kb/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-1.md

title	chunk	source	category	tags	date_saved	instance
Where Mathematics Comes From	2/3	https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From	reference	science, encyclopedia	2026-05-05T08:47:17.193195+00:00	kb-cron

== Example of metaphorical ambiguity == WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set

    A
    =
    {
    {
    ∅
    }
    ,
    {
    ∅
    ,
    {
    ∅
    }
    }
    }
    .
  

{\displaystyle A=\{\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}.}

Then recall two bits of standard terminology from elementary set theory:

The recursive construction of the ordinal natural numbers, whereby 0 is

    ∅
  

{\displaystyle \emptyset }

, and

    n
    +
    1
  

{\displaystyle n+1}

is

    n
    ∪
    {
    n
    }
    .
  

{\displaystyle n\cup \{n\}.}

The ordered pair (a,b), defined as

    {
    {
    a
    }
    ,
    {
    a
    ,
    b
    }
    }
    .
  

{\displaystyle \{\{a\},\{a,b\}\}.}

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.

== The Romance of Mathematics == The "Romance of Mathematics" is WMCF's light-hearted term for a perennial philosophical viewpoint about mathematics which the authors describe and then dismiss as an intellectual myth:

Mathematics is transcendent, namely it exists independently of human beings, and structures our actual physical universe and any possible universe. Mathematics is the language of nature, and is the primary conceptual structure we would have in common with extraterrestrial aliens, if any such there be. Mathematical proof is the gateway to a realm of transcendent truth. Reasoning is logic, and logic is essentially mathematical. Hence mathematics structures all possible reasoning. Because mathematics exists independently of human beings, and reasoning is essentially mathematical, reason itself is disembodied. Therefore, artificial intelligence is possible, at least in principle. It is very much an open question whether WMCF will eventually prove to be the start of a new school in the philosophy of mathematics. Hence the main value of WMCF so far may be a critical one: its critique of Platonism and romanticism in mathematics.

== Critical response ==

Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews of WMCF by mathematicians in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the WMCF's philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings. For example, Fermat's Last Theorem means exactly what it meant when Fermat initially proposed it in 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The metaphor and the conceptual strategy are not the same as the formal definition which mathematicians employ. However, WMCF points out that formal definitions are built using words and symbols that have meaning only in terms of human experience. Critiques of WMCF include the humorous:

It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it. — Joseph Auslander and the physically informed: