*explanatory indispensability argument*in support some of our mathematical beliefs. The challenge for everyone, though, is to articulate a reasonable form of IBE that accords with scientific practice, but which does not overgenerate beliefs in things which we reject. Despite its clarity, Lipton's discussion of IBE seems to me overly restrictive because he focuses only on causal explanations. Are there plausible principles for the use of IBE which allow non-causal explanations?

Here is one, but it results in problems for any use of IBE to justify our mathematical beliefs. I call it "Sensitivity", although perhaps this not the best label:

Here I am imagining an agent who is in doubt about the truth of some competing options A1, A2, A3. Suppose that A1 appears in our best explanation. Sensitivity tells us that this contribution of A1 to the explanation can only license belief in A1 when the way in which A1 contributes makes either A2 or A3 less likely.Sensitivity: A claim which appears in an explanation can receive support via IBE only when the explanatory contribution tells against some relevant epistemic possibilities.

This seems to me to be a very weak and plausible restriction on IBE. It is met by the standard atoms and electrons cases, and also by Woodward's non-causal explanation of the stability of planetary orbits. In my next post, I want to outline a case for the claim that sensitivity blocks the use of IBE to support mathematical claims.

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