four sided figure with all sides of the same length
Which of 1-4 is not true in every rhombus?
1
The two diagonals have the same length.
2
Each diagonal bisects the two angles of the rhombus
3
The two diagonals are perpendicular.
4
The opposite angles have the same measure
5
All of 1-4 are true in every rhombus.
Which of 1-4 is not true (in every rhombus)?
Let’s take a poll?
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Dr. Math,
What is the correct definition for a trapezoid? And why?
My questions come from the Math Department at Carroll
Middle School in SouthLake, Texas. Two of the math teachers
have found well-known publications with very different
definitions for a Trapezoid.
1) Trapezoid: Quadrilateral with at least 1 pair of sides parallel.
2) Trapezoid: A trapezoid is a quadrilateral with exactly one
pair of parallel sides.
Trape whatzis
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
What is a trapezoid, a trapezium?
Trape whatzis
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
What is a trapezoid, a trapezium?
A trapezoid (in North America) or trapezium (in Britain and
elsewhere) is a quadrilateral, which is defined as a shape with
four sides, which has a pair of parallel sides. Some authors
define it as a quadrilateral having exactly one pair of parallel
sides, so as to exclude parallelograms.
The exactly opposite concept, a quadrilateral that has no
parallel sides, is referred to as a trapezium in North America,
and as a trapezoid in Britain and elsewhere. (Wikipedia ?!?!?!)
defining Definition
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
What is a definition?
defining Definition
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
What is a definition?
An abbreviation
Definitions
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Can a definition be right or wrong?
Define: prime number.
Definitions
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Can a definition be right or wrong?
Define: prime number. Is 1 prime?
Is it just a matter of convention?
Theorem
Every natural number can be uniquely written as a product of
prime numbers.
km
n = p1k1 . . . pm
Thesis
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Formal language provides a precise way to describe
mathematical objects.
This ideal is a powerful tool for analyzing curriculum and
discourse.
Natural numbers
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
universe
0, 1, 2, 3, . . .
Language
Cues
operations
Diversions
+, ×, 0, 1
Relations
=, <
Patterns
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Consider a circle with n points on it. How many regions
will the circle be divided into if each pair of points is
connected by a chord?
Patterns
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Consider a circle with n points on it. How many regions
will the circle be divided into if each pair of points is
connected by a chord?
1
Is this question well-formed? That is does the answer
depend on the placement of the points.
2
Variant: What is the
maximum number of regions of a circle that you can make
by drawing
chords between n points on the circumference?
3
Guess the
formula? Be very careful.
Language
Cues
Diversions
Metaphysics or Epistemology
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
(a) The anti-realist often claims that the burden of proof is on
the epistemological front and challenges the realist by asking:
How is it that human beings are able to access information
about this mysterious platonic universe of atemporal, acausal
mathematical objects?
(b) The realist on the other hand, would often like to place the
burden of proof on explaining the universality of mathematics.
If mathematics is just a fiction, why is it so useful and universal
in nature?
Both the realist and the anti-realist would like to make us
believe that answering this question is such an impossible task
that the only reasonable conclusion is to adopt their point of
view.
Henrik Nordmark (on fom)
What is mathematics?
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Quoting ”Timothy Y. Chow” tchow at alum.mit.edu¿ Sat, 03
Mar 2007:
Some years ago it occurred to me that a possible definition of
mathematics is that anything that is *sufficiently precise* is
mathematics.
The term ”sufficiently precise” it itself not sufficiently precise
to count as mathematical, but perhaps it is sufficiently precise
to be a useful idea.
Thus mathematics, unlike most other fields of study, is
characterized not so much by its *subject matter* as by a
certain *threshold of precision*.
Sazonov
Let M be Mary!
Cemela
Summer
School
Mathematics
as language
Fact or
Metaphor?
John T.
Baldwin
Language
Cues
Diversions
Problem
The Golden Eagle Ferry is 150 feet long.
Cars are 20 feet long and Trucks are 30 feet long.
Each lane holds the same number of cars and trucks.
How many cars and trucks are in each lane?
Solution:
20C + 30T = 150.
Since C = T , I can write 20C + 30C = 150. So 50C = 150.
The last equation says 50 cars are 150 feet so each car is 3 feet
long.