Tuesday, 4 May 2010

The Indispensability Argument


Colyvan described the Indispensability argument as “the best argument for Platonism.”  Its basic structure, as per Colyvan:
    We ought to have ontological commitment to all the entities that are indispensable to our best scientific theories.  (Confirmational Holism)

    We ought to have ontological commitment to only the entities that are indispensable to our best scientific theories.  (Naturalism)

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    (Premise1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

    (Premise2) Mathematical entities are indispensable to our best scientific theories.

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    (Conclusion) We ought to have ontological commitment to mathematical entities.



Field objects to Premise2.  His argument has two parts.  The first is that mathematical theories don't have to be true to be useful, they need only be conservative. Conservatism refers to this result: if A is a consequence of T (a scientific theory) + M (Mathematics), then A would be a consequence of T alone (Brown, pg58).  Mathematics is a useful tool, but it is not indispensable.  The second part of Field's program is to demonstrate that our best scientific theories can be suitably nominalised. By nominalising a portion of Newtonian gravitational theory, he attempts to show that there is no need to assert the existence of mathematical entities in a scientific theory. This is not trivial example – the hope is this example can represent the greater case of all scientific theories.

Field’s objection does strike the Indispensability argument with a good blow.  However, as noted by Brown (pg. 59), “the notion of logical consequence that is needed is that of second-order logic” which is not recursively axiomatizable.  This means that the notion of consequence is not “nominalistically acceptable” since it involves being true in all models.  Also, if Maddy’s program pans out, then Field’s objection is irrelevant.


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