Benacerraf's identification problem

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In the

set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be".[1][2] Historically, the work became a significant catalyst in motivating the development of mathematical structuralism.[3]

The identification problem argues that there exists a fundamental problem in

intrinsic properties of these abstract mathematical objects, is impossible.[1] As a result, the identification problem ultimately argues that the relation of set theory to natural numbers cannot have an ontologically Platonic nature.[1]

Historical motivations

The historical motivation for the development of Benacerraf's identification problem derives from a fundamental problem of ontology. Since

systems–are such abstract objects. Contrarily, mathematical nominalism
denies the existence of any such abstract objects in the ontology of mathematics.

In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included

predicativism
. By the mid-20th century, however, these anti-Platonist theories had a number of their own issues. This subsequently resulted in a resurgence of interest in Platonism. It was in this historic context that the motivations for the identification problem developed.

Description

The identification problem begins by evidencing some set of elementarily-equivalent, set-theoretic models of the natural numbers.[1] Benacerraf considers two such set-theoretic methods:

Set-theoretic method I (using
Zermelo ordinals
)
0 = ∅
1 = {0} = {∅}
2 = {1} = {{∅}}
3 = {2} = {{{∅}}}
...
Set-theoretic method II (using
von Neumann ordinals
)
0 = ∅
1 = {0} = {∅}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}
...

As Benacerraf demonstrates, both method I and II reduce natural numbers to sets.

transitivity of identity, the search for true identity statements similarly fails.[1]
By attempting to reduce the natural numbers to sets, this renders a set-theoretic falsehood between the isomorphic structures of different mathematical systems. This is the essence of the identification problem.

According to Benacerraf, the philosophical ramifications of this identification problem result in Platonic approaches failing the ontological test.[1] The argument is used to demonstrate the impossibility for Platonism to reduce numbers to sets and reveal the existence of abstract objects.

See also

  • Benacerraf's epistemological problem

References

  1. ^ a b c d e f g h i Paul Benacerraf (1965), “What Numbers Could Not Be”, Philosophical Review Vol. 74, pp. 47–73.
  2. ^ Bob Hale and Crispin Wright (2002) "Benacerraf's Dilemma Revisited" European Journal of Philosophy, 10(1).
  3. ISBN 0195139305
  4. ISBN 0415401348

Bibliography
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