On page 5 of “Reason, Mathematics, and Modality”, Hartry Field considers an objection that claims to show that fictionalism about mathematics is unintelligible. Basically, the argument suggests that the reason we are fictionalist about the existence of Oliver Twist is that we don’t literally believe the story, because literal belief would involve certain results of certain investigations of 19th century London. However, for mathematics, there are no corresponding investigations whose results we reject, and thus the fictionalist says nothing different from what the platonist says. Field points out that this argument actually would only show that platonism and fictionalism are the same, and not that fictionalism is unintelligible.

However, Field argues that there are in fact differences between platonism and fictionalism. He says that fictionalism has both positive and negative content. The positive content is that “it commits one to abjuring all appeal to mathematical entities in explanations when the chips are down: it must be possible, for instance, to develop theoretical physics without any appeal to mathematical entities.” Its negative content is that “it avoids having to answer some questions that seem to need answering on a platonist view.”

It seems to me that these claims can be stated more succinctly by saying that the positive content of fictionalism is the conservativity of mathematics over purely concrete physical theories and the negative content is taking an instrumentalist view of abstract mathematical entities to avoid having to take a position on certain supposed facts of the matter about them. That is, fictionalism is just instrumentalism plus conservativity.

Conservativity is the (testable) claim that there are no observational or purely concrete claims in, say, physics that are decided by the addition of mathematical axioms making reference to a separate class of non-observable entities. That is, we should be able to come up with a purely concrete physical theory that only quantifies over concrete objects and makes every prediction of the current theories that do make reference to purely mathematical objects. (I believe this was the project of Field’s 1980 book “Science Without Numbers”, though he only tackled Newtonian physics there.)

Instrumentalism is the belief in all consequences of a theory while remaining agnostic about the theory itself. One may say that the Copenhagen Interpretation of quantum mechanics is instrumentalism about quantum mechanics, believing that it makes accurate predictions, but remaining agnostic as to whether the postulated particles are in one location or another until observed.

However, when phrased like this, instrumentalism about a conservative theory seems nearly pointless. Adopting the theory in a purely instrumental fashion adds no new predictive power. However, (as Field suggests) it may make many calculations much simpler.

Of course, for fictionalism to seem plausible, it seems that the theory over which mathematics is conservative should be more simply characterized in some way other than as “the observable consequences of the mathematical theory”. The Copenhagen Interpretation takes this sort of instrumentalist approach (which Steel refers to in “Does Mathematics Need New Axioms?” with regards to large cardinal axioms and their consistency), but Field tries to give a purely concretely stateable theory over which mathematics is conservative. Thus, the conservativity of the mathematical theory seems to be what makes instrumentalism plausible, rather than merely a syntactic move. For a non-conservative extension (that is, a mathematical theory whose purely concrete fragment has no direct characterization as of yet), instrumentalism seems like nothing more than a brazen opposition to the Quine-Putnam indispensability argument, denying commitment to the entities quantified over in one’s best theory. However, by giving a purely concrete theory over which the mathematical theory is conservative, Field manages to make the mathematical entities dispensable, so that the argument no longer applies. Thus, both conservativity and instrumentalism are necessary for the fictionalist view, although conservativity is a fact about theories and instrumentalism is an attitude.

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