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Science Nominalized
Abstract
I propose a way of formulating scientific laws and magnitude attributions which eliminates ontological commitment to mathematical entities. I argue that science only requires quantitative sentences as thus formulated, and hence that we ought to deny the existence of sets and numbers. I argue that my approach cannot plausibly be extended to the concrete "theoretical" entities of science.
... In 'Science Nominalized', Terence Horgan [1984] develops an if/thenist account of applied mathematics. Horgan's is, in the jargon, a kind of easy-road nominalism; the intended contrast is with 'hard-road' nominalists such as Hartry Field. ...
An undemanding claim ϕ sometimes implies, or seems to, a more demanding one ψ. Some have posited, to explain this, a confusion between ϕ and ϕ*, an analogue of ϕ that does not imply ψ. If-thenists take ϕ* to be If ψ then ϕ. Incrementalism is the form of if-thenism that construes If ψ then ϕ as the surplus content of ϕ over ψ (ϕ∼ψ). The paper argues that it is the only form of if-thenism that stands a chance of being correct.
... Horgan's counterfactual style of if-thenism runs into an analogous problem(Horgan 1984). Hellman suggests a "non-interference proviso": "we must stipulate from the outset that the only possibilities we entertain in employing the [modal operator] are such as to leave the actual world entirely intact."(Hellman ...
Indicative conditionals appear to lie on a continuum, with the subjective and information-based on one side, and the objective and fact-based on the other. Attempts to bring them all under the same theoretical umbrella usually start at the subjective end; conditionals get more objective as they come to be based in higher-quality, less parochial, information. I propose to go in the other direction, looking first for a class of “absolute” conditionals, then bringing in other conditionals by relaxing the constraints defining that class. (A plan of action is laid out at the end of section 4. The final footnote of each section sketches the contents of the next.)
Using ideas proposed in Aboutness and developed in ‘If-thenism’, Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo’s stated goal is to provide a reading of a sentence like ‘The number of planets is eight’ with a sort of content on which it fails to imply ‘Numbers exist’. After presenting Yablo’s framework, our paper raises a problem with his view that has gone virtually unnoticed in the literature. If we are right, then Yablo’s version of if-thenism cannot succeed.
This Element defends mathematical anti-realism against an underappreciated problem with that view-a problem having to do with modal truthmaking. Part I develops mathematical anti-realism, it defends that view against a number of well-known objections, and it raises a less widely discussed objection to anti-realism-an objection based on the fact that (a) mathematical anti-realists need to commit to the truth of certain kinds of modal claims, and (b) it's not clear that the truth of these modal claims is compatible with mathematical anti-realism. Part II considers various strategies that anti-realists might pursue in trying to solve this modal-truth problem with their view, it argues that there's only one viable view that anti-realists can endorse in order to solve the modal-truth problem, and it argues that the view in question-which is here called modal nothingism-is true.
Recent work in the philosophy of language attempts to elucidate the elusive notion of aboutness (Berto 2018; Lewis 1988; Fine 2017a, b; Hawke 2017; Moltmann 2018; Yablo 2014). A natural question concerning such a project has to do with its motivation: why is the notion of aboutness important? Stephen Yablo (2014) offers an interesting answer: taking into consideration not only the conditions under which a sentence is true, but also what a sentence is about opens the door to a new style of criticism of certain philosophical analyses. We might criticize the analysis of a given notion not because it fails to assign the right truth conditions to a class of sentences, but because it characterizes those sentences as being about something they are not about. In this paper, I apply Yablo’s suggestion to a case study. I consider meta-fictionalism, the view that the content of a mathematical claim S is ‘according to standard mathematics, S’. I argue, following Woodward (2013), that, on certain assumptions, meta-fictionalism assigns the right truth-conditions to typical assertoric utterances of mathematical statements. However, I also argue that meta-fictionalism assigns the wrong aboutness conditions to typical assertoric utterances of mathematical statements.
Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation.
I reply to comments on "If-Thenism" by Suki Finn, Katharina Felka, Amie Thomasson, Seahwa Kim, Daniel Dohrn, Gideon Rosen, Otavio Bueno, Brad Armour-Garb, Fred Kroon, Mary Leng, Joseph Ulatowski, Mark Colyvan, and Matteo Plebani.
I am hugely grateful for these provocative and illuminating comments. My thanks to all N commentators. I will have something to say about each contribution, but the overall organization will be thematic. A reminder first of the issues we’re wrestling with. [First paragraph]
The paper explores Stephen Yablo's suggestion that ‘If-Thenism’ in the philosophy of mathematics is best formulated as the thesis that the real content of a mathematical claim C is the result of subtracting the potentially problematic metaphysical commitments of mathematics (numbers exist) from C [Yablo 2017]. Yablo's proposal assumes that some propositions make others true. The present discussion assumes that propositions are coarse-grained sets of possible worlds and asks what Yablo's proposal looks like on that assumption. The conclusion is that the adequacy of the proposal turns on hard-to-settle questions about the truthmakers for propositions expressed by material conditionals.
Standard theories treat A→C and A→D as equivalent when C and D coincide on A. However, Yablo's incremental conditional does not behave in this way. Consider the following:
(1) a = b→Fa.
(2) a = b→Fb.
According to Yablo, the real content of (1) is Fb and the real content of (2) is Fa. In worlds where a = b is true, both (1) and (2) have the same truth-value. However, ‘Fa can come apart truth-value-wise from Fb in worlds where a isn't b.’ In this paper, I argue that this feature of Yablo's incremental conditional makes his if-thenism incompatible with fictionalism.
A ‘sceptical’ approach to easy arguments involves reducing our confidence in the supposedly uncontroversial premise with which the arguments begin. Here I address the question: if we accept Yablo's new version of a sceptical proposal, what difference might that make for the relevant meta-ontological debates? I argue that serious difficulties remain for even this ‘best’ version of a sceptical approach. Noting these difficulties might motivate us to look again at the alternative strategy—of reading the uncontroversial premise straightforwardly and thinking that doubts about the conclusion were based on artificial inflation or confusion.
In our paper, we mount a novel argument, which trades on recent work by Roy Sorensen [2016 Sorensen, R. 2016. Unicorn Atheism, Nous 50, DOI: 10.1111/nous.12161 [Google Scholar]], following work by Saul Kripke, against Yablo's preferred reading of if-thenism, which is an attempt to read problematically ontologically committing sentences in a way that does not carry such ontological commitments. Although our argument is directed at Yablo's proposed reading of if-thenism, if the argument is successful, other versions of if-thenism may be affected. After reviewing Sorensen's recent work and presenting our argument, we consider a possible way out and use this as a means for presenting some challenges to Yablo.
‘Abstract expressionist’ accounts of applied mathematics seek to avoid the apparent Platonistic commitments of our scientific theories by holding that we ought only to believe their mathematics-free nominalistic content. The notion of ‘nominalistic content’ is, however, notoriously slippery. Yablo's account of non-catastrophic presupposition failure offers a way of pinning down this notion. However, I argue, its reliance on possible worlds machinery begs key questions against Platonism. I propose instead that abstract expressionists follow Geoffrey Hellman's lead in taking the assertoric content of empirical science to be irreducibly modal, using the ‘non-interference’ of mathematical objects as justification for detaching nominalistic consequences.
This paper suggests a focal meaning account of indicative conditionals, with incremental, or surplus content, conditionals at the center.
There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or non-causal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a liberalized version of Field’s response—and I discuss its significance. I argue that if it succeeds, it provides a new refutation of the indispensability argument; and that, even if it fails, its failure may bolster some of the fictionalist responses to the indispensability argument already under discussion. In addition, I use grounding to reply to a recent challenge to these responses.
Although Hale and Resnik are correct in their specific objection to my proposal for nominalizing science, the proposal can be saved by means of a simple and plausible modification.
We argue that Horgan's program for nominalizing science fails, because its translation of quantitative statements destroys the inferential structures of explanations, predictions and retrodictions of nonquantitative scientific facts.
Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims
to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical
science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that
this attribution is mistaken. Quine’s argument for the existence of abstract mathematical objects differs from the argument
which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of
abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam’s writings reveals some neglected
arguments for platonism which may be superior to the indispensability argument.
It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option.
En reponse a l'argument de l'indispensabilite elabore par Quine et Putnam en faveur du platonisme, l'A. defend la these de l'antiplatonisme en developpant une approche fictionnaliste de l'indispensable application des mathematiques a la science empirique
This book is a study of the concept of necessity. In the first three chapters, I clarify and defend the distinction between modality de re and modality de dicto. Also, I show how to explain de re modality in terms of de dicto modality. In Ch. 4, I explicate the concept of a possible world and define what it is for an object x to have a property P essentially. I then use the concept of an essential property to give an account of essences and their relationship to proper names. In Ch. 6, I argue that the Theory of Worldbound Individuals—even when fortified with Counterpart Theory—is false. Chapters 7 and 8 address the subject of possible but non‐existent objects; I argue here for the conclusion that there is no good reason to think that there are any such objects. In Ch. 9, I apply my theory of modality to the Problem of Evil in an effort to show that the Free Will Defense defeats this particular objection to theism. In Ch. 10, I present a sound modal version of the ontological argument for the existence of God. Finally, in the appendix, I address Quinean objections to quantified modal logic.
I hope that some people see some connection between the two topics in the title. If not, anyway, such connections will be developed in the course of these talks. Furthermore, because of the use of tools involving reference and necessity in analytic philosophy today, our views on these topics really have wide-ranging implications for other problems in philosophy that traditionally might be thought far-removed, like arguments over the mind-body problem or the so-called ‘identity thesis’. Materialism, in this form, often now gets involved in very intricate ways in questions about what is necessary or contingent in identity of properties — questions like that. So, it is really very important to philosophers who may want to work in many domains to get clear about these concepts. Maybe I will say something about the mind-body problem in the course of these talks. I want to talk also at some point (I don’t know if I can get it in) about substances and natural kinds.
According to the doctrine of nominalism, abstract entities-such as numbers, functions, and sets-do not exist. The problem this normally poses for a description of the physical world is as follows: any such description must include a physical theory, physical theories are assumed to require mathematics, and mathematics is replete with references to abstract entities. How, then, can nominalism reasonably be maintained? In answer, Hartry Field shows how abstract entities ultimately are dispensable in describing the physical world and that, indeed, we can "do science without numbers." The author also argues that despite the ultimate dispensability of mathematical entities, mathematics remains useful, and that its usefulness can be explained by the nominalist. The explanation of the utility of mathematics does not presuppose that mathematics is true, but only that it is consistent. The argument that the nominalist can freely use mathematics in certain contexts without assuming it to be true appears early on, and it first seems to license only a quite limited use of mathematics. But when combined with the later argument that abstract entities ultimately are dispensable in physical theories, the conclusion emerges that even the most sophisticated applications of mathematics depend only on the assumption that mathematics is consistent and not on the assumption that it is true.
Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/43392/1/11098_2004_Article_BF00646421.pdf
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