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Okay, something basic about the background of Field’s program is really confusing me.
The motivation seems to come from Quine’s statement, in “On What There Is,” that
The variables of quantification, “something”, “nothing”, “everything”, range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.
Which makes sense. If you say “every x has property y” it would be weird if you didn’t think “an x” was a thing. (Well, it seems weird at first glance, anyway.)
But how on earth is this a problem for science? The statements of Newtonian physics (say) aren’t assertions about things that are true for (say) all numbers. They’re about physical variables (lengths and durations), or – if you must – predictions, or whatever. But if I expect a theory of physics to make assertions to me about numbers, I am looking for something very strange out of it.
To be specific, I’m expecting these theories to ultimately assert things like
“for every distance d (where d could be measured numerically, if you swing that way), it is true that […]”
rather than
“for any real number r, consider a distance of r units; it is true that […]”
That is, if it forces me to believe in anything, it should force me to believe in things like distances, not in numbers. (What would it be like for a Newtonian physicists to believe in numbers but not distances? I envision myself sitting forever beside the inert Real Line, possessing a set of physical laws which cannot be applied because there is no space or time or mass. “Well, this theory is a set of statements about real numbers, ultimately.” Really? Could have fooled me!)
The failure of Field’s program, if you think it failed, should be taken to show that “if you swing that way” is not really appropriate here; i.e., there’s not really any other way to swing.
I think you have an overly restrictive view of which statements have ontological commitments - or at least a view that would be heterodox in some circles. Remember my natural history museum example? Say I’m a small child tugging at my mother’s shirtsleeves and whispering excitedly, “Those bones used to belong to a dinosaur in Brazil!” Here are some things to whose existence I seem to have committed:
- bones
- dinosaur
- Brazil
To see why, see how ridiculous these sentences sound:
- There’s no such thing as bones, and those bones used to belong to a dinosaur in Brazil!
- There’s no such thing as dinosaurs, and those bones used to belong to a dinosaur in Brazil!
- There’s no such thing as Brazil, and those bones used to belong to a dinosaur in Brazil!
But, unless a hard-road nominalization process is viable (and I gave some reasons in the previous post that people think it isn’t), you seem to want to put scientists in the position of saying “There’s no such things as functions and real numbers, and the state of the world at time t2 is related to the state of the world at time t1 by [some mechanism using functions and real numbers].” Embarrassingly, I don’t know enough about how scientists would talk about these things to give a good example, and it’s not fleshed out much here.
The statement in the last paragraph doesn’t seem bad to me? Or, it’s exactly as bad as “there’s no such thing as the number 3, and here are 3 apples.” Which sounds strange when you put it that way, but there are people – roughly, non-Platonists of various kinds – who would defend the idea that “there is no such thing as a the number 3″ (because numbers are not objects, say) even though we can count things. The details of the phrasing matter: there certainly is a number 3 in the sense that we mean something when we say “3,” but that isn’t sufficient for there to be “such a thing as” the number 3.
(Admittedly, the statement with the 3 apples isn’t a quantification. But we could turn it into one, like “there is no such thing as a natural number, but for any natural number n, if I have n apples, then [something],” which sounds awkward at first glance, but substantively doesn’t seem any worse than the 3 apples statement.)
ETA: in other words, it seems like the physicist’s statement is only awkward/ridiculous if Platonism is true. But if we can just act like Platonism is uncontroversially true, then, well, goodbye Philosophy of Mathematics!
