Quasi-Concrete Objects

One of the central claims of Charles Parsons' remarkable new book Mathematical Thought and Its Objects is that abstract objects come in two flavors: pure and quasi-concrete. Independently of the epistemological consequences of this division, we can ask how cogent and well-motivated it really is. Abstract objects are introduced using the standard negative tests: "an object is abstract if it is not located in space and time and does not stand in causal relations" (1). Quasi-concrete objects are abstract objects that have an additional feature:

What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete "representation," such that different objects of the kind in question are distinguishable by having different representations (34).

A central example of this sort of objects is expression-types. Each token represents a type and we individuate a given type by the tokens that represent it. It is clear how the token is concrete and also fairly clear how its representation of the type is intrinsic. One could say that the token stands in the relation it does to the type solely in virture of its intrinsic features.

Still, the situation is less clear with a second central case:

Although sets are in general not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership (35).

The first objection that Parsons notes to this proposal is that the representation relationship is too different because "one element can hardly represent the set as a whole". But it seems to me that a more serious objection focuses on the intrinsic nature of the membership relation. For concrete objects do not stand in any intrinsic relationship to sets. That is, a concrete object is not the member of a set solely in virtue of its intrinsic features. If we drop this intrinsic-ness test, then the motivation for carving out the quasi-concrete objects escapes me.

It is true that an impure set stands it an essential relation to its members, and so we might say that it also stands in an intrinsic relation to its members. But this it to reverse the direction of representation that Parsons originally invokes.

Kinds of A Priori Justification

In A Priori Justification, Casullo presents a minimal conception of the a priori as simply nonexperiential justification. This, in turn, is explained negatively as justification that does not arise due to the operation of the five senses. This clearly leaves open the possibility that there are several different kinds of a priori justification, but nearly all advocates for the a priori that I can find seem to assume that there must be a single, unified source.

A notable exception is Pap's 1944 article (Phil. Rev. 53: 465-484), although he weakens the plausibility of his three-fold distinction by concluding that the different kinds of a priori can easily intermingle.

Why not take a harder line and insist that there are different kinds of nonexperiential justification? One thought would be that some a priori justification is absolute because it is tied to conditions on concept possession, as with Peacocke, while some a priori justification is relative because it is tied to constitutive frameworks, as with Michael Friedman. Maybe this is the best way for the defender of the a priori to take on the radical empiricist.

Do We Know the Cause of Aerodynamic Lift?

We are all familiar with the upward force experienced by an airplane as it travels down the runway and through the air. But it is not entirely clear if we know the cause of this lift. Let's distinguish three tests for knowing the cause of some phenomenon:

(1) We can bring the phenomenon about with regularity and in a wide variety of circumstances.
(2) We have a scientific model which allows us to predict that the phenomenon will occur in these circumstances.
(3) We have a scientific model which includes accurate representations of the fundamental physical processes responsible for the phenomenon in these circumstances and this model allows us to predict that the phenomenon will occur in these circumstances.

(1) and (2) seem to me to be inadequate unless we adopt a non-standard account of causation. If I understand McCabe correctly, then he is arguing that we lack (3). This is because we believe that particle-to-particle interactions are the fundamental physical processes responsible for the generation of lift, but none of the models that we can work with accurately represent these processes.

As McCabe and Flatow explain, the common explanation in terms of a difference in pressure, known as Bernoulli's principle, fails. What is less clear is how the textbook explanation in terms of circulation relates to particle-to-particle interactions and how tenuous this relation can be consistent with a claim to know the cause.

The Philosophy of Applied Mathematics?

Here I try to make a start on what the philosophy of applied mathematics might look like. I present it as a potential development from the turn to mathematical practice, familiar from Maddy and Corfield. Then I turn to the extended example of Prandtl's boundary layer theory before concluding with some potential epistemic, semantic and metaphysical implications for the philosophy of mathematics as a whole.

Suggestions for improvement or pointers to similar work by others is appreciated.

Vienna and the Vienna Circle

Here are the slides from a presentation I gave in April to the European Studies group at Purdue. The basic point is that a study of the historical context of a philosopher can be essential to understanding the content of their position. One can concede this role to context without thinking that the correctness of one's philosophical views are determined by the context. Extended debate about the importance of context for the history of philosophy has developed out of Soames' book. See here for Soames' various replies to criticisms.

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.

Welcome!

This is a new philosophy blog, with a focus on philosophy of mathematics, philosophy of science and the history of analytic philosophy. I will look to post links of interest to people working in these areas along with updates on my own work.