Transcript Document

Mathematical Association of America
MathFest: The Klein Project
Pittsburgh, PA
Issues in the Transition from
Concrete to Formal Mathematics
7 August 2010
Susanna S. Epp
[email protected]
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“Elementary Mathematics from an Advanced Standpoint”
by Felix Klein (1908, 1924)
When thinking about the relation between developments in
advanced mathematics and K-12 education, Klein says that two
questions should customarily be addressed:
1. “How much of all this should be taken up by the schools?”
2. “What should the teacher and what should the pupils know?”
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“Modern” Logic for Mathematics Instruction
 Introduction of quantifiers  and 
 Description of a variable as a placeholder, similar to a pronoun
 Distinction between free and bound variables
 Formulation of inference rules for quantified statements,
concept of “natural deduction”
 Careful thought about problems that arise when variables are
used to express statements involving both  and 
 History: Quantifiers were introduced formally in the late 19th
century. Significant further development with relevance to
mathematics education occurred into at least the 1950s. - Frege,
C. S. Peirce, Schröder, Peano, Hilbert, Ackermann, Whitehead,
Russell, Gödel, Gentzen, Jaśkowski, Tarski, Quine, Church, Copi,
Montague, Suppes, et al.
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The Transition to Operations with Letters
Felix Klein: This represents such a long step in abstraction that
one may well declare that real mathematics begins with
operations with letters.
A. N. Whitehead: “The ideas of ‘any’ and ‘some’ are introduced to
algebra by the use of letters.…it was not till within the last few
years that it has been realized how fundamental any and some are to
the very nature of mathematics…”
Alfred Tarski: “Without exaggeration it can be said that the
invention of variables constitutes a turning point in the history of
mathematics; with these symbols man acquired a tool that prepared
the way for the tremendous development of the mathematical
science and for the solidification of its logical foundations.”
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Variables as Placeholders/Pronouns
G. Frege (1893): The letter ‘x’ serves only to hold places open for a numeral
that is to complete the expression… This holding-open is to be understood
as follows: all places at which ‘ξ’ stands must be filled always by the same
sign, never by different ones. I call these places argument-places…
W. V. Quine (1950?): The variables remain mere pronouns, for crossreference; just as ‘x’ in its recurrences can usually be rendered ‘it’ in verbal
translations, so the distinctive variables ‘x’,’y’, ‘z’, etc., correspond to the
distinctive pronouns ‘former’ and ‘latter’, or ‘first’, ‘second’, and ‘third’, etc.
A. Church (1956): …a variable is a symbol whose meaning is like that of a
proper name or constant except that the single denotation of the constant
is replaced by the possibility of various values of the variable.
P. Halmos (1977): ‘He who hesitates is lost’. In pedantic mathematese this
can be said as follows: ‘For all X, if X hesitates, then X is lost’.
S. Pinker (1994 – adapted to this example):The “He” in “He who hesitates is
lost” does not refer to any particular person or group of people; it is simply
a placeholder indicating that the “he” who is lost is the same as the “he”
who hesitates.
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The Concept of Variable: a Unified Approach
We use a variable as a placeholder when we want to talk about a
quantity but either -i.e, a “mathematical John Doe”
Case 1: We know or hypothesize that it has certain values but we
don’t know what those values are.
Ex: an unknown quantity,
-- either to be found if possible (e.g., solving an equation)
-- or to be reasoned with (e.g., when its existence is implied
by a definition or deduced in a proof)
Case 2: We don't want to restrict it to a particular, concrete value
because we want whatever we say about it to be equally true for all
elements in a given set.
Ex: -- a symbol used to express an object in a universal statement
(e.g., identity, function definition)
-- a generic element in a proof
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Are Variables “Variable Quantities”?
Alfred Tarski (1941): “As opposed to the constants, the variables do not possess any
meaning by themselves. …The ‘variable number’ x could not possibly have any
specified property … the properties of such a number would change from case to
case … entities of such a kind we do not find in our world at all; their existence
would contradict the fundamental laws of thought.”
W. V. Quine ( 1950): “Care must be take, however, to divorce this traditional word
of mathematics [variable] from its archaic connotations. The variable is not best
thought of as somehow varying through time, and causing the sentence in which it
occurs to vary with it. Neither is it to be thought of as an unknown quantity,
discoverable by solving equations.
A. Church (1956): Mathematical writers do speak of “variable real numbers,” or
oftener “variable quantities,” but it seems best not to interpret these phrases
literally. Objections … have been clearly stated by Frege and need not be repeated
here at length. The fact is that a satisfactory theory has never been developed on
this basis, and it is not easy to see how it might be done.
Comment: Sometimes a variable is defined to be “a quantity that can change.”
Indeed, the very word “variable” suggests changeability. But it is not the x or the
y that changes; it is the values that that may be put in their places.
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Variables Used in Functional Relationships
Ex: “y = 2x + 1”  How should a student interpret this? Cf. x + y = y + x
For each possible change in the value of x the equation defines a
corresponding change in the value of y. But: when we speak of “the value
of x ” or “the value of y ” we mean the values that are put in their places.
Ex: “the function f (x) = 2x + 1”  What does this mean?
The relation defined by corresponding to any given real number the real
number obtained by multiplying that number by 2 and adding 1.
I.e., no matter what real number is placed in box , f () = 2 + 1.
Problem: Saying “the function f (x) = 2x + 1” conflates the function (as a
relation) with its value at x.
Ex: The slope of x 2 at x = 3 is
In essence, we ask students to
d ( x 2 ) 
learn that what we mean is


 [2x ]
 6.
different from what we say.
x

3
 dx  x 3
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Variables Used to Express Unknown Quantities
When we say, “Solve 4  3x  x ,” what we mean is “Find all numbers
(if any) that can be substituted in place of x in the equation 4  3x  x
in a way that makes its left-hand side equal to its right-hand side.”
The role of x as a placeholder in a situation like this is sometimes
highlighted by replacing x by an empty box: 4  3   .
For comparison with U.S., see: TIMSS video Hong Kong 4
http://www.rbs.org/catalog/pubs/pd57.php/
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The Logic of Equation Solving
“When we solve an equation, we operate with the unknown(s)
“as if it were a known quantity. . .A modern mathematician is
so used to this kind of reasoning that his boldness is now
barely perceptible to him.”
--Jean Dieudonné (1972)
Given an equation, we ask:
Is it true
for some value(s) of the variable(s)?  direct proof
for no value(s) of the variable(s)?
 proof by contradiction
for all value(s) of the variable(s)?
 generalizing from the
generic particular
Start the same way in all three cases:
Suppose there is a value of the variable that satisfies the equation,
and deduce properties it must satisfy.
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Variables Used to Express Universal Statements
Q: Why is the
answer always 5?
A:
2(n  7)  4
n  5
2
Comment: These
students didn’t
know how to
simplify the
expression on the
left.
Davis, S. and Thompson, D. R. To encourage "algebra for all," start an algebra
network. The Mathematics Teacher. Apr 1998. Vol. 91 (#4), p. 282
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Variables Used to Express Universal Statements
Davis, S. and Thompson, D. R. To encourage "algebra for all," start an algebra
network. The Mathematics Teacher. Apr 1998. Vol. 91 (#4), p. 282
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Variables Used to Express Universal Statements
Ex: The distributive property for real numbers states that for all
real numbers a, b, and c, ac + bc = (a + b)c. This means that no matter
what real numbers are substituted in place of a, b, and c, the two
sides of the equation ac + bc = (a + b)c will be equal.
Comment: To learn to apply the distributive property in a broad range
of situations, it is important to understand that the a, b, and c are
just placeholders (aka dummy variables). They could be replaced by
any three letters. Or we could represent the property by writing
  ( + ) =    +  
where any three real numbers (or expressions that can represent real
numbers) can be placed in the boxes.
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Bound Variables and Scope
Example: An integer is even if, and only if, there is an integer k so
that the integer equals 2k.
Problem: Bound variables that jump beyond their bounds.
Proposed proof that the sum of any two even integers is even:
Suppose m and n are any even integers. For an integer to be even
means that there is an integer k so that the integer equals 2k . Thus
m + n = 2k + 2k = 4k … etc.
Auxiliary Definitions: For an integer to be even means that
• there exists an integer a so that the integer equals 2a.
• there exists an integer m so that the integer equals 2m.
• it equals twice some integer.
• it equals 2  , for some integer that can be placed into the box.
• it equals 2  (some integer).
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Existential Instantiation
If we know that an object exists, then we may give it a name
as long as we are not already using the name for another
object in our current discussion.
Two main uses:
1. In applying a statement of the form x (y such that P (x,y ))
[Ex: Every even integer equals twice some integer.]
2. When existence is hypothesized
[Ex: Does there exist a number x such that the LHS of this
equation equals its RHS?]
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The Dependence Rule*
*Arsac & Durand-Guerrier, 2005
In the sentence
“For all x in set D, there exists a y in set E such that…”
the value of y depends on the value of x.
“Proof: Suppose n is any odd integer. By definition of odd,
n = 2k + 1 for any integer k…”
Theorem: For all functions f : X  Y and g :Y  Z, if f and g
are onto, then so is g of.
“Proof : By definition of onto, given any y in Y, there is an x
in X with f (x) = y. Also by definition of onto, given any z in
Z, there is a y in Y with g (y) = z. So g of (x) = g (f (x)) = g (y)
= z, and so g of is onto.”
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Universal Generalization
(Generalizing from the Generic Particular)
If we can prove that a property is true for a particular, but
arbitrarily chosen, element of a set, then we can conclude
that the property is true for every element of the set.
i.e., a generic element of the set
“Mathematics, as a science, commenced when first
someone, probably a Greek, proved propositions about
‘any’ things or about ‘some’ things without
specification of definite particular things.”
Alfred North Whitehead (1861-1947)
Ex: Is a sum of odd integers always even?
Answer: Yes. Suppose m and n are any [particular but arbitrarily
chosen] odd integers. We will show that m + n is even.
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Universal Instantiation
If a property is true for all elements of a set, then
it is true for any particular element of the set.
Two main uses:
1. Every time we do algebra
Ex: Simplify
k2k+2 + (k + 2)2k+2
= (k + (k + 2))2k+2
Etc.
2. Extensively in explanation/justification/proof
Ex: Is a sum of odd numbers always even?
“... So m + n = 2(r + s + 1), where r + s + 1 is an integer, and this
number is even because any integer that can be written as
2(some integer) is even.”
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Existential Generalization
If we know that a certain property is true for a
particular object, then we may conclude that
“there exists an object for which the property is true.”
Main use: Counterexamples
Example: True or false? The quantity n 2 + n + 41 is always prime. (Euler)
Answer: False, because 412 + 41 + 41 = (41)(43), which is not prime.
That is: There exists an integer n such that n 2 + n + 41 is not prime.
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What We Say vs. What We Mean
What we say
the value of x
as the value of x increases
as the value of x increases, the value of y increases
where x is any real number
Let n be any even integer.
By definition of even, n = 2k for some integer k.
the function x2
where x is some real number that satisfies the
given property
A general linear function is a function of the form
f(x) = ax + b where a and b are any real numbers.
What we mean
the quantity that is put in place of x
as larger and larger numbers are put in place of x
If larger and larger numbers are put in place of x, the
corresponding numbers that are put in place of y become larger
and larger
for all possible substitutions of real numbers in place of x
Imagine substituting an integer in place of n but do not assume
anything about its value except that it is an even integer.
By definition of even, there is an integer we can substitute in
place of k so that the equation n = 2k will be true. (Note that
there is only one such integer; its value is n/2.)
the function that relates each real number to the square of that
number. In other words, for each possible substitution of a real
number in place of x, the function corresponds the square of that
number.
There is a real number that will make the given property true if
we substitute it in place of x.
A general linear function is a function defined as follows: for all
substitutions of real numbers in place of a and b, the function
relates each real number to a times that number plus b. Or: the
function is the set of ordered pairs where any real number can be
substituted in place of the first element of the pair and the second
element of the pair is a times the first number plus b.
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Some References
1. Church, A. (1956), Introduction to Mathematical Logic, vol. 1.
2. Frege, G. (1893), The Foundations of Arithmetic.
3. Klein, F. (1908, 1924), Elementary Mathematics from an Advanced
Standpoint.
4. Pinker, S. (1994), The Language Instinct.
5. Quine, W. V. (1950, 1982), Methods of Logic.
6. Tarski, A. (1941, 1965), Introduction to Logic and to the
Methodology of Deductive Sciences.
7. Whitehead, A. N. (1911, 1958), An Introduction to Mathematics.
Also: Epp, S., What Is a Variable – Draft article
(http://condor.depaul.edu/~sepp/WhatIsAVariable.pdf)
E-mail: [email protected]
Thank you!
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