Where Mathematics Comes From - Example of Metaphorical Ambiguity

Example of Metaphorical Ambiguity

WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set . Then recall two bits of standard terminology from elementary set theory:

1. The recursive construction of the ordinal natural numbers, whereby 0 is, and n+1 is n {n}.
2. The ordered pair (a,b), defined as .

By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics.

While (1) and (2) above are admittedly canonical, especially within the consensus set theory known as the Zermelo–Fraenkel axiomatization, WMCF does not let on that they are but one of several definitions that have been proposed since the dawning of set theory. For example, Frege, Principia Mathematica, and New Foundations (a body of axiomatic set theory begun by Quine in 1937) define cardinals and ordinals as equivalence classes under the relations of equinumerosity and similarity, so that this conundrum does not arise. In Quinian set theory, A is simply an instance of the number 2. For technical reasons, defining the ordered pair as in (2) above is awkward in Quinian set theory. Two solutions have been proposed:

• A variant set-theoretic definition of the ordered pair more complicated than the usual one;
• Taking ordered pairs as primitive.