The Inverse Indispensability Argument

Most philosophers take it that the truth term plays the role of a predicate.  Since predicates denote properties this provides prima facie reason to think that truth is a property; many claims that we take to be correct appear to be ascriptions of a truth property to sentences or propositions.  However, as Quine famously pointed out, we would require the truth predicate for certain expressive functions—viz. undertaking commitments without the need to express them explicitly—whether or not there is a property of truth, and this fact constitutes an undercutting defeater for the prima facie reason to think that truth is a property.  In other words, the indispensability of a truth predicate (for purposes other than attributing a property of truth to sentences) undercuts the reason to think that ‘is true’ denotes a property.

Mathematical talk refers to quantifies over abstract mathematical objects.  Since referring terms denotes objects, mathematical talk provides prima facie reason to think that mathematical objects exist.; many claims that we take to be correct appear to be descriptions of an abstract realm of mathematical objects.   However, we would require reference (or apparent reference) to mathematical objects in order to describe concrete systems (in order to model physical phenomena mathematically) whether or not there are mathematical objects, and this fact constitutes an undercutting defeater for the prima facie reason to think that there are mathematical objects.  In other words, the indispensability of mathematics (for purposes other than describing a realm of abstract mathematical objects) undercuts the reason to think that mathematical terms denote extant abstract mathematical objects.

Two Kinds of Indispensability Argument

Continuing on the theme of nominalism and pragmatism…

The Putnam of yore took it that mathematical objects exist and is credited along with Quine as being an early proponent of the indispensability argument.  There are though two very different kinds of indispensability argument that Putnam made.  The first runs like this:

[Q]uantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.  This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes. [Philosophy of Logic: 347]

Why be a platonist?  Because, according to the argument, nominalism is inconsistent with physics.  One big problem with the argument is that there is plenty that gets quantified over in the sciences that we don’t take to exist; especially idealised versions of physical systems, the stock examples being frictionless surfaces, continuous fluids and the like.  So nominalism’s being “inconsistent with physics” in this sense isn’t a big deal, since the (clearly true) claims that fluids are not continuous, that there are no frictionless planes etc. are also “inconsistent with physics”.  (Penelope Maddy in Naturalism in Mathematics and Mary Leng in Mathematics and Reality both make this kind of point.)

Putnam also made a very different kind of indispensability argument, often conflated with the first, that goes like this:

I believe that the positive argument for realism has an analogue in the case of mathematical realism.  Here too, I believe, realism is the only philosophy that doesn’t make a success of science a miracle.  [Philosophy of Logic: 73]

There is an important shift from looking flatly to what entities are quantified over in our best scientific theories to looking at what quantification over these entities can be used to achieve.  This pragmatic spin is in fact necessary because quantification over mathematical objects is not indispensable simpliciter (if such a notion even makes sense), but indispensable for certain ends.  We could do without quantification over mathematical objects; we might just also have to do without iPhones, air travel and so on, if we did.  As Sellars famously said (in ‘A Semantical Solution to the Mind-Body Problem’) ‘[c]learly human beings could dispense with all discourse, though only at the expense of having nothing to say’.  So, if the indispensability of quantification over mathematical objects is supposed to be a problem for nominalism, it must be because talk of mathematical objects must be made use of to achieve certain ends; in which case what is at issue are mathematical practices.  The best explanation for the success of mathematical practices must involve the existence of mathematical objects, or so the thought goes.  But here’s the kicker: mathematical objects, because they are acausal, changeless and not subject to any events, cannot be invoked to explain any practices.  So the best explanation of the success of science needn’t invoke mathematical objects.