The historical motivation for the development of Benacerraf's identification problem derives from a fundamental problem of ontology. Since Medieval times, philosophers have argued as to whether the ontology of mathematics contains abstract objects. In the philosophy of mathematics, an abstract object is traditionally defined as an entity that: (1) exists independent of the mind; (2) exists independent of the empirical world; and (3) has eternal, unchangeable properties.^[4] Traditional mathematical Platonism maintains that some set of mathematical elements–natural numbers, real numbers, functions, relations, 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 intuitionism, formalism, and 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.

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.^[1] Benacerraf formulates the dilemma as a question: which of these set-theoretic methods uniquely provides the true identity statements, which elucidates the true ontological nature of the natural numbers?^[1] Either method I or II could be used to define the natural numbers and subsequently generate true arithmetical statements to form a mathematical system. In their relation, the elements of such mathematical systems are isomorphic in their structure. However, the problem arises when these isomorphic structures are related together on the meta-level. The definitions and arithmetical statements from system I are not identical to the definitions and arithmetical statements from system II. For example, the two systems differ in their answer to whether 0 ∈ 2, insofar as ∅ is not an element of {{∅}}. Thus, in terms of failing the 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.

• Benacerraf's epistemological problem

• Benacerraf, Paul (1973) "Mathematical Truth", in Benacerraf & Putnam Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edition. 1983, pp. 403–420.
• Hale, Bob (1987) Abstract Objects. Oxford: Basil Blackwell. ISBN 0631145931