Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

140 paolo mancosuIf this is correct, we have something in the vicinity of an indispensabilityargument. In ‘Platonism’ Dummett rejects a similar argument on accountof the fact that ‘real numbers and ordinals do not act on each other oron anything else; so there is nothing which is left unaccounted for if wesuppose them not to be there’ (Dummett, 1978, p.204). But it is evident thatsuch rejection is based on the questionable assumption that all explanationsmust be causal. The argument presented above would of course leave acommitted predicativist unimpressed (for the explanation in question wouldbe a derivation which uses tools not available to the predicativist). However,just as standard indispensability arguments address those who are realists abouttheoretical entities in science, so here the intended audience for the argumentwould consist of those who are realists about a certain realm of mathematicalentities (say, the natural numbers) and in addition are not already committedto a foundational position (such as predicativism) which forbids entertainingthe entities being postulated by the explanation. Reconstructed as such theargument is useful in providing rational grounds for the acceptance of themathematical entities appealed to in the explanation. Its strength becomesevident when one considers that the argument goes through even if it turnsout that the entities in question are in principle eliminable on account of thefact that the explained result is derivable within a narrower framework (as inthe case where you have a theory T ′ which is a conservative extension ofT). However, if this derivation results in a loss of explanatory power then westill have good reasons to believe in the entities in question. This of courseleaves us with the question of when a derivation is explanatory. And thisparallels the situation we discussed in the case of mathematical explanations ofphysical phenomena. I should point out, however, that I am not endorsing theindispensability argument for mathematics I have been considering but that Ido find it of interest.The original form of the indispensability argument relied on a form ofconfirmational holism. This left the argument open to the objection, raisedforcefully by Maddy, that scientific practice proceeds otherwise or to theobjection that other accounts of confirmation block the conclusion (Sober,1993). In response, advocates like Colyvan and Baker have argued thatexplanatory considerations lead to platonism even if we drop confirmationalholism. But, as I pointed out, nobody really has an account of mathematicalexplanations of scientific phenomena.In addition to being of independent interest, the move to mathematicalexplanations of mathematical facts is justified also by the following twoconsiderations. First, it is conceivable that whatever account we will endup giving of mathematical explanations of scientific phenomena, it won’t be