N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).

11. Conclusion

In our earlier sections we studied the problem of translating into nominalistic language certain nonsyntactical sentences that had appeared to be explicable only in platonistic terms. In 5-10 we have been concerned with giving such a translation for syntax. This syntax enables us to describe and deal with many formulas (of the object language) for which we have no direct nominalistic translation. For example, the formula that is the full expansion in our object language of "(n)(n + n = 2n)" will contain variables calling for abstract entities as values; and if it cannot be translated into nominalistic language, it will in one sense be meaningless for us. But, taking that formula as a string of marks, we can determine whether it is indeed a proper formula of our object language, and what consequence-relationships it has to other formulas. We can thus handle much of classical logic and mathematics without in any further sense understanding, or granting the truth of, the formulas we are dealing with.

The gains that seem to have accrued to natural science from the use of mathematical formulas do not imply that those formulas are true statements. No one, not even the hardiest pragmatist, is likely to regard the beads of an abacus as true; and our position is that the formulas of platonistic mathematics are, like the beads of an abacus, convenient computational aids which need involve no question of truth. What is meaningful and true in the case of platonistic mathematics as in the case of the abacus is not the apparatus itself, but only the description of it: the rules by which it is constructed and run. These rules we do understand, in the strict sense that we can express them in purely nominalistic language. The idea that classical mathematics can be regarded as mere apparatus is not a novel one among nominalistically minded thinkers; but it can be maintained only if one can produce, as we have attempted to above, a syntax that is itself free from platonistic commitments.

At the same time, every advance we can make in finding direct translations for familiar strings of marks will increase the range of the meaningful language at our command.