Transcript The Square Root of 2, p, and the King of France

The Square Root of 2, p, and the
King of France:
Ontological and Epistemological
Issues
Encountered (and Ignored) in
Introductory Mathematics Courses
Martin E. Flashman*
([email protected])
Humboldt State University and
Occidental College
Dedicated to the memory of Jean van Heijenoort.
Abstract
• Students in many beginning college level courses are presented
with proofs that the square root of 2 is irrational along with
statements about the irrationality and transcendence of p.
• In Bertrand Russell’s 1905 landmark article ”On Denoting” one of
the central examples was the statement,
“The present King of France is bald.”
• In this presentation the author will discuss both the ontological and
epistemological connections between these examples in trying to
find a sensible and convincing explanation for the difficulties that are
usually ignored in introductory presentations; namely,
• what is it that makes the square root of 2 and p numbers? and
• how do we know anything about them?
• If time permits the author will also discuss the possible value in
raising these issues at the level of introductory college mathematics.
• Dedicated to the memory of Jean van Heijenoort.
Apology
This work is the result of many
years of thought- but is still
only a preliminary attempt to
record some of these thoughts
and connect them to some
historic and contemporary
philosophical approaches.
Pre-Calculus Course Questions
• What is a number?
– Students give some examples of numbers
• Different ways to describe and represent
numbers3 , sqt(2), i, pi, e, …
• Different ways to use numbers
• Compare numbers:
3
5
• What is a function? !
Bertrand Russell
“On Denoting” Mind,1905
• An attempt to resolve issues related to the
meaning in discourse of terms of
denotation.
• A response to
– the simplistic response that any noncontradictory description denotes something
that exists.
– Frege’s response that such terms have two
aspects: meaning and denotation.
Key examples
Russell’s Examples
1. The author of
Waverly is Scott.
2. The present king
of England is bald.
3. The present king
of France is bald.
Mathematics Examples
1. The square root of
4 is 2.
2. The square root of
4 is rational.
3. The square root of
2 is rational.
4.p
rational.
is not
Russell’s Theory for Denoting
“… a phrase is denoting solely in virtue of its
form. We may distinguish three cases:
(1) A phrase may be denoting, and yet not
denote anything; e.g., `the present King of
France'.
(2) A phrase may denote one definite object;
e.g., `the present King of England' denotes
a certain man.
(3) A phrase may denote ambiguously; e.g. `a
man' denotes not many men, but an
ambiguous man.”
The importance of context
• “…denoting phrases never have any
meaning in themselves, but that every
proposition in whose verbal expression
they occur has a meaning.”
Interpretation of indefinite
denoting phrases
• Conditional forms, conjunctions and
assertions about statements explain
apparent indefinite denoting phrases.
• Example [Russell]:`All men are mortal'
means ` ``If x is human, x is mortal'' is
always true.' This is what is expressed in
symbolic logic by saying that `all men are
mortal' means ` ``x is human'' implies ``x is
mortal'' for all values of x'.
Denoting: Syntax and Semantics
• In a formal mathematical context (Tarski) we can
distinguish
– the syntax of an expression: how symbols of an
expression are organized in a formal context.
– Examples: 2, the square root of 2
– the semantics of an expression: a correspondence
in a context between an expression and an object of
the context.
– Examples: the integer 2, the positive real number
whose square is 2.
Existence, Being, and Uniqueness
• Russell: When a denoting phrase uses “the”, the
use entails uniqueness.
• Russell presents a linguistic transformation
to produce “a reduction of all propositions in
which denoting phrases occur to forms in
which no such phrases occur.”
• As a consequence, Russell tries to resolve
confusion and apparent paradoxes (Frege)
from the use of denoting phrases that have
meaning with no denotation.
Russell on meaning and denotation
• “…a denoting phrase is essentially part of a
sentence, and does not, like most single
words, have any significance on its own
account.”
• “ … if 'C' is a denoting phrase, it may happen
that there is one entity x (there cannot be
more than one) for which the proposition `x
is identical with 'C' ‘ is true. We may then say
that the entity x is the denotation of the
phrase 'C'.”
Denoting:Primary and Secondary
Occurrence - Context Examples
• Primary:
`One and only one man wrote Waverley,
and George IV wished to know whether
Scott was that man'.
• Secondary:
`George IV wished to know whether one
and only one man wrote Waverley and
Scott was that man'
Distinguishing the use of a
denoting phrase in propositions.
• Russell: “… all propositions in which `the
King of France' has a primary occurrence
are false: the denials of such propositions
are true, but in them `the King of France'
has a secondary occurrence.”
Return to Key examples for
discussion
Russell’s examples
1. The author of
Waverly is Scott.
2. The present king
of England is bald.
3. The present king
of France is bald.
Mathematics Examples
1. The square root of
4 is 2.
2. The square root of
4 is rational.
3. The square root of
2 is rational.
Application to Square Roots
1. The square root of 4 is 2.
• One and only one positive integer has its
square equal to 4, and the proposition is
true if the number 2 is that number.
• The proposition is true if one and only
one positive integer has its square equal
to 4 and the number 2 is that number.
Application to Square Roots
2. The square root of 4 is rational.
• One and only one positive integer has
its square equal to 4, and the proposition
is true if that number is rational.
• The proposition is true if one and only
one positive integer has its square
equal to 4 and that number is rational.
Notice either interpretation of the
proposition is true.
Application to Square Roots
3. The square root of 2 is rational.
• One and only one positive integer has its
square equal to 2, and the proposition is
true if that number is rational.
• The proposition is true if one and only one
positive integer has its square equal to 2
and that number is rational.
Denoting and Knowing
• How does one identify a meaning with it
denotation? Context.
• How does one determine the truth/falsity
of a statement that uses a denoting
phrase? Context and Usage.
• How is a statement that uses a denoting
phrase a “proposition”?
Propositions in context give meaning to
the denoting phrase!
Contexts for the Mathematical
Examples
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Counting contexts (units)
Geometric contexts
Measurement contexts (units)
Comparative contexts (Ratios)
Analytic contexts (Platonist)
Algorithmic contexts (Procedural)
Formal contexts (Symbolic)
Structural Contexts (Conceptual)
Set Theory / Logic Contexts (Reductions)
Philosophical Questions
• How are the contexts for mathematics
articulated?
– A process of development? Dynamic
– A process of discovery? Static
• How do we know the truth of mathematical
propositions?
– The truth of mathematical propositions is
intrinsically connected to their context by their
denoting phrases.
Pedagogical consequences
• Awareness of the issues related to denoting should
increase with greater familiarity and experience with a
variety of mathematical contexts.
• With greater maturity and at appropriate levels, students
should be made more aware of philosophical issues
related to existence, uniqueness, and the dependence
on context in the study of mathematics.
• The consequence of greater awareness of these issues
might be seen in increased conceptual flexibility and new
approaches to understanding and solving problems
through the articulation of new contexts.
Time!
• Questions?
• Responses?
• Further Communication by e-mail:
[email protected]
• These notes will be available at
http://www.humboldt.edu/~mef2
Thanks-
The end!