By Karlis Podnieks

The Quinean indispensability argument, as put by Mark Colyvan (2001: The Indispensability of Mathematics, Oxford University Press, 192 pp., p.1): "... mathematical entities are indispensable to our best physical theories and therefore share the ontological status of scientific entities."

Of course, one may take several different positions with respect to the ontological status of scientific entities such as, for example, quarks (quarks can't be observed even in principle). Do quarks "really exist", or are they only a (currently successful) theoretical construct used by physicists in their models? Perhaps, the "least committed" position could be the formalist one: let us define the "real existence" of some scientific entity as its invariance in future scientific theories. If quarks will be retained as a construct in our best future physical theories, then one may think of quarks as "really existing". Even from such a very formalistic point of view, the Quinean argument seems quite reasonable. Indeed, if some mathematical entity is indispensable to our best physical theories, then shouldn't we believe, that this entity "exists" in the same sense as quarks are believed to exist?

However, imagine two mathematical entities E1 and E2, such that the existence of E1 contradicts the existence of E2. Can both of such entities be indispensable to our best physical theories?

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... Applying to set theory the above-mentioned formalistic explanation of the existence of quarks, we could say: if, for a long time in the future, set theorists will continue their believing in AC, then one may think of a unique "world of sets" as existing in the same sense as quarks are believed to exist.

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Full text:
K. Podnieks. Indispensability Argument and Set Theory.
The Reasoner, Vol. 2, N 11, November 2008, pp. 8-9
(available online).