Benacerraf: What Numbers Could Not Be

 Without commentary, here's a link to Benacerraf's original paper and a cut-paste job from the Stanford Encyclopedia of Philosophy.

4. Structuralism and Nominalism

Benacerraf’s work motivated philosophers to develop both structuralist and nominalist theories in the philosophy of mathematics (Reck & Price 2000). And since the late 1980s, combinations of structuralism and nominalism have also been developed.

4.1 What Numbers Could Not Be

As if saddling platonism with one difficult problem were not enough (section 3.4), Benacerraf formulated a challenge for set-theoretic platonism (Benacerraf 1965). The challenge takes the following form.

There exist infinitely many ways of identifying the natural numbers with pure sets. Let us restrict, without essential loss of generality, our discussion to two such ways:

I:0123⋮II:0123⋮=∅={∅}={{∅}}={{{∅}}}=∅={∅}={∅,{∅}}={∅,{∅},{∅,{∅}}}$$\begin{array}{rl}\mathrm{I}:& \\ 0& =\varnothing \\ 1& =\{\varnothing \}\\ 2& =\{\{\varnothing \}\}\\ 3& =\{\{\{\varnothing \}\}\}\\ ⋮& \\ & \\ \mathrm{I}\mathrm{I}:& \\ 0& =\varnothing \\ 1& =\{\varnothing \}\\ 2& =\{\varnothing ,\{\varnothing \}\}\\ 3& =\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}\\ ⋮& \end{array}$$

The simple question that Benacerraf asks is:

Which of these consists solely of true identity statements: I or II?

It seems very difficult to answer this question. It is not hard to see how a successor function and addition and multiplication operations can be defined on the number-candidates of I and on the number-candidates of II so that all the arithmetical statements that we take to be true come out true. Indeed, if this is done in the natural way, then we arrive at isomorphic structures (in the set-theoretic sense of the word), and isomorphic structures make the same sentences true (they are elementarily equivalent). It is only when we ask extra-arithmetical questions, such as ‘1∈3$1\in 3$?’ that the two accounts of the natural numbers yield diverging answers. So it is impossible that both accounts are correct. According to story I, 3={{{∅}}}$3=\{\{\{\varnothing \}\}\}$, whereas according to story II, 3={∅,{∅},{∅,{∅}}}$3=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}$. If both accounts were correct, then the transitivity of identity would yield a purely set theoretic falsehood.

Summing up, we arrive at the following situation. On the one hand, there appear to be no reasons why one account is superior to the other. On the other hand, the accounts cannot both be correct. This predicament is sometimes called labelled Benacerraf’s identification problem.

The proper conclusion to draw from this conundrum appears to be that neither account I nor account II is correct. Since similar considerations would emerge from comparing other reasonable-looking attempts to reduce natural numbers to sets, it appears that natural numbers are not sets after all. It is clear, moreover, that a similar argument can be formulated for the rational numbers, the real numbers… Benacerraf concludes that they, too, are not sets at all.

PS: Here is some commentary from Stewart Shapiro, in "Philosophy of Mathematics: Structure and Ontology".

In an earlier paper, Benacerraf [1965] raises another problem for realism in ontology (see also Kitcher [1983, chapter 6]). It is well known that virtually every field of mathematics can be reduced to, or modeled in, set theory. Matters of economy suggest that there be a single type of object for all of mathematics—sets. Why have numbers, points, functions, functionals, and sets when sets alone will do? However, there are several reductions of arithmetic to set theory. If natural numbers are mathematical objects, as the realist contends, and if all mathematical objects are sets, then there is a fact concerning which sets the natural numbers are. 

According to one account, due to von Neumann, the natural numbers are finite ordinals. Thus, 2 is {φ, {f}}, 4 is {f, {f}, {f, {f}}, {f, {f}, {f, {f}}}}, and so 2 ∈ 4. 

According to Zermelo’s account, 2 is {{f}}, 4 is {{{{f}}}}, and so 2 ∉ 4. 

Moreover, there seems to be no principled way to decide between the reductions. Each serves whatever purpose a reduction is supposed to serve. So we are left without an answer to the question of whether 2 is really a member of 4 or not. Will the real 2 please stand up? What, after all, are the natural numbers? Are they finite von Neumann ordinals, Zermelo numerals, or other sets? 

From these observations and questions, Benacerraf and Kitcher conclude that numbers are not objects, against realism in ontology. This conclusion, I believe, is not warranted. It all depends on what it is to be an object, a matter that is presently under discussion. Benacerraf’s and Kitcher’s conclusion depends on what sorts of questions can legitimately be asked about objects and what sorts of questions have determinate answers waiting to be discovered.