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If we believe that the mathematical assumptions utilized by those theories are true, and also believe any non-mathematical assumptions we make use of, then we have no difficulty justifying our belief in any empirical consequences we validly derive from those assumptions: if the premises of our arguments are true then the truth of the conclusions we derive will be guaranteed as a matter of logic. On the other hand, though, if we do not believe the mathematical premises in our empirical arguments, what reason have we to believe their conclusions?
Different versions of mathematical fictionalism take different approaches to answer this question, depending on how realist they wish to be about our scientific theories. We will consider these combinations separately. If our mature scientific theories include, as Putnam himself contends that they do, statements whose approximate truth would require the existence of mathematical objects, then the combination of scientific realism with mathematical fictionalism seems impossible.
If one accepts scientific realism so-formulated, then what prospects are there for mathematical fictionalism? The only room for manoeuvre comes with the notion of a mature scientific theory. Certainly, in formulating the claims of our ordinary scientific theories we make use of sentences whose literal truth would require the existence of mathematical objects.
But we also make use of sentences whose literal truth would require the existence of ideal objects such as point masses or continuous fluids, and we generally do not take our use of such sentences to commit us to the existence of such objects.
That is, we are not committed to point masses or continuous fluids because these theoretical fictions can be dispensed with in our best formulation of these mature theories. Hartry Field, who wishes to combine scientific realism with mathematical fictionalism, thinks that the same can be said for the mathematical objects to which our ordinary scientific theories appear to be committed: in our best expressions of those theories, sentences whose literal truth would require the existence of such objects can be dispensed with.
P1 Scientific Realism : We ought to believe that the sentences used to express our best mature scientific theories, when taken at face value, are true or approximately true. C Mathematical Platonism : We ought to believe in the existence of mathematical objects. For an alternative formulation of the argument, and defence, see Colyvan In his defence of mathematical fictionalism, Field rejects P2, arguing that we can dispense with commitments to mathematical objects in our best formulations of our scientific theories.
In Science without Numbers Field makes the case for the dispensability of mathematics in Newtonian science, sketching how to formulate the claims of Newtonian gravitational theory without quantifying over mathematical objects. But Field is a fictionalist about mathematics, not a mere skeptic about mathematically-stated theories. The claim that we can dispense with mathematics in formulating the laws of our best scientific theories is, therefore, only the beginning of the story for Field: he also wishes to explain why it is safe for us to use our ordinary mathematical formulations of our scientific theories in our day-to-day theorizing about the world.
A mathematically-stated empirical theory P is a conservative extension of a nominalistically stated theory N just in case any nominalistically stated consequence A of P is also a consequence of the nominalistic theory N. Or, put another way, suppose we have an ordinary mathematically-expressed, and therefore platonistic scientific theory P and a nominalistically acceptable reformulation of that theory, N.
If this reformulation is successful, then every nonmathematical fact about the nonmathematical realm implied by P will also be implied by N. Why, if we have a pair of such theories, N and P, does this give us license to believe the nonmathematical consequences of P, a theory whose truth we do not accept?
Simply because those consequences are already consequences of the preferred nominalistic theory N, which as scientific realists we take to be true or approximately true. Our confidence in the truth of the nonmathematical consequences we draw from our mathematically stated scientific theories piggy backs on our confidence in the truth of the nonmathematical theories those theories conservatively extend.
But why, we may ask, should we bother with the mathematically-infused versions of our scientific theories if these theories simply extend our literally believed nominalistic theories by adding a body of falsehoods? Nominalistically stated theories are unwieldy, and arguments from nominalistically stated premises to nominalistically stated conclusions, even when available, can be difficult to find and impractically long to write down. With the help of mathematics, though, such problems can become tractable.
In short, following Carl G. Thus, in the establishment of empirical knowledge, mathematics as well as logic has, so to speak, the function of a theoretical juice extractor: the techniques of mathematical and logical theory can produce no more juice of factual information than is contained in the assumptions to which they are applied; but they may produce a great deal more juice of this kind than might have been anticipated upon a first intuitive inspection of those assumptions which form the raw material for the extractor.
Hempel : Furthermore, if our nominalistic theories employ second-order logic in their formulation, some of these consequences can only be extracted with the help of mathematics as noted by Field n. However, it does suggest a further indispensability argument that questions our license to use mathematics in drawing inferences if we do not believe that mathematics to be true.
Michael D. Even if Field can dispense with mathematics in stating the laws of our scientific theories, the focus this argument places on the indispensability of mathematics in derivations, i. If we stick with second-order formulations of our nominalistic theories, then premise 2 of this argument is right in claiming that mathematics is indispensable to drawing out some of the consequences of these theories in the sense that, for any consistent such theory and any sound derivation system for such a theory there will be semantic consequences of those theories that are not derivable within those theories relative to that derivation system.
But does the use of mathematics in uncovering the consequences of our theories require belief in the truth of the mathematics used? Arguably, our reliance on, e. See, e. Field , though, has expressed some concerns about the second-order version of Newtonian gravitational theory developed in Science without Numbers. This, however, rather complicates the account of applications given in Science without Numbers , since Field can no longer claim that our ordinary mathematically stated scientific theories conservatively extend their nominalistically stated counterparts.
As Field concedes, at best we can have partial representation theorems that allow for some match between our mathematical and nonmathematical claims, without straightforward equivalence. As Alasdair Urquhart points out, Newtonian science is very convenient in that, since it assumes that spacetime has the structure of R 4 , it becomes easy to find claims about relations between spacetime points that correspond to mathematical claims expressed in terms of real numbers.
But contemporary science takes spacetime to have non-constant curvature, and with the lack of an isomorphism between spacetime and R 4 , the prospects for finding suitable representation theorems to match mathematical claims with claims expressed solely in terms of qualitative relations between spacetime points are less clear.
Furthermore, as David Malament notes, Newtonian science is likewise convenient in that its laws primarily concern spacetime points and their properties the mass concentrated at a point, the distance between points, etc. But many of our best scientific theories such as classical Hamiltonian mechanics or quantum mechanics are standardly expressed as phase space theories, with their laws expressing relations between the possible states of a physical system.
An analogous approach to that of Science without Numbers would dispense with mathematical expressions of these relations in favour of nonmathematical expressions of the same — but this would still leave us with an ontology of possibilia , something that would presumably be at least as problematic as an ontology of abstract mathematical objects. It is not enough to rely on the sketch provided by Science without Numbers : mathematical fictionalists who wish to remain scientific realists in the Putnam-Boyd sense must show how they plan to dispense with mathematics in those cases where the analogy with Newtonian science breaks down at a minimum, explaining how to deal with spacetime of non-constant curvature, phase space theories, and the probabilistic properties of quantum mechanics.
But there clearly remains much to be done to show that mathematical assumptions can be dispensed with in favour of nominalistically acceptable alternatives. What accounts for the trustworthiness of those theories is not that they are true in their mathematical and nonmathematical parts , but that they are correct in the picture they paint of the nonmathematical realm. Perhaps those, too, may be successful not because they are true in all their parts, but simply because they are correct in the picture they paint of the nonmathematical world?
The contribution mathematical assumptions might be making to our theoretical success would then not depend on the truth of those assumptions, but merely on their ability to enable us to represent systems of nonmathematical objects. Mathematics provides us with an extremely rich language to describe such systems; maybe it is even indispensable to this purpose.
But if all that the mathematics is doing in our scientific theories is enabling us to form theoretically amenable descriptions of physical systems, then why take the indispensable presence of mathematical assumptions used for this purpose as indicative of their truth? This view depends on holding that there is a nominalistic content to our empirical theories even if this content cannot be expressed in nominalistic terms , and that it is reasonable to believe just this content believing that, as we might say, our empirical theories are nominalistically adequate , not that they are true.
Similar claims about the value of our mathematically stated scientific theories residing in their accurate nominalistic content rather than in their truth can be found in the work of Joseph Melia , Stephen Yablo and Mary Leng , though of these only Leng explicitly endorses fictionalism. Difficulties arise in characterising exactly what is meant by the nominalistic content of an empirical theory or the claim that such a theory is nominalistically adequate.
That is, fictionalists must explain why they can reasonably rely on our ordinary scientific theories in meeting standard scientific goals such as the goals of providing predictions and explanations, if they do not believe in the mathematical objects posited by those theories. Balaguer attempts such an explanation by means of his principal of causal isolation , the claim that there are no causally efficacious mathematical objects. According to Balaguer ,. Empirical science knows , so to speak, that mathematical objects are causally inert.
That is, it does not assign any causal role to any mathematical entity. Thus, it seems that empirical science predicts that the behaviour of the physical world is not dependent in any way on the existence of mathematical objects. Not all predictions predict by identifying a cause of the phenomenon predicted, and, perhaps more crucially, not all explanations explain causally.
Balaguer suggests that, were there no mathematical objects at all, physical objects in the physical world could still be configured in just the ways our theories claim. But if there were no mathematical objects, would that mean that we would lose any means of explaining why the world is configured the way it is? If mathematical posits are essential to some of our explanations of the behaviour of physical systems, and if explanations have to be true in order to explain, then a fictionalist who does not believe in mathematical objects cannot reasonably endorse the kinds of explanations usually provided by empirical science.
Alan Baker summarises the argument as follows:. Let us suppose that this means that sentences whose literal truth would require the existence of mathematical objects are present amongst the explanandans of some explanations that we take to be genuinely explanatory. Two lines of response suggest themselves: first, we may challenge the explanatoriness of such explanations, arguing that candidate such explanations are merely acting as placeholders for more basic explanations that do not assume any mathematics effectively rejecting 2 ; Melia , may be interpreted as taking this line; Bangu does so more explicitly.
On the other hand, we may accept that some such explanations are genuinely explanatory, but argue that the explanatoriness of the mathematics in these cases does not depend on its truth effectively rejecting 1 ; this line is taken by Leng , who argues that explaining the behaviour of a physical system by appeal to the mathematical features of a mathematical model of that system does not require belief in the existence of the mathematical model in question, only that the features that the model is imagined to have are appropriately tied to the actual features of the physical system in question.
On the other hand, though, it is less clear that an explanation that appeals to the mathematically described structure of a physical system loses its explanatory efficacy if one is merely pretending or imagining that there are mathematical objects that relate appropriately to them. An alternative fictionalist option is the combination of mathematical fictionalism with constructive empiricism, according to which we should only believe that our best scientific theories are empirically adequate , correct in their picture of the observable world, remaining agnostic about the claims that those theories make about unobservables.
While Bas van Fraassen is standardly viewed as presenting constructive empiricism as agnosticism about the unobservable physical world, it would seem straightforward that any reason for epistemic caution about theories positing unobservable physical entities should immediately transfer to a caution about theories positing unobservable mathematical entities.
As Gideon Rosen puts the point, abstract entities. Experience cannot tell us whether they exist or what they are like. The theorist who believes what his theories say about the abstract must therefore treat something other than experience as a source of information about what there is. The empiricist makes it his business to resist this. So it would seem that just as he suspends judgment on what his theory says about unobservable physical objects, he should suspend judgment on what they say about the abstract domain.
Despite their natural affinity, the combination of mathematical fictionalism with constructive empiricism is not as straightforward as it may at first seem. Van Fraassen 64 adopts the semantic view of theories, and holds that a. This might be something mundane, like the way we talk about the sun setting in the west it is the earth that moves , or it could be something much deeper, like engaging in talk that is ostensibly about numbers despite believing that numbers do not literally exist.
Rather than regard such behaviour as self-defeating, a "fictionalist" is someone who thinks that this kind of discourse is entirely appropriate, even helpful, so long as we treat what is said as a useful fiction, rather than as the sober truth. Within contemporary analytic philosophy, fictionalism has been on the scene for well over a decade and has matured during that time, growing in popularity. There are now myriad competing views about fictionalism and consequently the discussion has branched out into many more subdisciplines of philosophy.
Yet there is widespread disagreement on what philosophical fictionalism actually amounts to and about how precisely it ought to be pursued. This volume aims to guide these discussions, collecting some of the most up-to-date work on fictionalism and tracing the view's development over the past decade. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others.
Metaphysics as a Fiction , by Gideon Rosen 2. Fictionalism and Reasons, by Chris Daley 4. Against Hermeneutic Fictionalism , by David Liggins 5. Fictionalism: Morality and Metaphor , by Richard Joyce 6. Should the Mathematical Fictionalist be a Moral Fictionalist too? He is on the editorial board at The American Philosophical Quarterly , and he will be starting as chair of his department in Fall He is on the editorial board of the Australasian Journal of Philosophy and is a subject editor for 20th Century Philosophy for the Stanford Encyclopedia of Philosophy.
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