633:"Ernie and Johnny are both brought up on set theory. When the time comes to learn arithmetic, Ernie is told, to his delight, that he already knows about the numbers; they are 0 (called 'zero'), {0} (called 'one'), (0, {0}} (called 'two'), and so on. His teachers define the operations of addition and multiplication on these sets, and when all the relabelling is done, Ernie counts and does arithmetic just like his schoolmates. Johnny's story is exactly the same, except that he is told that the Zermelo ordinals are the numbers. He also counts and does arithmetic in agreement with his schoolmates, and with Ernie. The boys enjoy doing sums together, learning about primes, searching for perfect numbers, and so on. But Ernie and Johnny are curious little boys; they want to know everything they can about these wonderful things, the numbers. In the process, Ernie discovers the surprising fact that one is a member of three. In fact, he generalizes, if n is bigger than m, then m is a member of n. Filled with enthusiasm, he brings this fact to the attention of his favourite playmate. But here, sadly, the budding mathematical collaboration breaks down. Johnny not only fails to share Ernie's enthusiasm, he declares the prized theorem to be outright false! He won't even admit that three has three members!"
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109:. 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. Traditional mathematical Platonism maintains that some set of mathematical elements–
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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
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As
Benacerraf demonstrates, both method I and II reduce natural numbers to sets. 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? Either method I
599:{\displaystyle {\begin{aligned}&0=\varnothing \\&1=\{\,0\,\}=\{\,\varnothing \,\}\\&2=\{\,0,1\,\}=\{\,\varnothing ,\{\,\varnothing \,\}\,\}\\&3=\{\,0,1,2\,\}=\{\,\varnothing ,\{\,\varnothing \,\},\{\,\varnothing ,\{\,\varnothing \,\}\,\}\,\}\\&\,\,\,\,\,\,\vdots \end{aligned}}}
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number of ways of identifying the natural numbers with pure sets, no particular set-theoretic method can be determined as the "true" reduction. Benacerraf infers that any attempt to make such a choice of reduction immediately results in generating a meta-level, set-theoretic falsehood, namely in
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According to
Benacerraf, the philosophical ramifications of this identification problem result in Platonic approaches failing the ontological test. The argument is used to demonstrate the impossibility for Platonism to reduce numbers to sets and reveal the existence of abstract objects.
619:, the search for true identity statements similarly fails. 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.
148:. 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.
349:{\displaystyle {\begin{aligned}&0=\varnothing \\&1=\{\,0\,\}=\{\,\varnothing \,\}\\&2=\{\,1\,\}=\{\,\{\,\varnothing \,\}\,\}\\&3=\{\,2\,\}=\{\,\{\,\{\,\varnothing \,\}\,\}\,\}\\&\,\,\,\,\,\,\vdots \end{aligned}}}
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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
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Benacerraf, Paul (1973) "Mathematical Truth", in
Benacerraf & Putnam Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edition. 1983, pp. 403–420.
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to the one chosen. The identification problem argues that this creates a fundamental problem for
Platonism, which maintains that mathematical objects have a real, abstract existence.
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The identification problem begins by evidencing some set of elementarily-equivalent, set-theoretic models of the natural numbers. Benacerraf considers two such set-theoretic methods:
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of these abstract mathematical objects, is impossible. As a result, the identification problem ultimately argues that the relation of set theory to natural numbers cannot have an
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and published in 1965 in an article entitled "What
Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the development of
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Platonic set-theory is arguing that the Platonic attempt to identify the "true" reduction of natural numbers to pure sets, as revealing the
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The historical motivation for the development of
Benacerraf's identification problem derives from a fundamental problem of ontology. Since
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In the late 19th and early 20th century, a number of anti-Platonist programs gained in popularity. These included
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their answer to whether 0 ∈ 2, insofar as ∅ is not an element of {{∅}}. Thus, in terms of failing the
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denies the existence of any such abstract objects in the ontology of mathematics.
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The identification problem argues that there exists a fundamental problem in
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697:"Benacerraf's Dilemma Revisited"
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791:. Oxford: Basil Blackwell.
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52:mathematical structuralism
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68:. Since there exists an
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617:transitivity of identity
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363:Set-theoretic method II
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48:set-theoretic Platonism
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87:intrinsic properties
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138:intuitionism
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152:Description
827:Set theory
806:Categories
797:0631145931
745:0415401348
725:0195139305
653:References
612:isomorphic
131:nominalism
590:⋮
562:∅
552:∅
538:∅
528:∅
476:∅
466:∅
424:∅
392:∅
340:⋮
312:∅
264:∅
224:∅
192:∅
142:formalism
123:relations
119:functions
79:identical
66:pure sets
641:See also
103:Medieval
70:infinite
59:reducing
46:against
365:(using
165:(using
127:systems
30:In the
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144:, and
38:is a
793:ISBN
762:ISBN
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706:(1).
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