This is known as the indispensability argument, because the mid-twentieth century philosopher Willard Van Orman Quine used it to argue (roughly) that mathematical entities were "indispensable" posits of empirical science, and hence we have empirical evidence that mathematical entities exist.
There are a number of objections to this argument in §4 of that Stanford Encyclopedia entry.
One of van Fraassen's arguments against scientific realism is also applicable to the indispensability argument. He's the realist argument:
the positive argument for realism is that it is the only philosophy that doesn't make the success of science a miracle. That terms in mature scientific theories typically refer (this formulation is due to Richard Boyd), that the theories accepted in a mature science are typically approximately true, that the same term can refer to the same thing even when it occurs in different theories—these statements are viewed by the scientific realist not as necessary truths but as part of the only scientific explanation of the success of science, and hence as part of any adequate scientific description of science and its relations to its objects.
And van Fraassen's objection:
The explanation provided is a very traditional one—adequatio ad rem, the ‘adequacy’ of the theory to its objects, a kind of mirroring of the structure of things by the structure of ideas—Aquinas would have felt quite at home with it.
Well, let us accept for now this demand for a scientific explanation of the success of science. Let us also resist construing it as merely a restatement of Smart's ‘cosmic coincidence’ argument, and view it instead as the question why we have successful scientific theories at all. Will this realist explanation with the Scholastic look be a scientifically acceptable answer? I would like to point out that science is a biological phenomenon, an activity by one kind of organism which facilitates its interaction with the environment. And this makes me think that a very different kind of scientific explanation is required.
I can best make the point by contrasting two accounts of the mouse who runs from its enemy, the cat. St. Augustine already remarked on this phenomenon, and provided an intentional explanation: the mouse perceives that the cat is its enemy, hence the mouse runs. What is postulated here is the ‘adequacy’ of the mouse's thought to the order of nature: the relation of enmity is correctly reflected in his mind. But the Darwinist says: Do not ask why the mouse runs from its enemy. Species which did not cope with their natural enemies no longer exist. That is why there are only ones who do.
In just the same way, I claim that the success of current scientific theories is no miracle. It is not even surprising to the scientific (Darwinist) mind. For any scientific theory is born into a life of fierce competition, a jungle red in tooth and claw. Only the successful theories survive—the ones which in fact latched on to actual regularities in nature.
This is from The Scientific Image, pp 39-40.
As I read this, van Fraassen argues that we don't need to explain the predictive success of scientific theories. We've constructed them to make good predictions, and basically if and when they no longer make good predictions we abandon them and formulate new ones. There's nothing remarkable about the phenomenon that scientific theories designed to be good for making empirical predictions are indeed good for making empirical predictions. Nothing follows about "the nature of reality."
The same argument can be made in response to Feynman's argument. We've developed and deployed mathematical formalisms that have been useful for the scientific task at hand. If and when they're no longer useful, we develop new formalisms. There's nothing remarkable about the phenomenon that mathematical formalisms designed to be good for certain tasks are indeed good for doing those tasks. Nothing follows about "the nature of reality."
