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

136 paolo mancosuQuine and Goodman according to which ‘we cannot say what the worldwould be like without numbers, because describing any thinkable experience(except for utter emptiness) presupposes their existence’. (1978b, p.20) Suchreasoning, however, can be blocked by arguing that the true statement ‘thereare two cows in the field’ does not commit the speaker to the existence of thenumber two, for the apparent reference to the number two can be explainedaway using a standard elimination of the number term by using existentialquantifications. Of course, this does not mean that in more complicatedphysical statements the situation does not agree with the characterization givenby Quine and Goodman.Whether one agrees or not with Steiner on the many issues raised by hisposition, it is important to point out that his account had the merit of addressingthe problem of when the mathematics plays an ‘essentially’ explanatory rolein the explanation of a natural phenomenon and when it does not. Theissue has resurfaced in Baker (2005), where Steiner (1978b) is however notmentioned. Baker proposes a new line on the indispensability argument inwhich mathematical explanations play a central role. There are several versionsof the indispensability argument, but the general strategy runs as follows.Mathematics is indispensable for our best science. We ought to believe ourbest scientific theories and therefore we ought to accept the kind of entitiesour best theories quantify over. There are several ways to question the cogencyof this line of argument but the key feature related to Baker’s discussion isthe following. Many versions of the argument rely on a holistic conception ofscientific theories according to which ontological commitment is determinedusing all the existentially quantified sentences entailed by the theory. Noparticular attention is given to an analysis of how different components of thetheory might be responsible for different posits and to the roles that differentposits might play. Baker proposes a version of the indispensability argumentwhich does not depend on holism. His contribution takes its start from a debatebetween Colyvan (2001, 2002)andMelia(2000, 2002) which saw both authorsagreeing that the prospects for a successful platonist use of the indispensabilityargument rests on examples from scientific practice in which the postulationof mathematical objects results in an increase of those theoretical virtues whichare provided by the postulation of theoretical entities. Both authors agree thatamong such theoretical virtues is explanatory power. Baker believes that suchexplanations exist but also argues that the cases presented by Colyvan (2001)fail to be genuine cases of mathematical explanations of physical phenomena.Most of his article is devoted to a specific case study from evolutionary biologyand concerns the life-cycle of the so-called ‘periodical’ cicada. It turns out thatthree species of such cicadas ‘share the same unusual life-cycle. In each species