Meanings of Symbolism in Advanced Mathematical Thinking
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1 PME-28 Bergen, Norway June 2004 Meanings of Symbolism in Advanced Mathematical Thinking David Tall University of Warwick, U.K.
2 FORMAL WORLD Three Worlds of Mathematics of DEFINITION and FORMAL PROOF A complete ordered field as a discrete set of elements satisfying specified axioms EMBODIED WORLD of PERCEPTION, EXPERIMENT and THOUGHT EXPERIMENT A continuous line that can be traced with a finger PERCEPTION ACTIO N SYMBOLIC WORLD of CALCULATION and SYMBOL MANIPULATION Numbers as decimal symbols that can be used for accurate calculation REFLECTION The Real Numbers
3 Concept images, formal proof & embodiment THOUGHT EXPERIMENTS FORMAL DEDUCTIONS NATURAL Embodied (& proceptual) FORMAL Formal Pinto, PhD, 1998, Tall, ESM, 2002
4 Example: Concept Definition: An equivalence relation on a set A satisfies: (i) x ~ x, x A, (ii) x ~ y implies y ~ x, x, y A, Axioms (iii) x ~ y and y ~ z implies x ~ z, x, y, z A. Structure Theorem: An equivalence relation on a set A is equivalent to a partition of A. A Formal Embodiment
5 Metaphors & Met-afores (or Met-befores) A met-before is a personal construction, part of the concept image, that resonates in new contexts. Examples After each whole number there s a next, (not true for fractions) Addition makes bigger, (not true for integers) + means add, e.g. 2+3 is 5, so 2+3x is... 5x? 1 m equals 100 cm, (1 m is worth 100 cms) so 1P = 6S. Letters stand for numbers... a = 1, b = 2, etc, so 2c is 23. A met-before includes both helpful previous experiences and conflicting ones (eg epistemological obstacles). A theory of met-befores reveals positive aspects supporting expansion of knowledge and negative aspects requiring reconstruction.
6 An (attempted) Logical Approach Start with Relations then: order relations, equivalence relations, functions (all as special cases of relations) Every year at Warwick, students declare this to be the most difficult part of the foundations course. A major problem is the transitive law: a ~ b and b ~ c implies a ~ c, a, b, c A. This has a different embodiment for order and for equivalence... For order, a < b and b < c implies a < c and a, b, c must be different.
7 Example A = {a, b, c} where a ~ a, b ~ b, a ~ b, b ~ a (and no others). Is this an equivalence relation on A? Solution: No, because c ~ c is not true. Common Student Solution: No, because the transitive law is false. Logically the transitive law is true in this case, but the students believe it false because there are not three different elements in the right relationship to be able to state the law. The met-before of (strict) order relation interferes with transitive law in equivalence relation.
8 Embodiment of axiomatic systems We build axiomatic systems from lists of axioms with the symbols being given meaning based on logical deduction. Definition: An equivalence relation on a set A satisfies: (i) x ~ x, x A, (ii) x ~ y implies y ~ x, x, y A, (iii) x ~ y and y ~ z implies x ~ z, x, y, z A. OR: An equivalence relation on a set A satisfies: (i) x ~ x, x A, (ii) x ~ y implies y ~ x, x, y A, (iii)* x ~ z and y ~ z implies x ~ y, x, y, z A. These are identical systems, but (iii) and (iii)* are not identical and have different embodiments
9 Embodiment of axiomatic systems We build axiomatic systems from lists of axioms with the symbols being given meaning based on logical deduction. To give shape to whole theories, we prove structure theorems that give meaning to structures as a whole. This meaning often links to an embodiment that suggests new theories. These structure theorems are the result of deductions from a combination of axioms and the same theorem may be given by different combinations. However, individual axioms may occupy different roles in different combinations of axioms and the embodiment from one system may cause cognitive conflict when used in another. Understanding the interplay between human embodiment and mathematical logic is an essential factor in the cognitive use of symbols.
10 Symbolic Cognition Symbolic cognition is about cognitive meanings of symbols as a whole, not only about the meanings of individual symbols, which are parts of the whole. Symbolic cognition has many aspects and many details which we are sharing in this working group. To understand the cognition of symbols, however, we need to reach the heart of human meaning for those symbols. I place this meaning in the foundation of human being, as we grow as individuals in a community of others, with experiences we learn to share and to extend to new contexts. In particular I focus on human growth through increasing sophistication of perception, action and reflection to construct meanings for mathematics and its symbols.
11 PME-28 Bergen, Norway June 2004 Meanings of Symbolism in Advanced Mathematical Thinking David Tall University of Warwick, U.K.
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