However, there is a revised version of the indispensability argument, defended best by Alan Baker, which evade all of these criticisms. Baker’s argument avoids them by arguing that mathematics plays more than just a representational role in science, but also plays an explanatory role. This revised argument also helps platonists in the fight against hard road nominalistm too. Although some of our theories can be nominalized, they do so by losing explanatory power. The argument is analogous to the scientific realist’s argument for quirks and electrons via their explanatory power. Baker’s example comes from biology. One of the puzzles facing biologists is why a certain insect, called the periodical cicada, has prime life cycles. A common explanation is that prime lifecycles cut down on intersection with predators. In this explanation, a property of numbers themselves, that prime numbers have the least multiples, plays an essential part of the explanation and does genuine explanatory work. Just like the scientific realist would argue, you can't have a genuine explanation without the explanans existing. Sometimes we posit entities that don’t play a causal explanatory role, but a mere logical explanatory role. For example, regardless whether you think this is the best explanation, some use the multiverse to explain the fine-tuning of the universe for intelligent life. It is argued that if there are multiple universes, a fine-tuned universe is really like a win of a cosmic lottery. Notice this explanation is purely logical, there is no causal contact between other universes for this explanation to work. Alan Baker suggests mathematical explanations are like that--they provide logical, not causal explanations.
1. It seems to me very difficult to run an indispensability argument out there as far as abstract objects go. Basically, indispensability says, "If you get rid of this, everything goes crazy." The problem, as I see it, is that this smuggles in causality, at least if causation is counterfactual. If you say, "Take X away, and you lose Y," then X is a cause of Y. If not, how could that conditional hold? You could say that Y is a part of X, and thus if you took X away you'd lose Y, but this is a rather trivial case and not one of causation (unless one wants to consider everything self-caused, but I digress). Balaguer points this out, more or less, in his book. Abstracta are not indispensable; you can take them away and it won't change anything at all. If per impossibile removing them caused external changes, that would mean they had some causal impact. But, ex hypothesi, they don't. Therefore, indispensability arguments really only make sense for concreta, for things which can stand in causal relations.
Explanations are tricky things. What is a good explanation in one person's eyes might not be good in another's. One also has to consider the whole explanation-truth relation: do good explanations have to be realist, or can they be fictionalist?
In the original indispensability argument, mathematical objects were idealizations and place holders--so it's not too hard to see how we could just retain the nominalized content and dispense with the platonic content. Let's specifically talk about the explanatory argument and specifically the case of the periodical cicada. Alan Baker points out that a property of numbers, primeness, is an essential part of the explanation. Only when you combine this piece of pure mathematics with a biological law and ecosystem constraints to you yield an explanation of the lifecycles of the cicada.
Without the piece of number theory, the explanation cannot get off the ground. Assuming talk of counterpossibles is meaningful, if there were no numbers, then this biological phenomenon would be inexplicable because there would be no property of primeness to help explain the data. So, yes, if you took away mathematical entities, then there is no reason why the cicada's lifecycle must be 13 or 17 years--a loss in explanatory power.
You're assuming a causal account of explanation, which is very contentious at best and question begging against the platonist at worst. Since we are dealing with a new type of relationship between objects, I don't see any reason to suspect other examples of this type of explanation should be available. Nevertheless, that crucial dependence between explanadum and explans still exists in my example.
We have to look at what makes an explanation a good one and Alan Baker reports that most biologists accept this explanation of the lifecycle of cicadas. An object can't do genuine explanatory work if it doesn't exist--this is the same point scientific realists make. If you retort that the difference between scientific unobservables and abstracta is their causal role, I'd repeat that in both cases if the objects didn't exist and have their properties, they couldn't account for the data.