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Discover the evolution of quantifiers and variables in mathematics education, from Frege to Quine, emphasizing the significance of symbols as placeholders. Learn about the historical context, implications for K-12 education, and the concept of variables as pronouns or placeholders.
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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 sepp@depaul.edu
“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?”
“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.
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.”
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 cross-reference; 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.
The Concept of Variable: a Unified Approach We use a variable as a placeholder when we want to talk about a quantity but either -- 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 i.e, a “mathematical John Doe”
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.
Variables Used in Functional Relationships How should a student interpret this? Cf. x + y = y + x Ex:“y = 2x + 1” For each possible change in the value of x the equation defines a corresponding change in the value of y. Ex:“the functionf(x) = 2x + 1” 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 x2 at x = 3 is But:when we speak of “the value of x” or “the value of y” we mean the values that are put in their places. What does this mean? In essence, we ask students to learn that what we mean is different from what we say.
Variables Used to Express Unknown Quantities When we say, “Solve ,” what we mean is “Find all numbers (if any) that can be substituted in place of x in the equation 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: . For comparison with U.S., see: TIMSS video Hong Kong 4 http://www.rbs.org/catalog/pubs/pd57.php/
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.
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 Q:Why is the answer always 5? A: Comment: These students didn’t know how to simplify the expression on the left.
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
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.
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: • Supposemand n are any even integers. For an integer to be even means that there is an integer k so that the integer equals2k. 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).
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?]
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 gof. “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 gof (x) = g(f(x)) = g(y) = z, and so gof is onto.”
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.
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 k2k+2 + (k + 2)2k+2 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.” = (k + (k + 2))2k+2 Etc.
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 n2 + 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 n2 + n + 41 is not prime.
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: sepp@depaul.edu Thank you!
