The Square Root of 2, p, and the King of France: Ontological and Epistemological Issues Encountered

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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
• (flashman_at_axe.humboldt.edu)
• Humboldt State University and
• Occidental College
• Dedicated to the memory of Jean van Heijenoort.

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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 Russells 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.

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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.

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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? !

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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
  non-contradictory description denotes something
  that exists.
• Freges response that such terms have two
  aspects meaning and denotation.

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Key examples

• Russells Examples
• The author of Waverly is Scott.
• The present king of England is bald.
• The present king of France is bald.

• Mathematics Examples
• The square root of 4 is 2.
• The square root of 4 is rational.
• The square root of 2 is rational.
• p is not rational.

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Russells Theory for Denoting

• a phrase is denoting solely in virtue of its
  form. We may distinguish three cases
• 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.

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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.

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Interpretation of indefinite denoting phrases

• Conditional forms, conjunctions and assertions
  about statements explain apparent indefinite
  denoting phrases.
• Example RussellAll 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'.

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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.

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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.

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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'.

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DenotingPrimary and Secondary Occurrence -
Context Examples

• PrimaryOne and only one man wrote Waverley,
  and George IV wished to know whether Scott was
  that man'.
• SecondaryGeorge IV wished to know whether one
  and only one man wrote Waverley and Scott was
  that man'

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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.

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Return to Key examples for discussion

• Russells examples
• The author of Waverly is Scott.
• The present king of England is bald.
• The present king of France is bald.

• Mathematics Examples
• The square root of 4 is 2.
• The square root of 4 is rational.
• The square root of 2 is rational.

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Application to Square Roots

• 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.

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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.

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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.

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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!

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Contexts for the Mathematical Examples

• 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)

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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.

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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.

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Time!

• Questions?
• Responses?
• Further Communication by e-mail
• flashman_at_humboldt.edu
• These notes will be available at
• http//www.humboldt.edu/mef2

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Thanks-

• The end!