Fictionalist Applicability without Dispensability

If Balaguer's earlier arguments worked, all that would remain, in order for him to show that antiplatonism is tenable is an antiplatonist response to the Quine-Putnam indispensability argument, which claims that, since our physical theories are true and use mathematics, the mathematics is true and, moreover, that we cannot patch up the physical theories to dispense with the mathematics.

Balaguer has already argued that he thinks that physical theories can be patched up, but he doesn't find that argument particularly convincing because it has to be prosecuted one physical theory at a time. He therefore tries to give an argument that shows, wholesale, that an antiplatonist can allow that our best physical theories include (ineliminable) uses of mathematics.

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Side remark. There is a traditional way of arguing that mathematics is dispensable that Balaguer ignores. It is worth discussing briefly. The idea is that one just replaces any physical theory by the set of all of its nonmathematical? purely physical? consequences. There is an obvious objection, which is that that theory would just be an infinite collection of sentences not suitably related, but a theorem of Craig (using what he called "the method of pleonasms") shows that if a theory is recursively axiomatizable (that is, intuitively, given by a fully specified, completely decidable, procedure), and the set of sentences that are nonmathematical is recursively enumerable, then the "physical part" of the theory is also recursively axiomatizable.

Field, Balaguer, and other advocates of antiplatonism pretty generally accept that "Craigifications" of theories don't do what is required, because, among other reasons, they don't have the explanatory power of the original theories: The general principles that lead to systematic predictions are, usually, mathematical, and there is no reason to think that they have counterparts in the "Craigified" theories.

Consider, for example, predictions made on the basis of, say, Newton's law of gravitation. One ends up with only the predictions.

I have a different, less common, objection to taking the physical parts of a theory: It is not clear what counts as purely physical, and it is even less clear that, if there is such a distinction to be made, whether it can be made in the way that is usually assumed, which is designating a "nominalistic vocabulary," and even whether the set of such sentences is recursively enumerable. Here are some examples of problems: Are statements about the equator of a rotating body nominalistically acceptable? Is any nominalized counterpart of a quantum-mechanical probability nominalistically acceptable? In addition, there are ongoing debates within, for example, quantum electrodynamics (qed), about whether particular quantities are mathematical or physical, for example the gauge fields of qed (Aharonov-Bohm).

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Balaguer distinguishes between two possible kinds of indispensability, relative and absolute. Relative indispensability is indispensability in some particular theory. Absolute indispensability is indispensability for the very possibility of doing science. The distinction plays no role in his further argument, but it is interesting because it suggests two very different kinds of ways in which one might argue for indispensability.

All the antiplatonist needs to explain is applicability in order to make a prima facie case for the tenability of antiplatonism. No one has a good account of indispensability, in part because there is not general agreement about whether mathematics actually is indispensable, and the challenge that was specific to antiplatonism was to see how a physical theory can usefully employ false statements (namely, the mathematical ones).

My take on Balaguer's answer is heavily influenced by work of Patrick Dieveney. Quine emphasizes that it is intellectually dishonest to use a theory while claiming not to believe it, at least without admitting an obligation to say what you actually believe. (It is, for example, perfectly ok to use geocentric astronomy for navigating a ship at sea.) That is the challenge Balaguer's antiplatonism must meet. (I owe that formulation to Dieveney. It is not Balguer's, and it was published after Balaguer wrote his book.)

Balaguer's proposed antiplatonist theory is fictionalist. It cannot endorse the standard view that our best physical theories are true. Why not? Since he is allowing that thoses theories use mathematics, if they were true, platonism would follow. Balaguer's fictionalist is therefore in the uncomfortable position of claiming that all of our scientific theories are wrong. What the fictionalist claims instead is that they are true enough, that the important bits are true. Here it is (nominalistic scientific realism): "what empirical science entails about the physical world … is true …, while its platonistic content … is fictional" (131).

It isn't clear what that means in the case of a physical theory in which mathematics is used. It becomes clearer when one looks at Balaguer's argument for it:

our empirical theories do not simply express some nominalistic facts and some platonistic facts; rather, they express mixed facts.

appreciate the full significance of the causal inertness of mathematical objects.

is true in virtue of facts about [the nominalistic part] and [the mathematical part] that are entirely independent of one another, that is, that hold or don't hold independently of one another

this is beyond doubt … we are forced to say that while (A) does express a mixed fact, it does not express a bottom-level mixed fact; that is, the mixed fact that (A) expresses supervenes on more basic facts that are not mixed.

133

Balaguer takes it to be "beyond doubt" that there are no "bottom level" facts that mix mathematics and nominalist parts because mathematical entities are causally inert.

Whenever someone says that anything is "beyond doubt" warning sirens should go off in your head: that is a written counterpart of waving your hands vigorously.

There are many serious gaps in the argument.

• Why is causal relation the only kind of relation that can make facts inseparable? It seems to me, for example, that semantic relations can make facts inseparable. There are facts expressed in physical theories about physical objects that are only expressed using some mathematics. They don't seem to have any meaningful counterparts without the mathematics. Consider Balaguer's sentence (A):
• The physical system S is forty degrees Celsius.

There just isn't any way to express that in the ordinary ways in which physicists speak that does not involve numbers. The example is tendentious, of course, since it is a use of mathematics that Field has shown us how to eliminate. There are lots of statements where it is not so easy. Consider, for example, the statement in Newtonian mechanics that momentum is conserved. It is part of Newtonian mechanics that parts of systems have associated a quantity of momentum. How is the statement that momentum is conserved separable into mathematical and nominalistic facts? Field shows how to eliminate such statements: instead of associating a mathematical quantity, momentum, with each part of a system, he associates a point in space with each such part, points along an ersatz number line. The conservation then says something about operations with distances along the ersatz number line. The Field version gives us an entirely physical fact (let us grant), but it doesn't show us which part of that fact is the nominalistic part of the original. I can't see what that part might be.

Even if we suppose that all "basic facts" are either mathematical or physical, how does that give one reason to believe that "mixed facts" are not "bottom level" facts, that they must supervene on basic facts? There is some metaphysical belief in "basic facts," "mixed facts," and "bottom-level facts" underlying the claim, and since Balaguer doesn't tell us anything else about what such things are, I have no idea what the beliefs are. It doesn't seem to me that there is any coherent notion of a bottom-level fact or a basic fact. Our epistemic situation and our physical theories are just not set up in that way.

Which of these is a basic fact?

• The kinetic energy of S is of a certain (physical) quantity. (How is kinetic energy nominalistically acceptable?)
• The mercury in the thermometer in S has its meniscus near an engraving in the following shape: 40.
• It appears to the physicist standing near what he takes to be S that he has sense data of the sort characteristically produced by what he takes to be a thermometer that has the meniscus of its mercury to be near an engraving in the following shape: 40.
• The temperature of S is the temperature we conventionally describe as 40 degrees Celsius.

I could go on forever. The fact that Balaguer has given us no clue indicates that his argument is, to say the least, lacking in sufficient detail

Of course, there is a fifty-year-long tradition of denying that one can make sense of the idea of a basic scientific fact along any of the lines I've just suggested and that there are reasons in principle for thinking that no other version could work either.

Forget about basic. I raised concerns above about whether there is even a clear distinction between mathematical facts and purely physical facts. Balaguer needs such a distinction to claim that mixed facts supervene on facts of those two kinds.

Since Balaguer grants that there may be no nominalized versions of our best scientific theories, he grants that it may not be possible to express the separate mathematical and physical facts, and so he is committed to only being committed to a part of a theory such that that part is not expressible. It seems to me that the Quinean insistence on intellectual honesty applies to such a view.

Balaguer's defense of his fictionalist position relies heavily on the idea, which he adopts without argument, that there are nominalist and mathematical components of our scientific theories.

Since that idea is critical for his position, it is worth strongly emphasizing that it is a consequence of his position, one that he explicitly draws, that the separate nominalist component of a scientific may not be separately expressible. He talks about nominalist content as if that is what matters, but, by his own lights, there need be no sentences of any language, even in principle, that can be used to state that content. The closest one can come, he allows, in at least some cases, may be to state a "mixed" fact (not actually a fact, since it is false) and claim that the nominalistic part of it holds up its part of the bargain. He never spells out that metaphor, but what it seems to me to mean is that the "mixed" statement would have been true if the mathematical part had been true. That makes a certain amount of sense, but it is hard to see how to analyze in any detail without becoming committed to the possible existence of mathematical objects, which, for Balaguer, would be just as bad (in fact, it is the same thing as) the actual existence of mathematical objects. (I am not claiming that someone might not argue that possible mathematical objects are somehow better than actual mathematical objects, just making the ad hominem point that Balaguer denies that, and hence owes us a lot more explanation.)

-- ShaughanLavine - 19 Apr 2007 - 24 Apr 2007