Six Papers in Mind About Mathematical Fictionalism

The October 2012 issue of Mind (posted today here) has an extended discussion section where mathematical fictionalists of various stripes respond to Colyvan's earlier article "There is No Easy Road to Nominalism". The discussion concludes with a detailed reply by Colyvan. While I am a fan of neither Colyvan's explanatory indispensability argument nor its fictionalist critics, I look forward to reading this discussion and engaging with it soon!

The contents:

Jody Azzouni Taking the Easy Road Out of Dodge Mind (2012) 121(484): 951-965

Otávio Bueno An Easy Road to Nominalism Mind (2012) 121(484): 967-982

Mary Leng Taking it Easy: A Response to Colyvan Mind (2012) 121(484): 983-995

David Liggins Weaseling and the Content of Science Mind (2012) 121(484): 997-1005

Stephen Yablo Explanation, Extrapolation, and Existence Mind (2012) 121(484): 1007-1029

Mark Colyvan Road Work Ahead: Heavy Machinery on the Easy Road Mind (2012) 121(484): 1031-1046

New Book: Mary Leng's Mathematics and Reality

Mary Leng's book is now out! From the back cover:

Mary Leng offers a defence of mathematical fictionalism, arguing that we have no reason to believe that there are any mathematical objects. In mounting this defence, she responds to the indispensability argument for the existence of mathematical objects ... In response to this argument, Leng offers an account of the role of mathematics in empirical science that does not assume that the mathematical hypotheses used in formulating our scientific hypotheses are true.

It's great to see an extended defence of fictionalism out there. In my book manuscript I argue that fictionalism can't work, so I will be reading this quickly to see what options the fictionalist has available! There will eventually be a symposium on this book in the journal Metascience and hopefully other reviews will appear soon.

Colyvan Blocks the "Easy Road" to Nominalism

In a paper posted on his webpage listed as forthcoming in Mind, Mark Colyvan launches a new offensive against fictionalists like Azzouni, Melia and Yablo. They present a non-platonist interpetation of the language of mathematics and science that, they argue, does not require the "hard road" that Field took. Recall that Field tried to present non-mathematical versions of our best scientific theories. As Colyvan describes the current situation, though, "There are substantial technical obstacles facing Field's project and thse obstacles have prompted some to explore other, easier options" (p. 2). Colyvan goes on to argue that, in fact, these fictionalists do require the success of Field's project if their interpretations are to be successful.

I like this conclusion a lot, and it is actually superficially similar to what I argued for in my 2007 paper "A Role for Mathematics in the Physical Sciences". But what I argued is that Field's project is needed to specify a determinate content to mixed mathematical statements (p. 269). Colyvan takes a different and perhaps more promising route. He argues that without Field's project in hand, the fictionalist is unable to convincingly argue that apparent reference to mathematical entities is ontologically innocent. This is especially difficult given the prima facie role of mathematics in scientific explanation:

The response [by Melia] under consideration depends on mathematics playing no explanatory role in science, for it is hard to see how non-existent entities can legitimately enter into explanations (p. 11, see also p. 14 for Yablo).

I have noted this explanatory turn in debates about indispensability before, but here we see Colyvan moving things forward in a new and interesting direction. Still, I continue to worry that we need a better positive proposal for the source of the explanatory contributions from mathematics, especially if it is to bear the weight of defending platonism.

Under the ruler, faster than the ruler?

One of the highlights of the recent workshop on Models and Fiction hosted by the Institute of Philosophy in London was Deena Skolnick Weisberg's presentation on her recent psychological study of how the imagination is deployed in fiction. One fairly robust finding for adults is that we tend to 'import' claims that we believe into the fiction even when they are not mentioned in a story. For example, when asked "is 2+2=4 true in the story you just heard?" most adults said "yes". Skolnick also observed a slightly reduced tendency to import beliefs that we recognize as contingent such as "is Obama President in the story you just heard?".

For me an equally interesting phenomenon is one that Skolnick just had time to mention: known ways in which our imaginative powers fail to track what will happen in the world. A dramatic example of this is found in this YouTube video. It is worthwhile just trying to predict what will happen! Apparently it is not clear what the systematic error that we tend to make is, but I take this example to complicate some attempts to identify scientific modelling with fiction.

Models and Fiction

In a forthcoming paper "Models and Fiction", Roman Frigg gives an argument for the view that scientific models are best understood as fictional entities whose metaphysical commitments are “none” (17). I think this argument is a new and important one, but I don’t agree with it. Frigg first considers the view that models are abstract structures. He points out that an abstract mathematical structure, by itself, is not a model because there is nothing about it that ties it to any purported target system. But "in order for it to be true that a target system possesses a particular structure, a more concrete description must be true of the system as well" (5). The problem is that this more concrete description is not a true description of the abstract structure and it is not a true description of the target system either in the case if idealization. So, for these descriptions to do their job of linking the abstract structure to their targets, they must be descriptions of "hypothetical systems", and it is these systems that Frigg argues are the models after all.

My objection to this argument is that there are things besides Frigg’s descriptions that can do the job of linking abstract structures to target systems. A weaker link is a relation of denotation between some parts of the abstract structure and features of the target systems. This, of course, requires some explanation, but a denotation or reference relation, emphasized, e.g. by Hughes, need not involve a concrete description of any hypothetical system.

(Cross-posted with It's Only a Theory.)

Meyer on Field-style Reformulations of Statistical Mechanics

Glen Meyer offers an in-depth discussion of Field's program to nominalize science with special emphasis on the challenges encountered with classical equilibrium statistical mechanics (CESM). He makes a number of excellent points along the way, but what I like most is his focus on the prevalence of an appeal to what some call "surplus" mathematical structure, i.e. mathematics that has no natural physical interpretation. As he argues, Field could reconstruct configuration spaces for point particles using physical points in space-time, but would face difficulties extending this approach to phase spaces and probability distributions on phase spaces.

One novelty of the paper is a distinction between interpretation and representation. Mathematical theories have some mathematical terms with a semantic reference and a representational role, but other mathematical terms may have a semantic reference with no representational role. When idealizations involve this latter kind of term, Field-style reformulations are in trouble. For example, Meyer discusses the need to treat certain discrete quantities as continuous in the derivation of the Maxwell-Boltzmann distribution law:

The intended ('intrinsic') interpretations of axioms describing a certain structure forces that structure to represent, as it were, in its entirety, i.e., that this structure be exemplified in the subject matter of the theory. Any introduction of the idealization above at the nominalistic level will therefore force us to adopt assumptions about the physical world that the platonistic theory, despite its use of this idealization, does not make. Unlike the case of point particles, this idealization is not part of the nominalistic content of the platonistic theory and therefore does not belong in any nominalistic reformulation. Without it, however, we have no way of recovering this part of CESM (p. 37).

Here we have a derivation that ordinary theories can ground, but that Field-style nominalistic theories cannot. I agree with Meyer here, but it raises the further issue: why is it so useful to make these sorts of non-physical idealizations? It may just be a pragmatic issue of convenience, or perhaps there is something deeper to say about how the mathematics contributes without representing? (See Batterman's recent paper for one answer.)

Rayo's "On Specifying Truth-Conditions"

This is a long, innovative and frustrating article (Phil. Review 117 (2008): 385-443), at least for someone like me who is not at all inclined to fictionalism or other non-standard approaches to mathematical language. But for those, like Rayo, who think that “expressions with identical syntactic and inferential roles can perform different semantic jobs” (415fn22) this paper may represent the current state of the art.

Rayo aims to defend the stability or coherence of a form of noncommittalism according to which the use of mathematical language does not commit one to the existence of any abstract mathematical objects. An important negative point that Rayo makes early on is that the noncommittalist is unable to accomplish this task by “translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleeze” (385). Still, the positive program is to set out a method of specifying associated truth conditions for these sentences so that these conditions do not involve abstract objects, but only the world being a certain way. The key innovation, if I am not misunderstanding the paper, is to employ the full range of semantic machinery in explaining how the world has to be for a given sentence to be correct. Rayo carefully explores the issue of whether this is legitimate, and concludes that if one begins as a noncommittalist, then one will feel entitled to appeal to numbers and functions in one’s semantic theory. As a result, the noncommittalist can claim an internally coherent or stable package of views.

While much of the interest of the paper is in the details, the following frank admission by Rayo struck me as worth highlighting:

Whatever its plausibility as an explanation of how the standard arithmetical axioms might be rendered meaningful in such a way that their truth is knowable a priori, it should be clear that the arithmetical stipulation is not very plausible as an explanation of how the axioms were actually rendered meaningful or how it is that we actually acquire a priori knowledge of their truth (431-432).

A committalist seems well within her rights to wonder, then, what the point is of carving out a stable position like Rayo’s if it has no bearing on our actual mathematical knowledge or on the actual contribution that mathematics makes in applied contexts.

Nominalistic Content

In "A Role for Mathematics in Phyiscal Theories" I argued that fictionalists have a problem when it comes to specifying the nominalistic content of our best scientific theories. David Liggins recently drew my attention to Gideon Rosen's explanation of nominalistic adequacy in his 2001 article "Nominalism, Naturalism, Epistemic Relativism". Does this approach block my argument?

Here is Rosen's account:

Let's say the concrete content of a world W is the largest wholly concrete part of W: the aggregate of all of the concrete objects that exist in W ... S is nominalistically adequate iff the concrete core of the actual world is an exact intrinsic duplicate of the concrete core of some world at which S is true -- that is, just in case things are in all concrete respects as if S were true (p. 75).

Here is a summary of my challenge to fictionalism: (i) the fictionalist must present something like Field's axioms if he is to explain which parts of the full content get into the nominalistic content. But (ii) giving these axioms would involve taking a stand on features of the concrete world that went beyond our evidence for the mathematical scientific theory. So, (iii) there was no epistemically responsible way for the fictionalist to specify how the nominalistic content differed from the full content.

A fictionalist might agree with the demand in (i), but think that Rosen's approach resolves the issue without appealing to Field-style axioms. I am not sure how this will work, though. If we use the real numbers to represent temperature, how does Rosen's test apply? For example, suppose we consider a law about thermal expansion. If that is part of my theory, what does it mean to say that the law is nominalistically adequate? Let's take two potential things that may or may not get in there: (a) instantiated temperatures are dense, (b) there is no lowest temperture. Both of these can be expressed in a nominalistic language provided we have Field's temperature predicates around. So, I think they are about the concrete world and should be relevant to nominalistic content.

Now I suggest that even if we accept Rosen's test, this is no help in resovling the question of the nominalistic adequacy of the law. I do not know if (a) and (b) are part of the nominalistic content of the law or how this is determined. This is the sense in which the commitments are indeterminate for the fictionalist. (I am not saying I explained this very well in the paper, but this is at least how I am thinking about it now.)

Suppose a fictionalist responded that whatever indeterminacy there is for the nominalistic content also arises for the full content. So, there is nothing here to tell against fictionalism and in favor of some kind of realism. My view is that a realist who can accept the mapping account can specify the full content with reference to these mappings. For this law, it would be something like "For any iron bar, if the temperature were to be increased by amount t, then the length of the bar would increase by alpha * t". Here the antecedent and the consequent involve mappings between objects with physical properties and mathematical objects. This clearly does not require (a) or (b). So, because we can apeal to mappings or relationships between physical properties and mathematical objects, we can resolve some apparent indeterminacies in the full content.

Why can't the fictionalist say the same thing? Maybe he can, but it seems that no fictionalists have explained how this would work beyond some toy examples involving counting. So, maybe the best way to see my discussion is as a challenge to the fictionalist to explain how she can match the realist in giving determinate contents to our scientific statements and theories. On my story, the commitment to realism comes in explaining the full content. The fictionalist either needs to bring in this explanation or else directly specify the nominalistic content by other means. I have not shown that both of these strategies are hopeless, but I think the burden is on the fictionalist to work it out.

Another reply is that the fictionalist need not satisfy my demand to explain how the full content relates to the nominalistic content or to clarify the nominalistic content directly. This is the reply I try to deal with in the article by saying that the fictionalist must explain what he is committing himself to in accepting a given statement or theory. Otherwise, he is not facing up to Quine's challenge on ontological commitment.