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* 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 non- contradictory 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 is not rational.

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 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 These notes will be available at

Thanks- The end!